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--- abstract: 'We investigate the class of visibly pushdown languages in the sliding window model. A sliding window algorithm for a language $L$ receives a stream of symbols and has to decide at each time step whether the suffix of length $n$ belongs to $L$ or not. The window size $n$ is either a fixed number (in the fixed-size model) or can be controlled by an adversary in a limited way (in the variable-size model). The main result of this paper states that for every visibly pushdown language the space complexity in the variable-size sliding window model is either constant, logarithmic or linear in the window size. This extends previous results for regular languages.' author: - Moses Ganardi bibliography: - 'refs.bib' title: Visibly Pushdown Languages over Sliding Windows --- Introduction ============ ##### The sliding window model. A [sliding window algorithm (SWA)]{} is an algorithm which processes a stream of data elements $a_1 a_2 a_3 \cdots$ and computes at each time instant $t$ a certain value that depends on the suffix $a_{t-n+1} \cdots a_t$ of length $n$ where $n$ is a parameter called the [window size]{}. This streaming model is motivated by the fact that in many applications data elements are outdated or become irrelevant after a certain time. A general goal in the area of sliding window algorithms is to avoid storing the window content explicitly (which requires $\Omega(n)$ bits) and to design space efficient algorithms, say using polylogarithmic many bits in the window size $n$. A prototypical example of a problem considered in the sliding window model is the <span style="font-variant:small-caps;">Basic Counting</span> problem. Here the input is a stream of bits and the task is to approximate the number of 1’s in the last $n$ bits (the [active window]{}). In [@DatarGIM02], Datar, Gionis, Indyk and Motwani present an approximation algorithm using $O(\frac{1}{\epsilon} \log^2 n)$ bits of space with an approximation ratio of $\epsilon$. They also prove a matching lower bound of $\Omega(\frac{1}{\epsilon} \log^2 n)$ bits for any deterministic (and even randomized) algorithm for <span style="font-variant:small-caps;">Basic Counting</span>. Other works in the sliding window model include computing statistics [@ArasuM04; @BabcockDMO03; @BravermanO07], optimal sampling [@BravermanOZ12] and various pattern matching problems [@BreslauerG14; @CliffordFPSS15; @CliffordFPSS16; @CliffordS16]. There are two variants of the sliding window model, cf. [@ArasuM04]. One can think of an adversary who can either insert a new element into the window or remove the oldest element from the window. In the [fixed-size]{} sliding window model the adversary determines the window size $n$ in the beginning and the initial window is set to $a^n$ for some default known element $a$. At every time step the adversary inserts a new symbol and then immediately removes the oldest element from the window. In the [variable-size]{} sliding window model the window size is initially set to $n = 0$. Then the adversary is allowed to perform an arbitrary sequence of insert- and remove-operations. A remove-operation on an empty window leaves the window empty. We also mention the timestamp-based model where every element carries a timestamp (many elements may have the same timestamp) and the active window at time $t$ contains only those elements whose timestamp is at least $t-t_0$ for some parameter $t_0$ [@BravermanOZ12]. Both the fixed-size and the timestamp-based model can be simulated in the variable-size model. ##### Regular languages. In a recent series of works we studied the membership problem to a fixed regular language in the sliding window model. It was shown in [@GanardiHL16] that in both the fixed-size and the variable-size sliding window model the space complexity of any regular language is either constant, logarithmic or linear (a [space trichotomy]{}). In a subsequent paper [@GHKLM18] a characterization of the space classes was given: A regular language has a fixed/variable-size SWA with $O(\log n)$ bits if and only if it is a finite Boolean combination of regular left ideals and regular length languages. A regular language has a fixed-size SWA with $O(1)$ bits if and only if it is a finite Boolean combination of suffix testable languages and regular length languages. A regular language has a variable-size SWA with $O(1)$ bits if and only if it is empty or universal. ##### Context-free languages. A natural question is whether the results above can be extended to larger language classes, say subclasses of the context-free languages. More precisely, we pose the questions: (i) Which language classes have a “simple” hierarchy of space complexity classes (like the space trichotomy for the regular languages), and (ii) are there natural descriptions of the space classes? A positive answer to question (i) seems to be necessary to answer question (ii) positively. In [@GJL18] we presented a family of context-free languages $(L_k)_{k \ge 1}$ which have space complexity $\Theta(n^{1/k})$ in the variable-size model and $O(n^{1/k}) \setminus o(n^{1/k})$ in the fixed-size model, showing that there exists an infinite hierarchy of space complexity classes inside the class of context-free languages. Intuitively, this result can be explained with the fact that a language and its complement have the same sliding window space complexity; however, the class of context-free languages is not closed under complementation (in contrast to the regular languages) and the analysis of co-context-free languages in this setting seems to be very difficult. Even in the class of deterministic context-free languages, which is closed under complementation, there are example languages which have sliding window space complexity $\Theta((\log n)^2)$ [@GJL18]. ##### Visibly pushdown languages. Motivated by these observations in this paper we will study the class of [visibly pushdown languages]{}, introduced by Alur and Madhusudan [@AlurM04]. They are recognized by [visibly pushdown automata]{} where the alphabet is partitioned into [call letters]{}, [return letters]{} and [internal letters]{}, which determine the behavior of the stack height. Since visibly pushdown automata can be determinized, the class of visibly pushdown languages turns out to be very robust (it is closed under Boolean operations and other language operations) and to be more tractable in many algorithmic questions than the class of context-free languages [@AlurM04]. In this paper we prove a space trichotomy for the class of visibly pushdown languages in the variable-size sliding window model, stating that the space complexity of every visibly pushdown language is either $O(1)$, $\Theta(\log n)$ or $O(n) \setminus o(n)$. The main technical result is a growth theorem (Theorem \[thm:rational-dichotomy\]) for rational transductions. A natural characterization of the $O(\log n)$-class as well as a study of the fixed-size model are left as open problems. Let us mention some related work in the context of streaming algorithms for context-free languages. Randomized streaming algorithms were studied for subclasses of context-free languages (DLIN and LL$(k)$) [@BabuLRV13] and for Dyck languages [@MagniezMN14]. A streaming property tester for visibly pushdown languages was presented by Fran[ç]{}ois et al. [@FrancoisMRS16]. Preliminaries ============= We define $\log n = \lfloor \log_2 n \rfloor$ for all $n \ge 1$, which is the minimum number $k$ of bits required to encode $n$ elements using bit strings of length [at most]{} $k$. If $w = a_1 \cdots a_n$ is a word then any word of the form $a_i \cdots a_n$ ($a_1 \cdots a_i$) is called [suffix (prefix)]{} of $w$. A prefix (suffix) $v$ of $w$ is [proper]{} if $v \neq w$. A [factor]{} of $w$ is any word of the form $a_i \cdots a_j$. A [factorization]{} of $w$ is formally a sequence of possibly empty factors $(w_0, \dots, w_m)$ with $w = w_0 \cdots w_m$. We call $w_0$ the [initial]{} factor and $w_1, \dots, w_m$ the [internal]{} factors. The [reversal]{} of $w$ is $w^{\mathsf{R}}= a_n a_{n-1} \cdots a_1$. For a language $L \subseteq \Sigma^$ we denote by ${\mathsf{Suf}}(L)$ the set of suffixes of words in $L$. If $L = {\mathsf{Suf}}(L)$ then $L$ is [suffix-closed]{}. ##### Automata. An [automaton]{} over a monoid $M$ is a tuple $A = (Q,M,I,\Delta,F)$ where $Q$ is a finite set of [states]{}, $I \subseteq Q$ is a set of [initial states]{}, $\Delta \subseteq Q \times M \times Q$ is the [transition relation]{} and $F \subseteq Q$ is the set of [final states]{}. A [run]{} on $m \in M$ from $q_0$ to $q_n$ is a sequence of transitions of the form $\pi = (q_0,m_1,q_1)(q_1,m_2,q_2)\cdots (q_{n-1},m_n,q_n) \in \Delta^$ such that $m = m_1 \cdots m_n$. We usually depict $\pi$ as $q_0 \xrightarrow{m_1} q_1 \xrightarrow{m_2} q_2 \, \cdots \, q_{n-1} \xrightarrow{m_n} q_n$, or simply $q_0 \xrightarrow{m} q_n$. It is [initial]{} if $q_0 \in I$ and [accepting]{} if $q_n \in F$. The [language]{} defined by $A$ is the set ${\mathsf{L}}(A)$ of all elements $m \in M$ such that there exists an initial accepting run on $m$. A subset $L \subseteq M$ is [rational]{} if $L = {\mathsf{L}}(A)$ for some automaton $A$. We only need the case where $M$ is the free monoid $\Sigma^$ over an alphabet $\Sigma$ or where $M$ is the product $\Sigma^ \times \Omega^$ of two free monoids. In these cases we change the format and write $(Q,\Sigma,I,\Delta,F)$ and $(Q,\Sigma,\Omega,I,\Delta,F)$, respectively. Subsets of $\Sigma^$ are called [languages]{} and subsets of $\Sigma^* \times \Omega^$ are called [transductions]{}. Rational languages are usually called [regular languages]{}. In this paper we will also use [right automata]{}, which read the input from right to left. Formally, a right automaton $A = (Q,M,F,\Delta,I)$ has the same format as a (left) automaton where the sets of initial and final states are swapped. Runs in right automata are defined from right to left, i.e. a run on $m \in M$ from $q_n$ to $q_0$ is a sequence of transitions of the form $(q_0,m_1,q_1)(q_1,m_2,q_2)\cdots (q_{n-1},m_n,q_n) \in \Delta^$ such that $m = m_1 \cdots m_n$. In the graphic notation we write the arrows from right to left. It is initial (accepting) if $q_n \in I$ ($q_0 \in F$). ##### Right congruences. For any equivalence relation $\sim$ on a set $X$ we write $[x]_\sim$ for the $\sim$-class containing $x \in X$ and $X/{\sim} = \{[x]_\sim \mid x \in X\}$ for the set of all $\sim$-classes. The [index]{} of $\sim$ is the cardinality of $X/{\sim}$. We denote by $\nu_\sim \colon X \to X/{\sim}$ the function with $\nu_\sim(x) = [x]_\sim$. A subset $L \subseteq X$ is [saturated]{} by $\sim$ if $L$ is a union of $\sim$-classes. An equivalence relation $\sim$ on the free monoid $\Sigma^$ over some alphabet $\Sigma$ is a [right congruence]{} if $x \sim y$ implies $xz \sim yz$ for all $x,y,z \in \Sigma^$. The [Myhill-Nerode right congruence]{} $\sim_L$ of a language $L \subseteq \Sigma^$ is the equivalence relation on $\Sigma^$ defined by $x \sim_L y$ if and only if $x^{-1} L = y^{-1} L$ where $x^{-1}L = \{ z \mid xz \in L \}$. It is indeed the coarsest right congruence on $\Sigma^$ which saturates $L$. We usually write $\nu_L$ instead of $\nu_{\sim_L}$. A language $L \subseteq \Sigma^$ is regular iff $\sim_L$ has finite index. ##### Rational transductions. Rational transductions are accepted by automata over $\Sigma^* \times \Omega^$, which are called finite state transducers. In this paper, we will use a slightly extended but equivalent definition. A [transducer]{} is a tuple $A = (Q,\Sigma,\Omega,I,\Delta,F,o)$ such that $(Q,\Sigma^ \times \Omega^,I,\Delta,F)$ is an automaton over $\Sigma^ \times \Omega^$ and a [terminal output function]{} $o \colon F \to \Omega^$. To omit parentheses we write runs $p \xrightarrow{(x,y)} q$ in the form $p \xrightarrow{x \mid y} q$ and depict $o(q) = y$ by a transition $q \xrightarrow{\mid y}$ without input word and target state. If $\pi$ is a run $p \xrightarrow{x \mid y} q$ we define ${\mathsf{out}}(\pi) = y$ and ${\mathsf{out}}_F(\pi) = y \, o(q)$. The transduction defined by $A$ is the set ${\mathsf{T}}(A)$ of all pairs $(x,{\mathsf{out}}_F(\pi))$ such that $\pi$ is an initial accepting run $p \xrightarrow{x \mid y} q$. Since the terminal output function can be eliminated by ${\varepsilon}$-transitions, a transduction is rational if and only if it is of the form ${\mathsf{T}}(A)$ for some transducer $A$. In this paper we will mainly use [rational functions]{}, which are partial functions $t \colon \Sigma^* \to \Omega^$ whose graph $\{(x,t(x)) \mid x \in {\mathrm{dom}}(t) \}$ is a rational transduction. A transducer $A$ is [trim]{} if every state occurs on some accepting run. If every word $x \in \Sigma^$ has at most one initial accepting run $p \xrightarrow{x \mid y} q$ for some $y \in \Omega^$ then $A$ is [unambiguous]{}. If $\Delta \subseteq Q \times \Sigma \times \Omega^ \times Q$ then $A$ is [real-time]{}. It is known that every rational function is defined by a trim unambiguous real-time transducer [@Berstel79 Corollary 4.3]. If $A$ is unambiguous and trim then for every word $x \in \Sigma^$ and every pair of states $(p,q) \in Q^2$ there exists at most one run from $p$ to $q$ with input word $x$. Therefore, the state pair $(p,q)$ and the input word $x$ uniquely determine the run (if it exists) and we can simply write $p \xrightarrow{x} q$. Similarly to [@WeberK95], we define for a real-time transducer $A$ the parameter ${\mathsf{iml}}(A) = \max \left(\{ \|y\| \mid (q,a,y,p) \in \Delta \} \cup \{ \|o(q)\| \mid q \in Q \} \right)$. For every run $\pi$ on a word $x \in \Sigma^$ we have $\|{\mathsf{out}}(\pi)\| \le {\mathsf{iml}}(A) \cdot \|x\|$ and $\|{\mathsf{out}}_F(\pi)\| \le {\mathsf{iml}}(A) \cdot (\|x\|+1)$. The following closure properties for rational transductions are known [@Berstel79]: The class of rational transductions is closed under inverse, reversal and composition where the [inverse]{} of $T$ is $T^{-1} = \{ (y,x) \mid (x,y) \in T \}$, the [reversal]{} of $T$ is $T^{{\mathsf{R}}} = \{ (x^{\mathsf{R}},y^{\mathsf{R}}) \mid (x,y) \in T \}$, and the composition of two transductions $T_1,T_2$ is $T_1 \circ T_2 = \{ (x,z) \mid \exists y: (x,y) \in T_1 \text{ and } (y,z) \in T_2 \}$. If $T \subseteq \Sigma^* \times \Omega^$ is rational and $L \subseteq \Sigma^$ is regular then the restriction $\{ (x,y) \in T \mid x \in L \}$ is also rational. If $K \subseteq \Sigma^$ is regular (context-free) and $T \subseteq \Sigma^ \times \Omega^$ is rational then $TK = \{ y \in \Omega^ \mid (x,y) \in T \text{ for some } x \in K \}$ is also regular (context-free). A [right transducer]{} is a tuple $A = (Q,\Sigma,\Omega,F,\Delta,I,o)$ such that $(Q,\Sigma^* \times \Omega^,F,\Delta,I)$ is a right automaton over $\Sigma^ \times \Omega^$ and a [terminal output function]{} $o \colon F \to \Omega^$. We depict $o(q) = y$ by a transition $\xleftarrow{\mid y} q$. If $\pi$ is a run $q \xleftarrow{x \mid y} p$ we define ${\mathsf{out}}(\pi) = y$ and ${\mathsf{out}}_F(\pi) = o(q) \, y$. All other notions on transducers are defined for right transducers in a dual way. ##### Growth functions. A function $\gamma \colon {\mathbb{N}}\to {\mathbb{N}}$ grows [polynomially]{} if $\gamma(n) \in O(n^k)$ for some $k \in {\mathbb{N}}$; we say that $\gamma$ grows [exponentially]{} if there exists a number $c > 1$ such that $\gamma(n) \ge c^n$ for infinitely many $n \in {\mathbb{N}}$. A function $\gamma(n)$ grows exponentially if and only if $\log \gamma(n) \notin o(n)$. We will define a generalized notion of growth. Let $t \colon \Sigma^* \to Y$ be a partial function and let $X \subseteq {\mathrm{dom}}(t)$ be a language. The [$t$-growth]{} of $X$ is the function $\gamma(n) = \|t(X \cap \Sigma^{\le n})\|$, i.e. it counts the number of output elements on input words from $X$ of length at most $n$. The [growth]{} of $X$ is simply the $\mathrm{id}_X$-growth of $X$, i.e. $\gamma(n) = \|X \cap \Sigma^{\le n}\|$. It is known that every context-free language has either polynomial or exponential growth [@Ginsburg1966]. Furthermore, a context-free language $L$ has polynomial growth if and only if it is [bounded]{}, i.e. $L \subseteq w_1^* \cdots w_k^$ for some words $w_1, \dots, w_k$ [@Ginsburg1966]. We need the fact that if $L$ is a bounded language and $K$ is a set of factors of words in $L$ then $K$ is bounded [@GS1964 Lemma 1.1(c)]. Visibly pushdown languages ========================== A [pushdown alphabet]{} is a triple $\tilde \Sigma = (\Sigma_c,\Sigma_r,\Sigma_{\mathit{int}})$ consisting of three pairwise disjoint alphabets: a set of [call letters]{} $\Sigma_c$, a set of [return letters]{} $\Sigma_r$ and a set of [internal letters]{} $\Sigma_{\mathit{int}}$. We identify $\tilde \Sigma$ with the union $\Sigma = \Sigma_c \cup \Sigma_r \cup \Sigma_{\mathit{int}}$. The set of [well-matched]{} words $W$ over $\Sigma$ is defined as the smallest set which contains $\{{\varepsilon}\} \cup \Sigma_{\mathit{int}}$, is closed under concatenation, and if $w$ is well-matched, $a \in \Sigma_c$, $b \in \Sigma_r$ then also $awb$ is well-matched. A word is called [descending (ascending)]{} if it can be factorized into well-matched factors and return (call) letters. The set of descending words is denoted by $D$. A [visibly pushdown automaton (VPA)]{} has the form $A = (Q,\tilde \Sigma,\Gamma,\bot,q_0,\delta, F)$ where $Q$ is a finite state set, $\tilde \Sigma$ is a pushdown alphabet, $\Gamma$ is the finite stack alphabet containing a special symbol $\bot$ (representing the empty stack), $q_0 \in Q$ is the initial state, $F \subseteq Q$ is the set of final states and $\delta = \delta_c \cup \delta_r \cup \delta_{\mathit{int}}$ is the transition function where $\delta_c \colon Q \times \Sigma_c \to (\Gamma \setminus \{\bot\}) \times Q$, $\delta_r \colon Q \times \Sigma_r \times \Gamma \to Q$ and $\delta_{\mathit{int}} \colon Q \times \Sigma_{\mathit{int}} \to Q$. The set of [configurations]{} ${\mathrm{Conf}}$ is the set of all words $\alpha q$ where $q \in Q$ is a state and $\alpha \in \bot (\Gamma \setminus \{\bot\})^$ is the [stack content]{}. We define $\delta \colon {\mathrm{Conf}}\times \Sigma \to {\mathrm{Conf}}$ for each $p \in Q$ and $a \in \Sigma$ as follows: - If $a \in \Sigma_c$ and $\delta(p,a) = (\gamma,q)$ then $\delta(\alpha p,a) = \alpha \gamma q$. - If $a \in \Sigma_{\mathit{int}}$ and $\delta(p,a) = q$ then $\delta(\alpha p, a) = \alpha q$. - If $a \in \Sigma_r$, $\delta(p,a,\gamma) = q$ and $\gamma \in \Gamma \setminus \{\bot\}$ then $\delta(\alpha \gamma p, a) = \alpha q$. - If $a \in \Sigma_r$ and $\delta(p,a,\bot) = q$ then $\delta(\bot p) = \bot q$. As usual we inductively extend $\delta$ to a function $\delta \colon {\mathrm{Conf}}\times \Sigma^* \to {\mathrm{Conf}}$ where $\delta(c,{\varepsilon}) = c$ and $\delta(c,wa) = \delta(\delta(c,w),a)$ for all $w \in \Sigma^$ and $a \in \Sigma$. The [initial]{} configuration is $\bot q_0$ and a configuration $c$ is [final]{} if $c \in \Gamma^ F$. A word $w \in \Sigma^$ is [accepted]{} from a configuration $c$ if $\delta(c,w)$ is final. The VPA $A$ [accepts]{} $w$ if $w$ is accepted from the initial configuration. The set of all words accepted by $A$ is denoted by ${\mathsf{L}}(A)$; the set of all words accepted from $c$ is denoted by ${\mathsf{L}}(c)$. A language $L$ is a [visibly pushdown language (VPL)]{} if $L = {\mathsf{L}}(A)$ for some VPA $A$. To exclude some pathological cases we assume that $\Sigma_c \neq \emptyset$ and $\Sigma_r \neq \emptyset$. In fact, if $\Sigma_c = \emptyset$ or $\Sigma_r = \emptyset$ then any VPL over that pushdown alphabet would be regular. One can also define nondeterministic visibly pushdown automata in the usual way, which can always be converted into deterministic ones [@AlurM04]. This leads to good closure properties of the class of all VPLs, as closure under Boolean operations, concatenation and Kleene star. The set $W$ of well-matched words forms a submonoid of $\Sigma^$. Notice that a VPA can only see the top of the stack when reading return symbols. Therefore, the behavior of a VPA on a well-matched word is determined only by the current state and independent of the current stack content. More precisely, there exists a monoid homomorphism $\varphi \colon W \to Q^Q$ into the finite monoid of all state transformations $Q \to Q$ such that $\delta(\alpha p, w) = \alpha \varphi(w)(p)$ for all $w \in W$ and $\alpha p \in {\mathrm{Conf}}$. Sliding window algorithms and main results ========================================== In our context a [streaming algorithm]{} is a deterministic algorithm $A$ which reads an input word $a_1 \cdots a_m \in \Sigma^$ symbol by symbol from left to right and outputs after every prefix either $1$ or $0$. We view $A$ as a deterministic (possibly infinite) automaton whose states are encoded by bit strings and thus abstract away from the actual computation, see [@GHKLM18] for a formal definition. A [variable-size sliding window algorithm]{} for a language $L \subseteq \Sigma^$ is a streaming algorithm $A$ which reads an input word $a_1 \cdots a_m$ over the extended alphabet $\overline{\Sigma} = \Sigma \cup \{ \downarrow \}$. The symbol $\downarrow$ is the operation which removes the oldest symbol from the window. At time $0 \le t \le m$ the algorithm has to decide whether the [active window]{} ${\mathrm{wnd}}(a_1 \cdots a_t)$ belongs to $L$ which is defined by $$\begin{aligned} {2} {\mathrm{wnd}}(\varepsilon) & = \varepsilon & \qquad {\mathrm{wnd}}(u \! \downarrow) & = \varepsilon \text{ if } {\mathrm{wnd}}(u) = \varepsilon \\ {\mathrm{wnd}}(ua) & = {\mathrm{wnd}}(u) a & {\mathrm{wnd}}(u \! \downarrow) & = v \text{ if } {\mathrm{wnd}}(u) = av\end{aligned}$$ for $u \in \Sigma^$, $a \in \Sigma$. For example, a variable-size sliding window algorithm $A$ for the language $L_a = \{ w \in \{a,b\}^ \mid w \text{ contains } a \}$ maintains the window length $n$ and the position $i$ (from the right) of the most recent $a$-symbol in the window (if it exists): We initialize $n := 0$ and $i := \infty$. On input $a$ we increment $n$ and set $i := 1$, on input $b$ we increment both $n$ and $i$. On input $\downarrow$ we decrement $n$, unless $n = 0$, and then set $i := \infty$ if $i > n$. The [space complexity]{} of $A$ is the function which maps $n$ to the maximum number of bits used when reading an input $a_1 \cdots a_m$ where the window size never exceeds $n$, i.e. $\|{\mathrm{wnd}}(a_1 \cdots a_t)\| \le n$ for all $0 \le t \le n$. Notice that this function is monotonic. For every language $L$ there exists a space optimal variable-size sliding window algorithm [@GHKLM18a Lemma 3.1] and we write $V_L(n)$ for its space complexity. Clearly we have $V_L(n) \in O(n)$. For example the example language $L_a$ above satisfies $V_{L_a}(n) \in O(\log n)$ because the algorithm above only maintains two numbers using $O(\log n)$ bits. The main result of this paper states: \[thm:vpl-trichotomy\] If $L$ is a visibly pushdown language then $V_L(n)$ is either $O(1)$, $\Theta(\log n)$ or $O(n) \setminus o(n)$. In the rest of this section we will give an overview of the proof of Theorem \[thm:vpl-trichotomy\]. ##### Suffix expansions. Let $\sim$ be an equivalence relation on $\Sigma^$. The [suffix expansion]{} of $\sim$ is the equivalence relation $\approx$ on $\Sigma^$ defined by $a_1 \cdots a_n \approx b_1 \cdots b_m$ if and only if $n = m$ and $a_i \cdots a_n \sim b_i \cdots b_n$ for all $1 \le i \le n$. Notice that $\approx$ saturates each subset $\Sigma^{\le n}$. Furthermore, if $\sim$ is a right congruence then so is $\approx$ since $\|u\| = \|v\|$ implies $\|ua\| = \|va\|$ and $a_i \cdots a_n \sim b_i \cdots b_n$ implies $a_i \cdots a_na \sim b_i \cdots b_na$. We also define suffix expansions for partial functions $t \colon \Sigma^* \to Y$ with suffix-closed domain ${\mathrm{dom}}(t)$. The [suffix expansion]{} of $t$ is the total function $\cev{t} \colon {\mathrm{dom}}(t) \to Y^$ defined by $\cev{t}(a_1 \cdots a_n) = t(a_1 \cdots a_n) \, t(a_2 \cdots a_n) \, \cdots \, t(a_{n-1} a_n) \, t(a_n)$ for all $a_1 \cdots a_n \in \Sigma^$. Here the range of $\cev{t}$ is the free monoid (alternatively, the set of all sequences) over $Y$. If $\sim$ is an equivalence relation on $\Sigma^$ then its suffix expansion $\approx$ is the [kernel]{} of $\cev{\nu}_\sim$, i.e. $x \approx y$ if and only if $\cev{\nu}_\sim(x) = \cev{\nu}_\sim(y)$. The space complexity in the variable-size model is captured by the suffix expansion $\approx_L$ of the Myhill-Nerode right congruence $\sim_L$ or alternatively by the suffix expansion $\cev{\nu}_L$ of $\nu_L$. \[thm:psiL\] For all $\emptyset \subsetneq L \subsetneq \Sigma^$ we have $V_L(n) = \log \|\Sigma^{\le n}/{\approx_L}\| = \log \|\cev{\nu}_L(\Sigma^{\le n})\|$. In particular, $V_L(n) = \Omega(\log n)$ for every non-trivial language. If $L$ is empty or universal, then $V_L(n) \in O(1)$ and otherwise $V_L(n) = \Omega(\log n)$. Hence to prove Theorem \[thm:vpl-trichotomy\] it suffices to show that either $V_L(n) \in O(\log n)$ or $V_L(n) \notin o(n)$ holds for every VPL $L$. If $L$ is a regular language and $A$ is the minimal DFA of $L$ with state set $Q$, one can identify $\nu_L(x)$ with the state $q \in Q$ reached on input $x$. Hence, $\cev{\nu}_L(x)$ is represented by a word over $Q$. Using the transition monoid of $A$ one can show that $\cev{\nu}_L \colon \Sigma^* \to Q^$ is rational (in fact [right-subsequential]{}, see Section \[sec:rat-dichotomy\]) and hence the image $\cev{\nu}_L(\Sigma^) \subseteq Q^$ is regular [@GHKLM18a Lemma 4.2]. Since the growth of $\cev{\nu}_L(\Sigma^)$ is either polynomial or exponential this implies that $V_L(n) \in O(\log n)$ or $V_L(n) \notin o(n)$. ##### Restriction to descending words. The approach above for regular languages can be extended to visibly pushdown languages $L$ if we restrict ourselves to the set $D$ of descending words. If a VPA with state set $Q$ reads a descending word $x \in D$ from the initial configuration it reaches some configuration $\bot q$ with empty stack. Notice that there may be distinct configurations $\bot p \neq \bot q$ with ${\mathsf{L}}(\bot p) = {\mathsf{L}}(\bot q)$, in which case we need to pick a single representative. Since every suffix of $x$ is again descending we can represent $\cev{\nu}_L(x)$ by a word $\sigma_0(x) \in Q^$ and in fact we will prove that $S_0 = \sigma_0(D)$ is a context-free language (Lemma \[lem:wmf-mon\]). By the growth theorem for context-free languages the growth of $S_0$ is either polynomial or exponential. If $S_0$ grows exponentially we obtain an exponential lower bound on $\|\cev{\nu}_L(\Sigma^{\le n})\|$ (Lemma \[lem:S-unbounded\]). Hence, the interesting case is that $S_0$ has polynomial growth, i.e. $S_0$ is bounded. ##### Representation by rational functions. In order to simulate a VPA by a finite automaton on arbitrary words we will “flatten” the input word in the following way. The input word $w$ is factorized $w = w_0 w_1 \cdots w_m$ into a descending prefix $w_0$, and call letters and well-matched factors $w_1, \dots, w_m$. The descending prefix $w_0$ is replaced by $\sigma_0(w_0)$ and each well-matched factor $w_i$ is replaced by a similar information $\sigma_1(w_i)$ which describes the behavior of the VPA on the factor $w_i$ and on each of its suffixes. The set ${\mathrm{Flat}}$ of all flattenings is a context-free language. Furthermore, there exists a rational function $\nu_f$ such that, if a flattening $s$ represents a word $w \in \Sigma^$ then $\nu_f(s)$ is a configuration representing the Myhill-Nerode class $\nu_L(w)$ (Proposition \[prop:nu-f\]). Hence, we can reduce proving the main theorem to the question whether the $\cev{\nu}_f$-growth of ${\mathrm{Flat}}$ is always either polynomial or exponential. This question is resolved positively as follows. We prove that for every rational function $t$ with suffix-closed domain $X = {\mathrm{dom}}(t)$ the $\cev{t}$-growth of $X$ is either polynomial or exponential (Theorem \[thm:rational-dichotomy\]). In the case that $S_0$ has polynomial growth we can overapproximate ${\mathrm{Flat}}$ by a regular superset ${\mathrm{RegFlat}}$. If the $\cev{\nu}_f$-growth of ${\mathrm{RegFlat}}$ is polynomial then the same holds trivially for the subset ${\mathrm{Flat}}$. If the $\cev{\nu}_f$-growth of ${\mathrm{RegFlat}}$ is exponential then the proper choice of ${\mathrm{RegFlat}}$ ensures that ${\mathrm{Flat}}$ also has exponential $\cev{\nu}_f$-growth (Proposition \[prop:transf-growth\]). ##### Dichotomy for rational functions. The main technical result of this paper states that for every rational function $t \colon \Sigma^* \to \Omega^$ with suffix-closed domain $X = {\mathrm{dom}}(t)$ the $\cev{t}$-growth of $X$ is either polynomial or exponential. We emphasize that the range of $\cev{t}$ is not $\Omega^$ but the free monoid over $\Omega^$ (consisting of all finite sequences of words over $\Omega$). There are in fact two reasons for exponential $\cev{t}$-growth: (i) The image $t(X)$ has exponential growth, and (ii) $X$ contains a so called linear fooling set. We need these lower bounds in the more general setting where $X \subseteq {\mathrm{dom}}(t)$ is a context-free subset, namely $X = {\mathrm{Flat}}$. \[prop:exp-image\] Let $t \colon \Sigma^ \to \Omega^$ be rational with suffix-closed domain. If $X \subseteq {\mathrm{dom}}(t)$ is context-free and $t(X)$ has exponential growth then $X$ has exponential $t$-growth and exponential $\cev{t}$-growth. \[ex:rat-trans\] Consider the transduction $f \colon \{a,b\}^ \to a^$ defined by $$f = \{ (a^n,a^n) \mid n \in {\mathbb{N}}\} \cup \{ (a^n b w,a^n) \mid n \in {\mathbb{N}}, w \in \{a,b\}^ \},$$ which projects a word over $\{a,b\}$ to its left-most (maximal) $a$-block and is rational. Its image $\cev{f}(\{a,b\}^)$ can be identified with the set of all sequences of natural numbers which are concatenations of monotonically decreasing sequences of the form $(k,k-1, \dots, 0)$. There are exactly $2^n$ of such sequences of length $n$ and hence $\{a,b\}^$ has exponential $\cev{f}$-growth. A [linear fooling scheme]{} for a partial function $t \colon \Sigma^* \to Y$ is a tuple $(u_2,v_2,u,v,Z)$ where $u_2,v_2,u,v \in \Sigma^$ and $Z \subseteq \Sigma^$ such that $u_2$ is a suffix of $u$ and $v_2$ is a suffix of $v$, $\|u_2\| = \|v_2\|$, $\{u_2,v_2\}\{u,v\}^Z \subseteq {\mathrm{dom}}(t)$ and for all $n \in {\mathbb{N}}$ there exists a word $z_n \in Z$ of length $\|z_n\| \le O(n)$ such that $t(u_2 w z_n) \neq t(v_2 w z_n)$ for all $w \in \{u,v\}^{\le n}$. The set $\{u_2,v_2\}\{u,v\}^Z$ is called a [linear fooling set]{} for $t$. Notice that the definition implies that $u_2 \neq v_2$ and hence $u$ is not a suffix of $v$, and vice versa, i.e. $\{u,v\}$ is a [suffix code]{}. Therefore $\{u,v\}^n$ contains $2^n$ words of length $O(n)$ and thus $\{u_2,v_2\}\{u,v\}^$ has exponential growth. \[prop:fooling\] Let $t \colon \Sigma^ \to \Omega^$ be a partial function with suffix-closed domain. If $X \subseteq {\mathrm{dom}}(t)$ contains a linear fooling set for $t$ then the $\cev{t}$-growth of $X$ is exponential. Let $(u_2,v_2,u,v,Z)$ be a linear fooling scheme with $\{u_2,v_2\}\{u,v\}^Z \subseteq X$. Let $n \in {\mathbb{N}}$ and let $z_n \in Z$ with the properties from the definition. Consider two distinct words $w, w' \in \{u,v\}^n$. Without loss of generality the words have the form $w = w_1 u w_2$ and $w' = w_3 v w_2$ for some $w_1,w_2,w_3 \in \{u,v\}^$. Hence $w$ has the suffix $u_2 w_2$ and $w'$ has the suffix $v_2 w_2$, which are suffixes of the same length. By assumption we have $t(u_2 w_2 z_n) \neq t(v_2 w_2 z_n)$ and hence also $\cev{t}(w z_n) \neq \cev{t}(w' z_n)$. This implies that $\|\cev{t}(u_2 \{u,v\}^n z_n)\| \ge 2^n$ for all $n \in {\mathbb{N}}$. Since all words in $u_2 \{u,v\}^n z_n \subseteq X$ have length $O(n)$ there exists a number $c > 1$ such that $\|\cev{t}(X \cap \Sigma^{\le cn})\| \ge 2^n$ for sufficiently large $n$. The following dichotomy theorem will be proved in Section \[sec:rat-dichotomy\]. \[thm:rational-dichotomy\] Let $t \colon \Sigma^ \to \Omega^$ be rational and with suffix-closed domain $X = {\mathrm{dom}}(t)$. If $X$ contains no linear fooling set for $t$ and $t(X)$ is bounded then the $\cev{t}$-growth of $X$ is polynomial. Otherwise the $\cev{t}$-growth of $X$ is exponential. Reduction to transducer problem =============================== Fix a VPA $A = (Q,\tilde \Sigma,\Gamma,\bot,q_0,\delta, F)$ and let $\emptyset \subsetneq L = {\mathsf{L}}(A) \subsetneq \Sigma^$ for the rest of this section. ##### Monotonic factorization. A factorization of $w = w_0 w_1 \cdots w_m \in \Sigma^$ into factors $w_i \in \Sigma^$ is [monotonic]{} if $w_0$ is descending (possibly empty) and for each $1 \le i \le m$ the factor $w_i$ is either a call letter $w_i \in \Sigma_c$ or a non-empty well-matched factor. If $w_0 w_1 \cdots w_m$ is a monotonic factorization then $w_i' w_{i+1} \cdots w_j$ is a monotonic factorization for any $0 \le i \le j \le m$ and suffix $w_i'$ of $w_i$. To see that every word $w \in \Sigma^$ has at least one monotonic factorization consider the set of non-empty maximal well-matched factors in $w$ (maximal with respect to inclusion). Observe that two distinct maximal well-matched factors in a word cannot overlap because the union of two overlapping well-matched factors is again well-matched. Since every internal letter is well-matched the remaining positions contain only return and call letters. Furthermore, every remaining call letter must be to the right of every remaining return letter, which yields a monotonic factorization of $w$. Figure \[fig:sc\] shows a monotonic factorization $w = w_0 w_1 \cdots w_m$ where the descending prefix $w_0$ is colored red and call letters $w_i$ are colored green. The [stack height function]{} for the word $w$ increases (decreases) by one on call (return) letters and stays constant on internal letters. ##### Representation of Myhill-Nerode classes. To apply Theorem \[thm:psiL\] we need a suitable description of the $\sim_L$-classes. We follow the approach in [@BaranyLS06] of choosing length-lexicographic minimal representative configurations. Since their definition slightly differs from ours (according to their definition, a VPA may not read a return letter if the stack contains $\bot$ only) we briefly recall their argument (in the appendix). Let ${\mathrm{rConf}}= \{ \delta(\bot q_0,w) \mid w \in \Sigma^ \}$ be the set of all [reachable]{} configurations in $A$, which is known to be regular [@BouajjaniEM97; @Buechi64]. Two configurations $c_1,c_2 \in {\mathrm{rConf}}$ are [equivalent]{}, denoted by $c_1 \sim c_2$, if ${\mathsf{L}}(c_1) = {\mathsf{L}}(c_2)$. By fixing arbitrary linear orders on $\Gamma$ and $Q$ we can consider the length-lexicographical order on ${\mathrm{rConf}}$ and define the function ${\mathsf{rep}}\colon {\mathrm{rConf}}\to {\mathrm{rConf}}$ which chooses the minimal representative from each $\sim$-class, i.e. for all $c \in {\mathrm{rConf}}$ we have ${\mathsf{rep}}(c) \sim c$ and for any $c' \in {\mathrm{rConf}}$ with $c \sim c'$ we have ${\mathsf{rep}}(c) \le_{\mathit{llex}} c'$. The set of representative configurations is denoted by ${\mathrm{Rep}}= {\mathsf{rep}}({\mathrm{rConf}})$. \[lem:rep-rat\] The function ${\mathsf{rep}}$ is rational. Finally we define $\nu_A \colon \Sigma^* \to {\mathrm{Rep}}$ by $\nu_A(w) = {\mathsf{rep}}(\delta(\bot q_0,w))$ for all $w \in \Sigma^$. It represents $\sim_L$ in the sense that ${\mathsf{L}}(\nu_A(w)) = w^{-1} {\mathsf{L}}(A)$ for all $w \in \Sigma^$ and hence $\nu_A(u) =\nu_A(v)$ if and only if $u \sim_L v$. Therefore we have $V_L(n) = \log \|\cev{\nu}_A(\Sigma^{\le n})\|$ by Theorem \[thm:psiL\]. ##### Flattenings. Since we cannot compute $\nu_A$ using a finite state transducer we choose a different representation of the input. Define the alphabet $\Sigma_f = \Sigma_c \cup Q \cup Q^Q$. A [flattening]{} is a word $s_0 s_1 \cdots s_m \in \Sigma_f^$ where $s_0 \in Q^$ and $s_i \in \Sigma_c \cup Q^Q Q^$ for all $1 \le i \le m$. Notice that the factorization $s = s_0 s_1 \cdots s_m$ is unique. The set of all flattenings is ${\mathrm{AllFlat}}= Q^ (\Sigma_c \cup Q^Q Q^)^$. We define a function $t_f \colon {\mathrm{AllFlat}}\to {\mathrm{rConf}}$ as follows. Let $s = s_0 s_1 \cdots s_m \in \Sigma_f^$ be a flattening and we define $t_f(s)$ by induction on $m$: - If $s_0 = {\varepsilon}$ then $t_f(s_0) = \bot q_0$. If $s_0 = q_1 \cdots q_n \in Q^+$ then $t_f(s_0) = \bot q_1$. - If $s_m \in \Sigma_c$ then $t_f(s_0 \cdots s_m) = \delta(t_f(s_0 \cdots s_{m-1}), s_m)$. - If $s_m = \tau q_2 \cdots q_m \in Q^Q Q^$ and $t_f(s_0 \cdots s_{m-1}) = \alpha q$ then $t_f(s) = \alpha \tau(q)$. Define the function $\nu_f \colon {\mathrm{AllFlat}}\to {\mathrm{Rep}}$ by $\nu_f = {\mathsf{rep}}\circ t_f$. \[lem:t-f-rat\] The functions $t_f$ and $\nu_f$ are rational. We first define a transducer $A_1$ which handles flattenings where the initial factor is empty. Let $A_1 = (Q,\Sigma_f,Q \cup \Gamma,\{q_0\},\Delta',Q,o)$ with the following transitions: - $p \xrightarrow{q \mid {\varepsilon}} p$ for all $p,q \in Q$ - $p \xrightarrow{a \mid \gamma} q$ for all $\delta(p,a) = (\gamma,q)$ where $a \in \Sigma_c$ - $p \xrightarrow{\tau \mid {\varepsilon}} \tau(p)$ for all $p \in Q, \tau \in Q^Q$ and $o(q) = q$. For each $p \in Q$ let $t_p$ be the rational function defined by $A_1$ with the only initial state $p$. One can easily show that for all $s \in {\mathrm{AllFlat}}$ we have $t_f(s) = \bot t_{q_0}(s)$ and $t_f(q_1 \cdots q_k s) = \bot t_{q_1}(s)$ for all $q_1 \cdots q_k \in Q^+$. Hence we can prove that $t_f$ is rational by providing a transducer for $t_f$: First it verifies whether the input word belongs to the regular language ${\mathrm{AllFlat}}\subseteq \Sigma_f^$. Simultaneously, it verifies whether the input word starts with a state $q \in Q$. If so, it memorizes $q$ and simulates $A_1$ on $s'$ from $q$, and otherwise $A_1$ is directly simulated on $s$ from $q_0$. Since ${\mathsf{rep}}$ is rational by Lemma \[lem:rep-rat\], $\nu_f$ is also rational. If $w = a_1 \cdots a_n \in D$ is a descending word then $\delta(\bot q_0, w) = \bot p$ for some $p \in Q$. By definition of $\nu_A$ there exists a state $q \in Q$ with $\nu_A(w) = \bot q$. Since each suffix of $w$ is also descending we have $\cev{\nu}_A(w) = \bot q_1 \, \bot q_2 \cdots \bot q_n$ for some $q_1, \dots, q_n \in Q$. We define $\sigma_0(w) = q_1 \cdots q_n \in Q^$, i.e. we remove the redundant $\bot$-symbols from $\cev{\nu}_A(w)$. If $w$ is non-empty and well-matched we additionally define $\sigma_1(w) = \tau q_2 \cdots q_n \in Q^Q Q^$ where $\tau = \varphi(w)$. We define the sets $S_0 = \sigma_0(D)$ and $S_1 = \sigma_1(W \setminus \{{\varepsilon}\})$. Notice that $S_0$ is exactly the set of proper suffixes of words from $S_1$ since descending words are exactly the (proper) suffixes of well-matched words. We say that $s = s_0 s_1 \cdots s_m \in {\mathrm{AllFlat}}$ [represents]{} a word $w \in \Sigma^$ if there exists a monotonic factorization $w = w_0 w_1 \cdots w_m \in \Sigma^$ such that $s_0 = \sigma_0(w_0)$, and for all $1 \le i \le m$ if $w_i$ is well-matched, then $s_i = \sigma_1(w_i)$, and if $w_i \in \Sigma_c$ then $s_i = w_i$. Since a word may have different monotonic factorizations, it may also be represented by many flattenings. We define the suffix-closed set ${\mathrm{Flat}}= S_0 (\Sigma_c \cup S_1)^$, containing all flattenings which represent some word. \[prop:nu-f\] If $s \in {\mathrm{AllFlat}}$ represents $w \in \Sigma^$ then $\nu_f(s) = \nu_A(w)$. Therefore, $\nu_f({\mathrm{Flat}}) = {\mathrm{Rep}}$ and $V_L(n) = \log \|\cev{\nu}_f({\mathrm{Flat}}\cap \Sigma_f^{\le n})\|$. \[lem:wmf-mon\] The languages $S_0$ and $S_1$ are context-free. Since $S_0$ is the set of all proper suffixes of words from $S_1$ it suffices to consider $S_1$. We will prove that $\{ w \otimes \sigma_1(w) \mid w \in W \}$ is a VPL over the pushdown alphabet $(\Sigma_c \times \Sigma_f,\Sigma_r \times \Sigma_f,\Sigma_{\mathit{int}} \times \Sigma_f)$. Since the class of context-free languages is closed under projections it then follows that $S_1$ is context-free. A VPA can test whether the first component $w = a_1 \cdots a_n$ is well-matched and whether the second component has the form $\tau q_2 \cdots q_n \in Q^Q Q^$. Since VPLs are closed under Boolean operations, it suffices to test whether $\tau \neq \varphi(w)$ or there exists a state $q_i$ with $\nu_A(a_i \cdots a_n) \neq \bot q_i$. To guess an incorrect state we use a VPA whose stack alphabet contains all stack symbols of $A$ and a special symbol $\#$ representing the stack bottom. We guess and read a prefix of the input word and push/pop only the special symbol $\#$ on/from the stack. Then at some point we store the second component $q_i$ in the next symbol and simulate $A$ on the remaining suffix. Finally, we accept if and only if the reached state is $q$ and ${\mathsf{rep}}(\bot q) \neq \bot q_i$. Similarly, we can verify $\tau$ by testing whether there exists a state $p \in Q$ with $\varphi(w)(p) \neq \tau(p)$. \[lem:S-unbounded\] The language $S_0$ is bounded if and only if $S_1$ is bounded. If $S_0$ is not bounded then the $\cev{\nu}_A$-growth of $\Sigma^$ is exponential and therefore $V_L(n) \notin o(n)$. Assume that $S_0 \subseteq s_1^ \cdots s_k^$ is bounded. Since $S_1 \subseteq \bigcup \{ \tau S_0 \mid \tau \in Q^Q \}$ we have $S_1 \subseteq \tau_1^ \cdots \tau_m^* s_1^* \cdots s_k^$ for any enumeration $\tau_1, \dots, \tau_m$ of $Q^Q$. Conversely, if $S_1$ is bounded then each word in $S_0$ is a factor, namely a proper suffix, of a word from $S_1$. Therefore $S_0$ must be also bounded. If the context-free language $S_0 = \sigma_0(D) \subseteq Q^$ is not bounded then its growth must be exponential. Recall that $\cev{\nu}_A(w)$ and $\sigma_0(w)$ are equal for all $w \in D$ up to the $\bot$-symbol. Hence $\|\cev{\nu}_A(\Sigma^{\le n})\| \ge \|\cev{\nu}_A(D \cap \Sigma^{\le n})\| = \|\sigma_0(D \cap \Sigma^{\le n})\| = \|S_0 \cap Q^{\le n}\|$, which proves the growth bound. ##### Bounded overapproximation. By Lemma \[lem:S-unbounded\] we can restrict ourselves to the case that $S_0$ and $S_1$ are bounded languages, which will be assumed in the following. We define $\Psi(a_1 \cdots a_n) = \{ (a_1,n), (a_2,n-1) \dots, (a_n,1) \}$ and $\Psi(L) = \bigcup_{w \in L} \Psi(w)$. \[lem:apx\] Let $K$ be a bounded context-free language. Then there exists a bounded regular superset $R \supseteq K$ such that $\{ \|w\| \mid w \in K \} = \{ \|w\| \mid w \in R \}$ and $\Psi(K) = \Psi(R)$, called a [bounded overapproximation]{} of $K$. We use Parikh’s theorem [@Parikh66], which implies that for every context-free language $K \subseteq \Sigma^$ the set $\{\|w\| \mid w \in K\}$ is [semilinear]{}, i.e. a finite union of arithmetic progressions, and hence $\{v \in \Sigma^ \mid \exists w \in K : \|v\|=\|w\|\}$ is a regular language. Assume that $K \subseteq w_1^* \cdots w_k^$ for some $w_1, \dots, w_k \in \Sigma^$. We define $$R = (w_1^* \cdots w_k^) \cap \{ v \in \Sigma^ \mid \exists w \in K: \|v\|=\|w\| \} \cap \{ w \in \Sigma^* \mid \Psi(w) \subseteq \Psi(K) \}.$$ Clearly, $K$ is contained in $R$ and it remains to verify that the third part is regular. It suffices to show that for each $a \in \Sigma$ the set $P_a = \{ i \mid (a,i) \in \Psi(K) \}$ is semilinear because then an automaton can verify the property $\Psi(w) \subseteq \Psi(K)$. Consider the transducer $$T_a = \{ (a_1 \cdots a_n,\Box^{n-i+1}) \mid a_1 \cdots a_n \in \Sigma^, \, a_i = a \}.$$ It is easy to see that $T_a$ is rational and $T_aK = \{ \Box^i \mid i \in P_a \}$. The claim follows again from Parikh’s theorem. For each $\tau \in Q^Q$ let $R_\tau$ be a bounded overapproximation of $\tau^{-1} S_1$ and let $R_1 = \bigcup_{\tau \in Q^Q} (\tau R_\tau)$. Let $R_0 = \bigcup_{\tau \in Q^Q} {\mathsf{Suf}}(R_\tau)$, which is the set of all proper suffixes of words in $R_1$. Both $R_0$ and $R_1$ are also bounded languages. Finally, set ${\mathrm{RegFlat}}= R_0 (\Sigma_c \cup R_1)^$, which is the same as ${\mathsf{Suf}}((\Sigma_c \cup R_1)^)$ and is suffix-closed. According to the definition of bounded overapproximations we can approximate a word $v = \tau q_2 \cdots q_k \in R_1$ in two possible ways: Firstly, define ${\mathsf{apx}}_\ell(v)$ to be any word of the form ${\mathsf{apx}}_\ell(v) = \tau p_2 \cdots p_k \in S_1$ with $\|v\| = \|{\mathsf{apx}}_\ell(v)\|$. Secondly, for any position $2 \le i \le k$ define ${\mathsf{apx}}_i(v)$ to be any word ${\mathsf{apx}}_i(v) = \tau s' q_i p_{i+1} \cdots p_k \in S_1$ where $s', p_{i+1} \cdots p_k \in Q^$. If $r = r_0 r_1 \cdots r_m \in {\mathrm{RegFlat}}$ then we can replace any internal factor $r_i \in R_1$ by ${\mathsf{apx}}_\ell(r_i)$ or any ${\mathsf{apx}}_j(r_i)$ without changing the value of $\nu_f(r)$. \[prop:flat-reg-eq\] $\nu_f({\mathrm{Flat}}) = \nu_f({\mathrm{RegFlat}}) = {\mathrm{Rep}}$. \[prop:transf-growth\] If ${\mathrm{RegFlat}}$ contains a linear fooling set for $\nu_f$ then also ${\mathrm{Flat}}$ contains a linear fooling set for $\nu_f$. If $L = \emptyset$ or $L = \Sigma^$ then $V_L(n) \in O(1)$. Now assume $\emptyset \subsetneq L \subsetneq \Sigma^$, in which case we have $V_L(n) = \Omega(\log n)$. Furthermore we know that $V_L(n) = \log \|\cev{\nu}_f({\mathrm{Flat}}\cap \Sigma_f^{\le n})\|$ by Proposition \[prop:nu-f\]. If the constructed language $S_0$ is not bounded then $V_L(n) \notin o(n)$ by Lemma \[lem:S-unbounded\]. Now assume that $S_0$ is bounded, in which case we can construct the regular language ${\mathrm{RegFlat}}$. By Theorem \[thm:rational-dichotomy\] the $\cev{\nu}_f$-growth of ${\mathrm{RegFlat}}$ is either polynomial or exponential (formally, we have to restrict the domain of $\nu_f$ to the regular language ${\mathrm{RegFlat}}$). If the $\cev{\nu}_f$-growth of ${\mathrm{RegFlat}}$ is polynomial then the same holds for its subset ${\mathrm{Flat}}$, and hence $V_L(n) \in O(\log n)$. If the $\cev{\nu}_f$-growth of ${\mathrm{RegFlat}}$ is exponential then by Theorem \[thm:rational-dichotomy\] either the image $\nu_f({\mathrm{RegFlat}})$ is not bounded or ${\mathrm{RegFlat}}$ contains a linear fooling set for $\nu_f$. By Proposition \[prop:flat-reg-eq\] we have $\nu_f({\mathrm{RegFlat}}) = \nu_f({\mathrm{Flat}}) = {\mathrm{Rep}}$. Hence, if ${\mathrm{Rep}}$ has exponential growth then Proposition \[prop:exp-image\] implies that ${\mathrm{Flat}}$ has exponential $\cev{\nu}_f$-growth and hence $V_L(n) \notin o(n)$. If ${\mathrm{RegFlat}}$ contains a linear fooling set for $\nu_f$ then also ${\mathrm{Flat}}$ contains one by Proposition \[prop:transf-growth\]. By Proposition \[prop:fooling\] the $\cev{\nu}_f$-growth of ${\mathrm{Flat}}$ is exponential and hence $V_L(n) \notin o(n)$. Dichotomy for rational functions {#sec:rat-dichotomy} ================================ In this section we will prove Theorem \[thm:rational-dichotomy\]. Let $t \colon \Sigma^* \to \Omega^$ be a rational function with suffix-closed domain $X = {\mathrm{dom}}(t)$. By Proposition \[prop:exp-image\] the interesting case is where the image $t(X)$ is polynomial growing, i.e. a bounded language. There are two further necessary properties in order to achieve polynomial $\cev{t}$-growth. Since we apply the rational function to all suffixes, it is natural to consider right transducers, reading the input from right to left. The first property states that $t$ has to resemble so called right-subsequential functions, which are defined by deterministic finite right transducers. Here we will make use of a representation of rational functions due to Reutenauer and Schützenberger, which decomposes the rational function $t$ into a right congruence ${\mathcal{R}}_t$ and a right-subsequential transducer $B$ [@ReutenauerS91]. Secondly, we demand that $B$ is well-behaved, which means that, roughly speaking, the output produced during a run inside a strongly connected component only depends on its entry state and the length of the run. We will prove that in fact these properties are sufficient for the polynomial $\cev{t}$-growth and in all other cases $X$ contains a linear fooling set. ##### The case of finite-index right congruences. Let $\sim$ be a finite index right congruence on $\Sigma^$ and $\approx$ its suffix expansion. We will characterize those finite index right congruences $\sim$ where $\Sigma^{\le n}/{\approx}$ is polynomially bounded, which can be viewed as a special case of Theorem \[thm:rational-dichotomy\] since $\nu_\sim \colon \Sigma^* \to \Sigma^/{\sim}$ is rational. First assume that $\sim$ is the Myhill-Nerode right congruence $\sim_L$ of a regular language $L$. Since $\log \|\Sigma^{\le n}/{\approx}\|$ is exactly the space complexity $V_L(n)$ by Theorem \[thm:psiL\], this case was characterized in [@GHKLM18] using so called critical tuples in the minimal DFA for $L$. We slightly adapt this definition for right congruences. A [critical tuple]{} in a right congruence $\sim$ is a tuple of words $(u_2, v_2, u, v) \in (\Sigma^)^4$ such that $\|u_2\| = \|v_2\| \ge 1$, there exist $u_1, v_1 \in \Sigma^$ with $u = u_1u_2$, $v = v_1v_2$, and $u_2 w \not \sim v_2 w$ for all $w \in \{u,v\}^$. \[prop:ct\] If $\sim$ has a critical tuple then $\|\Sigma^{\le n}/{\approx}\|$ grows exponentially and there exists a critical tuple $(u_2, v_2, u, v)$ in $\sim$ such that $u_2 u \sim u_2 w u$ and $v_2 u \sim v_2 w u$ for all $w \in \{u,v\}^$. If $(u_2, v_2, u, v)$ is critical tuple in a right congruence $\sim$ then we claim that $\|\Sigma^{\le n}/{\approx}\|$ grows exponentially. Let $n \in {\mathbb{N}}$ and let $w \neq w' \in \{u,v\}^n$. There exists a word $z \in \{u,v\}^$ such that $w$ and $w'$ have the suffixes $u_2 z$ and $v_2 z$ of equal length. By the definition of critical tuples we have $u_2 z \not \sim v_2 z$, which implies $w \not \approx w'$. Therefore $\|\Sigma^{\le cn}/{\approx}\| \ge 2^n$ where $c = \max \{\|u\|,\|v\|\}$. The second part is based on the proof of [@GHKLM18a Lemma 7.4]. Let $\equiv$ be the syntactic congruence on $\Sigma^$ defined by $x \equiv y$ if and only if $\ell x \sim \ell y$ for all $\ell \in \Sigma^$. Since $\sim$ is a right congruence $\equiv$ is a congruence on $\Sigma^$ of finite index satisfying $\equiv \, \subseteq \, \sim$. Define the monoid $M = \Sigma^/{\equiv}$. It is known that there exists a number $\omega \in {\mathbb{N}}$ such that $m^\omega$ is idempotent for all $m \in M$, i.e. $m^\omega \cdot m^\omega = m^\omega$. Now let $(u_2, v_2, u, v)$ be a critical tuple and define $u' = (v^\omega u^\omega)^\omega$ and $v' = (v^\omega u^\omega)^\omega v^\omega$. Since $u_2$ is a suffix of $u'$, $v_2$ is a suffix of $u'$ and $u',v' \in \{u,v\}^$ the tuple $(u_2,v_2,u',v')$ is again a critical tuple in $\sim$. Furthermore we have $u'u' = (v^\omega u^\omega)^\omega (v^\omega u^\omega)^\omega \equiv (v^\omega u^\omega)^\omega = u'$ and $v'u' = (v^\omega u^\omega)^\omega v^\omega (v^\omega u^\omega)^\omega \equiv (v^\omega u^\omega)^\omega = u'$, and therefore $u' \equiv wu'$ for all $w \in \{u',v'\}^$. Since $\equiv$ is a congruence this implies $u_2 u' \equiv u_2 w u'$ and $v_2 u' \equiv v_2 w u'$ for all $w \in \{u',v'\}^$, and thus also $u_2 u' \sim u_2 w u'$ and $v_2 u' \sim v_2 w u'$, which concludes the proof. \[thm:log\] Let $L \subseteq \Sigma^$ be regular. Then $V_L(n) \in O(\log n)$ if and only if $\|\Sigma^{\le n}/{\approx_L}\|$ is polynomially bounded if and only if $\sim_L$ has no critical tuple. The first equivalence follows from Theorem \[thm:psiL\]. By Proposition \[prop:ct\] the existence of a critical tuple in $\sim$ implies exponential growth of $\|\Sigma^{\le n}/{\approx}\|$. Now assume that $V_L(n) \notin O(\log n)$. By [@GHKLM18 Lemma 7.2] there exist words $u_2,v_2,u,v \in \Sigma^$ such that $u_2$ is a suffix of $u$, $v_2$ is a suffix of $v$, $\|u_2\| = \|v_2\|$ and $u_2 w \not \sim_L v_2 w'$ for all $w,w' \in \{u,v\}^$ (one needs the fact that $x \sim_L y$ if and only if $x$ and $y$ reach the same state in the minimal DFA for $L$). Since in particular $u_2 w \not \sim_L v_2 w$ for all $w \in \{u,v\}^$ the tuple $(u_2, v_2, u, v)$ constitutes a critical tuple. We generalize this theorem to arbitrary finite index right congruences (Theorem \[thm:log-cong\]). Given equivalence relations $\sim$ and $\sim'$ on a set $X$, we say that $\sim'$ is [coarser]{} than $\sim$ if $\sim \, \subseteq \, \sim'$, i.e. each $\sim'$-class is a union of $\sim$-classes. The [intersection]{} $\sim \cap \sim'$ is again an equivalence relation on $X$. \[lem:ct-coarse-intersect\] Let $\sim$ and $\sim'$ be right congruences. (a) If $\sim'$ is coarser than $\sim$ and $\sim$ has no critical tuple, then $\sim'$ also has no critical tuple. (b) If $\sim$ and $\sim'$ have no critical tuple then $\sim \cap \sim'$ is also a right congruence which has no critical tuple Closure under coarsening is clear because the property “$\sim$ has no critical tuple” is [positive]{} in $\sim$: $\forall u = u_1u_2 \, \forall v = v_1v_2 (\|u_2\| = \|v_2\| \to \exists w \in \{u,v\}^ : u_2 w \sim v_2 w)$. Consider two right congruences $\sim$, $\sim'$ which have no critical tuples. One can verify that their intersection $\sim \cap \sim'$ is again a right congruence. Let $u = u_1u_2$ and $v = v_1v_2$ with $\|u_2\| = \|v_2\|$. Because $\sim$ has no critical tuple there exist a word $w \in \{u,v\}^$ with $u_2 w \mathrel{\sim} v_2 w$. Now consider the condition for the words $u_1(u_2w)$ and $v_1(v_2w)$. Because $\sim'$ has no critical tuple there exists a word $x \in \{uw, vw\}^$ such that $u_2 wx \mathrel{\sim'} v_2 wx$. Since $\sim$ is a right congruence we also have $u_2 wx \mathrel{\sim} v_2 wx$ and thus $u_2 wx \mathrel{(\sim \cap \sim')} v_2 wx$. This proves that $\sim \cap \sim'$ has no critical tuple. \[thm:log-cong\] $\|\Sigma^{\le n}/{\approx}\|$ is polynomially bounded if and only if $\sim$ has no critical tuple. Let $u_1, \dots, u_m$ be representatives from each $\sim$-class. Observe that $\sim \, = \bigcap_{i=1}^m \sim_{[u_i]_\sim}$ because $\sim$ saturates each class $[u_i]_\sim$ and $\bigcap_{i=1}^m \sim_{[u_i]_\sim}$ also saturates each class $[v]_\sim$. Let us write $\sim_i$ instead of $\sim_{[u_i]_\sim}$ and let $\approx_i$ be its suffix expansion $\approx_{[u_i]_\sim}$. Then we have $\sim \, = \bigcap_{i=1}^m \sim_i$ and $\approx \, = \bigcap_{i=1}^m \approx_i$. This implies that $$\label{eq:poly-related} \max_{1 \le i \le m} \|\Sigma^{\le n}/{\approx_i}\| \le \|\Sigma^{\le n}/{\approx}\| \le \prod_{i=1}^m \|\Sigma^{\le n}/{\approx_i}\|.$$ ($\Rightarrow$): If $\|\Sigma^{\le n}/{\approx}\|$ is polynomially bounded then the same holds for $\|\Sigma^{\le n}/{\approx_i}\|$ for all $1 \le i \le k$ by . By Theorem \[thm:log\] $\sim_{[u_i]_{\sim}}$ has no critical tuple for all $1 \le i \le k$ and therefore Lemma \[lem:ct-coarse-intersect\](b) implies that $\sim \, = \bigcap_{i=1}^m \sim_{[u_i]_\sim}$ has no critical tuple. ($\Leftarrow$): If $\sim$ has no critical tuple then each congruence $\sim_i$ has no critical tuple by Lemma \[lem:ct-coarse-intersect\](a) because $\sim_i$ is coarser than $\sim$. Theorem \[thm:log\] implies that $\|\Sigma^{\le n}/{\approx_i}\|$ is polynomially bounded for all $1 \le i \le k$. By also $\|\Sigma^{\le n}/{\approx}\|$ is polynomially bounded. ##### Regular look-ahead. A result due to Reutenauer and Schützenberger states that every rational function $f$ can be factorized as $f = r \circ \ell$ where $\ell$ and $r$ are [left- and right-subsequential]{}, respectively [@ReutenauerS91]. A rational function is left- or right-subsequential if the input is read in a deterministic fashion from left to right and right to left, respectively. In the literature the order of the directions is usually reversed, i.e. one decomposes $t$ as $f = r \circ \ell$. Often this is described by the statement that every rational function is (left-)subsequential with regular look-ahead. Furthermore, this decomposition is canonical in a certain sense. We follow the notation from the survey paper [@FiliotR16]. A [right-subsequential transducer]{} $B = (Q,\Sigma,\Omega,F,\Delta,\{q_{\mathit{in}}\},o)$ is a real-time right transducer which is [deterministic]{}, i.e. $q_{\mathit{in}}$ is the only initial state and for every $p \in Q$ and $a \in \Sigma$ there exists at most one transition $(p,a,y,q) \in \Delta$. Clearly, right-subsequential transducers define rational functions, the so called [right-subsequential functions]{}, but not every rational function is right-subsequential. Let ${\mathcal{R}}$ be a right congruence on $\Sigma^$ with finite index. The [look-ahead extension]{} is the injective function $e_{\mathcal{R}}\colon \Sigma^ \to (\Sigma \times \Sigma^/{\mathcal{R}})^$ defined by $$e_{\mathcal{R}}(a_1 \cdots a_n) = (a_1,[{\varepsilon}]_{\mathcal{R}}) (a_2,[a_1]_{\mathcal{R}}) (a_3,[a_1a_2]_{\mathcal{R}}) \cdots (a_n,[a_1 \cdots a_{n-1}]_{\mathcal{R}}).$$ Let $f \colon \Sigma^* \to \Omega^$ be a partial function. The partial function $f[{\mathcal{R}}] \colon (\Sigma \times \Sigma^/{\mathcal{R}})^* \to \Omega^$ with ${\mathrm{dom}}(f[{\mathcal{R}}]) = e_{\mathcal{R}}({\mathrm{dom}}(f))$ is defined by $f[{\mathcal{R}}](e_{\mathcal{R}}(x)) = f(x)$. Furthermore we define a right congruence ${\mathcal{R}}_f$ on $\Sigma^$. For this we need the distance function $\\|x,y\\| = \|x\| + \|y\| - 2\|x \wedge y\|$ where $x \wedge y$ is the longest common suffix of $x$ and $y$. Equivalently, $\\|x,y\\|$ is the length of the reduced word of $xy^{-1}$ in the free group generated by $\Sigma$. Notice that $\\|\cdot,\cdot\\|$ satisfies the triangle inequality. We define $u \mathrel{{\mathcal{R}}_f} v$ if and only if (i) $u \sim_{{\mathrm{dom}}(f)} v$ and (ii) $\{ \\|f(uw),f(vw)\\| \mid uw,vw \in {\mathrm{dom}}(f) \}$ is finite. One can verify that ${\mathcal{R}}_f$ is a right congruence on $\Sigma^$. As an example, recall the rational transduction $f$ from Example \[ex:rat-trans\]. The induced right congruence ${\mathcal{R}}_f$ has two classes, which are $a^$ and $a^b\{a,b\}^$. A partial function $f \colon \Sigma^* \to \Omega^$ is rational if and only if ${\mathcal{R}}_f$ has finite index and $f[{\mathcal{R}}_f]$ is right-subsequential. For the rest of the section let $B = (Q,\Sigma \times \Sigma^/{{\mathcal{R}}_t},\Omega,F,\Delta,\{q_{\mathit{in}}\},o)$ be a trim right-subsequential transducer for $t[{\mathcal{R}}_t]$. One obtains an unambiguous real-time right transducer $A$ for $t$ by projection to the first component, i.e. $A = (Q,\Sigma,\Omega,F,\Lambda,\{q_{\mathit{in}}\},o)$ where $\Lambda = \{ (q,a,y,p) \mid (q,(a,b),y,p) \in \Delta \}$. Notice that every run $q \xleftarrow{x \mid y} p$ in $A$ induces a corresponding run $q \xleftarrow{(x,z) \mid y} p$ in $B$ for some $z \in (\Sigma^/{{\mathcal{R}}_t})^$ and that this correspondence is a bijection between the sets of all runs in $A$ and $B$. We need two auxiliary lemmas which concern the right congruence ${\mathcal{R}}_t$. \[lem:linear-distance\] Let $u,v,w \in \Sigma^$ with $uw,vw \in X$. If $u \mathrel{{\mathcal{R}}}_t v$ then $\\|t(uw),t(vw)\\| \le O(\|u\|+\|v\|)$. Two partial functions $t_1, t_2 \colon \Sigma^ \to \Omega^$ are [adjacent]{} if $\sup \{\\|t_1(w),t_2(w)\\| \mid w \in {\mathrm{dom}}(t_1) \cap {\mathrm{dom}}(t_2) \} < \infty$ where $\sup \emptyset = -\infty$. We remark that two functions are adjacent in our definition if and only if their reversals are adjacent according to the original definition [@ReutenauerS91]. Notice that $u \mathrel{{\mathcal{R}}_t} v$ if and only if $u \sim_X v$ and the functions $w \mapsto t(uw)$ and $w \mapsto t(vw)$ are adjacent. \[lem:short-witness\] Let $t_1, t_2 \colon \Sigma^ \to \Omega^$ be rational functions which are not adjacent. Then there are words $x,y,z \in \Sigma^$ such that $xy^z \subseteq {\mathrm{dom}}(t_1) \cap {\mathrm{dom}}(t_2)$ and $\\|t_1(x y^k z),t_2(x y^k z)\\| = \Omega(k)$. In particular, for each $k \in {\mathbb{N}}$ there exists a word $x \in {\mathrm{dom}}(t_1) \cap {\mathrm{dom}}(t_2)$ of length $\|x\| \le O(k)$ such that $\\|t_1(x),t_2(x)\\| \ge k$. \[prop:crit-tup\] If ${\mathcal{R}}_t$ has a critical tuple then $X$ contains a linear fooling set. Let $(u_2,v_2,u,v)$ be a critical tuple in ${\mathcal{R}}_t$ with $u = u_1u_2$ and $v = v_1v_2$. By Proposition \[prop:ct\] we can assume that $u_2 u \mathrel{{\mathcal{R}}_t} u_2 w u$ and $v_2 u \mathrel{{\mathcal{R}}_t} v_2 w u$ for all $w \in \{u,v\}^$. By assumption we know that $(u_2 u, v_2 u) \notin {\mathcal{R}}_t$. Furthermore, we claim that $u_2 u \sim_X v_2 u$: Let $z \in \Sigma^$ and assume that $u_2 u z \in X$. Then $u_2 v_1 v_2 u z \in X$ because $u_2 u \sim_X u_2 v_1 v_2 u$, and thus $v_2 u z \in X$ because $X$ is suffix-closed. The other direction follows by a symmetric argument. Let $n \in {\mathbb{N}}$ and define $$N = \max_{x \in \{u_2,v_2\}} \max_{w \in \{u,v\}^{\le n}} \sup \{ \\|t(x u z),t(x w u z)\\| \mid x u z,x w u z \in X \} < \infty.$$ By Lemma \[lem:linear-distance\] we have $N \le O(n)$. Since $(u_2 u, v_2 u) \notin {\mathcal{R}}_t$ and $u_2 u \sim_X v_2 u$, the functions $z \mapsto t(u_2 u z)$ and $z \mapsto t(v_2 u z)$ are not adjacent. By Lemma \[lem:short-witness\] there exists a word $z_n \in (u_2u)^{-1}X$ with $\\|t(u_2 u z_n), t(v_2 u z_n) \\| \ge 2N+1$ and $\|z_n\| \le O(N) \le O(n)$. We claim that $t(u_2 w u z_n) \neq t(v_2 w u z_n)$ for all $w \in \{u,v\}^{\le n}$: By the triangle inequality we have $$\begin{aligned} 2N+1 &\le \\| t(u_2 u z_n), t(v_2 u z_n) \\| \\ &\le \\| t(u_2 u z_n), t(u_2 w u z_n) \\| + \\| t(u_2 w u z_n), t(v_2 w u z_n) \\| + \\| t(v_2 w u z_n) , t(v_2 u z_n) \\| \\ &\le 2N + \\| t(u_2 w u z_n), t(v_2 w u z_n) \\| \end{aligned}$$ which implies $\\| t(u_2 w u z_n), t(v_2 w u z_n) \\| \ge 1$ and in particular $t(u_2 w u z_n) \neq t(v_2 w u z_n)$. We have proved that for each $n \in {\mathbb{N}}$ there exists a word $z_n$ of length $O(n)$ such that $t(u_2 w u z_n) \neq t(v_2 w u z_n)$ for all $w \in \{u,v\}^{\le n}$. If $Z$ is the set of all constructed $z_n$ for $n \in {\mathbb{N}}$ then $\{u_2,v_2\} \{u,v\}^ u Z \subseteq X$ and $(u_2,v_2,u,v, uZ)$ is a linear fooling scheme. ##### Well-behaved transducers. Let $(Q,\preceq)$ be the quasi-order defined by $q \preceq p$ iff there exists a run from $p$ to $q$ in $A$ or equivalently in $B$. Its equivalence classes are the strongly connected components (SCCs) of $A$ and $B$. A word $w \in \Sigma^$ is [guarded]{} by a state $p \in Q$ if there exists a run $q' \xleftarrow{w} p$ in $A$ such that $p \preceq q'$, i.e. $p$ and $q'$ belong to the same SCC. Notice that the set of all words which are guarded by a fixed state $p$ is suffix-closed. A run $q \xleftarrow{w} p$ in $A$ is [guarded]{} if $w$ is guarded by $p$. We say that $A$ is [well-behaved]{} if for all $p \in Q$ and all guarded accepting runs $\pi, \pi'$ from $p$ with $\|\pi\| = \|\pi'\|$ we have ${\mathsf{out}}_F(\pi) = {\mathsf{out}}_F(\pi')$. \[prop:wb\] If $A$ is not well-behaved then $X$ contains a linear fooling set. Assume there exist states $p,q,r,q',r' \in Q$, and accepting runs $q \xleftarrow{u_2} p$ and $r \xleftarrow{v_2} p$ with $\|u_2\| = \|v_2\|$ and ${\mathsf{out}}_F(q \xleftarrow{u_2} p) \neq {\mathsf{out}}_F(r \xleftarrow{v_2} p)$. Furthermore let $p \xleftarrow{u_1} q' \xleftarrow{u_2} p$, $p \xleftarrow{v_1} r' \xleftarrow{v_2} p$ and $p \xleftarrow{s} q_{\mathit{in}}$ be runs. Let $u = u_1u_2$ and $v = v_1v_2$ and consider any word $w \in \{u,v\}^$. Since $t(u_2ws) = {\mathsf{out}}_F(q \xleftarrow{u_2} p) \, {\mathsf{out}}(p \xleftarrow{ws} q_{\mathit{in}})$ and $t(v_2ws) = {\mathsf{out}}_F(r \xleftarrow{v_2} p) \, {\mathsf{out}}(p \xleftarrow{ws} q_{\mathit{in}})$, we have $t(u_2ws) \neq t(v_2ws)$. This shows that $(u_2,v_2,u,v,\{s\})$ is a linear fooling scheme. If $\pi$ is a non-empty run $p \xleftarrow{a_1 \cdots a_n} q$ in $A$ and $p \xleftarrow{(a_1,\rho_1) \cdots (a_n,\rho_n)} q$ is the corresponding run in $B$ then we call $\rho_1$ the [key]{} of $\pi$. The following lemma justifies the name, stating that $\pi$ is determined by the state $q$, the word $a_1 \cdots a_n$ and the key $\rho_1$. \[lem:key\] If $p \xleftarrow{w} q$ and $p' \xleftarrow{w} q$ are non-empty runs in $A$ with the same key then the runs must be identical. Assume that $w = a_1 \cdots a_n$ and let $p \xleftarrow{(a_1,\rho_1) \cdots (a_n,\rho_n)} q$ and $p' \xleftarrow{(a_1,\rho'_1) \cdots (a_n,\rho'_n)} q$ be the corresponding runs in $B$ with $\rho_1 = \rho_1'$. We proceed by induction on $n$. If $n = 1$ then this statement is trivial because $B$ is deterministic. Now assume $n \ge 2$ and let $p \xleftarrow{a_1} r \xleftarrow{a_2 \cdots a_n} q$ and $p' \xleftarrow{a_1} r' \xleftarrow{a_2 \cdots a_n} q$. Since $B$ is trim there exist an accepting run on $e_{{\mathcal{R}}_t}(u)$ from $p$ and an accepting run on $e_{{\mathcal{R}}_t}(u')$ from $p'$ for some words $u,u' \in \Sigma^$. By definition of $t[{\mathcal{R}}_t]$ we have $[u]_{{\mathcal{R}}_t} = \rho_1 = \rho'_1 = [u']_{{\mathcal{R}}_t}$ and therefore $\rho_2 = [ua_1]_{{\mathcal{R}}_t} = [u'a_1]_{{\mathcal{R}}_t} = \rho_2'$. By induction hypothesis we know that the runs $r \xleftarrow{a_2 \cdots a_n} q$ and $r' \xleftarrow{a_2 \cdots a_n} q$ are identical. Since $p \xleftarrow{(a_1,\rho_1)} r$ and $p' \xleftarrow{(a_1,\rho_1')} r'$ and $B$ is deterministic we must have $p = p'$. Let $\pi$ be any run on a word $y \in \Sigma^$. If $\pi$ is not guarded, we can factorize $\pi = \pi' \pi''$ such that $\pi''$ is the shortest suffix of $\pi$ which is unguarded, and then iterate this process on $\pi'$. This yields unique factorizations $\pi = \pi_0 \pi_1 \cdots \pi_m$ and $y = y_0 y_1 \cdots y_m$ where $\pi_i$ is a run on $y_i$ from a state $q_i$ to a state $q_{i-1}$ such that $y_i$ is the shortest suffix of $y_0 \cdots y_i$ which is not guarded by $q_i$ for all $1 \le i \le m$ and $\pi_0$ is guarded. The factorization $\pi = \pi_0 \pi_1 \cdots \pi_m$ is the [guarded factorization]{} of $\pi$. \[prop:poly\] Assume that $t(X)$ is bounded, $A$ is well-behaved and ${\mathcal{R}}_t$ has no critical tuple. Then the $\cev{t}$-growth of $X$ is polynomially bounded. We will describe an encoding of $\cev{t}(w)$ for $w \in X$ using $O(\log \|w\|)$ bits. For each word $w \in \Sigma^$ and each state $q \in Q$ we define a tree $T_{q,w}$ recursively, which carries information at the nodes and edges. If $w$ is guarded by $q$ then $T_{q,w}$ consists of a single node labelled by the pair $(q,\|w\|)$. Otherwise let $w = uv$ such that $v$ is the shortest suffix of $w$ which is not guarded by $q$. Then $T_{q,w}$ has a root which is labelled by the tuple $(q,\|w\|,\|v\|,\cev{\nu}_{{\mathcal{R}}_t}(u))$. For each run $p \xleftarrow{v} q$ we attach $T_{p,u}$ to the root as a direct subtree. The edge is labelled by the pair $(\rho,{\mathsf{out}}(p \xleftarrow{v} q))$ where $\rho$ is the key of $p \xleftarrow{v} q$. By Lemma \[lem:key\] distinct outgoing edges from the root are labelled by distinct keys. The tree $T_{q,w}$ can be encoded using $O(\log \|w\|)$ bits: Since we have $p \prec q$ for every unguarded run $p \xleftarrow{v} q$ the tree $T_{q,w}$ has height at most $\|Q\|$ and size at most $\|Q\|^{\|Q\|}$. All occurring numbers have at most magnitude $\|w\|$, and the states and keys can be encoded by $O(1)$ bits. The output words ${\mathsf{out}}(p \xleftarrow{v} q)$ are factors of words from the bounded language $t(X)$ and have length at most ${\mathsf{iml}}(A) \cdot \|v\|$. Thus they can be encoded using $O(\log \|w\|)$ bits. The node label $\cev{\nu}_{{\mathcal{R}}_t}(u)$ can be encoded using $O(\log \|w\|)$ bits by Theorem \[thm:log-cong\] since ${\mathcal{R}}_t$ has no critical tuple. Let $w = xy \in \Sigma^$, $q \in Q$ and let $\pi$ be an accepting run on $y$ from $q$. We show that $T_{q,w}$ and $\|y\|$ determine ${\mathsf{out}}_F(\pi)$ by induction on the length of the guarded factorization $\pi = \pi_0 \pi_1 \cdots \pi_m$. Since $T_{q_{\mathit{in}},w}$ determines the length $\|w\|$, the tuple $\cev{t}(w)$ is determined by $T_{q_{\mathit{in}},w}$ for all $w \in X$. If $m = 0$ then $y$ is guarded by $q$. Since $A$ is well-behaved ${\mathsf{out}}_F(\pi)$ is determined by $q$ (which is part of the label of the root of $T_{q,w}$) and $\|y\|$ only. Now assume $m \ge 1$ and suppose that $\pi_i$ is a run $q_{i-1} \xleftarrow{y_i} q_i$ for all $1 \le i \le m$ with $q_m = q$. Then $y_m$ is the shortest suffix of $w$ which is not guarded by $q$. The root of $T_{q,w}$ is labelled by $(q,\|y_m\|,\cev{\nu}_{{\mathcal{R}}_t}(x y_0 \cdots y_{m-1}))$. Since $\|y_m\|$ and $\|y\|$ are known, we can also determine $\|y_0 \cdots y_{m-1}\|$. From $\cev{\nu}_{{\mathcal{R}}_t}(x y_0 \cdots y_{m-1})$ and $\|y_0 \cdots y_{m-1}\|$ we can then determine $[y_0 \cdots y_{m-1}]_{{\mathcal{R}}_t}$, which is the key of $\pi_m$. By Lemma \[lem:key\] we can find the unique edge which is labelled by $([y_0 \cdots y_{m-1}]_{{\mathcal{R}}_t},{\mathsf{out}}(\pi_m))$. It leads to the direct subtree $T_{q_{m-1},x y_0 \cdots y_{m-1}}$ of $T_{q,w}$. By induction hypothesis $T_{q_{m-1},x y_0 \cdots y_{m-1}}$ and $\|y_0 \cdots y_{m-1}\|$ determine ${\mathsf{out}}_F(\pi_0 \cdots \pi_{m-1})$. Finally, we can determine ${\mathsf{out}}_F(\pi_0 \cdots \pi_m) = {\mathsf{out}}_F(\pi_0 \cdots \pi_{m-1}) \, {\mathsf{out}}(\pi_m)$, concluding the proof. Now we can prove Theorem \[thm:rational-dichotomy\]: If $X$ contains no linear fooling set for $t$ then $A$ must be well-behaved by Proposition \[prop:wb\] and ${\mathcal{R}}_t$ has no critical tuple by Proposition \[prop:crit-tup\]. If additionally $t(X)$ is bounded then the $\cev{t}$-growth of $X$ is polynomially bounded by Proposition \[prop:poly\]. Otherwise, if either $X$ contains a linear fooling set or $t(X)$ is not bounded then the $\cev{t}$-growth of $X$ is exponential by Proposition \[prop:fooling\] and by Proposition \[prop:exp-image\]. Proof of Proposition \[prop:exp-image\] ======================================= Since $\cev{t}(x)$ determines $t(x)$ we have $\|\cev{t}(X \cap \Sigma^{\le n})\| \ge \|t(X \cap \Sigma^{\le n})\|$ for all $n \in {\mathbb{N}}$. It suffices to show that every non-empty preimage $t^{-1}(\{y\})$ contains at least one word of length $O(\|y\|)$ in $X$, i.e. there exists a number $c > 0$ such that $t(X) \cap \Omega^{\le n} \subseteq t(X \cap \Sigma^{\le cn})$ for sufficiently large $n \in {\mathbb{N}}$. Then, if by assumption $\|t(X) \cap \Omega^{\le n}\|$ grows exponentially, then so does $\|t(X \cap \Sigma^{\le n})\|$ and also $\|\cev{t}(X \cap \Sigma^{\le n})\|$. Let us now prove the claim, for which we need to define context-free grammars over arbitrary monoids. A context-free grammar over a monoid $M$ has the form $G = (N,S,\to_G)$ where $N$ is a finite set of nonterminals (which is disjoint from $M$), $S$ is the starting nonterminal, and $\to_G \, \subseteq N \times (M * N^)$ is a finite set of productions where $M N^$ is the free product of the monoids $M$ and $N^$. A derivation tree for $m \in M$ is a node-labelled rooted ordered tree with the following properties: - Inner nodes are labelled by nonterminals $A \in N$. - Leaves are labelled by monoid elements $m \in M$. - If a node $s$ has children $s_1, \dots, s_k$ where $v$ is labelled by $A$ and $s_1, \dots, s_k$ are labelled by $\alpha_1, \dots, \alpha_k$ then there exists a production $A \to_G \alpha_1 \cdots \alpha_k$. - If $m_1, \dots, m_\ell$ are the labels of the leaves read from left to right then $m = m_1 \cdots m_\ell$. The language ${\mathsf{L}}(A)$ generated by a nonterminal $A \in N$ is the set of all elements $m \in M$ such that there exists a derivation tree for $m$ whose root is labelled by $A$. The language ${\mathsf{L}}(G)$ generated by $G$ is the language ${\mathsf{L}}(S)$. We first construct from a context-free grammar $G = (N,S,\to_G)$ for $X \subseteq \Sigma^$ a context-free grammar $H = (N',S',\to_H)$ for $t\|_X = \{ (x,t(x)) \mid x \in X \}$ over the product monoid $\Sigma^ \times \Omega^$. We can assume that ${\varepsilon}\notin X$ and that $G$ is in Chomsky normal form, i.e. each rule has the form $A \to a$ where $A \in N$ and $a \in \Sigma$, or $A \to BC$ where $A,B,C \in N$. Let $R = (Q,\Sigma,\Omega,I,\Delta,F,o)$ be a real-time transducer for $t$. We define $N' = \{ S' \} \cup \{ A_{p,q} \mid A \in N, \, p,q \in Q \}$ and $\to_H$ contains the productions - $A_{p,q} \to_H (a,y)$ for all productions $A \to_G a$ and transitions $p \xrightarrow{a \mid y} q$ in $R$, - $A_{p,q} \to_H B_{p,r} C_{r,q}$ for all productions $A \to_G BC$ and $p,q,r \in Q$, - $S' \to_H S_{p,q} \, ({\varepsilon},o(q))$ for all $(p,q) \in I \times F$. One can verify that for all $A \in N$ and $p,q \in Q$ the language ${\mathsf{L}}(A_{p,q})$ is the set of all pairs $(x,y) \in {\mathsf{L}}(A) \times \Omega^$ such that $p \xrightarrow{x \mid y} q$ in $R$, and that ${\mathsf{L}}(H) = t\|_X$. Now let $A \in N$, $p,q \in Q$ and $(x,y) \in {\mathsf{L}}(A_{p,q})$ with the property that $\|x\| = \min \{\|x'\| \mid (x',y) \in {\mathsf{L}}(A_{p,q}) \}$. Consider a derivation tree $T$ for $(x,y)$ whose root is labelled by $A_{p,q}$. If $s$ is a node in $T$ which derives $(u,v)$ then we define the [weight]{} of $s$ to be $\|v\|$. Clearly, the weight of an inner node is the sum of the weights of its children. If $(s_1, s_2, \dots, s_k)$ is a path in $T$ such that all nodes $s_i$ on the path have the same length then $k \le \|N'\|$. Assume that $k > \|N'\|$. There exist two nodes $s_i \neq s_j$ with $i < j$ which are labelled by the same nonterminal from $N'$. The subtrees rooted in $s_i$ and $s_j$ are derivation trees for pairs $(u,v)$ and $(u',v)$ for some $u,u' \in \Sigma^$ and $v \in \Omega^$ where $u'$ is a proper factor of $u$. We can then replace the subtree rooted in $s_i$ by the subtree rooted in $s_j$ and obtain a derivation tree for a pair $(x',y)$ with $\|x'\| < \|x\|$, contradiction. Set $c = \|N'\|$. By the claim above every subtree whose root has weight 0 has depth at most $c$ and hence its size is at most $C = 2^c-1$. Define $D = c+(c-1)C$. The derivation tree $T$ has $O(\|y\|)$ nodes. We prove by induction on $\|y\|$ that, if $\|y\| \ge 1$ then $T$ has at most $(2\|y\|-1)D$ nodes. The root of $T$ has weight $\|y\|$. Let $(s_1, \dots, s_k)$ be the maximal path starting in the root whose nodes have weight $\|y\|$. We know that $k \le c$. If $s_i'$ is the sibling of $s_{i-1}$ for $2 \le i \le k$, then $s_i'$ has weight $0$ and the subtree rooted in $s_i'$ has at most $C$ nodes. 1. Assume that $s_k$ is a leaf. Then $T$ consists of at most $D = c + (c-1)C \le (2\|y\|-1)D$ nodes, namely $k \le c$ nodes on the path $(s_1, \dots, s_k)$ and $c-1$ many subtrees with at most $C$ nodes. 2. Assume that $s_k$ has two children $s_{k+1}$ and $s'_{k+1}$ and let $w$ and $w'$ be the weights of $s_{k+1}$ and $s'_{k+1}$, respectively. We have $\|y\| = w + w'$ and $1 \le w,w' < \|y\|$. By induction hypothesis the subtrees rooted in $s_{k+1}$ and $s'_{k+1}$ have at most $(2w-1)D$ and $(2w'-1)D$ nodes, respectively. Therefore $T$ has in total at most $D + (2w-1)D + (2w'-1)D \le (2\|y\|-1)D$ nodes. This concludes the proof of the claim. Now let $y \in t(X) \cap \Omega^{\le n}$ and $x \in X$ be any word with $t(x) = y$. There exists an initial accepting run $p \xrightarrow{x \mid y'} q$ with $y = y' \, o(q)$. As shown above there exists a word $x'$ with $p \xrightarrow{x' \mid y'} q$ and $x' \le O(\|y'\|) \le O(n)$, which concludes the proof. Proof of Lemma \[lem:rep-rat\] ============================== In [@BaranyLS06] it was observed that $\sim$ can be recognized by a synchronous 2-tape automaton. The [convolution]{} of two words $u = a_1 \cdots a_m, v = b_1 \cdots b_n \in \Omega^$ is the word $u \otimes v = c_1 \cdots c_\ell$ of length $\ell = \max(m,n)$ over the alphabet $(\Omega \cup \{\Box\})^2$ where $c_i = (a_i,b_i)$ if $1 \le i \le \min(m,n)$, $c_i = (a_i,\square)$ if $m < i \le n$ and $c_i = (\square,b_i)$ if $n < i \le m$. Similarly, one can define an associative operation $\otimes$ on $k$-tuples of words. A $k$-ary relation $R \subseteq (\Omega^)^k$ is [synchronous rational]{} if $\otimes R = \{ \otimes(u_1, \dots, u_k) \mid (u_1, \dots, u_k) \in R\}$ is a regular language over $(\Omega \cup \{\Box\})^k$. The set of synchronous rational relations is known to be closed under first-order operations and, in particular, under Boolean operations, cf. [@KhoussainovN94]. Clearly, every binary synchronous rational relation is a rational transduction. The equivalence relation $\sim^{\mathsf{R}}$ is synchronous rational. We present a right automaton which recognizes the complement of $\sim^{\mathsf{R}}$. It reads two configurations $\alpha p$ and $\beta q$ synchronously which are aligned to the right, from right to left. The automaton stores a pair of states of $A$, starting with the pair $(p,q)$. It then guesses a word $w$ by its monotonic factorization which witnesses that $w$ belongs to exactly one of the languages ${\mathsf{L}}(\alpha p)$ and ${\mathsf{L}}(\beta q)$. Notice that it suffices to read the maximal descending prefix of $w$ and test whether the reached state pair $(p',q')$ belongs to some fixed set of state pairs since the remaining ascending suffix cannot access the stack contents of the reached configurations. To simulate $A$ on a descending prefix in each step the automaton either guesses a return symbol and removes the top most stack symbol of both configurations (or leaves $\bot$ at the top), or guesses a state transformation $\tau \in \varphi(W)$ which only modifies the current state pair. It is well-known that $\le_{\mathit{llex}}$ is a synchronous rational relation. By the closure properties of synchronous rational relations the function ${\mathsf{rep}}$ is rational. Proof of Proposition \[prop:nu-f\] ================================== Let $w = w_0 w_1 \cdots w_m \in \Sigma^$ be a monotonic factorization and let $s = s_0 s_1 \cdots s_m \in {\mathrm{Flat}}$ be the associated flattening. We prove $t_f(s) \sim \delta(\bot q_0, w)$ by induction on $m$. - If $m = 0$ and $s_0 = {\varepsilon}$ then $t_f(s) = \bot q_0 = \delta(\bot q_0,{\varepsilon})$. - If $m = 0$ and $s_0 = q_1 \cdots q_k \in Q^+$ then $t_f(s) = \bot q_1$ and $\nu_A(w) = {\mathsf{rep}}(\delta(\bot q_0,w)) = \bot q_1$. - If $m \ge 1$ and $s_m \in \Sigma_c$ then $s_m = w_m$. By induction hypothesis we know that $t_f(s_0 \cdots s_{m-1}) \sim \delta(\bot q_0,w_0 \cdots w_{m-1})$. Since $\delta(\bot q_0,w) = \delta(\delta(\bot q_0,w_0 \cdots w_{m-1}),w_m)$ and $t_f(s) = \delta(t_f(s_0 \cdots s_{m-1}),s_m)$ we obtain $\delta(\bot q_0,w) \sim t_f(s)$. - If $m \ge 1$ and $s_m = \tau q_2 \cdots q_k \in Q^Q Q^$ then $w_m$ is well-matched and $\varphi(w_m) = \tau$. Assume that $t_f(s_0 \cdots s_{m-1}) = \alpha p$ and $\delta(\bot q_0,w_0 \cdots w_{m-1}) = \beta q$. By induction hypothesis we know that $\alpha p \sim \beta q$. Since $t_f(s) = \alpha \tau(p) = \delta(\alpha p, w_m)$ and $\delta(\bot q_0,w) = \delta(\beta q,w_m)$ we obtain $t_f(s) \sim \delta(\bot q_0,w)$. Since $\nu_f = {\mathsf{rep}}\circ t_f$ and $\nu_A(w) = {\mathsf{rep}}(\delta(\bot q_0,w))$ we have $\nu_f(s) = \nu_A(w)$. \[lem:state-transd\] Let $w = w_0 w_1 \cdots w_m \in \Sigma^$ be a monotonic factorization with empty initial factor $w_0 = {\varepsilon}$ and let $s = s_0 s_1 \cdots s_m \in \Sigma_f^$ be the associated flattening. If $\delta(\bot p,w) = \bot \alpha q$ then $p \xrightarrow{s \mid \alpha} q$ in $A_1$ and hence $t_p(s) = \alpha q$. Proof by induction on $m$. If $m = 0$ then $w = s = {\varepsilon}$, $p = q$ and $\alpha = {\varepsilon}$. For the induction step assume $\delta(\bot p, w_1 \cdots w_{m-1}) = \bot \alpha q$ and $\delta(\bot \alpha q, w_m) = \bot \alpha \alpha_1 q_1$. By induction hypothesis the run of $A_1$ on $s$ has the form $p \xrightarrow{s_1 \cdots s_{m-1} \mid \alpha} q \xrightarrow{s_m \mid \alpha_2} q_2$. We do a case distinction. If $w_m \in \Sigma_c$ then $\delta(q,w_m) = (\alpha_1,q_1)$. Since $s_m = w_m$ and by definition of $A_1$ we have $\alpha_1 = \alpha_2$ and $q_1 = q_2$. Otherwise $w_m \in W \setminus \{{\varepsilon}\}$ and $\alpha_1 = {\varepsilon}$. The word $s_m = \sigma_1(w_m)$ starts with $\tau = \varphi(w_m)$ and we have $\tau(q) = q_1$. By definition of $A_1$ we indeed have $q_2 = \tau(q)$ and $\alpha_2 = {\varepsilon}$. We define the following total function $t_f \colon \Sigma_f^* \to (Q \cup \Gamma)^$. Let $s \in \Sigma_f^$ be an input word and let $q_1 \cdots q_k \in Q^$ be the maximal prefix of $s$ from $Q^$, say $s = q_1 \cdots q_k s'$ for some $s' \in \Sigma_f^$. Then we define $$t_f(s) = \begin{cases} t_{q_0}(s), & \text{if $k = 0$,} \\ t_{q_1}(s'), & \text{if $k \ge 1$.} \\ \end{cases}$$ It is easy to see that $t_f$ is rational by providing a transducer for $t_f$. It verifies whether $s$ starts with a state $q \in Q$. If so, it memorizes $q$ and simulates $A_1$ on $s'$ from $q$, and otherwise $A_1$ is directly simulated on $s$ from $q_0$. Now let $w = w_0 w_1 \cdots w_m$ be a monotonic factorization and $s = s_0 s_1 \cdots s_m \in \Sigma_f^$ be the associated flattening. We claim that $\delta(\bot q_0, w) \sim \bot t_f(s)$. If $w_0 = {\varepsilon}$ then $s_0 = {\varepsilon}$ and $s$ does not start with a state from $Q$. In this case we have $\delta(\bot q_0, w) = \bot t_{q_0}(s) = \bot t_f(s)$ by Lemma \[lem:state-transd\]. If $w_0 \neq {\varepsilon}$ then $s_0$ starts with some state $q_1 \in Q$. By definition of $\sigma_0$ we have $\delta(\bot q_0, w_0) \sim \bot q_1$ and thus $\delta(\bot q_0, w) \sim \delta(\bot q_1, w_1 \cdots w_m)$. By Lemma \[lem:state-transd\] we have $\delta(\bot q_1, w_1 \cdots w_m) = \bot t_{q_1}(s_1 \cdots s_m) = \bot t_f(s)$, which proves the claim. Finally, we can set $\nu_f(s) = {\mathsf{rep}}(\bot t_f(s))$ for all $s \in \Sigma_f^$. Proof of Proposition \[prop:flat-reg-eq\] ========================================= By Proposition \[prop:nu-f\] we know $\nu_f({\mathrm{Flat}}) = {\mathrm{Rep}}$. Clearly $\nu_f({\mathrm{Flat}}) \subseteq \nu_f({\mathrm{RegFlat}})$ and it remains to show the other inclusion. Consider a word $r \in {\mathrm{RegFlat}}$ which does not have a non-empty prefix from $R_0$, say $r = u_1 v_1 u_2 v_2 \cdots v_m u_{m+1}$ where $u_1, \dots, u_{m+1} \in \Sigma_c^$ and $v_1, \dots, v_m \in R_1$. Then $r' = u_1 \, {\mathsf{apx}}_\ell(v_1) \, u_2 \, {\mathsf{apx}}_\ell(v_2) \cdots {\mathsf{apx}}_\ell(v_m) \, u_{m+1}$ belongs to ${\mathrm{Flat}}$ and $\nu_f(r) = \nu_f(r')$. Now assume that $r$ has a non-empty prefix $q_1 \cdots q_k \in R_0$. We do the replacements above and the following. By definition $q_1 \cdots q_k$ is a proper suffix of some word $x = \tau p_2 \cdots p_{i-1} q_1 \cdots q_k \in R_1$. Let $y = {\mathsf{apx}}_i(x) \in S_1$ which has a proper suffix of the form $q_1 q_2' \cdots q_k'$ belonging to $S_0$. We can replace $q_1 \cdots q_k$ by $q_1 q_2' \cdots q_k'$ in $r$ and obtain a word $r' \in {\mathrm{Flat}}$ with $\nu_f(r) = \nu_f(r')$. Proof of Proposition \[prop:transf-growth\] =========================================== Assume that $(u_2,v_2,u,v,Z)$ is a linear fooling scheme for $\nu_f$ with $\{u_2,v_2\} \{u,v\}^Z \subseteq {\mathrm{RegFlat}}$. We first ensure that $\{u,v\} \cup Z \subseteq (\Sigma_c \cup R_1)^$. Assume that $u, v \in Q^$ and hence $\{u_2,v_2\} \{u,v\}^ \subseteq Q^$ is contained in the set of prefixes of words in $R_0$. Since $R_0$ is bounded by assumption also $\{u_2,v_2\} \{u,v\}^$ must be bounded, which contradicts the fact that $\{u_2,v_2\} \{u,v\}^$ has exponential growth. Without loss of generality, assume that $u = u_3 u_4$ such that $u_4$ either starts with a call letter $a \in \Sigma_c$ or a transformation $\tau \in Q^Q$. We claim that $(u_2 u_3,v_2 u_3, u_4 u u_3, u_4 v u_3, u_4 Z)$ is a linear fooling scheme for $\nu_f$. It has the following properties: - $\{ u_2 u_3,v_2 u_3\} \{u_4 u u_3, u_4 v u_3\}^ u_4 Z \subseteq {\mathrm{RegFlat}}$, - $u_2u_3$ is a suffix of $u_4 u u_3$, - $v_2 u_3$ is a suffix of $u_4 v u_3$, - $\|u_2 u_3\|=\|v_2 u_3\|$. Also, we know that for every $n \in {\mathbb{N}}$ there exists a word $z_n \in Z$ with $\|z_n\| \le O(n)$ and $\nu_f(u_2 w z_n) \neq \nu_f(v_2 w z_n)$ for all $w \in \{ uu,uv \}^{\le n}\{u\}$ and thus, by factoring out the first $u_3$- and the last $u_4$-factor, we have $\nu_f(u_2 u_3 w u_4 z_n) \neq \nu_f(v_2 u_3 w u_4 z_n)$ for all $w \in \{ u_4 u u_3,u_4 v u_3 \}^{\le n}$. Hence we have verified the conditions of a linear fooling scheme. It has the desired properties that $\{u_4 u u_3, u_4 v u_3\} \cup u_4 Z \subseteq (\Sigma_c \cup R_1)^$ because $u_4$ starts with a call letter or a transformation. Now let $(u_2,v_2,u,v,Z)$ be a linear fooling scheme with $\{u,v\} \cup Z \subseteq (\Sigma_c \cup R_1)^$. We replace occurring factors from $R_1$ by factors from $S_1$ while maintaining the values $\nu_f(u_2wz)$ and $\nu_f(v_2wz)$ for $w \in \{u,v\}^$ and $z \in Z$. 1. First, in each word $z \in Z \subseteq (\Sigma_c \cup R_1)^$ we replace each $R_1$-factor $v$ by ${\mathsf{apx}}_\ell(v)$ which ensures that $Z \subseteq (\Sigma_c \cup S_1)^$. 2. Next consider $u$ and $v$, and assume that $u = u_1 u_2$ and $v = v_1v_2$ for some $u_1,v_1 \in \Sigma_f^$. Let us consider $R_1$-factors which cross the factorization $u = u_1 u_2$ or $v = v_1 v_2$, respectively. If $u_2$ starts with some state we can factorize $u_1$ and $u_2$ as $u_1 = u_3 \tau q_2 \cdots q_{i-1}$ and $u_2 = q_i \cdots q_k u_4$ where $u_3, u_4 \in (\Sigma_c \cup R_1)^$ and $\tau q_2 \cdots q_k \in R_1$. Let ${\mathsf{apx}}_i(\tau q_2 \cdots q_k) = \tau s' q_i p_{i+1} \cdots p_k \in S_1$. We replace $u_1$ by $u_3 \tau s'$ and $u_2$ by $q_i p_{i+1} \cdots p_k u_4$. Notice that the length of $u_2$ has not changed (this maintains $\|u_2\| = \|v_2\|$) and the first state of $u_2$ has not changed either (this maintains the values $\nu_f(u_2 w z)$). If $v_2$ starts with some state we do the analogous replacements for $v_1$ and $v_2$. 3. Finally, each $R_1$-factor $v$ in $u_1$, $u_2$, $v_1$ and $v_2$ is replaced by ${\mathsf{apx}}_\ell(v)$. One can verify that the obtained tuple $(u_2,v_2,u,v,Z)$ is again a linear fooling scheme for $\nu_f$ satisfying $\{u_2,v_2\}\{u,v\}^ Z \subseteq {\mathrm{Flat}}$. Proof of Lemma \[lem:linear-distance\] ====================================== Suppose that $w = a_1 \cdots a_m$. Since ${\mathcal{R}}_t$ is a right congruence we know that $ua_1 \cdots a_i \mathrel{{\mathcal{R}}}_t v a_1 \cdots a_i$ for all $0 \le i \le m$. By definition of the look-ahead extension the words $e_{{\mathcal{R}}_t}(uw)$ and $e_{{\mathcal{R}}_t}(vw)$ have the common suffix $$s = \binom{a_1}{[u]_{{\mathcal{R}}_t}} \binom{a_2}{[ua_1]_{{\mathcal{R}}_t}} \cdots \binom{a_m}{[u a_1 \cdots a_{m-1}]_{{\mathcal{R}}_t}}.$$ The initial accepting runs of $B$ on $e_{{\mathcal{R}}_t}(uw)$ and $e_{{\mathcal{R}}_t}(vw)$ have the form $$q \xleftarrow{e_{{\mathcal{R}}_t}(u)} p \xleftarrow{s} q_{\mathit{in}} \quad \text{and} \quad r \xleftarrow{e_{{\mathcal{R}}_t}(v)} p \xleftarrow{s} q_{\mathit{in}}$$ and thus $t(uw)$ and $t(vw)$ share the suffix ${\mathsf{out}}(p \xleftarrow{s} q_0)$. This implies $$\\|t(uw),t(vw)\\| \le \|{\mathsf{out}}_F(q \xleftarrow{e_{{\mathcal{R}}_t}(u)} p)\| + \|{\mathsf{out}}_F(r \xleftarrow{e_{{\mathcal{R}}_t}(v)} p)\| \le {\mathsf{iml}}(A) \cdot (\|u\|+\|v\|+2),$$ proving the statement. Proof of Lemma \[lem:short-witness\] ==================================== Assume that $t_1$ and $t_2$ are not adjacent. By [@ReutenauerS91 Proof of Proposition 1.] there exist words $x,y,z \in \Sigma^$ and $u_1,u_2,v_1,v_2,w_1,w_2 \in \Omega^$ such that $t_1(x y^k z) = u_1v_1^kw_1$, $t_2(x y^k z) = u_2v_2^kw_2$ for all $k \in {\mathbb{N}}$, and $\sup \{ \\|u_1v_1^kw_1,u_2v_2^kw_2\\| \mid k \in {\mathbb{N}}\}= \infty$. By the triangle inequality we have $$\begin{aligned} \\|v_1^kw_1,v_2^kw_2\\| &\le \\|v_1^kw_1,u_1v_1^kw_1\\| + \\|u_1v_1^kw_1,u_2v_2^kw_2\\| + \\|u_2v_2^kw_2,v_2^kw_2\\| \\ &= \|u_1\| + \\|u_1v_1^kw_1,u_2v_2^kw_2\\| + \|u_2\| = \\|t_1(x y^k z),t_2(x y^k z)\\| + \|u_1\| + \|u_2\|. \end{aligned}$$ We prove that $\\|v_1^kw_1,v_2^kw_2\\| \ge \Omega(k)$ which implies that $\\|t_1(x y^k z),t_2(x y^k z)\\| \ge \Omega(k)$. If both $v_1 = v_2 = {\varepsilon}$ then $$\sup_{k \in {\mathbb{N}}} \\|v_1^kw_1,v_2^kw_2\\| = \\|w_1,w_2\\| < \infty,$$ which contradicts $\sup \{ \\|u_1v_1^kw_1,u_2v_2^kw_2\\| \mid k \in {\mathbb{N}}\}= \infty$. If $\|v_1\| \neq \|v_2\|$ then $$\\|v_1^kw_1,v_2^kw_2\\| \ge \big\| \|v_1^kw_1\| - \|v_2^kw_2\| \big\| = \Omega(k).$$ Now assume $\|v_1\| = \|v_2\| \ge 1$. Since $$\\|v_1^kw_1,v_2^kw_2\\| = \|v_1^kw_1\| + \|v_2^kw_2\| - 2\|v_1^kw_1 \wedge v_2^kw_2\| \ge \Omega(k) -2\|v_1^kw_1 \wedge v_2^kw_2\|$$ it suffices to show that $\sup_k \|v_1^kw_1 \wedge v_2^kw_2\| < \infty$. Towards a contradiction assume that $\sup_k \|v_1^kw_1 \wedge v_2^kw_2\| = \infty$. Then, for every $k \in {\mathbb{N}}$ there exists $K \in {\mathbb{N}}$ such that $\|v_1^Kw_1 \wedge v_2^Kw_2\| \ge \max \{ \|v_1^kw_1\|,\|v_2^kw_2\| \}$. If $\|v_1^kw_1\| \ge \|v_2^kw_2\|$ then $v_1^k w_1$ is a suffix of $v_1^Kw_1 \wedge v_2^Kw_2$ and otherwise $v_2^k w_2$ is a suffix of $v_1^Kw_1 \wedge v_2^Kw_2$. This shows that for all $k \in {\mathbb{N}}$ either $v_1^k w_1$ is a suffix of $v_2^kw_2$, or vice versa, and therefore $\|v_1^kw_1 \wedge v_2^kw_2\| = \min \{\|v_1^kw_1\|,\|v_2^kw_2\|\}$. Since $\|v_1\| = \|v_2\|$ we obtain $$\\|v_1^kw_1,v_2^kw_2\\| = \|v_1^kw_1\| + \|v_2^kw_2\| - 2\min \{\|v_1^kw_1\|,\|v_2^kw_2\|\} = \|w_1\| + \|w_2\| - 2 \min\{\|w_1\|,\|w_2\|\}$$ contradicting $\sup_k \\|v_1^kw_1,v_2^kw_2\\| = \infty$.
--- abstract: 'In this study we derive the Finite element apriori error estimate for the monodomain cardiac electric model in conjunction with the generic form for a class of nonlinear ionic models. The analysis establishes a $o(h^2+k)$ space-time convergence. Further the computational realization of the same for different reduced ionic test models is presented.' author: - \| Meena Pargaei and B.V. Rathish Kumar\ Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur\ Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur title: Finite element apriori error estimate for a class of Cardiac Electric models --- [*Keywords:* Apriori Estimate; Finite Element Method; ODE-PDE system.]{} Introduction ============ Cardiac electrophysiology is a branch of medicine and biology. Electrophysiological models of heart describe how the electrical potential is generated in every part of the heart. These models consist of coupled ODE-PDE system. Electrical activity at the cell level is described by these ODEs and PDEs describe the same at the tissue level. One of these models is the Bidomain model which is described by two degenerate non-linear parabolic reaction- diffusion equations which are coupled with a non-linear system of ODE ionic models. The other popular model is the, Monodomain model, which is a simplified form of the Bidomain model. It consists of a non-linear parabolic reaction-diffusion equation together with a non-linear system of ODE ionic models. While both finite difference method (FDM)and Finite element method (FEM) have been used to solve these models [@fem; @fdm1; @fdm2; @luca].The space-time convergence analysis through apriori error estimation has not been reported for these models. In this paper we derive the $L^2$ apriori error estimate for the FE analysis of Monodomain model with general form of ionic models. In the next section we will describe the cardiac electric models. In section (3) finite element formaulation is presented. In section (4) we derive the $L^2$ apriori error estimate for semi-discrete and fully discrete system. In section (5) a numerical example with space-time convergence results have been presented. Cardiac Electric Model ====================== Cardiac tissue is considered as the overlapping of the intra and extracellular continuous domains such that each point in the intracellular myocardium is also in the extracellular and the two domains are connected with continuous cellular membrane. Bidomain model(BDM) [@luca] is the characterization of such cardiac tissue and it consists of a set of mathematical equations which describe the propagation of intra and extracellular electric potentials in cardiac tissue. The monodomain model(MDM) [@luca], simplified version of the BDM, with the capacity to provide significant information similar to that of BDM, is used to calculate the action potential. This model consists of a parabolic reaction-diffusion equation coupled with a system of ODE’s which is given as $$\label{mv} \frac{\partial v}{\partial t}- div(D(x)\nabla v) - I_{ion}(v,w)=I_{app} \hspace{1.5cm} \Omega \times (0,T)$$ $$\label{mw} \frac{\partial w}{\partial t}-g(v,w)=0 \hspace{5cm} \Omega \times (0,T)$$ $$\label{in} v(x,0)= v_0(x,0), \hspace{5mm} w(x,0)=w_0(x,0) \hspace{2cm} \Omega$$ $$\label{bndry} n^T D(x) \nabla v =0 \hspace{5cm} \partial \Omega \times (0,T)$$ where $v$ and $w$ are the action potential and gating variables respectively. $g(v,w)$ and $I_{ion}$ is given by the models at the cell level, called ionic models. Ionic Models ------------ Hodgkin and Huxely [@book] in 1952 gave the first mathematical model that describes accurately the action potential waveform. This model is complex in nature. There are various complex ionic models such as Luo Rudy 1 [@luo], Beeler Reuter ,LRd [@book]. For large spatial and temporal investigation of any phenomena, various reduced ionic models FitzHugh Nagumo Model(FHNM), Roger-McCulloch Model(RMM) ,Aliev Panfilov Model(APM), Mitchell Schaeffer Model(MSM) [@book; @fhn; @panfilov; @McCulloch; @ms] have been described to substantially provide the action potential at low cost. Galerkin Finite Element Method (GFEM) formulation ================================================= Weak formulation of the system (\[mv\]-\[bndry\]) is obtained, find $v(t) \in H^1(\Omega)$ [@book], $w(t) \in L^2(\Omega) or L^{\infty}(\Omega)$[@book] and $\psi_1 \in H^1(\Omega) $ , $\psi_2 \in L^2(\Omega)$, $$\begin{aligned} (v_t,\psi_1) - (\nabla.(D(x)\nabla v), \psi_1)=(I_{ion},\psi_1) \\ (w_t,\psi_2)=(g,\psi_2)\end{aligned}$$ Integration by parts together with boundary condition gives $$\begin{aligned} \label{weak} (v_t,\psi_1) - (D(x)\nabla v, \nabla \psi_1)=(I_{ion},\psi_1) \\ (w_t,\psi_2)=(g,\psi_2)\end{aligned}$$ Consider $\{T_h\}_h$ be a member of a family of quasi-uniform triangularizations of $\Omega$ with ${max}_{\tau \in T_h} diam \tau \leq h $ and let $S_h$ be the corresponding finite dimensional space of continuous functions on $\Omega$ which reduces to linear functions in each of the triangles of $T_h$. Let $\{\phi_i\}_{i=1}^N$ be the basis functions of $S_h$ so that we can write $v(x,t)= \sum_{i=1}^{N}v_i(t)\phi_i(x)$ and $w(x,t)= \sum_{i=1}^{N}w_i(t)\phi_i(x)$ and equation (\[weak\]) becomes $$\begin{aligned} \Big (\sum_{i=1}^{N}v_{i,t}\phi_i(x),\phi_i \Big)+ \Big (D(x)\nabla \sum_{i=1}^{N}v_i\phi_i,\nabla \phi_i\Big)=(I_{ion},\phi_j) , j=1,2,...,N.\end{aligned}$$ The matrix form of the system is given as $$\begin{aligned} M v_t+A v= I_{ion}, \hspace{1cm}Mw_t=G,\end{aligned}$$ where $M_{ij}=((\phi_i,\phi_j))$ , $A_{ij}=(\nabla \phi_i,\nabla \phi_j)$, $I_{ion}=\Big((I_{ion},\phi_1),...,(I_{ion},\phi_N)\Big)$ and $G=\Big((g,\phi_1),...,(g,\phi_N)\Big)$. $L^2$ error estimate for the semi-discrete problem ================================================== Let $\Omega$ be a plane convex domain with smooth boundary and consider the problem (\[mv\]-\[bndry\]). Assume that $D(x)$ is symmetric and uniformly positive definite, i.e. there exist $\alpha>0$ such that $\forall x \in R^3 , \forall \xi \in R^3$, $\xi^T D(x)\xi\geq \alpha{\mid\xi\mid}^2$. Introducing the inner product $(\begin{bmatrix} \alpha_1 & \alpha_2 \end{bmatrix}, \begin{bmatrix} \beta_1 & \beta_2 \end{bmatrix})_X:=( \alpha_1,\beta_1)+(\alpha_2 , \beta_2)$, and the associated norm $\parallel . \parallel_X$, where $(. ,.)$ is the standard $L^2$- inner product. Let $u={\begin{bmatrix} v & w \end{bmatrix}}^t$ , $ G = \begin{bmatrix} \nabla & 0 \\ 0 & 0 \end{bmatrix} $ , $Gu = G\begin{bmatrix} v \\ w \end{bmatrix} = \begin{bmatrix} \nabla v \\ 0 \end{bmatrix}$. So,the weak form of the problem becomes Find $u \in H^1(\Omega) \times L^2(\Omega)$ such that $$\label{wf} (u_t,\psi)_X + (D Gu, G \psi)_X=(F(u),\psi)_X$$ where $F(u)= {\begin{bmatrix} I_{ion}(u) & g(u) \end{bmatrix}}^t$. Let $\pi_h \colon H^1(\Omega) \times L^2(\Omega) \longrightarrow S_h \times S_h $ be the projection and $u_h={\begin{bmatrix} v_h & w_h \end{bmatrix}}^t$ be the solution of the finite element formulation $$\label{fe} (u_{h,t},\psi)_X + (D Gu_h, G \psi)_X=(\pi_h f(u_h),\psi)_X , \forall \psi \in S_h \times S_h$$ \[1\] Let $u$ be the solution of the problem $(\ref{mv}-\ref{in})$, and $u_h$ be the solution of the problem in semi-discrete case. Then, if $I_ion(v,w)$ and $g(v,w)$ is Lipschitz continuous in $v$ and $w$ and $D(x)$ is symmetric and positive definite, we have $$\begin{aligned} \label{semidiscrete} \parallel u_h(T)-u(T) \parallel_X \leq \parallel u_{0,h}-u_0 \parallel_X + C h^2 \parallel u_0 \parallel_X + Ch^2 \int_{0}^{T}(\parallel (F-\pi_h F)(u) \parallel_X+ \parallel u \parallel_X +\parallel u_t \parallel_X )dt \end{aligned}$$ Proof: Decompose the error $$\begin{aligned} \label{err} u-u_h = (u - R_hu)+(R_hu-u_h). \end{aligned}$$ where $R_hu$ is the elliptic projection of the $u$ defined as,$$(D(x) G(R_hu - u) , G\psi)_X =0, \hspace{0.5cm} \forall \psi \in S_h \times S_h .$$ Now we will bound the $\theta= R_hu-u_h$ and $\rho=(u-R_hu)$ separately. In order to bound $\theta$, note that, $$\begin{aligned} (\theta_t,\psi)_X+(D(x)G\theta,G\psi)_X=(F(u)-\pi_h F(u_h),\psi)_X-(\rho_t,\psi)_X,\end{aligned}$$ Choose $\psi=\theta$,and applying Chauchy-schwartz and lipschitz continuity of F along with boundedness of $L^2$ projection, we get $$\begin{aligned} \frac{d}{dt} \parallel \theta \parallel_X \leq \parallel (F-\pi_h F)(u) \parallel_X+ M \parallel u-u_h \parallel_X + \parallel \rho_t \parallel_X.\end{aligned}$$ After integration we get, $$\begin{aligned} \label{theta} \parallel \theta(T) \parallel_X \leq \parallel \theta(0) \parallel_X+ \int_{0}^{T} M(\parallel \theta \parallel_X+\parallel \rho \parallel_X)+\parallel (F-\pi_h F)(u) \parallel_X + \parallel \rho_t \parallel_X dt.\end{aligned}$$ Now apply Gronwall’s lemma, and the following bounds for $\parallel \rho \parallel_X$ and $\parallel \rho_t \parallel_X$ taken from the elliptic theory [@vidar] $$\begin{aligned} \label{rho} \parallel \rho \parallel_X \leq C h^2 \parallel u \parallel_X, \parallel \rho_t \parallel_X \leq C h^2 \parallel u_t \parallel_X,\end{aligned}$$ Also, $$\begin{aligned} \label{theta0} \parallel \theta(0) \parallel_X \leq \parallel u_{0,h}-u_0 \parallel_X + \parallel R_hu_0-u_0 \parallel_X \leq \parallel u_{0,h}-u_0 \parallel_X + C h^2 \parallel u_0 \parallel_X\end{aligned}$$ Using (\[theta\]-\[theta0\]) in \[err\], we arrive at the estimate \[semidiscrete\]. $L^2$ error estimate for the fully discrete problem --------------------------------------------------- Let $k$ be the time step, $t_n = nk$, and let $U^n$ be the approximation of $u(t_n)$ in $S_h \times S_h $. We will use backward Euler Galerkin scheme and linearize the problem \[wf\] by replacing $U^n$ by $U^{n-1}$ to obtain $$\label{wft} (\bar{{\partial}_t} U^n , \psi)_X+ (D(x)GU^n,G\psi)_X = (F(U^{n-1}),\psi)_X, \hspace{1cm} \forall \chi \text{in} S_h.$$ where \[deltU\][$\bar{{\partial}_t} U^n = \frac {1} {k} (U^n-U^{n-1}) $]{}. \[2\] Let $U^n$ and $u$ be solutions of \[wft\] and \[mv\], \[mw\] respectively. Then, if $I_{ion}(v,w)$ and $g(v,w)$ are Lipschitz continuous in $v$ and $w$ and $D(x)$ is symmetric and uniformly positive definite, we have $\Arrowvert U^n - u(t_n) \Arrowvert_X \leq C \Arrowvert u_{0,h}-u_0 \Arrowvert_X + C(u) (h^2+k) \forall t_n \in \bar{J}$ Proof. $u^n = u(t_n)$ , $$U^n-u^n = (U^n-\widetilde{U}^n) + (U^n-u^n) = \theta ^n + \rho ^n$$ where $\widetilde{U}^n$ is the elliptic projection of $u^n$ defined as $$(D(x) G(\widetilde{U}^n - u^n) , G\psi)_X = (DG\rho^n , G\psi)_X =0$$ $\rho^n$ will be bounded by Elliptic theory. Now we need to bound only $\theta^n$. For $\psi \in S_h \times S_h$, $$\begin{aligned} (\bar{{\partial}_t} \theta^n , \psi)_X+ (D(x)G \theta^n,G\psi)_X = ((\bar{{\partial}_t} U^n , \psi)_X+ (D(x)GU^n,G\psi)_X - (\bar{{\partial}_t} \widetilde{U}^n , \psi)_X - (D(x)G\widetilde{U}^n,G\psi)_X\end{aligned}$$ $$\begin{aligned} =(F(U^{n-1}),\psi)-(u_t^n,\psi)_X-({\overline{{\partial}_t}}\tilde{U}^n-u_t^n,\psi)_X-(D(x)G\tilde{U}^n,G\psi)_X-(D(x)Gu^n,G\psi)_X+(D(x)Gu^n,G\psi)_X\end{aligned}$$ $$\begin{aligned} =(F(U^{n-1}),\psi)_X-(F(u^n),\psi)_X-({\overline{{\partial}_t}}(\tilde{U}^n-u^n),\psi)_X -(D(x)G(\tilde{U}^n-u^n),G\psi)_X-({\overline{{\partial}_t}}u^n-u_t^n,\psi)_X\end{aligned}$$ Now using Lipschitz continuity of $F$ $$\begin{aligned} \parallel F(U^{n-1})-F(u^n) \parallel_X \leq C\parallel U^{n-1}-u^n \parallel_X \leq C (\parallel U^{n-1}-u^{n-1} \parallel_X + \parallel u^{n-1}-u^{n} \parallel_X )\end{aligned}$$ $$\label{lipf} \leq C(\parallel \theta^{n-1} \parallel_X + \parallel \rho \parallel_X +k \parallel {\overline{{\partial}_t}}u^n \parallel_X)$$ Take $\psi = \theta^n$ and use the ellipticity of D and (\[lipf\]) we get, $$\begin{aligned} \frac{1}{2} {\overline{{\partial}_t}} {\parallel \theta^{n} \parallel}_X^2 + \mu {\parallel G\theta^{n} \parallel}_X ^2 \leq C (\parallel \theta^{n-1} \parallel_X + \parallel \rho^{n-1} \parallel_X +k \parallel {\overline{{\partial}_t}}u^n \parallel_X)(\parallel \theta^{n} \parallel_X) \end{aligned}$$ $$\begin{aligned} + (\parallel {\overline{{\partial}_t}}\rho^n \parallel_X +\parallel {\overline{{\partial}_t}}u^n-u_t^n \parallel_X)\parallel \theta^{n} \parallel_X\end{aligned}$$ $$\begin{aligned} {\overline{{\partial}_t}} {\parallel \theta^{n} \parallel}_X^2 \leq C{ (\parallel \theta^{n-1} \parallel}_X^2 + {\parallel \rho^{n-1} \parallel}_X ^2 +k {\parallel {\overline{{\partial}_t}}u^n \parallel}_X^2) + {\parallel {\overline{{\partial}_t}}\rho^n \parallel_X}^2+ {\parallel {\overline{{\partial}_t}}u^n-u_t^n \parallel_X}^2 + C {\parallel \theta^{n} \parallel_X}^2 \end{aligned}$$ $$\begin{aligned} {\overline{{\partial}_t}} {\parallel \theta^{n} \parallel}_X^2 \leq C({\parallel \theta^{n} \parallel}_X^2 + {\parallel \theta^{n-1}\parallel}_X^2+ T_h)\end{aligned}$$ Using the definition \[deltU\] $$\begin{aligned} {\parallel \theta^{n} \parallel_X}^2(1-Ck) \leq {\parallel \theta^{n} \parallel}_X^2(1+Ck) +Ck T_h\end{aligned}$$ $$\begin{aligned} {\parallel \theta^{n} \parallel}_X^2 \leq {\parallel \theta^{n} \parallel_X}^2(1+Ck)^2 +Ck(1+Ck) T_h\end{aligned}$$ $$\begin{aligned} {\parallel \theta^{n} \parallel}_X^2 \leq (1+Ck)^{n+1} {\parallel \theta^{0} \parallel}_X^2 +Ck \sum_{j=1}^n (1+Ck)^{n-j+1} T_j\end{aligned}$$ $$\begin{aligned} {\parallel \theta^{n} \parallel}_X^2 \leq C {\parallel \theta^{0} \parallel}_X^2 +Ck \sum_{j=1}^n T_j\end{aligned}$$ $$\begin{aligned} \parallel \rho^j \parallel_X = \parallel{ \tilde U}^j -u^j \parallel_X \leq Ch^2 {\parallel u(jk)\parallel}_X \leq C(u) h^2\end{aligned}$$ $$\begin{aligned} \parallel \overline{{\partial}_t} \rho^j \parallel_X = \parallel k^{-1} {\int_{(j-1)k}^{jk}} {\rho_t ds} \parallel_X \leq C(u)h^2\end{aligned}$$ $$\begin{aligned} \parallel {\overline{{\partial}_t}}u^n-u_t^n \parallel_X = \parallel k^{-1} {\int_{(j-1)k}^{jk}} (s-(j-1)k)u_{tt}(s)ds \parallel_X \leq C(u)k\end{aligned}$$ We have $T_j \leq C(u)(h^2+k)^2$. Hence,$\parallel \theta^n \parallel_X \leq C \parallel \theta^0 \parallel_X +C(u)(h^2+k)$. Numerical Test and Discussion ============================= We consider the monodomain model with the following different ionic models in a square domain $[-1.25,1.25]^2$. - FHNM: $I_{ion} = u(u-0.1)(1-u)-w , g(v,w)=u-2w$, - RMM: $I_{ion} = u(u-0.1)(1-u)-vw , g(v,w)=u-2w$, - APM: $I_{ion}=-ku(u-a)(u-1)-uw ,$ $g(u,w)=\epsilon' (-ku(u-1-a)-w) $, where $\epsilon'=\epsilon_0 + \mu_1 w/(u+\mu_2)$, - MSM: $I_{ion}=-\frac{w}{\tau_{in}}u^2(u-1)-\frac{u}{\tau_{out}},$ $g(u,w)= \begin{cases} \frac{1-w}{\tau_{open}} & u\leq u_{gate},\\ \frac{-w}{\tau_{close}} & u>u_{gate}.\\ \end{cases}$ . We solved it using linear finite elements in space and Backward Euler in time and the nonlinear terms $I_{ion}$ and $g$ are linearized by taking values at the previous time step. We compute the $L^2$ norm of the error and the space and time rate of convergence. Initial conditions for $v$ and $w$ are chosen to be 0.2 , 0.1 respectively for all the cases and $I_{app}=0 , dt=h^2 , Dx[n]=h,DT[n]=dt.$ Space rate of convergence(sroc) = $log(L^2(e_{n-1})/L^2(e_n))/log(Dx[n-1])/Dx[n]$, time rate of convergence(troc) = $log(L^2(e_{n-1})/L^2(e_{n})/log(DT[n-1])/DT[n]$, where $e_n : n^{th}$ level error. h 1/8 1/16 1/32 1/64 1/128 ------------ ----------- ------------ ------------- ------------- ------------- FHNM error 0.0153718 0.00418786 0.0010467 0.000261422 6.53429e-05 FHNM sroc - 1.876 2.00037 2.0014 2.00027 FHNM troc - 0.937999 1.00018 1.0007 1.00014 RMM error 0.0293018 0.00587156 0.00132513 0.000321176 7.93853e-05 RMM sroc - 2.31917 2.14761 2.04469 2.01642 RMM troc - 1.15959 1.07381 1.02235 1.00821 MSM error 0.0123688 0.0030894 0.000772382 0.000193005 4.82713e-05 MSM sroc - 2.00131 1.99994 2.0000 2 MSM troc - 1.00065 0.999971 1.00004 0.999998 APM error 0.0110065 0.00273321 0.000681779 0.000170299 4.25639e-05 APM sroc - 2.00969 2.00322 2.00123 2.000037 APM troc - 1.00484 1.00161 1.00062 1.00018 : $L^2$ norm of the error and the space and time rates of convergence for different ionic models.[]{data-label="table1"} From Table \[table1\] it is clear that as the grid resolution is increased not only the error in $L^2$ norm decreases but also the theortically predicted sroc and troc are realized in all the cases. Conclusion ========== For the Monodomain CEM in a generic framework for ionic models an apriori error estimate under fem approach has been theortically established and computationally verified. Acknowledgement {#acknowledgement .unnumbered} =============== We would like to thank to the DST for the support through Inspire Fellowship. R. Aliev, A. Panfilov, A simple two-variable model of cardiac excitation, Chaos,solutions and Fractals. 7 :293-201, 1996. J.S. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line stimulating nerve axon. Proc. IRE, 2061-2071, 1962. J.M. Roger, A.D. McCulloch, A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Engr. 41(8) :743-757, 1994. Franzone PC, Pavarino L, A parallel solver for reaction-diffusion systems in computational electrocardiology, Math. Models Methods App. 14 :883-911, 2004. M. Hanslien, K.H. Karlsen, A. Tveito, On a finite difference scheme for a Beeler-Reuter based model of cardiac electrical activity, International Journal of Numerical analysis and Modeling. 3(4) :395-412, 2006. P.C. Franzone, L. Pavarino, S. Scacch, Mathematical Cardiac Electrophysiology, Springer, 2014. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Second ed., Springer-Verlag, 2006. I.S. Hasan, D. C. Paul, and T. N. Kwong, Three-Dimensional Finite-Difference Bidomain Modeling of Homogeneous Cardiac Tissue on a Data-Parallel Computer, IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. 44(2), 1997. P. C. Franzone, L. Pavarino, B. Taccardi, Simulating patterns of excitation, repolarization and action potential duration with cardiac Bidomain and Monodomain models, Math. Biosci. 197(1) :35-66, 2005. C. Luo, Y. Rudy, A model of the ventricular cardiac action potential. depolarization, repolarization, and their interaction, Circ. Res. 68 :1501–1526, 1991. C.C. Mitchell, D.G. Schaeffer, A Two-Current Model for the Dynamics of Cardiac Membrane, Bull. Math. Biol 65(5) :767-793, 2003 .
--- abstract: 'We propose a novel approach to reconstruct the surface magnetic helicity density on the Sun or sun-like stars. The magnetic vector potential is determined via decomposition of vector magnetic field measurements into toroidal and poloidal components. The method is verified using data from a non-axisymmetric dynamo model. We apply the method to vector field synoptic maps from Helioseismic and Magnetic Imager (HMI) on board of Solar Dynamics Observatory (SDO) to study evolution of the magnetic helicity density during solar cycle 24. It is found that the mean helicity density of the non-axisymmetric magnetic field of the Sun evolves in a way which is similar to that reported for the current helicity density of the solar active regions. It has predominantly the negative sign in the northern hemisphere, and it is positive in the southern hemisphere. Also, the hemispheric helicity rule for the non-axisymmetric magnetic field showed the sign inversion at the end of cycle 24. Evolution of magnetic helicity density of large-scale axisymmetric magnetic field is different from that expected in dynamo theory. On one hand, the mean large- and small-scale components of magnetic helicity density display the hemispheric helicity rule of opposite sign at the beginning of cycle 24. However, later in the cycle, the two helicities exhibit the same sign in contrast with the theoretical expectations.' author: - 'Valery V. Pipin' - 'Alexei A. Pevtsov' - Yang Liu - 'Alexander G. Kosovichev' title: Evolution of Magnetic Helicity in Solar Cycle 24 --- Introduction {#sec:intro} ============ Magnetic helicity is an integral measure of topological properties of the magnetic field in closed volume $\rm V$: $$H_{M}=\int\mathbf{A}\cdot\mathbf{B}d\mathrm{V},\label{eq:def}$$ where $\mathbf{A}$ is the magnetic vector potential, $\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}$, and the $\mathbf{B}$ is confined to the volume $\mathrm{V}$. Locally, it can be characterized by a number of parameters such as linkage, twist and writhe of the field lines [@2018JPhA51W5501B]. In astrophysical dynamos, magnetic helicity is commonly accounted as a nonlinear constraint of turbulent generation of large-scale magnetic field [@pouquet-al:1975b; @kleruz82; @brsu05]. Computation of magnetic helicity on the Sun requires knowledge of vector magnetic field in a 3D region, but observations are usually taken in a shallow layer of the solar atmosphere (typically, in the photosphere). Thus, early studies turned to calculation of so-called helicity proxies, such as, for example, vertical components of current helicity density $J_z \cdot B_z$ or $\alpha = J_z / B_z$ (a measure of magnetic twist). @see1990SoPh and @pev95ApJ found that the current helicity density and twist in solar active regions follow $\it {the~ hemispheric~ helicity~ rule}$ with predominantly negative values in the northern hemisphere and positive values in the southern hemisphere. The hemispheric preference for the helicity sign was later confirmed by several researchers [e.g., @long1998ApJ; @baoZh98; @hagsak05]. @choud2004ApJ proposed that due to interaction of the large-scale toroidal field from a previous solar cycle and the poloidal field of a new cycle, the hemispheric helicity could reverse sign at the beginning of each cycle. @hagsak05 reported the presence of such sign reversal in the Okayama Observatory Solar Telescope (OAO) and the Mitaka Solar Flare Telescope (SFT) observations. @Pevtsov.etal2008 examined the periods of sign reversals observed by different instruments and found no agreement among the datasets during these periods. They concluded that at least some reversals could be due to statistical nature of the hemispheric helicity rule. Later, @zetal10 showed that hemispheric helicity rule of the solar active regions evolve with time and found its reversals at the beginning of Solar Cycles 22 and 23. @sok2013 found inversions of the current helicity density in active regions at the beginning and at the end of the solar cycle 23. These early findings are based on vector magnetograms of active regions only. Systematic full disk observations of vector magnetic field became available in 2009 from Vector Spectromagnetograph (VSM) on Synoptic Optical Long-term Investigations of the Sun (SOLIS) platform [@Balasubramaniam.Pevtsov2011] and in 2010 from Helioseismic and Magnetic Imager [HMI, @Scherrer.etal2012] on board Solar Dynamics Observatory [SDO, @Pesnell.etal2012]. Prior to these observations, calculations of current helicity relied on vector magnetic field reconstructed from rotational modulation of the observed longitudinal (line-of-sight) field [@2000ApJ...528..999P]. Later studies demonstrated that the sign of helicity of large-scale magnetic fields is opposite to the sign of helicity of active regions [@PP14; @2017ApJ836.21B]. The large-scale helicity was also found to evolve during solar cycles similarly to helicity of active regions. The solar dynamo theory predicts bi-helical properties of magnetic fields [@black-bran:02; @brsu05]. In this theory the sign of magnetic helicity density of large-scale field should correspond to the sign of the $\alpha$-effect, while sign of magnetic helicity density of the small-scale field would result from the magnetic helicity conservation, and thus, it would be opposite to large-scale helicity. In the framework of this model, small-scale magnetic field corresponds to active regions and the large-scale stands for the global axisymmetric components of the solar magnetic activity. The bi-helical properties were studied recently by [@2017ApJ836.21B] and [@2018arXiv180404994S] using two-scale approximation and the vector magnetic field measurements from SDO/HMI and SOLIS. The results from two different instruments appear to be inconclusive in respect to bi-helical nature of solar magnetic field. Other predictions of mean-field dynamo models include the existence of polar and equatorial branches in the time-latitude diagram of magnetic helicity evolution. According to @pip13M, those branches represent the transport of magnetic helicity flux to the polar regions, both on large and small scales. In this paper we present the first observational evidence of polar branches of the large-scale magnetic helicity in solar cycle 24. Section \[sec:method\] describes our method of calculation of magnetic helicity. Section \[sec:bench\] verifies the proposed methodology using synthetic data from mean-field dynamo model calculations. Section \[sec:obs\_helicity\] presents the derivation of magnetic helicity using observed magnetic fields, and Section \[sec:discussion\] discusses our findings. The method {#sec:method} ========== To determine the magnetic helicity density, we employ a decomposition of the vector magnetic field into toroidal and poloidal components using scalar potentials $S$ and $T$ [@KR80; @2018JPhA51W5501B]: $$\begin{aligned} \mathbf{B} & = & \boldsymbol{\nabla}\times\left(\hat{\mathbf{r}}T\right)+\boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\left(\hat{\mathbf{r}}S\right) = \nonumber \\ & = & -\frac{\hat{r}}{r}\Delta_{\Omega}S+\hat{\theta}\left(\frac{1}{\sin\theta}\frac{\partial T}{\partial\phi}-\frac{\sin\theta}{r}\frac{\partial F_{S}}{\partial\mu}\right)+\hat{\phi}\left(\sin\theta\frac{\partial T}{\partial\mu}+\frac{1}{r\sin\theta}\frac{\partial F_{S}}{\partial\phi}\right) \label{eq:pol_tor}\end{aligned}$$ where ${\displaystyle \Delta_{\Omega}=\frac{\partial}{\partial\mu}\sin^{2}\theta\frac{\partial}{\partial\mu}+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}}$, $\mu=\cos\theta$ and $\theta$ is the polar angle, and $F_{S}=\partial\left(rS\right)/\partial r$. Three components of vector magnetic field [radial $r$, meridional $\theta$, and zonal $\phi$, see, @Virtanen.etal2019] are then represented by three independent variables, $S$ , $T$, and $F_{S}$: $$\begin{aligned} B_{r} & = & -\frac{1}{r}\Delta_{\Omega}S,\label{eq:S}\\ B_{\theta} & = & \frac{1}{\sin\theta}\frac{\partial T}{\partial\phi}-\frac{\sin\theta}{r}\frac{\partial F_{S}}{\partial\mu},\label{eq:fs}\\ B_{\phi} & = & \sin\theta\frac{\partial T}{\partial\mu}+\frac{1}{r\sin\theta}\frac{\partial F_{S}}{\partial\phi}\label{eq:T}\end{aligned}$$ To determine a unique solution of Equations \[eq:S\]–\[eq:T\] we apply the following gauge (see, e.g., @KR80): $$\int_{0}^{2\pi}\int_{-1}^{1}Sd\mu d\phi=\int_{0}^{2\pi}\int_{-1}^{1}Td\mu d\phi=\int_{0}^{2\pi}\int_{-1}^{1}F_{S}d\mu d\phi=0.\label{eq:norm}$$ Note, that in the case of potential magnetic field $T=0$, and $F_{S}$ is determined by $S$. Therefore, the system of Equations \[eq:S\]–\[eq:T\] represents a least-squares problem. Hereafter we consider the general case of nonpotential magnetic fields on the solar surface. Eqs.(\[eq:S\]), (\[eq:fs\]) and (\[eq:T\]) can be transformed in: $$\begin{aligned} -\frac{1}{r}\Delta_{\Omega}S & = & B_{r}\\ \Delta_{\Omega}T & = & \frac{\partial}{\partial\mu}\sin\theta B_{\phi}+\frac{1}{\sin\theta}\frac{\partial B_{\theta}}{\partial\phi},\\ \frac{1}{r}\Delta_{\Omega}F_{S} & = & \frac{1}{\sin\theta}\frac{\partial B_{\phi}}{\partial\phi}-\frac{\partial}{\partial\mu}\sin\theta B_{\theta}\end{aligned}$$ Reconstruction and differentiating is done in the spectral spherical harmonic space using the SHTools [@sht2018]. After finding solutions for $S$, $T$ and $F_{S}$ we can determine components of vector potential $\mathbf{A}$, $$\begin{aligned} \mathbf{A} & = & \hat{\mathbf{r}}T+\boldsymbol{\nabla}\times\left(\hat{\mathbf{r}}S\right) = \nonumber \\ & = & \hat{\mathbf{r}}T+\frac{\hat{\theta}}{\sin\theta}\frac{\partial S}{\partial\phi}+\hat{\phi}\frac{\sin\theta}{r}\frac{\partial S}{\partial\mu}.\end{aligned}$$ Below we demonstrate the method using output of a dynamo model, where we have complete information about the distribution of vector magnetic field, its vector potential and magnetic helicity density. Dynamo Model Benchmark {#sec:bench} ====================== We use results of the non-axisymmetric $\alpha^{2}\Omega$ dynamo model developed recently by @2018ApJ867.145P. The model simulates the solar-type dynamo, in which the surface magnetic activity is governed by the dynamo-generated axisymmetric toroidal magnetic field. The dynamo parameters of the model are below threshold of the dynamo instability of large-scale non-axisymmetric magnetic fields. This resembles the situation for solar-type dynamos [@rad86AN]. In order to mimic emergence of solar active regions, we take into account the Parker’s magnetic buoyancy instability, which produces bipolar regions from the toroidal magnetic field at random latitudes and random moment of time under the condition that the magnetic field strength exceeds a critical threshold. The reader can find a detailed description of the model and the code in @2018ApJ867.145P and @2dspy. Figure \[snap\] shows snapshots of the magnetic field components, as well as the magnetic and current helicity densities. The snapshots are taken during the maximum of the dynamo cycle. The magnetic helicity has predominantly positive sign in the northern hemisphere and the negative sign in the southern hemisphere. In this case, the hemispheric helicity rule of the large-scale magnetic field follows from the dynamo theory which predicts that the sign of the magnetic helicity of the large-scale field corresponds to the sign of the $\alpha$-effect in a given hemisphere [@bl-br2003; @2018arXiv180404994S]. ![\[snap\]a) Snapshot of the radial magnetic field distribution (color image), streamlines show the horizontal (toroidal and poloidal) magnetic field; b) snapshot for the magnetic helicity density; c) the radial component of the current helicity density $H_{C}=B_{r}\left(\boldsymbol{\nabla}\times\mathbf{B}\right)_{r}$.](b){width="100.00000%"} Magnetic Helicity Density Derived From Observations {#sec:obs_helicity} =================================================== Observational Data ------------------ We apply the formalism described in Section \[sec:method\] to a set of intermediate resolution (360 by 720 pixels) vector magnetic fields synoptic maps from HMI/SDO. The dataset includes 116 Carrington rotations (CR) from CR2097 (May 2010) to CR2214 (March 2019). The HMI synoptic maps are calculated in Carrington longitude (degrees) – sine (latitude) coordinate grid. The pixel size is 0.5 degree in longitude and 1/180 in sine latitude. Method for producing the HMI synoptic maps is described in details by [@2017SoPh292.29L]. For these data, the 180 degree ambiguity in the horizontal field direction was resolved by the HMI team using a combination of a minimum energy criterium (for pixels with stronger fields) and random disambiguation (for weak field pixels). Additional details about the HMI data reduction and the disambiguation procedure can be found in the above cited paper. [@2017SoPh292.29L] found that with the chosen combination of disambiguation methods the noise level of the vector magnetic field in the synoptic charts varies from $\pm 10$G during the solar maximum to $\pm 20$G during the solar minimum. Results ------- ![\[synm\]Synoptic maps of the vector magnetic field and the reconstructed potential from SDO/HMI for CR 2156.](brh2156){width="95.00000%"} As the first example, we consider results of reconstruction of the vector potential for the synoptic maps of CR2156. This Carrington rotation is characterized by strong active region NOAA AR 12192 that emerged in the southern hemisphere. Figure \[synm\] shows the synoptic maps of the vector magnetic field and the reconstructed potentials $\mathrm{S}$, $\mathrm{A_{r}=rT}$. Distribution of the vector potential reveals a large-scale non-axisymmetric pattern associated with this active region. Interesting that the vector-potential components show inverse sign relative to the corresponding magnetic field components in the core of the active region. This results in predominantly negative magnetic helicity in AR 12192, which is shown in Figure \[recon\]. We notice that the east part of the southern hemisphere shows the background magnetic helicity of the positive sign which corresponds to the basic hemispheric helicity of the active regions [@Pevtsov.etal2014]. The large area of the negative magnetic helicity density around NOAA AR 12192 resembles the situation demonstrated in our dynamo model. Thus, it can be speculated that the origin of this active region is related to large-scale magnetic field located in a shallow subsurface layer. ![\[recon\]Distribution of the magnetic helicity density $\mathbf{A}\cdot\mathbf{B}$ for CR 2156.[]{data-label="fig:cr2156"}](abh2156){width="50.00000%"} Solving Equations \[eq:S\]–\[eq:T\] for each Carrington rotation from CR2097 to CR2214 we derived the distributions of the vector potential and magnetic helicity density for solar cycle 24. The large-scale distributions of magnetic field and helicity density are obtained by means of the azimuthal averaging of the synoptic maps for each Carrington rotation. To represent the mean signal, where we filtered out the time variations with period less than 2 year. Also, we employ the Gaussian smoothing with a FWHM equal 5 pixels in latitude and 10 CR time. Figure \[bhmi\] shows the time-latitude diagrams for the axisymmetric components of magnetic field and vector potential. The time-latitude diagrams of the magnetic field evolution are in agreement with [@2018MNRAS480477V]. The reconstructed potentials disagree with results of [@PP14], who used profiles of $\bar{B}_r$ from different longitudinal distances from the central meridian to reconstruct $\bar{B}_{\phi}$ and $\bar{A}_{r}$. Also, the obtained $\bar{B}_{\phi}$ disagrees with our results at the polar-ward side of the sunspot activity zone, where the large-scale toroidal magnetic field is present. The sign of this field is opposite to the sign of toroidal field in the solar active regions. Such bi-modal structure of the axisymmetric toroidal magnetic field affects the distribution of $\bar{A}_{r}$. The sign of the polar-ward side of the axisymmetric toroidal magnetic field can hardly be explained by surface effect of the differential rotation acting on the meridional component of the axisymmetric magnetic field. From Figure \[bhmi\] it is clear that the direction of the meridional component is opposite to the one which might produce the polar-side toroidal magnetic field. Also, in our case, the evolution of $\bar{A}_{\phi}$ differs from results of [@sten88; @bl-br2003] and [@PP14] as well, where they found symmetric $\bar{A}_{\phi}$ profiles relative to the equator. The difference is likely due to the rather asymmetric development of Cycle 24 in the northern and southern hemispheres. Figure \[hhmi\]a shows the time-latitude diagrams of the magnetic helicity density of the large-scale field, $\bar{\mathbf{A}}\cdot\bar{\mathbf{B}}$. The magnetic helicity of the large-scale field is positive in the high-latitude zone of the northern hemisphere at the beginning of Cycle 24, and it changes the sign to negative after the polar field reversal. A similar situation is observed in the southern hemisphere. Figure \[hhmi\]b shows the time-averaged hemispheric helicity sign rule together with uncertainty bars. The uncertainties are estimated by using differences of the original and smoothed signals. It is seen that in average over the cycle 24 the large scale magnetic field shows the same hemispheric sign as the “small-scale” magnetic field. However if we would restrict the averaging period to the first half of the cycle 24, i.e., by CR2097–2156 we get, in general, the positive magnetic helicity density in the northern hemisphere and negative in the southern one (except the low latitudes). Therefore at this period of time the bi-helical property can be confirmed with some reservations. This agrees with the two-scale analysis of [@2017ApJ836.21B] as well as with results of [@PP14] (cf., Fig9a there). Also, Figures \[hhmi\]c and d support this conclusion. Figure \[hhmi\]c shows evolution of “small-scale” magnetic helicity density, i.e., $\mathbf{\overline{a\cdot\mathbf{b}}}=\overline{\mathbf{{A}\cdot\mathbf{{B}}}}- \mathbf{\overline{A}\cdot\mathbf{\overline{B}}}$. The definition of small-scale helicity, $\mathbf{\overline{a\cdot\mathbf{b}}}$, includes magnetic fields from all range of scales except the axisymmetric magnetic field. A more accurate analysis of the magnetic helicity distribution over the scales can be done by using the two-scale analysis introduced by @2017ApJ836.21B. Averaged over the cycle $\mathbf{\overline{a\cdot\mathbf{b}}}$, shows the hemispheric helicity rule for the solar active regions. It shows the large uncertainty bars which are mostly caused by fluctuations in magnetic flux emergence. It is also can be seen that for the first half of the cycle 24 the $\mathbf{\overline{a\cdot\mathbf{b}}}$ holds the hemispheric helicity rule. The violation of the rule in the time-averaged signal is likely caused by the magnetic activity in the Southern hemisphere during 2014–2015 years. In particular, the active region NOAA 12192 strongly violates the hemispheric helicity rule as we see in Figure \[fig:cr2156\]. In the end of our observational period, the $\mathbf{\overline{a\cdot\mathbf{b}}}$ shows inversion of helicity sign in the low latitudes. This agrees with the results for the current helicity density evolution in cycle 23 [e.g., @zetal10]. We find that the standard error for $\bar{\mathbf{A}}\cdot\bar{\mathbf{B}}$/R is less than 1 G$^2$ and has maximum at the sunspot formation latitudes. Similarly, the standard error of $\mathbf{\overline{a\cdot\mathbf{b}}}/R$ is less than 100 G$^2$. [@2017SoPh292.29L] estimated the maximum uncertainty of the vector magnetic field measurement about $\sim$ 10–20 G. Then, we can conclude that the standard deviation of uncertainty of the determined the normalized magnetic vector potential components is less than 10 G. A more accurate estimate of errors for the magnetic helicity density on each synoptic map requires additional analysis, and it will be done in a separate paper. ![\[bhmi\]The time-latitude evolution of the large-scale magnetic field : $\bar{B}_r$, $\bar{B}_{\phi}$, and $\bar{B}_{\theta}$, as well as the large-scale vector-potential components $\bar{A}_r$ and $\bar{A}_{\phi}$, the total magnetic helicity density $\overline{\mathbf{{A}\cdot\mathbf{{B}}}}$.](bhmi){width="95.00000%"} ![\[hhmi\] Magnetic helicity in Solar Cycle 24: a) The time-latitude evolution of the large-scale magnetic helicity density, $\mathbf{\overline{A}\cdot\mathbf{\overline{B}}}$; b) the mean latitudinal profile of the large-scale magnetic helicity density with 95$\%$ confidence interval for the standard error, the red line shows the average profile for the first half of the cycle including CR2097-2156; c) shows the same as (a) for the azimuthal averaging of the small-scale magnetic helicity density $\mathbf{\overline{a\cdot\mathbf{b}}}=\overline{\mathbf{{A}\cdot\mathbf{{B}}}}- \mathbf{\overline{A}\cdot\mathbf{\overline{B}}}$; d) shows the same as (b) for the $\mathbf{\overline{a\cdot\mathbf{b}}}$.](AB){width="95.00000%"} Discussion and Conclusions {#sec:discussion} ========================== Here, we propose a novel approach to reconstruct the surface magnetic helicity density on the Sun and sun-like stars. The magnetic vector potential is determined via decomposition of vector magnetic field in the toroidal and poloidal components. The method is applied to study the evolution of magnetic helicity density of large- and small-scale fields in solar cycle 24. Our results show that at the beginning if cycle 24, the large- and small-scale magnetic helicities are opposite in sign to each other, and both show the hemispheric asymmetry in sign. This is in agreement with the magnetic helicity conservation theory [@pouquet-al:1975a], which predicts the sign of variations of the small-scale magnetic helicity density, $\mathbf{\overline{a\cdot\mathbf{b}}}$ should be opposite to the sign of the large-scale helicity. However, in the declining phase of cycle 24 the large-scale polar magnetic field shows the negative (positive) magnetic helicity in the North (South), which is the same in sign as the helicity of solar active regions (small-scale field). Apparently, in the polar regions the large-scale helicity, $\mathbf{\overline{A}\cdot\mathbf{\overline{B}}}$, changed the sign near the solar maximum. The dynamo models predict that both $\mathbf{\overline{A}\cdot\mathbf{\overline{B}}}$ and $\mathbf{\overline{a\cdot\mathbf{b}}}$ can change sign in course of the solar cycle due to other sources of the magnetic helicity evolution like the turbulent eddy diffusivity of the large-scale magnetic field [@kleruz82; @pip13M; @sok2013]. Another possible reason is the magnetic helicity flux escape from the solar surface [@2000JGR...10510481B]. We also note that dynamo models of [@pip13M] predict another polar inversion of the large-scale magnetic helicity models during solar minima. Thus, we expect that the magnetic helicity of large-scale (polar field) will reverse its sign again around the solar minimum, which would restore the hemispheric helicity rule to the same condition as it was at the beginning of cycle 24. The amplitude of magnetic helicity of large-scale fields is significantly (an order of magnitude) smaller as compared with helicity of small-scale (active region) fields. Thus, active regions appear to be the major contributors to magnetic helicity observed on the visible solar surface (photosphere). This is further supported by the fact that the emergence of a single (large) active region could affect the hemispheric preference in sign of helicity for some Carrington rotations. In our data we see that the magnetic helicity density of the large- and small-scale fields often show the same sign in each hemisphere. In our opinion this can be due to a bi-modal distribution of the large-scale toroidal magnetic field on the solar surface. Indeed, we see the two components of the toroidal magnetic field: one is near equator and another is in the mid latitudes. The near equatorial toroidal field is originated from sunspots. The origin of the mid-latitude toroidal field is unclear. This component of the large-scale field can results into mono-helical over the scales magnetic field distributions in the each hemisphere. The evolution of the large-scale magnetic field and its vector potential results in a complicated magnetic helicity evolution. In the data, it is seen that the southern hemisphere was close to the bi-helical state during the growing phase of the solar cycle, from 2010 to 2012. The northern hemisphere, except the polar region, shows the same sign of the magnetic helicity density for the large- and small-scale field during that period. Another source of violation of the hemispheric helicity rule is the asymmetric about equator development of solar cycle 24. This results in the equatorial parity breaking in the large-scale magnetic field and vector potential components. In particular, the obtained evolution of $\bar{A}_{\phi}$ is different from previous results by @black-bran:02 and [@PP14]. The hemispheric helicity rule and bi-helical distributions of the solar magnetic fields are essential properties of the solar dynamo operating in the convection zone. We find that in solar cycle 24 these properties of solar dynamo show a complicated evolution in the way which is not expected in any current dynamo model. The suggested method of the magnetic helicity reconstruction can be applied to other stars where the low-degree modes of the vector magnetic field distributions, such as determined in [@2018MNRAS480477V], can be used to calculate the magnetic vector potential and magnetic helicity density of the large-scale stellar magnetic field. This work was borne out of discussions held among the authors during “Solar Helicities in Theory and Observations: Implications for Space Weather and Dynamo Theory” Program at Nordic Institute for Theoretical Physics (NORDITA) in 4–29 March 2019. JSOC/HMImaps: Courtesy of NASA/SDO HMI science team. AAP acknowledges partial support by NASA 80NSSC17K0686 and NSF AGS-1620773 grants. AGK was partially supported by NASA grants NNX14AB70G and NNX17AE76A. VVP conducted this work as a part of FR II.16 of ISTP SB RAS, also the support of the RFBR grant 19-52-53045 GFEN-a is acknowledged. natexlab\#1[\#1]{}\[1\][[\#1](#1)]{} \[1\][doi: [](http://doi.org/#1)]{} \[1\][[](http://ascl.net/#1)]{} \[1\][[](https://arxiv.org/abs/#1)]{} , K. S., & [Pevtsov]{}, A. 2011, in , Vol. 8148, Solar Physics and Space Weather Instrumentation IV, 814809 , S., & [Zhang]{}, H. 1998, , 496, L43, , M. A., & [Hornig]{}, G. 2018, Journal of Physics A Mathematical General, 51, 495501, , M. A., & [Ruzmaikin]{}, A. 2000, , 105, 10481, , E. G., & [Brandenburg]{}, A. 2002, , 579, 379, —. 2003, , 584, L99, , A., [Petrie]{}, G. J. D., & [Singh]{}, N. K. 2017, , 836, 21, , A., & [Subramanian]{}, K. 2005, , 417, 1, , A. 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--- abstract: 'We present 7mm Mopra observations of the dense molecular gas towards the W28 supernova remnant (SNR) field, following a previous 12mm line survey of this region. These observations take advantage of the 7mm beam size to probe the dense and disrupted gas in the region at $\sim$1 arcmin scales. Our observations are focused towards the north-eastern (NE) HESS J1801-233 and southern HESS J1800-240B TeV gamma-ray sources, with slightly less observations towards HESS J1800-240A & C. Using the CS(1-0) transition we reveal multiple regions of dense gas, $n_{\rm H_2} \sim10^{5}$. We report the discovery of dense gas towards HESS J1800-240C, at the site of a 1720MHz OH maser. The NE molecular cloud is known to be disrupted, many 1720MHz OH masers and broad CO line emission are detected at the rim of W28. Here we reveal this shock interaction region contains generally extended clumpy CS, as well as clumpy SiO and CH$_3$OH emission with broad line profiles. The FWHM of the molecular lines extend up to 18on the W28 side of the NE cloud. The detection of SiO towards maser clumps OH C, D, E & F provide further evidence of the shocked conditions in the NE cloud. Several other lines associated with star formation are also detected towards the southern source, notably the energetic complex G5.89-0.39. The spatial match of dense gas with the TeV emission further supports the CR origin for the gamma-rays. We estimate the mass of several extended dense clouds within the field and predict the TeV flux from the dense cloud components. The predicted fluxes are on the order of $10^{-14}$–$10^{-13}$phs$^{-1}$, which should be detectable and possibly resolved by a future TeV instrument, such as the Cherenkov Telescope Array.' title: 'A 7mm line survey of the shocked and disrupted molecular gas towards the W28 field TeV gamma-ray sources.' --- \[firstpage\] ISM: clouds – regions – ISM: supernova remnants – molecular data – gamma-rays: observations – supernovae: individual: W28. Introduction ============ W28 is a striking example of TeV (10$^{12}$ eV) gamma-ray emission spatially overlapping with molecular gas [@hess_w28]. W28 is part of the increasing list of sources with spatial overlap between TeV or GeV (10$^{9}$ eV) gamma-ray emission and molecular gas (e.g. HESS J1745-290/SNR G359.1-0.5 [@hess_1745], HESS J1714-385/CTB 37A [@hess_ctb37a], HESS J1923+141/SNR G49.2-0.7 [@feinstein], IC 443 [@ic443_magic; @ic443_veritas]), in addition to the central molecular zone (CMZ) towards the Galactic centre region [@hess_CMZ]. W28 is a mixed morphology old age ($>10^{4}$yr; @kaspi) supernova remnant (SNR) with dimensions of 50$^\prime \times 45^\prime$. W28 is estimated to be at a distance of 1.2 to 3.3kpc (e.g. @goudis [@lozinskaya; @motogi]). It has been shown to exhibit non-thermal radio emission [@dubner], thermal X-ray emission [@rho2002], and, more recently gamma-ray emission at TeV [@hess_w28] and GeV [@fermi_w28; @agile] energies. Several CO surveys in the (1-0), (2-1) and (3-2) lines reveal massive molecular clouds to the north-east (NE) and to the south (S) of the SNR [@arikawa; @reach; @torres; @hess_w28; @nanten21]. Most of the CO emission appears centred at a local standard of rest velocity () similar to that inferred for W28 $\sim$7(corresponding to a distance $\sim 2$kpc) based on studies [@velazquez]. The molecular clouds NE of W28 are known to exhibit broad CO line emission. It has been argued that W28 has disrupted much of this gas, giving rise to its broad velocity distribution [@arikawa; @reach; @torres]. The most likely mechanism for this type of disruption would be the SNR shock interacting with the molecular gas, as is the case for other sources e.g. HESS J1745-290/SNR G359.1-0.5 [@hess_1745], HESS J1714-385/CTB 37A [@hess_ctb37a], HESS J1923+141/SNR G49.2-0.7 [@feinstein] and IC 443 [@ic443_magic; @ic443_veritas]. All of these sources similarly display 1720MHz OH masers and are mature SNRs (age $>10^{4}$yr). Our 3 observations [@me] revealed that the W28 SNR shock has disrupted the dense core of the NE cloud, and strong 3(3,3) and (6,6) detections with broad line widths (FWHM $>10$), suggest that this region is warm and turbulent. The shocked NE cloud also provides an opportunity to study the diffusion and propagation of CRs into molecular clouds. Additionally, the effects of a SNR shock propagating through a dense molecular cloud can be studied and search for sites of star formation triggered by the passing shock. The southern clouds harbour sites of high mass star formation, containing multiple regions, G6.225-0.569, G6.1-0.6 [@lockman; @kuchar] and the ultracompact region (UC) G5.89-0.39, (see e.g @harvey [@kimkoo2001]). A recently detected 1720MHz OH maser towards a candidate SNR G5.71-0.08 [@brogan] may also suggest that there is another SNR shock with molecular gas interaction occurring in one of the southern clouds. A key question is whether the W28 SNR has disrupted the southern clouds and is responsible for the TeV emission, or whether other high energy processes are at play. The star formation activity present in the southern clouds may play a part in the production of TeV emission, as theory into protostellar particle acceleration may suggest [@araudo]. The the TeV gamma-ray emission detected towards the NE cloud has strong evidence supporting the hadronic emission mechanism. In order to model leptonic scenarios, uncomfortably low magnetic fields and gas densities are required [@yamazaki2006; @fujita2009; @gabici2010; @fermi_w28]. These requirements contradict with the observed molecular gas densities $n_{\rm H_2} > 10^{4}$ [@me] and the magnetic field enhancements seen in dense molecular clouds [@crutcher]. The key question relates to the origin of the TeV particles. Provided the hadronic emission scenario, a population of high energy particles are required to interact with the dense clouds. For the NE cloud, the SNR shock is the obvious source of accelerated particles. However, the same can not be conclusively said about the southern sources. CRs accelerated by W28 in the past may have propagated south to illuminate the southern clouds. Additionally, multiple regions and sites of star formation, including the energetic UCcomplex G5.89-0.39, could assist in CR acceleration towards the southern TeV sources. Since the common and abundant gas tracer, CO, with a critical density of 10$^{2}$, rapidly becomes optically thick towards clumps and cores, probing the molecular cloud density profile can be impaired. Ideal tracers of dense gas are those with a lower abundance (10$^{-5}\times$CO) and higher critical density (10$^{4}$-$10^{5}$). Using the Mopra telescope, we [@me] conducted broad scale ($\sim1.5^{\circ}$square) observations of the W28 field in a 12mm line survey. Taking advantage of the 23GHz 3 inversion lines, the cold dense interiors of the molecular clouds were probed and revealed dense gas spatially consistent with both the CO gas and TeV emission. In order to further probe the structure and details of the W28 field molecular clouds, we have continued to use the Mopra radio telescope in a 7mm line survey to observe the CS(1-0) line, as well as other lines including SiO(1-0), Class I CH$_{3}$OH masers and cyanopolyynes HC$_{n}$N ($n=3,5,7$), to trace the dense gas, the presence of shocks, outflows, and disrupted gas, and sites of high-mass star formation. The 43GHz CS(1-0) line is a useful warm, dense gas tracer, and, as CS also exists in various isotopologue forms, the optical depth and column density can be constrained. The higher frequency of 7mm lines compared to our previous 12mm observations, results in a smaller beam FWHM ($\sim1$ arcminute). This makes CS an ideal follow-up tracer to determine the properties of the dense and disrupted gas toward the molecular clouds in the W28 region. Mopra Observations and Data Reduction {#sec:obs} ===================================== Observations were performed with the Mopra radio telescope in April of 2009 and March of 2010 and utilised the UNSW Mopra wide-band spectrometer (MOPS) in zoom mode. Mopra is a 22m single-dish radio telescope located $\sim$450km north-west of Sydney, Australia ($31^\circ 16^\prime 04^{\prime\prime}$S, $149^\circ 05^\prime 59^{\prime\prime}$E, 866m a.s.l.). The 7mm receiver operates in the 30-50GHz range and, when coupled with MOPS, allows an instantaneous 8GHz bandwidth. This gives Mopra the ability to cover 40% of the 7mm band and simultaneously observe many spectral lines. The zoom mode of MOPS allows observations in up to 16 windows simultaneously, where each window is 137.5MHz wide and contains 4096 channels in each of two polarisations. At 7mm this gives MOPS an effective bandwidth of $\sim$1000with resolution of $\sim$0.2. Across the whole 7mm band, the beam FWHM varies from 1.37$^{\prime}$ (31GHz) to 0.99$^{\prime}$ (49GHz) [@mopra_beam]. Table \[tab:lines\] lists the lines which MOPS was tuned to receive. ------------------------ ------------ ---------- -------------- Molecular Line Frequency Detected Detected Map Deep Spectra $^{30}$SiO(1-0, $v=0$) 42373.365 – – SiO(1-0, $v=3$) 42519.373 Yes – SiO(1-0, $v=2$) 42820.582 Yes – $^{29}$SiO(1-0, $v=0$) 42879.922 – – SiO(1-0, $v=1$) 43122.079 Yes – SiO(1-0, $v=0$) 43423.864 Yes Yes CH$_{3}$OH-I 44069.476 Yes Yes HC$_{7}$N(40-39) 45119.064 – – HC$_{5}$N(17-16) 45264.75 Yes Yes HC$_{3}$N(5-4) 45488.839 Yes Yes 3CS(1-0) 46247.58 Yes Yes HC$_{5}$N(16-15) 47927.275 Yes Yes 34S(1-0) 48206.946 Yes Yes OCS(4-3) 48651.6043 – – CS(1-0) 48990.957 Yes Yes ------------------------ ------------ ---------- -------------- : Molecular lines and their corresponding rest frequencies which MOPS was tuned to receive. The final two columns indicate whether the line was detected in our mapping or deep pointing observations.\[tab:lines\] ‘On-the-fly’ (OTF) mapping observations were based on our earlier 12mm mapping of the W28 region. Although we mapped all four TeV sources, deeper observations towards the NE shocked cloud (HESS J1801-233) and southern cloud (HESS J1800-240B) were obtained than toward HESS J1800-240A & C. The observations were recorded alternating the scanning direction from RA to Dec. to reduce noise levels and to eliminate artificial stripes that can be introduced when only one scanning direction is used. Collectively, over both observation runs we obtained 9 passes on the NE cloud (HESS J1801-233), 9 passes towards the southern cloud (HESS J1800-240B) and 3 passes toward each of the HESS J1800-240A and HESS J1800-240C regions. Data were reduced using the standard ATNF packages , , and .[^1] For mapping data, was used to perform a bandpass calibration for each row using the preceding off-scan as a reference, and applied a linear fit to the baseline. re-gridded and combined all data from all mapping scans into a single data cube, with pixels ($\Delta x,\Delta y,\Delta z$)=(15, 15, 0.21). The mapping data were also weighted according to the relevant , Gaussian-smoothed based on the Mopra beam (1.2$^\prime$ FWHM and 3$^\prime$ cut-off radius), and pixel masked to remove noisy edge pixels. Data cubes were then converted into an scantable and had a $7^{{\rm th}}$ order polynomial function fit subtracted from each pixel in the cube before returning the final output fits files. Analysis of position-switched deep pointings employed with time-averaging, weighting by the relevant , and baseline subtraction using a linear fit after masking the 15 channels at each bandpass edge. In both mapping and position-switched data, the antenna temperature (corrected for atmospheric attenuation and rearward loss) was converted to the main beam brightness temperature (K), such that = /$\eta_{\rm mb}$ where $\eta_{\rm mb}$ is the Mopra main beam efficiency. Each line was corrected by the relevant $\eta_{\rm mb}$ following @mopra_beam. Due to the uneven mapping exposure across the W28 field, the RMS error in , (K), varies. On top of the exposure dependence, there is also a frequency dependence on across the 42-49GHz band. For this reason in Figure \[fig:exposure\] we present a CS(1-0) peak pixel map of the W28 field with dashed boxes outlining the exposure boundaries. The numbers presented in the white boxes are the averaged achieved (across the 42-49GHz band) within that boundary. We also show in Figure \[fig:whole\_velopix\] the velocity of the peak pixels to show the general velocity locations and distribution of the various clumps of CS(1-0) gas. We show the same contours on both Figures \[fig:exposure\] and \[fig:whole\_velopix\] for guidance. ![image](./figures/peakpix/CS_whole_peakpix1.eps){width="\textwidth"} ![image](./figures/peakpix/CS_whole_peakvel.eps){width="\textwidth"} For the detected lines, images of velocity-integrated intensity, position velocity (PV) and velocity dispersion () have been produced where possible (although not all are shown). Velocity-integrated intensity images (0$^{\rm th}$ moment) are integrated over a velocity range determined by the observed line width. This was chosen to encompass the bulk of the emission from the region of interest. Minimum contour levels were set based on the integrated of the emission. The integrated of each image was determined by creating additional 0$^{\rm th}$ moment maps in a velocity space either side of the velocity range of interest, ensuring the same number of channels were used. These additional moment maps were used to create pixel distribution histograms which were Gaussian fitted. In this way we estimated the integrated from the Gaussian fit results. On all images, the minimum contour levels are mentioned in terms of the raw value as well as . Generally the minimum accepted contour level on integrated maps is 2. PV plots were created by re-ordering the data cubes axes and Hanning smoothing the velocity axis (width $\sim3$) to improve image quality. The PV plots show the peak pixel along the declination axis for illustrative purposes. Intensity weighted velocity dispersion (2 moment), , maps were calculated for pixels above a reasonable threshold. [lcccccccccc]{} Core / & & & FWHM & $\tau$ & $^{}$ & $N$\[CS\] & $N$\[H$_{2}$\] & Mass & n$_{\rm H_{2}}$ & Virial Mass\ Region & \[$K$\] & \[\] & \[\] & & \[$K$\] & \[$\times10^{14}$\] & \[$\times10^{23}$\] & \[M$_{\odot}$\] & \[$\times10^{5}$\] & \[$\times10^{2}$M$_{\odot}$\]\ \ DP-1 & 1.1 & 13.1 & 18.1 & 1.0 & 46 & 16.5 & 4.1 & 790 & 6.9 & 133 – 279\ DP-2 & 2.4 & 7.4 & 11.2 & 1.6 & 46 & 28.8 & 7.2 & 1400 & 12.0 & 23 – 81\ DP-3$^{\dagger}$ & 1.4 & 9.0 & 2.2 & – & 20 & 0.1 & 2.2$\times10^{-2}$ & 77 & 0.5 & 1.2 – 4.3\ DP-4 & 9.5 & 9.3 & 4.0 & 2.8 & 33 & 61.1 & 15.3 & 2900 & 25.4 & 3.4 – 11.9\ DP-5$^{\ddagger}$ & 2.8 & 13.0 & 1.1 & 2.1 & 20 & 4.0 & 1.0 & 190 & 1.7 & 0.3 – 0.9\ \ &\ &\ &\ \ \ \ [lcccccccc]{} Region & & FWHM & $\tau^{}$ & $^{\dagger}$ & $N$\[CS\] & $N$\[H$_{2}$\] & Mass & n$_{\rm H_{2}}$\ & \[$K$\] & \[\] & & \[$K$\] & \[$\times10^{14}$\] & \[$\times10^{23}$\] & \[M$_{\odot}$\] & \[$\times10^{4}$\]\ \ NE Shocked Cloud$^{a}$ & 1.8 & 7.3 & 2.3 & 35 & 17.8 & 4.4 & $<$56000 & 6.1\ G5.89-0.39 Region$^{b}$ & 2.2 & 3.8 & 3.8 & 25 & 11.6 & 2.9 & 54000 & 2.7\ G5.89-0.39 NW Arm$^{c}$ & 0.8 & 3.5 & – & 20 & 1.4 & 0.3 & 4200 & 0.6\ G5.89-0.39 SE Ridge$^{d}$ & 0.7 & 3.1 & – & 16 & 2.7 & 0.7 & 12000 & 0.9\ G6.1-0.6 Region$^{e}$ & 0.7 & 5.1 & – & 25 & 1.8 & 0.5 & 6500 & 0.6\ G6.225-0.569 Region$^{f}$ & 1.5 & 3.4 & – & 21 & 1.8 & 0.5 & 1600 & 1.2\ \ \ \ \ \ \ \ \ Analysis and Results Overview {#sec:overview} ============================= Of the seventeen molecular transitions searched for (Table \[tab:lines\]), eleven were detected. From the mapping data we detected: CS, 34S and 3CS in the (1-0) transition; SiO(1-0) in several vibrational modes; CH$_{3}$OH; HC$_{3}$N(5-4) and HC$_{5}$N in the (17-16) and (16-15) transitions. CS Emission ----------- The CS(1-0) line was detected across the W28 field and is the most prominent. Along with the common CS isotopologue, $^{12}$C$^{32}$S we also detect other isotopologues, 34S and 3CS in the (1-0) transition. Unless otherwise indicated, CS refers to the common $^{12}$C$^{32}$S isotopologue. From our deep pointing observations (DP-1 to DP-5, locations shown in Figure \[fig:exposure\]) with $\sim$2 times lower than the mapping data, a Gaussian fit to the detected isotopologue line emission allowed the optical depth and column density to be estimated. The method assumed an elemental abundance ratio between CS and the rarer isotopologues of 22.5 for \[CS\]/\[34S\] and 75 for \[CS\]/\[3CS\] to calculate the optical depth using @zinchenko [Equation 1]. The upper state column density was then estimated via the method from @goldsmith_langer [Equation 9]. The total column density of all states was obtained by applying a local thermal equilibrium (LTE) approximation and using temperature estimations from our 3 observations [@me], assuming that the CS gas is tracing the same gas as our 3 observations. From these parameters, the LTE hydrogen mass, number density and virial mass were estimated via the methods outlined in @me, by assuming a compact-source size $r=0.2$pc and a molecular abundance ratio, [$\chi$]{}$_{\rm CS}$, of CS to H$_{2}$. CS molecular abundance ratios are known to vary dramatically from different sites in the Galaxy, ranging from $1\times10^{-8}$ [@frerking eg. Sgr B - Table 4] to $6\times10^{-11}$ [@linke eg. W43 - Table 3]. We adopted an abundance ratio of [$\chi$]{}$_{\rm CS}=4\times10^{-9}$ [@irvine] following @zinchenko which is typical for dense quiescent gas. This is likely the case for most of the clumps, with the exception of the NE cloud as later discussed in section \[sec:NEcloud\]. The results of our compact-source size LTE analysis and cloud mass/densities from deep pointing observations are displayed in Table \[tab:massdens\]. The LTE analysis method has also been applied to larger extended regions by utilising spectra from mapping data. For extended source calculations, we extracted and averaged the spectra for all pixels contained within an elliptical region. In this way, we apply the standard analysis outlined above to a single averaged spectra for the entire region. We used the same abundance ratio, but assumed the emission fills a volume defined by a prolate ellipsoid of volume $V=4/3 \times \pi\,r_{1}\,r_{2}^{2}$. Results of the extended source LTE analysis from mapping data are displayed in Table \[tab:extdmassdens\] SiO Emission ------------ We also tuned MOPS to include several different SiO(1-0) transitions arising from different vibrational modes ($v=0,1,2,3$), as well as the isotopologues $^{29}$SiO and $^{30}$SiO. The SiO molecule is a known and proven signpost for the presence of shocked and disrupted gas [@Schilke:1996; @Martin-Pintado:2000; @Gusdorf:2008a; @Gusdorf:2008b]. Our detection of SiO from several locations provides some insight into the disruptive conditions within the molecular clouds. Unless otherwise indicated, SiO refers to $^{28}$SiO in the $v=0$ mode. CH$_{3}$OH Emission ------------------- Multiple detections of CH$_{3}$OH masers towards the southern clouds are indicative of the star formation activity present towards the regions and surrounding environments. Additional detections of CH$_{3}$OH emission in the NE cloud is interesting. To date, no star formation tracers such as: H$_{2}$O masers, CH$_{3}$OH masers or infra-red emission have been detected towards this cloud. The CH$_{3}$OH emission in the NE cloud does not appear to correspond to stellar activity, as discussed later. Cyanopolyyne HC$_{n}$N ($n=3,5,7)$ Emission ------------------------------------------- Detections of the cyanopolyynes HC$_{3}$N(5-4), HC$_{5}$N(16-15) and HC$_{5}$N(17-16) towards the southern clouds are indicative of [activity in cold dense cores at the very early stages of high mass star formation]{}. Interestingly, additional detections of HC$_{3}$N(5-4) towards the shocked NE cloud are revealed. The sites of HC$_{3}$N(5-4) emission in the NE cloud do not appear to correspond to stellar activity. A more detailed discussion of the morphology and implications of the molecular transitions detected from each of the mapped regions follows in Section \[sec:specdisc\] Specific Discussion of Regions {#sec:specdisc} ============================== North-eastern Cloud / HESS J1801-233 {#ssec:northeast} ------------------------------------ Located at the northern boundary of HESS J1801-233 and nearby the giant region M20, we observe several molecular lines, which reveal dense gas and star formation activity. These molecular lines are all at a range $\sim20$, which is further away than the rest of the molecular emission seen in the NE cloud, $\sim10$, discussed later. This core was detected in our 12mm study and where it was referred to as Core1. Toward this northern most core is an IRAS source, IRAS 17589-2312 [@bronfman], and we adopt the IRAS nomenclature. The dense CS core seen here is likely part of a molecular bridge connecting down towards the peak of the HESS J1801-233 TeV emission seen in Figure \[fig:exposure\]. We note that in the peak pixel map, Figure \[fig:exposure\], the lowest contour level is $\sim10$. This gives an indication to the prominence and the extent of the CS(1-0) emission detected towards the NE cloud and the W28 clouds in general. Toward IRAS 17589-2312, we observe multiple Class I CH$_{3}$OH masers along the line of sight within the beam. The maser emission is relatively weak (peak $\sim$0.6K) compared to the other CH$_{3}$OH masers seen in the W28 field discussed later. This detection adds to the list of maser transitions which have been observed from this region: 6.7GHz CH$_{3}$OH [@fontani], 22GHz H$_{2}$O [@codella; @me] and now 44GHz CH$_{3}$OH. We also detect the cyanopolyyne HC$_{3}$N(5-4) line towards this core indicating hot gas phase chemistry probably driven by the central source. There is also a detection of SiO(1-0) which indicates shocks and/or outflows originating from IRAS 17589-2312. @lefloch label this core as TC5 and suggest that the star formation in this core is recent and could possibly have been triggered by the W28 SNR. Spectra for the line emission detected towards this core are shown in Figure \[fig:core1spec\]. Further south an extended dense region is spatially consistent with the TeV gamma-ray source HESS J1801-233. Towards the north-east rim of W28 several mm-line surveys [@wootten; @arikawa; @reach; @torres; @hess_w28; @nanten21; @me] have revealed the well-known SNR shock interaction site, traced by broad line measurements and the presence of many 1720MHz OH masers [@frail; @claussen]. Figure \[fig:intmultipanel\] contains integrated intensity images for emission detected towards this region. The CS(1-0) emission is spatially matched to the CO(2-1) emission measured by the Nanten telescope [@nanten21], and 3 emission measured by Mopra revealed a dense clump in the NE cloud [@me]. This spatial overlap is generally expected as CS will trace the denser ($n\sim$10$^{5}$) regions of the molecular clouds. An image showing the spatial relationship between the CO(2-1), CS(1-0) and 3(3,3) gas is the NE cloud is shown in Figure \[fig:densediffuse\]. The arcmin beam size allows the dense portion of the cloud to be probed in more detail than in our two arcmin beam 3 study. Figure \[fig:intmultipanel\] highlights the clumpy nature of the cloud. Several dense clumps are arranged in a north-east to south-west direction. We also note the minimum contour level for the integrated CS(1-0) emission in Figure \[fig:intmultipanel\] is set to 10. The CS(1-0) emission is very prominent throughout the NE cloud and likely extends towards the limits of our mapping. The line profiles of the CS and isotopologue emission are broad, with FWHM$\sim$18consistent with other molecular line observations from the region. Additionally, the asymmetry in the spectra provide evidence for the presence of shocks and gas disruption [@seta_1998]. Some CS line profiles indicate wings on the ‘positive’, or far side of the peak emission (e.g. CS(1-0) point 3/DP-1 on Figure \[fig:intmultipanel\]) and also on the ‘negative’, or nearer, side of the peak emission in others (e.g. CS(1-0) point 1/DP-2 on Figure \[fig:intmultipanel\]). In addition to the asymmetric line profiles, symmetric line profiles (or at least profiles with similar wings on both sides of the peak emission) are also seen towards this NE cloud (e.g. CS(1-0) point 2 on Figure \[fig:intmultipanel\]). These line profiles may suggest there are components of the shock travelling in different directions, the possibility of reverse shocks or may simply be a projection effect. ![image](./figures/spectra/core2_spectra_new.eps){width="\textwidth"} Several clumps of SiO(1-0) are detected and bounded by an arc of 1720MHz OH masers, further confirming that a shock has disrupted this molecular cloud. The peak of the SiO(1-0) emission (point 15 in Figure \[fig:intmultipanel\]) lies offset from the peak of the dense gas traced by CS(1-0) (point 2 on Figure \[fig:intmultipanel\]), which may suggest that the densest CS(1-0) core is shielded from the disruption of the shock. A class I CH$_{3}$OH maser transition is detected toward the eastern edge of the dense gas (point 13 on Figure \[fig:intmultipanel\]). This is the weakest of the CH$_{3}$OH masers detected, with a peak temperature of $\sim0.4$K. A second possible site of emission appears real from the DP-2 spectra (point 14/DP-2 on Figure \[fig:intmultipanel\]). Here the spectra is weak, peak 0.06K, but with broad FWHM 15. Both these CH$_{3}$OH emission regions are not associated with typical star forming cores. Rather, they appear to be a by-product of the SNR shock heating the gas in a similar manner as discussed, for the case of expanding shocks from regions in @voronkov. Similarly, the detection of HC$_{3}$N(5-4) from this region further indicates hot core gas phase chemistry. The emission is again broad, FWHM$\sim$12as for other lines detected from the region. The position velocity (PV) plot shown in Figure \[fig:core2pv\] highlights the CS(1-0) emission, especially from the western or W28 side of the dense cloud. The velocity at the peak pixel map shown in Figure \[fig:whole\_velopix\] reveals the existence of a velocity gradient, where the peak emission towards the south-western edge of the cloud has larger values which decrease towards the north-east. This gradient would suggest that the shock is disrupting the cloud from the south-western or W28 side. The velocity dispersion image for the CS(1-0) emission (for $>$0.5K in the -5 to 20range) from the north-eastern region is also shown in Figure \[fig:core2pv\]. The emission with the largest dispersion originates from the south-western side of the cloud, which was also the case for 3(3,3) emission in @me. The more disrupted emission is generally offset from the peaks seen in the integrated map, which are also shown on the PV plot (Figure \[fig:core2pv\]) as white contours. Interestingly one of the clumps seen in the CS(1-0) integrated map (Point 1/DP-2 in Figure \[fig:intmultipanel\]) appears disrupted and has a large value. This clump also corresponds to a peak in the SiO(1-0) integrated emission (Point 17/DP-2 in Figure \[fig:intmultipanel\]). ![image](./figures/pvplot/CS_core2_pvRA.eps){width="49.00000%"} ![image](./figures/veldisp/CS_c2_veldisp.eps){width="49.00000%"} We have taken two position switched deep pointings towards the NE cloud, specifically targeted towards the western side of the cloud where the most turbulent conditions are seen. The locations of the position switched observations are illustrated in Figure \[fig:exposure\]. DP-1 is towards the broadest and most disruptive region, and DP-2 is towards a disrupted core. From these deep spectra, we have been able to extract gas parameters, which are displayed in Table \[tab:massdens\]. We find that the compact-source size mass, assuming a core with radius $r=0.2$pc, is more than a factor 10 smaller than the mass obtained when using the virial theorem. This is to be expected as these observations are targeted to the most disrupted regions of the cloud which are not expected to be gravitationally bound. Additionally, the molecular abundance ratio may be quite different in this shocked region compared to the unshocked gas. South-eastern Cloud / HESS J1800-240A {#ssec:HESSJ1800-240A} ------------------------------------- The region associated with HESS J1800-240A contains two regions, G6.225-0.569 and G6.1-0.6 [@lockman; @kuchar] and IRAS 17588-2358. Integrated intensity maps of detected emission are shown in Figure \[fig:core4int\]. Dense gas traced by CS(1-0) populates this region in clumpy cores near the two regions. A core unrelated to the regions (point 3 in Figure \[fig:core4int\]) shows signs of extension and also displays a double peaked line profile towards its edge (point 4 in Figure \[fig:core4int\]). Further south, additional clumps are detected with $>5$ integrated . The line profiles of the CS emission towards these cores are also moderately broad, FWHM $\sim5$, but not to the extent of the emission seen in the NE cloud. In the NE corner of our mapping, near the region 6.225-0.569, we trace star formation activity, detecting a bright Class I CH$_{3}$OH maser (peak $\sim$18K), a HC$_{3}$N(5-4) emitting core and evidence of shocks/outflows with a detection of SiO(1-0). The location of these star formation tracers are offset from the catalogued position of G6.225-0.569. We note that this region, associated with HESS J1800-240A, has less exposure (a factor 3 lower than the north-eastern HESS J1801-233 and southern HESS J1800-240B regions). ![image](./figures/spectra/core4_spectra.eps){width="\textwidth"} Southern Cloud / HESS J1800-240B / G5.89-0.39 {#ssec:south} --------------------------------------------- Spatially well matched to HESS J1800-240B are three dense cores oriented in a south-east (SE) to north-west (NW) alignment (points 1, 2 & 3 in Figure \[fig:southCS\]), which were seen in our previous 12mm observations [@me labelled as Triple Core SE, Central and NW]. The SE and central sources are associated with the IR-bright and energetic UC region G5.89-0.39 and the NW source with a pulsating M type star, V5561 Sgr, a further 5 arcmin away. The extraordinary molecular outflow emanating from the complex G5.89-0.39 has been extensively studied from arcsec to arcmin scales in multiple molecular lines (e.g. @harvey [@churchwell; @zijlstra; @gomez; @choi; @acord; @claussen; @thompson; @kimkoo2001; @kimkoo2003; @sollins; @hunter; @me]) Outflow velocities $\pm70$are seen in CO lines and similar claims are made for SiO lines by @acord. @sollins argues that the outflow originates from a 1.3mm continuum source and not from Feldt’s candidate star [@feldt], as the star is not equidistant from the outflow lobes. The complex G5.89-0.39 is actually comprised of two active star formation regions, G5.89-0.39A to the east, and UCG5.89-0.39B approximately 2 arcmin to the west [@kimkoo2001]. G5.89-0.39A is also known as W28A2 with strong radio continuum emission, while G5.89-0.39B possesses the outflow. We observe SiO(1-0) over a wide region towards the three sources suggesting there are shocks or outflows. Multiple higher order SiO transitions on smaller scales are detected towards the G5.89-0.39B region and have been discussed by @acord. A new perspective that this work provides is the broad scale (15 arcmin) dense gas surrounding the regions. NW of the UC region G5.89-0.39B in CS(1-0) we see an extended arm feature with length $\sim6$ arcmin extending north towards W28 (e.g points 3 & 4 in Figure \[fig:southCS\]). The dense gas seen in this arm appears to have a slight velocity gradient, approximately $+0.3$arcmin$^{-1}$ from north to south. To the SE of the regions there are several dense clumps which trace the dense component of the cloud seen in CO(2-1) by @nanten21. Once again, these cores display broad CS(1-0) emission (FWHM$>$4) however, these clumps are at a slightly larger compared to the regions. Interestingly, most of the CS(1-0) emission from the southern cloud contains multiple clumps along the line of sight, all within $\sim$20of each other. This makes calculations difficult as the multiply clumped emission produces artificially broad results. Star formation tracers such as 44GHz class I CH$_{3}$OH masers and HC$_{3}$N(5-4) emission have been detected towards all three of the aligned sources. We also detect signs of warm gas phase chemistry via detections of HC$_{3}$N(5-4) towards dense cores SE of the regions (e.g point 22 in Figure \[fig:southCS\]) which lie in a dense molecular ridge (see Figure \[fig:exposure\]). We also detect the HC$_{5}$N(16-15) and (17-16) lines towards G5.89-0.39B. This would indicate that there is considerable energy available to excite the molecules into such a highly excited state. An additional site of stellar activity is seen towards our southernmost mapped region. At the base of the dense ridge seen in CS(1-0), we detect SiO emission in the $v=1,2,3$ modes of the (1-0) line. Here we are probably tracing the radiationally excited SiO emission from the variable star V5357Sgr. Spectra for the SiO emission are displayed in Figure \[fig:SiO123\]. ![image](./figures/spectra/southern_spectra_lrg.eps){height="0.92\textheight"} We have also taken position switched deep pointings towards G5.89-0.39B (DP-4) and towards the northern most point of the arm NE of G5.89-0.39 (DP-3). We have estimates of the mass and density from the CS isotopologue emission towards G5.89-0.39B. We do not detect any isotopologue emission from the extended arm position (DP-3) so we estimate the gas parameters under the assumption that the CS(1-0) emission is optically thin. The results of the compact-source analysis are displayed in Table \[tab:massdens\]. HESS J1800-240C / G5.71-0.08 {#ssec:OHmaser} ---------------------------- The CS(1-0) emission detected towards HESS J1800-240C shows two emitting regions along the line of sight, one with $\sim-25$and another with $\sim13$. This dense CS core is slightly offset from the peak of the TeV source and may provide a suitably dense region for CR interactions. The CS emission is also spatially matched with the SNR candidate G5.71-0.08 and the 1720MHz OH maser at $=13$ [@hewitt]. The OH maser may indicate that HESS J1800-240C is tracing another SNR/molecular cloud interaction. Figure \[fig:OHmaser\_multi\] is an integrated intensity image of the HESS J1800-240C region comparing our CS(1-0) emission and the CO(2-1) emission from @nanten21. Spectra from our 34S(1-0) and CS(1-0) deep pointings, the CO(2-1) spectra from @nanten21 and CO(1-0) spectra from @liszt at the position of the 1720MHz OH maser are also presented. The spectra presented in Figure \[fig:OHmaser\_multi\] reveal line-wings in the CO(2-1) and (1-0) profiles, with $\sim$0-8 and 20-30, indicative of turbulent conditions. The CS(1-0) profile also shows some structure in these ranges, which could be interpreted as multiple clumps along the line of sight, or evidence for line wings. There is also an absorption dip at $\sim10$. Similarly, absorption features are present in the 1720MHz OH spectrum at $=7$ and $-25$ [@hewitt], and may also be evident in the CO emission. This is likely due to absorption against a continuum source. Interestingly the width of the CS(1-0) line profile is narrow, with FWHM 1. This could suggest the central region of the core is rather quiescent. The two isotopologues of CS are detected in our deep position switched observation towards the new OH maser position and allow the mass and density to be estimated. For this CS core we estimate a mass of 190M$_{\odot}$ and a H$_{2}$ number density of 1.7$\times10^{5}$. Finally, we note that this mapped region associated with HESS J1800-240C has less exposure (a factor 1.5 to 3 lower) than the HESS J1801-233 and HESS J1800-240B regions. ![image](./figures/spectra/OHmaser_side.eps){width="95.00000%"} X-Ray & TeV Gamma-Ray Emission and discussion {#sec:HEdiscussion} ============================================= HESS J1801-233/NE Shocked Cloud {#sec:NEcloud} ------------------------------- The CS(1-0) emission seen towards the north-eastern cloud traces an extended dense region of molecular material. Using the emission within a region defined by $\int~\tmb~dv~\geq~10$($\sim40$) and finding the average CS(1-0), 34S(1-0) and 3CS(1-0) emission profiles within the region, we can estimate the gas parameters as an extended source. We apply this strict limit on the CS(1-0) emission to only accept pixels containing CS, 34S and 3CS emission, and not bias the optical depth result. Under these conditions, we estimate the CS emission fills an ellipse on the sky with physical dimensions of $3.1\times1.4$pc (at the estimated 2kpc distance). This elliptical region is indicated on Figure \[fig:whole\_velopix\]. Additionally, we assume the volume of the emission is defined by a prolate ellipsoid, where the third axis (in the $z$ direction) is the same as the minor axis of the ellipse on the sky. In this way, the ellipsoid has radii $3.1\times1.4\times1.4$pc. Using the method outlined in Section \[sec:overview\], we calculate the mass using @me [Equation 3] and include an additional factor $\eta_{\rm mb}/\eta_{{\rm xb}}=0.77$ to account for the Mopra extended beam efficiency $\eta_{\rm{xb}}=0.43$ [@mopra_beam]. This gives an estimated extended source mass of $5.6\times10^{4}$and a H$_{2}$ number density $6.1\times10^{4}$. The extended source mass and density estimates are presented in Table \[tab:extdmassdens\]. We compare our CS gas mass to other mass estimates of the region. Over the 0-12velocity range, the CO(1-0) mass has been estimated to be $\sim2\times10^{4}$, and over the wider 0-25velocity range $\sim5\times10^{4}$ [@hess_w28]. From CO(3-2) data @arikawa estimated there to be $2\times10^{3}$of shocked gas in the NE cloud and $4\times10^{3}$of unshocked gas. Additionally a conservative lower limit to the mass of gas traced by 3 inversion transitions has been estimated to be $>1300$ [@me]. At a glance the mass derived from CS is larger than previously published estimates. It is at least 10$\times$ the lower limit obtained via 3 observations. This is likely due to the considerable uncertainties in molecular abundance ratios, which are usually the largest sources of systematic error in mass/density calculations. While an order of magnitude mass discrepancy is not ideal, it has been shown for a sample of molecular cores, both hot and cold, that the core masses have equal or higher values from CS observations than those found via 3; in some cases discrepancies up to two orders of magnitude [@zhou]. We suggest the CS mass be an upper limit as the abundance in a shocked source is likely to be lower than that we have assumed applies in an undisturbed, dense molecular cloud. The fact that the morphological extent of the CS emission is similar to that in CO however, suggests that a quite high fraction of the molecular gas is in a high density state. The X-ray morphology of W28 is described as centre filled [@rho2002] whereby X-ray emission lies towards the centre of the SNR. Interestingly, there is a brightening of the X-ray emission towards the NE of the SNR. This so called ‘[ear]{}’ feature, labelled by @rho2002, is found at the boundary of the pre- and post-shocked molecular cloud seen in CO(1-0) and (3-2) [@arikawa]. XMM-Newton X-ray observations towards the NE shocked cloud have been discussed in @ueno2003 following earlier observations with ASCA and ROSAT [@rho2002]. Here, we have looked at additional XMM-Newton archival observations totalling $\sim$56ks of observation time. Data were reprocessed using the XMM SAS software suite (with standard quality cuts) and major proton flares were filtered out. Observation IDs used were 0135742401 (5.5ks post filter), 0145970101 (20ks - analysed by @ueno2003), and 0145970401 (30ks). The X-rays result mostly from thermally heated gas (post shocked) with $kT=0.3$keV [@ueno2003]. Figure \[fig:xray\] presents the XMM-Newton full EPIC camera (PN+MOS1+MOS2) image in the 0.2 to 10 keV band corrected for exposure in the blue channel. The red and green channels are the post-shocked CO(3-2) and pre-shocked CO(1-0) gas from @arikawa. The right panel of Figure \[fig:xray\] is a zoom of the X-ray ‘[ear]{}’ feature, revealing the boundary between the X-ray emission and the shocked gas. Also seen at this boundary are many 1720MHz OH masers [@claussen groups OH E & OH F, labelled on Figure \[fig:core2pv\]]. Interestingly, adjacent to the OH maser groups E and F we also detected SiO(1-0) at the 2level, further enhancing the evidence that there is shocked and disrupted dense gas in the region. ![image](./figures/xmm/3col_new.eps){width="\textwidth"} The cluster of masers (OH E & OH F) are possibly associated with the X-ray [‘ear’]{} (Figure \[fig:xray\]). What is particularly intriguing is that the OH E & F clusters of 1720MHz OH masers appear to lie on both sides of the shocked gas (Figure \[fig:xray\], red channel) with respect to W28. This may result from projection effects which are hard to disentangle or the possibility of an additional or reverse shock propagating inward towards W28. The OH E & F maser clusters lie in the 10-15velocity range (Figure \[fig:core2pv\]). This suggests that the post-shocked X-ray emitting gas may lie behind most of the NE dense molecular cloud, which peaks in the 5-10velocity range (Figure \[fig:core2pv\]). The broad molecular line emission observed towards the NE cloud would likely result from non-thermal components. The additional kinetic energy required to produce the broad line emission can be calculated. We determine the non-thermal energy as $W_{\rm kin}~=~1/2M(\Delta v_{\rm kin})^{2}$, where $M$ is the mass of the broad line gas, and $\Delta v_{\rm kin}$ is the line FWHM. Using the average CS(1-0) line FWHM=7.3and the average extended mass upper limit $M=5.6\times10^{4}$, we obtain $W_{\rm kin}<3.8\times10^{49}$erg, which is a few percent of the total $10^{51}$erg released in a typical SNR explosion. This upper limit is in agreement with our earlier estimate of the kinetic energy deposited into the dense NE cloud ($W_{\rm kin}>7\times10^{47}$erg) from our earlier 3 observations. Similarly, @arikawa estimated $\sim3\times10^{48}$erg of kinetic energy has been deposited into the 2000traced by shocked CO(3-2). The impact of CRs and SNR shocks on molecular clouds has been discussed extensively. @aharonian1994 discuss a molecular cloud (MC) being overtaken by a SNR shock. The ‘crushed cloud’ model from @uchiyama discusses the re-acceleration of particles within a molecular cloud, but is mainly applicable to GeV energies. @yamazaki2006 discuss three cases for VHE emission from old or evolving SNRs and molecular clouds (MC). This includes (i) emission from the SNR shock alone, (ii) the SNR shock colliding with a MC and (iii) CRs accelerated a distance away from a MC and diffusing through the interstellar medium (ISM) to illuminate the cloud. This latter case of CR acceleration and diffusion is also more recently discussed in @fermi_w28 [@gabici2010; @ohira_2010], with discussion specifically toward W28. @fujita2009 discuss the scenario of CRs accelerated by a SNR shock nearby or just outside the W28 MCs. Recent work by @inoue models the propagation of middle aged SNR shocks into a cloudy ISM, revealing the presence of multiple secondary shocks with total surface area comparable to the primary shock. These shocks are able to accelerate particles to a maximum energy $E_{{\rm max}}>10$TeV [@inoue]. Overall, all models can account for the TeV and GeV emission seen towards W28 and other similarly aged SNRs. They clearly favour the hadronic emission mechanism for TeV emission. Under a hadronic emission mechanism, TeV photons result from CR collisions producing $\pi^{0}$ mesons. The $\pi^{0}$s then decay into the TeV photons. The leptonic emission mechanism produces TeV photons by inverse Compton processes and/or electron non-thermal Bremsstrahlung. However, these leptonic processes are suppressed in evolved SNRs due to electron cooling via synchrotron losses at much earlier epochs. TeV gamma-rays are observed with energies in the 0.3-3TeV range [@hess_w28]. In the hadronic scenario, typically 17 percent of the CR particle’s energy is transferred to gamma-rays, therefore 0.3-3TeV gamma-rays trace a parent population of $\sim2-20$ TeV CRs at the source. The striking spatial overlap between the TeV emission and the dense gas towards HESS 1801-233, suggests that CRs are reaching the dense portions of the molecular cloud, and could help provide constraints for CR diffusion models [@gabici]. However present H.E.S.S. observations do not have sufficient angular resolution ($\sim10$arcmin FWHM) to test whether the CR spectrum varies towards the interior of the dense cores. Future TeV instruments with sufficient angular resolution (arcmin) are required before such spectral features may become apparent and help to discriminate between models of re-acceleration and diffusion. A future TeV instrument, such as the Cherenkov Telescope Array (CTA) [@CTAdesign], will have both improved angular resolution and sensitivity. The expected angular resolution could reach a few arcmin FWHM and the energy sensitivity is expected to be a factor of 10 better than H.E.S.S., achieving $\sim$few $\times10^{-14}$ergs$^{-1}$ in $\sim50$hr. Together, these improvements will permit TeV emission maps and detailed spectra with comparable spatial scales to the molecular gas maps and provide a platform to test CR diffusion models and probe re-acceleration models. The expected flux of TeV emission from a molecular cloud impacted by Galactic CRs (GCR) with particle flux following an integral power law $E^{-1.6}$ can be found from @aharonian_1991 [Equation 10]: $$F(\geq E_{\gamma})=2.85\times10^{-13}\,E^{-1.6}_{{\rm TeV}}\,\left( \frac{M_{5}}{d^{2}_{{\rm kpc}}} \right) \,k_{{\rm CR}} \hspace{2mm}{\small [{\rm ph\,cm}^{-2}\,{\rm s}^{-1}]} \label{eqn:TeVFlux}$$ where $M_{5}$ is the mass of the cloud (in $10^{5}$), $d_{{\rm kpc}}$ is the distance to the cloud (kpc) and $k_{{\rm CR}}$ is the CR density enhancement factor above the local solar-system value. The observed TeV spectral indices [@hess_w28] imply an incident proton particle spectral integral index of $\Gamma \sim 1.6$ (in a power law particle spectrum $\propto E^{-\Gamma}$) similar to GCRs. Thus Equation \[eqn:TeVFlux\] should be sufficient for initial estimates of the TeV fluxes arising from parts of the clouds. We note that further discussion of other systematic uncertainties concerning the predicted TeV fluxes from cloud clumps is left for section \[sec:conc\]. The CR enhancement factors have been estimated towards three of the four TeV sources in [@hess_w28 Table 2]. With this in mind, we can estimate the gamma-ray flux for emission from the dense cloud components traced by CS(1-0) emission. Using the NE shocked cloud extended source mass estimate from Table \[tab:extdmassdens\], we estimate $F(\geq\,1\,{\rm TeV}) = 5.1\times10^{-13}$phs$^{-1}$. The flux results for this, and other CS regions, are summarised in Table \[tab:highenergy\]. We note that this flux is $\sim15$% higher than that currently detected by H.E.S.S. This further supports our case for treating the CS mass estimate as an upper limit given the uncertainties in the CS abundance. The general picture though is that the predicted TeV flux is detectable and can be well studied by CTA in a reasonable observation time (10-50hr). CTA could then discriminate between the TeV emission from within the dense cloud component vs. that of the entire cloud as seen by H.E.S.S.. HESS 1800-240A & B ------------------ The southern TeV sources HESS J1800-204A and B are also associated with dense gas which could provide a suitable target for CR interactions. The dense regions of gas spatially consistent with HESS 1800-240A are treated as elliptical extended sources filling prolate ellipsoids with radii $1.4\times 0.7\times 0.7$pc for the gas nearby G6.225-0569 and $2.8\times 1.4\times 1.4$pc for the gas nearby G6.1-0.6. Similarly, the CS gas seen towards HESS J1800-240B is treated as three separate extended sources. The large clump associated with the G5.89-0.39 regions is assumed to fill a prolate ellipsoid with radii $2.4\times 2.1\times 2.1$pc. The CS emission from the NW arm, extending from V5561 Sgr, is assumed to fill an ellipsoid with radii $3.1\times 1.1\times 1.1$pc, and the SE molecular ridge is assumed to fill an ellipsoid with radii $3.5\times 1.4\times 1.4$pc. These elliptical regions are indicated on Figure \[fig:whole\_velopix\]. We detect only CS(1-0) emission from the NW arm and SE ridge regions, therefore, we assume the emission to be optically thin. The individual estimates for the mass and density of all the extended regions towards HESS J1800-240A & B are summarised in Table \[tab:extdmassdens\]. The three extended regions towards HESS J1800-240B suggest that $7.1\times10^{4}$of material is contained in dense clumps. The HESS J1800-240B southern cloud is estimated to have $10^{5}$based on the CO(1-0) emission [@hess_w28]. Based on our extended CS clump estimates, we find that $\sim70$% of the southern clouds mass is contained in the dense clumps surrounding the G5.89-0.39 regions. As these three dense extended regions all lie within the HESS J1800-240B source, the improved angular resolution of CTA could potentially resolve this emission into three separate sources. In this case, the expected gamma-ray flux can be found via Equation \[eqn:TeVFlux\]. The expected TeV flux for photon energy $E > 1$TeV for each of the dense cloud components are summarised in Table \[tab:highenergy\]. Typically the expected flux is $10^{-13}$ to $10^{-14}$ phs$^{-1}$, which could be detectable and well studied by CTA in a reasonable observation time (10-50hr). As for the NE cloud, we find that the total expected flux from the three HESS J1800-240B sources is higher than the detected level of TeV emission (by a factor of $\sim$2). Again, this could be explained by our choice of CS abundance being overestimated in this region which may be expected given likely internally shocked nature of the G5.89-0.39 complex. Additionally, our cloud-averaged CR density $k_{\rm CR}$ may not reflect smaller scale variations. The question of the origin of the CR population towards HESS J1800-240A and B is still uncertain, although diffusion of CRs from W28 seems to reasonably well explain the observed GeV to TeV gamma-ray spectra (e.g. @fujita2009 [@fermi_w28; @gabici2010; @li2010]). However, additional CRs accelerated inside the local star formation regions (from G5.89-0.39 in particular), and another potential SNR towards HESS J1800-240C (discussed shortly) remains a possibility (see e.g. @araudo). Our observation of the arcmin-scale dense NW arm and SE ridge features in the molecular cloud may signal additional sites of disruption which could arise from internal (star formation) and/or external (e.g. SNR shocks) forces. Further tests of these scenarios can come from future arcmin-scale studies of gamma-ray spectra throughout the clouds. HESS J1800-240C --------------- Towards HESS J1800-240C, we have discovered a dense molecular core. Unlike the other TeV sources in the W28 region, the dense gas lies slightly offset from the peak of the TeV emission. The CS gas is found in a dense, compact core, which has a narrow line width, FWHM 1. Additionally, there is absorption in the CS line profile at $\sim$10, which may also be evident in the CO(1-0) and (2-1) line profiles (see spectra in Figure \[fig:OHmaser\_multi\]). Based on the CO(1-0) emission from the NANTEN survey [@nanten10], we estimated the mass of the CO cloud. We assumed a CO abundance of [$\chi$]{}$_{\rm CO}$ = 1.5$\times10^{20}$ and assumed a spherical emission region with radius 0.1$^{\circ}$ centred on the HESS J1800-204C TeV source. We included emission from the 0-20range, which was selected to be consistent with the ranges used in @hess_w28. Under these assumptions, we estimated 1.4$\times10^{4}$of CO gas towards HESS J1800-240C. We estimated the CR enhancement factor $k_{\rm CR} = 35$, using Equation \[eqn:TeVFlux\], based on the CO mass and the detected TeV flux following @hess_w28. Assuming the value of $k_{\rm CR}=35$ we determined the expected TeV flux from just the dense CS core. The result is shown in Table \[tab:highenergy\] The determined CR enhancement of 35 is higher than towards the other TeV sources. Enhancements of this level are expected in the regions of CR accelerators, which could be expected as this region could be associated with both the W28 SNR as well as the local SNR candidate G5.71-0.08 which may be producing the OH maser. Under a purely hadronic emission scenario, the expected TeV flux from just the dense CS core is $F(E \geq 1\,{\rm TeV})\sim2\times10^{-15}$ phs$^{-1}$. This flux level would likely push the reasonable detection limits for CTA. [lcc]{} Region & $k_{{\rm CR}}$ & Expected Flux $F(E \geq 1\,{\rm TeV})$\ & & \[phs$^{-1}$\]\ NE Shocked Cloud & 13 & $<$5.1$\times10^{-13}$\ G5.89-0.39 Region & 18 & 7.0$\times10^{-13}$\ G5.89-0.39 NW Arm & 18 & 5.4$\times10^{-14}$\ G5.89-0.39 SE Ridge & 18 & 1.6$\times10^{-13}$\ G6.1-0.6 Region & 14$^{}$ & 6.5$\times10^{-14}$\ G6.225-0.569 Region & 14$^{}$ & 1.6$\times10^{-14}$\ HESS J1800-240C & 35$^{\dagger}$ & 1.9$\times10^{-15}$\ \ \ \ \ \ Summary and Conclusions {#sec:conc} ======================= Using the Mopra 22m telescope, we have conducted 7mm molecular line mapping covering dense and disrupted gas toward the W28 SNR field. We have followed up our previous 12mm line study of the molecular clouds towards the TeV gamma-ray peaks observed by the H.E.S.S. telescope, with higher resolution targeted observations towards the NE cloud/shock interaction, HESS J1801-233 and the bright and energetic UC-regions in the southern cloud HESS J1800-240B. Sites of shocks, outflows and disruption are revealed with SiO. In the majority of cases, SiO detections are towards sites of stellar activity and regions, however, SiO is also detected in the NE cloud/SNR shock interaction region with broad line profiles. Interestingly, in this shock region the SiO emission is bounded by clusters of 1720OH masers, indicating the downstream direction of the shock is towards W28. Other lines detected are 44GHz CH$_{3}$OH masers and cyanopolyynes HC$_{n}$N ($n=3,5,7$) which are again preferentially associated with star formation and regions. The CS cores are typically found towards CO peaks and trace conditions ranging from quiescent cloud cores to regions. The obvious exception is the NE cloud/SNR shock interaction region, which exhibits broad line emission FWHM$>10$in all detected lines. Based on CS isotopologue ratios from our position-switched deep pointing observations, we are able to estimate the upper state CS column density. Temperature estimates from our previous 3 observations allow the LTE molecular hydrogen mass and density to be estimated. Our results presented in Table \[tab:massdens\] assume a compact-source core radius $r=0.2$pc, however, scaling factors for various core radii are also included.\ We have estimated the mass of the extended dense NE cloud and several of the dense clumps in the southern clouds, assuming the CS isotopologue emission is contained within a prolate ellipsoid. Under these assumptions, we estimate there to be 5.6$\times10^{4}$, with H$_{2}$ number density 6.1$\times10^{4}$ contained towards the NE HESS J1801-233 TeV source. We find that $\sim4\times10^{49}$erg of kinetic energy is required to produce the broad emission in the CS line profile towards the NE cloud, which is a few percent of the typical $10^{51}$erg of kinetic energy released in a SNR. For the southern cloud towards HESS J1800-240B we estimate that $\sim70$% of the gas mass is contained in dense clumps surrounding the regions. The southern clouds show no signs of external disruption, providing no evidence that the W28 SNR shock has influenced the region. Although our CS masses are treated as upper limits, we note that a somewhat independent estimate of the CS mass for clumps not part of the NE Shocked Cloud comes from the virial theorem. For these cases the fact that the virial mass is similar to the LTE mass within a factor 2 to 3 suggests our choice of CS abundance is adequate for order of magnitude estimates. For the NE Shocked Cloud the virial theorem will likely not apply. However, as discussed in section \[sec:NEcloud\], the NE Shocked Cloud mass derived from our CS observations is 10% larger than the mass from CO observations, indicating that our CS abundance may be slightly overestimated here. Additional systematic uncertainties arise from the unknown spectrum of CRs entering the dense clouds, and, penetrating their interiors as a result of energy-dependent diffusion. Equation \[eqn:TeVFlux\] assumes that a CR spectral integral index of $\Gamma =-1.6$ as expected for Galactic CRs (GCRs) impacting passive clouds some distance away. The spectral indices of the TeV emission towards W28 are in fact found to be similar to GCRs so Equation \[eqn:TeVFlux\] should be adequate in this regard. The striking spatial match between the TeV emission and the dense molecular gas suggests that the CRs are able to penetrate some distance into the dense cloud core. However, the energy dependent diffusion of CRs into the cloud cores [@gabici] could easily suppress the predicted emission for low energies $E<1$ TeV below that suggested by Equation \[eqn:TeVFlux\]. The level of suppression is strongly dependent on the largely unknown magnetic field structure and turbulence in the clump and is a topic beyond the scope of this paper. Overall, we would like to emphasise that the uncertainties discussed above clearly motivate the need for new 1 arcmin angular resolution TeV observations (for example by CTA) in order to begin probing the diffusion properties of CR into dense molecular clouds. This work is part of our ongoing study into the molecular gas towards the W28 region, and follows our earlier 12mm line mapping campaign. Additional observations of the NE cloud SNR interaction region have been conducted with Mopra at 12mm in order to extract detailed 3 inversion spectra across the cloud core. Results of these observations will be the focus of a future paper to further understand the molecular gas properties in the interesting NE SNR shock/molecular cloud interaction region. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Cormac Purcell for his baseline fitting and removal script. This work was supported by Australian Research Council grants (DP0662810, DP1096533). The Mopra Telescope is part of the Australia Telescope and is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. The University of New South Wales Mopra Spectrometer Digital Filter Bank used for these Mopra observations was provided with support from the Australian Research Council, together with the University of New South Wales, University of Sydney, Monash University and the CSIRO. [99]{} Abdo A. A., (Fermi Collab.), 2010, ApJ 718, 348 Acciari V. A., (VERITAS Collab.), 2009, ApJ, 698, L133 Acord J. M., Walmsley C. M. & Churchwell E., 1997, ApJ, 475, 693 Aharonian F. A., 1991, Ap&SS, 180, 305 Aharonian F. A., Drury L. O’C. & Völk H. J., 1994, A&A, 285, 645 Aharonian F., (H.E.S.S. 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L. & Umana G., 1990, MNRAS, 246, 217 Zinchenko I., Forsström V., Lapinov A. & Mattila K., 1994, A&A, 288, 601 Additional Figures and Tables ============================= ![Spectra for the line emission detected towards the IRAS 17589-2312 star formation core from mapping data.[]{data-label="fig:core1spec"}](./figures/appendix/core1_spectra.eps){width="\columnwidth"} ![Spectra of the SiO(1-0) $v=1,2,3$ lines coming from the V5357Sgr Variable Star of Mira Cet type.[]{data-label="fig:SiO123"}](./figures/appendix/SiO_v123.eps){width="\columnwidth"} ![Image showing the relationship between the diffuse and dense gas in the NE cloud. The colour scale is the NANTEN2 CO(2-1) -10 to 25integrated intensity image. The black dased contours are the H.E.S.S. TeV emission (4 & 5$\sigma$ levels). The white dashed contours are the CS(1-0) -5 to 20integrated emission as seen on Figure \[fig:intmultipanel\]. The solid cyan contours are the 3(3,3) -5 to 20integrated emission (1,3,5,7,9,11,13levels) from @me.[]{data-label="fig:densediffuse"}](./figures/integrated/CO_CS_NH3.eps){width="\columnwidth"} [lcccc]{} Region & Peak & $\int$$dv$ & & FWHM\ & \[$K$\] & \[\] & \[\] & \[\]\ \ \ CH$_{3}$OH & 0.03 $\pm$ 0.003 & 0.55 $\pm$ 0.09 & 11.70 $\pm$ 1.072 & 18.74 $\pm$ 2.29\ SiO(1-0) & 0.09 $\pm$ 0.003 & 2.15 $\pm$ 0.12 & 13.12 $\pm$ 0.395 & 22.71 $\pm$ 0.98\ HC$_{3}$N(5-4) & 0.04 $\pm$ 0.006 & 0.61 $\pm$ 0.14 & 12.18 $\pm$ 1.215 & 15.23 $\pm$ 2.65\ 3CS(1-0) & 0.02 $\pm$ 0.003 & 0.58 $\pm$ 0.13 & 13.68 $\pm$ 1.778 & 25.66 $\pm$ 4.63\ 34S(1-0) & 0.07 $\pm$ 0.006 & 1.39 $\pm$ 0.19 & 12.30 $\pm$ 0.603 & 17.80 $\pm$ 2.05\ CS(1-0) & 1.06 $\pm$ 0.005 & 20.46 $\pm$ 0.16 & 13.05 $\pm$ 0.043 & 18.09 $\pm$ 0.11\ \ \ CH$_{3}$OH & 0.06 $\pm$ 0.004 & 0.98 $\pm$ 0.09 & 5.67 $\pm$ 0.458 & 15.31 $\pm$ 1.08\ SiO(1-0) & 0.13 $\pm$ 0.004 & 2.29 $\pm$ 0.11 & 6.10 $\pm$ 0.231 & 15.99 $\pm$ 0.58\ HC$_{3}$N(5-4) & 0.10 $\pm$ 0.007 & 1.16 $\pm$ 0.12 & 6.87 $\pm$ 0.361 & 10.40 $\pm$ 0.84\ 3CS(1-0) & 0.07 $\pm$ 0.006 & 0.79 $\pm$ 0.10 & 8.43 $\pm$ 0.352 & 10.49 $\pm$ 1.04\ 34S(1-0) & 0.20 $\pm$ 0.007 & 1.99 $\pm$ 0.12 & 7.55 $\pm$ 0.168 & 9.60 $\pm$ 0.45\ CS(1-0) & 2.37 $\pm$ 0.007 & 28.27 $\pm$ 0.13 & 7.37 $\pm$ 0.015 & 11.20 $\pm$ 0.04\ \ \ CS(1-0) & 1.39 $\pm$ 0.064 & 3.28 $\pm$ 0.22 & 8.98 $\pm$ 0.043 & 2.21 $\pm$ 0.11\ \ \ CH$_{3}$OH & 3.65 $\pm$ 0.014 & 4.46 $\pm$ 0.03 & 9.00 $\pm$ 0.002 & 1.15 $\pm$ 0.01\ SiO(1-0) & 0.25 $\pm$ 0.006 & 8.98 $\pm$ 0.05 & 12.35 $\pm$ 0.011 & 19.56 $\pm$ 0.02\ HC$_{5}$N(17-16) & 0.59 $\pm$ 0.005 & 5.15 $\pm$ 0.17 & 9.17 $\pm$ 0.152 & 3.29 $\pm$ 0.53\ HC$_{3}$N(5-4) & 4.12 $\pm$ 0.009 & 2.07 $\pm$ 0.04 & 9.09 $\pm$ 0.024 & 3.56 $\pm$ 0.05\ 3CS(1-0) & 0.40 $\pm$ 0.013 & 15.63 $\pm$ 0.07 & 9.35 $\pm$ 0.005 & 3.64 $\pm$ 0.01\ HC$_{5}$N(16-15) & 0.52 $\pm$ 0.009 & 1.53 $\pm$ 0.06 & 8.97 $\pm$ 0.037 & 3.10 $\pm$ 0.10\ 34S(1-0) & 1.15 $\pm$ 0.013 & 1.72 $\pm$ 0.06 & 9.29 $\pm$ 0.037 & 3.67 $\pm$ 0.08\ CS(1-0) & 9.50 $\pm$ 0.015 & 4.50 $\pm$ 0.09 & 9.26 $\pm$ 0.022 & 4.01 $\pm$ 0.06\ \ \ 34S(1-0) & 0.29 $\pm$ 0.022 & 0.31 $\pm$ 0.04 & 12.97 $\pm$ 0.037 & 1.01 $\pm$ 0.09\ CS(1-0) & 2.80 $\pm$ 0.022 & 3.34 $\pm$ 0.04 & 13.04 $\pm$ 0.004 & 1.12 $\pm$ 0.01\ [^1]: See <http://www.atnf.csiro.au/computing/software/> for more information on these data reduction packages.
--- author: - 'Sergio Colafrancesco,' - 'Mohammad Shehzad Emritte,' - Paolo Marchegiani title: 'The impact of non-Planckian effects on cosmological radio background' --- Introduction ============ The existence of a Planck spectrum for a system filled with a black body (BB) radiation assumes the presence of thermodynamic equilibrium between photons and matter. Interaction between photons and matter must be small enough for the gas of photons to be considered ideal and to avoid substantial absorption and irreversible attenuation of electromagnetic radiation in the system (we note here that the interaction between photons is extremely weak). At the same time, the presence of matter and the weak interaction between matter and radiation is necessary for the photon gas to be in equilibrium [@LL]. These conditions are valid for a BB radiation from a photon gas and hence are also expected to be valid for the cosmic microwave background (CMB) spectrum emerging at the epoch of recombination (at redshift $z_{rec} \approx 1400$) because the CMB spectrum is assumed and measured [@Fixsen1996] to be close to a Planckian BB spectrum.\ In this context, the dispersion relation for electromagnetic radiation in non-ionized media (e.g., a rarefied gases) is given by $$\omega = c k$$ where $k$ is the wavenumber, $c$ the speed of light and $\omega$ the angular frequency of the photon. However, in a medium which is ionized (e.g., an ionized plasma), the analogous dispersion relation is given by [@Triger2010] $$\omega^2 = {c^2 k^2 + \Omega_p^2}$$ where $\Omega_p =2 \pi \nu_p$ is the angular plasma frequency, with $\nu_p^2=\sum_{i=1}^{N} n_i e^2/4 \pi m_i \epsilon$ being the actual plasma frequency, where the sum is over all charged particles species. The existence of such a dispersion relation is due to the coupling between the electromagnetic waves (photons) and the collective behavior of the plasma [@Kittell1986]. Consequently, photons with frequencies lower than the plasma frequency $\nu_p$ do not exist in the final radiation spectrum. The presence of a non-zero value of the plasma frequency (in the following we use for our convenience the a-dimensional plasma frequency $x_p= h \nu_p/k_B T_{CMB}$, with $k_B$ being the Boltzmann constant and $T_{CMB}$ the CMB temperature today) will cause the intensity of the photon spectrum to be zero for frequencies less than $x_p$ and this fact will reflect as a modified blackbody spectrum subject to non-Planckian (NP) effects. A general form for such a NP spectrum has been derived by [@Triger2010] and [@Medvedev1998] for astrophysical plasma-photon systems, such as the CMB and $\gamma$-ray bursts atmospheres. We note here that the plasma frequency $x_p$ depends on the number density of all charged particles in the plasma, and hence deviation from Planck spectrum will also depend on this number density. In principle, the plasma frequency can be computed using knowledge of the cosmological recombination history. For example, a simple result obtained using a number density of 300 electrons/cm$^3$ is that the plasma frequency at recombination is $\approx 0.2$ MHz [@Triger2010]. However, other effects bring uncertainties on this result. In fact, in addition to the ionized plasma, dark matter annihilation can produce electrons and positrons which will perturb the ionization fraction [@Chen2004; @Padmanabhan2005; @Huetsi2009; @Galli2009; @Dvorkin2013]. It has been shown that the ionization fraction can increase due this effect by more than an order of magnitude providing a plasma frequency of order of $\approx 1.5$ MHz [@Dvorkin2013]. Furthermore, electrons and positrons (hereafter we refer to these particles simply as electrons) can also be produced by hadronic collisions [@Dermer1986], originating another additional source of free particles, and consequently an increase of the plasma frequency value. The CMB spectrum is also vulnerable to other distortions like the $\mu$ and $y$ distortions, related respectively to a frequency-dependent chemical potential and the Compton scattering of the CMB photons (see [@Tashiro2014] for a review and references therein). The plasma effects in principle could also create a distortion well before the recombination era. Since the free particles number density depends on redshift, the plasma frequency for the CMB will also depend on redshift as $\nu_p \propto (1+z)^{3/2}$. Other distortions that can occur in the early universe are the Silk damping of small-scale perturbations, the cooling of photons by electrons and baryons, the decay and annihilation of relic particles, primordial magnetic fields and evaporating primordial black holes (see [@ChlubaSunyaev2012; @Tashiro2014; @Chluba2014] and references therein). It is well known that some processes like Compton scattering, double Compton scattering and bremsstrahlung can change the photons energies or inject photons in the universe, in principle leading to a thermalization process, causing loss of information about the distortions occurring in the CMB [@ChlubaSunyaev2012; @Tashiro2014; @Procopio2009]. However this does not seem to be possible for distortions due to the plasma frequencies. In fact, the time scales of these processes [@Tashiro2014], $t_{comp} =1.23 \times 10^{29} / (1+z)^4$ s, $t_{double}=1.34 \times 10^{40} [x^3/(e^x -1 )] / (1+z)^5$ s and $t_{free}= 8.59 \times 10^{26} [x^3/(e^x -1 )] / (1+z)^{2.5}$ s, are larger than the plasma time scale defined as $t_p = 1/\nu_p$ [@Padmanabhan2000]. For example, at $z=1400$ and for a plasma frequency of 0.2 MHz corresponding to $x_p=2.5 \times 10^{-9}$, we obtain the values $t_{comp} =3.2 \times 10^{16}$ s, $t_{double} = 1.5 \times 10^7$ s, $t_{free}=72$ s and $t_p=6.4 \times 10^{-6}$ s; at $z= 10^6$ and a plasma frequency of 3 GHz corresponding to $x_p=6.64 \times 10^{-8}$, we obtain $t_{comp} =1.2 \times 10^5$ s, $t_{double} = 5.9 \times 10^{-5}$ s, $t_{free}=3.8 \times 10^{-5}$ s and $t_p=3.4 \times 10^{-10}$ s. The role of these processes, is, therefore, to eliminate the effects due to other distortions (see, e.g., [@Illarionov75a; @Illarionov75b; @Danese77; @Danese82; @Burigana91a; @Burigana91b; @Hu93a; @Hu93b]), and to lead the CMB spectrum at the recombination to have the BB shape modified by the plasma frequency. This is the spectrum we can measure, and from which we can derive our results about the plasma frequency. So far, only theoretical arguments on NP plasma effects on the CMB have been discussed and neither experimental limit has been derived nor observational strategies to probe these effects has been proposed. All these uncertainties make it desirable to obtain limits to the plasma frequency from observations, that can be used to constrain the information about the processes occurred during and before the recombination. In this paper we want to derive for the first time constraints on the value of $x_p$ from the spectral shape of the CMB using current observations. In order to check if there is evidence for a non-zero value of $x_p$ in the CMB spectrum, we therefore perform a comparison of the theoretical expectation for a NP CMB spectrum with the available observations at both low (radio) and high (microwave and mm.) frequencies. While the knowledge of the CMB spectrum is very precise in the microwave and mm. frequency regions after the COBE-FIRAS measurements [@Fixsen1996], its knowledge at lower radio frequencies (particularly at frequencies below 1 GHz where the NP effects are expected to emerge) is still quite uncertain [@Howell1967; @Sironi1990; @Sironi1991]. Therefore, in Sect.2 we will investigate first the limits on NP effects on the CMB spectrum and on the value of $x_p$ from the comparison with CMB measurements. We further stress that the modification on the CMB spectrum at low-$\nu$ has an effect on other observable quantities that depend on the CMB spectrum. In this paper we will consider two additional cosmological observables whose spectral shapes are modified by the presence of NP effects in the CMB spectrum: i) the Sunyaev-Zel’dovich (SZ) effect produced by the inverse Compton scattering (ICS) of CMB photons off the electrons residing in the atmospheres of cosmic structures like galaxy clusters, radio galaxies and galactic halos [@Colafrancesco2014], and ii) the temperature brightness change of the cosmological 21-cm background as produced by the evolution and reionization history of the primordial plasma during the Dark Ages (DA) and the epoch of reionization (EoR) [@Zaroubi2013]. Thus, in Sect.3 we study the imprint of the NP deviations on the CMB spectrum on the Sunyaev-Zel’dovich (SZ) effect in both thermal and non-thermal plasmas and we show that the main modifications to the standard SZ effect arise mainly in the very low-$\nu$ range where a very specific spectral shape of the modified SZ effect can be observed. We will show that at $\nu \simlt 500$ MHz these spectral features allow to measure directly the plasma frequency $x_p$. Then in Sect.4 we will explore the effect on NP deviations of the CMB spectrum on the temperature brightness change of the cosmological 21-cm line background emerging from the Dark Ages and Epoch of Reionization and we predict the brightness temperature modifications observable with the next generation radio telescopes. We finally discuss our results in Sect.5.\ Throughout the paper, we use a flat, vacuum–dominated cosmological model with $\Omega_m = 0.308$, $\Omega_{\Lambda} = 0.692$ and $H_0 =67.8$ km s$^{-1}$ Mpc$^{-1}$ [@Ade2015]. The spectral distribution of the CMB ==================================== The spectral distribution of a photon gas which is in equilibrium with matter (e.g rarefied gases) at temperature $T$ is given by the Planck distribution which can be written in terms of the a-dimensional frequency $x=h \nu/k_B T$ as $$\displaystyle I_0(x)=2 \frac{(k_B T)^3}{(h c)^2} \frac{x^3}{e^x-1} \; . \label{planckian}$$ For the case of the CMB, $T=T_{\text{CMB}}=2.725 \pm0.001$ K [@Fixsen2002] and $h$ is the Planck constant, $k_B$ is the Boltzmann constant and $c$ is the speed of light. In a ionized plasma with electron plasma frequency $x_p$, deviations from Planck distribution in eq.(\[planckian\]) are possible because photons are suppressed at frequencies lower than $x_p$. By taking in consideration this effect, we can write the generalized Planck distribution for the CMB photons as follows [@Triger2010] $$\displaystyle \tilde{I}_0(x;x_p)=2 \frac{(k_B T)^3}{(h c)^2} \frac{x^2}{e^x-1} \sqrt{x^2-x_p^2}\ \text{H}(x-x_p), \label{nonplanckian}$$ where $\text{H}(x-x_p)$ is the Heaviside step function (with values 1 for $x > x_p$ and 0 for $x \leq x_p$). The step-function $\text{H}(x-x_p)$ takes into account the fact that photons are suppressed at frequencies below $x_p$. The difference between the two distributions is more pronounced if $x_p$ is larger than 1 but nonetheless there are appreciable differences even for values of $x_p < 1$. The largest contrast between the generalized Planck distribution in eq.(\[nonplanckian\]) and the standard Planck distribution in eq.(\[planckian\]) is that the former has a cut-off frequency which depends on $x_p$. We also notice that the frequency at which the maximum of the CMB spectrum occurs shifts to higher frequencies as the cuff-frequency $x_p$ increases [@Triger2010] and that the CMB peak intensity decreases as the value of $x_p$ increases. ![The non-Planckian spectral distribution of the CMB for different values of the plasma frequency $\nu_p$ as derived from the fit to the data. The experimental data shown here are from [@Howell1967; @Sironi1990; @Sironi1991] and COBE [@Fixsen1996]. Other data in the range $\sim 1.3-50$ GHz obtained from ground-based, ballon-borne and from the COBE-DR experiment (see http://asd.gsfc.nasa.gov/archive/arcade/cmb\_spectrum.html) are not shown here but they lie almost exactly along the curves of the CMB spectrum shown in the plot.[]{data-label="CMBfull"}](fullspectrum){width="140mm" height="110mm"} ![The non-Planckian effects due to a finite value of the plasma frequency $\nu_p$ on the CMB spectrum at $\nu < 1$ GHz for different values of the upper limit on $\nu_p$ as obtained from the fit to the CMB spectrum data. Experimental data are from [@Howell1967; @Sironi1990; @Sironi1991].[]{data-label="CMBlow"}](cmblow){width="120mm" height="100mm"} In order to investigate the presence of non-Planckian effect on the CMB spectrum, we fit the NP distribution of the CMB spectrum in eq.(\[nonplanckian\]) to the CMB data [@Fixsen1996],[@Howell1967; @Sironi1990; @Sironi1991] and we minimize the $\chi^2$ with respect to the plasma frequency $x_p$; in principle one can also use the CMB temperature $T$ as a free parameter: however, we set its value to the one measured by COBE-FIRAS that is very accurate with uncertainty of order of $\approx 0.03 \%$ and it is also determined in the high-$\nu$ range of the CMB spectrum (around its maximum) where the NP effects are small. With this fitting procedure we obtain a value of $x_p=0$ for the minimum $\chi^2$. This means that a pure Planckian spectrum is best fitting the present data. However, the uncertainties in the data (in particular at low frequencies $\nu < 1$ GHz) allow us to set upper limits on the quantity $x_p$ which are $x_p=(3.63, 6.10, 7.36)\times10^{-3}$ for $1\sigma$, $2\sigma$ and $3\sigma$ confidence level (c.l.), corresponding to frequencies of 206, 346 and 418 MHz, respectively. In Fig \[CMBfull\] we show the NP effects on the CMB spectrum over the full frequency range. This result indicates that it is not possible at the 3$\sigma$ c.l. to distinguish between the Planck spectrum and the NP modified spectrum based on the current knowledge of the CMB spectrum. This fact opens hence an experimental window for the exploration of NP effects in the early universe. We then show in Fig \[CMBlow\] the CMB spectrum and its experimental uncertainties zooming around the low frequency region at $\nu <1$ GHz showing the modified CMB spectrum with the upper limits on $x_p$ at $1\sigma$, $2\sigma$ and $3\sigma$ c.l. These results show that the intensity of the CMB goes to zero below $\approx 206$ MHz, i.e. for the value of $x_p = 3.63 \times 10^{-3}$ at $1\sigma$ c.l., or below $\approx 418$ MHz, i.e. for the value of $x_p = 7.36 \times 10^{-3}$ at $3\sigma$ c.l. In Fig \[CMBpeak\], we show the relative CMB spectrum variations around the peak of the CMB where the COBE data [@Fixsen1996] are available (for the sake of illustration of the amplitude of the NP effects we zoom within the region 149–191 GHz around the maximum of the CMB spectrum). ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![The effect of non-Planckian distribution on the CMB spectrum in frequency range 143 GHz to 191 GHz is shown for different values of the plasma frequency $x_p$ as derived in Sect.2. Experimental data are from COBE [@Fixsen1996].[]{data-label="CMBpeak"}](cmb149 "fig:"){width="80mm" height="60mm"} ![The effect of non-Planckian distribution on the CMB spectrum in frequency range 143 GHz to 191 GHz is shown for different values of the plasma frequency $x_p$ as derived in Sect.2. Experimental data are from COBE [@Fixsen1996].[]{data-label="CMBpeak"}](cmb163 "fig:"){width="80mm" height="60mm"} ![The effect of non-Planckian distribution on the CMB spectrum in frequency range 143 GHz to 191 GHz is shown for different values of the plasma frequency $x_p$ as derived in Sect.2. Experimental data are from COBE [@Fixsen1996].[]{data-label="CMBpeak"}](cmb190 "fig:"){width="80mm" height="60mm"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- As shown in Fig \[CMBpeak\], the CMB spectrum intensity decreases with increasing values of $x_p$ and the frequency location of its maximum increases with increasing value of $x_p$. The amplitude of the CMB intensity variation due to NP effects is of order of $\approx$ mJy/arcmin$^2$ at low-$\nu$ while it is of order of $\approx 0.1$ mJy/arcmin$^2$ at high-$\nu$. Therefore, the best frequency region to look experimentally for these effects is at low radio frequencies, i.e. at $\nu \simlt 1000$ MHz. Our analysis shows that precise measurements of the CMB spectrum at low radio frequency have the power to set strong constraints to NP effects due to the existence of a finite value of the plasma frequency at recombination and that one can probe these effects with the next generation experiments like the SKA (SKA-LOW and SKA-MID [@Dewdneyetal2012]) and HERA [@Hera].\ Even though the CMB spectrum can be considered as a direct probe of NP effects in the early universe, its measurement at very low radio frequencies $\simlt 400$ MHz is challenging and it is subject to various experimental systematics and biases, and on the ability to subtract carefully radio foregrounds [@Oliveira2008]. The next generation low-$\nu$ radio telescopes (SKA, HERA) will increase our knowledge of the CMB spectrum in this frequency range, but the problem of foreground contamination and of the component separation of such diffuse signals at low frequency still remains. Therefore it is interesting to explore the use of other observable quantities that depend directly on the CMB spectrum, like, e.g., the modified SZ effect [@ColafrancescoMarchegiani2014a] and the modifications to the 21-cm brightness temperature change w.r.t. the CMB. In the following Sections we will discuss these two additional probes of NP effects in the early universe. The modified Sunyaev Zel’dovich effect ====================================== Another method which can be used to probe the existence of non-Planckian effect in the CMB is to use the modified SZ effect (see [@ColafrancescoMarchegiani2014a], and [@Birkinshaw1999; @Colafrancescoetal2003] for the general derivation of the SZ effect). The SZ effect is a spectral distortion of the CMB spectrum where the CMB photons interact with the electrons (thermal and non-thermal) residing in the atmosphere of galaxy clusters and other structures on large scales, like e.g. radio galaxies and galaxy halos. The SZ effect is a differential measure of the CMB spectrum on and off an area of the sky containing the selected cosmic structure and hence it is not affected by large-scale foregrounds in the observations at low-frequency (see [@Colafrancescoetal2015] for a discussion). The spectral distortion due to SZ effect on the CMB spectrum can be computed, at first order in the optical depth $\tau_e = \sigma_T \int d \ell n_e$, as follows [@Colafrancescoetal2003]: $$\displaystyle \Delta{I}(x)=\tau_e\bigg[\int ds P_1 (s){I}_0(x e^{-s}) - {I}_0(x)\bigg], \label{szeplanck}$$ where $P_1(s)$ is the single scattering redistribution function, obtained by summing the photon redistribution function $P_s(s,p)$, that gives the probability to have a logarithmic shift $s=\ln (\nu'/\nu)$ in the photon frequency due to the ICS process by an electron with a-dimensional momentum $p=\beta \gamma$, over the electron momentum distribution function $f_e (p)$, normalized as to have $\int_0^\infty f_e(p) dp=1$ (see [@Ensslin2000; @Colafrancescoetal2003]). This is written as $$\displaystyle P_1(s)=\int_0^\infty dp f_e (p) P_s (s,p). \label{p1s}$$ For the calculation of the standard SZ effect, the quantity $I_0(x)$ is a Planck spectrum in the case of the CMB. Since the SZ effect is related directly to the spectrum of the CMB (see eq.\[szeplanck\]), the existence of non-Planckian effect in the CMB background will be reflected directly in the SZ spectral distortion, and it is expected to be visible in particular around the cut-off frequency $x_p$. In order to compute the SZ effect for the case of a NP spectrum, eq.(\[nonplanckian\]) is inserted into eq.(\[szeplanck\]) instead of $I_0(x)$ as follows $$\displaystyle \Delta \tilde{I}(x;x_p)=\tau_e\bigg[\int ds P_1 (s) \tilde{I}_0(x e^{-s};x_p) - \tilde{I}_0(x;x_p)\bigg]. \label{eq.deltai}$$ One can notice directly from this equation that the final SZ spectral distortion in the case of NP plasma effect depends directly on $x_p$, hence on the plasma frequency $\nu_p$. In the context of the SZ effect modified by NP effects (that we refer here to as SZ$_{NP}$) we first compute the thermal spectral distortion for an electron plasma within galaxy clusters with $\tau_e =10^{-3}$ and with different electron temperatures of 5 keV, 10 keV, 15 keV and 20 keV. We show in Fig \[thermalsze\] the SZ$_{NP}$ spectrum for different values of $x_p$ derived from the analysis of the CMB spectrum in the previous Sect.2. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![The thermal SZ effect spectral distortions computed for galaxy clusters with increasing plasma temperature (see various panels) for a standard Planck distribution (solid line) and including the effect of non-Planckian distribution of photons for the values of the plasma frequency $x_p$ derived at $1\sigma$, $2\sigma$ and $3\sigma$ level. The cluster plasma optical depth is fixed to the value $\tau = 0.001$.[]{data-label="thermalsze"}](szeT5 "fig:"){width="80mm" height="75mm"} ![The thermal SZ effect spectral distortions computed for galaxy clusters with increasing plasma temperature (see various panels) for a standard Planck distribution (solid line) and including the effect of non-Planckian distribution of photons for the values of the plasma frequency $x_p$ derived at $1\sigma$, $2\sigma$ and $3\sigma$ level. The cluster plasma optical depth is fixed to the value $\tau = 0.001$.[]{data-label="thermalsze"}](szeT10 "fig:"){width="80mm" height="75mm"} ![The thermal SZ effect spectral distortions computed for galaxy clusters with increasing plasma temperature (see various panels) for a standard Planck distribution (solid line) and including the effect of non-Planckian distribution of photons for the values of the plasma frequency $x_p$ derived at $1\sigma$, $2\sigma$ and $3\sigma$ level. The cluster plasma optical depth is fixed to the value $\tau = 0.001$.[]{data-label="thermalsze"}](szeT15 "fig:"){width="80mm" height="75mm"} ![The thermal SZ effect spectral distortions computed for galaxy clusters with increasing plasma temperature (see various panels) for a standard Planck distribution (solid line) and including the effect of non-Planckian distribution of photons for the values of the plasma frequency $x_p$ derived at $1\sigma$, $2\sigma$ and $3\sigma$ level. The cluster plasma optical depth is fixed to the value $\tau = 0.001$.[]{data-label="thermalsze"}](szeT20 "fig:"){width="80mm" height="75mm"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![The non-thermal SZ spectral distortion for the case of non-thermal plasmas with increasing electron minimum momentum $p_1$ (see various panels) for a usual Planck distribution (solid line) of photons and the effect of non-Planckian distribution at $1\sigma$, $2\sigma$ and $3\sigma$ level computed in the case of a single power law for different minimum momentum $p_1$. We use here an optical depth of $\tau =0.001$.[]{data-label="nonthermalsze"}](szeNp05 "fig:"){width="80mm" height="77mm"} ![The non-thermal SZ spectral distortion for the case of non-thermal plasmas with increasing electron minimum momentum $p_1$ (see various panels) for a usual Planck distribution (solid line) of photons and the effect of non-Planckian distribution at $1\sigma$, $2\sigma$ and $3\sigma$ level computed in the case of a single power law for different minimum momentum $p_1$. We use here an optical depth of $\tau =0.001$.[]{data-label="nonthermalsze"}](szeNp1 "fig:"){width="80mm" height="75mm"} ![The non-thermal SZ spectral distortion for the case of non-thermal plasmas with increasing electron minimum momentum $p_1$ (see various panels) for a usual Planck distribution (solid line) of photons and the effect of non-Planckian distribution at $1\sigma$, $2\sigma$ and $3\sigma$ level computed in the case of a single power law for different minimum momentum $p_1$. We use here an optical depth of $\tau =0.001$.[]{data-label="nonthermalsze"}](szeNp5 "fig:"){width="85mm" height="80mm"} ![The non-thermal SZ spectral distortion for the case of non-thermal plasmas with increasing electron minimum momentum $p_1$ (see various panels) for a usual Planck distribution (solid line) of photons and the effect of non-Planckian distribution at $1\sigma$, $2\sigma$ and $3\sigma$ level computed in the case of a single power law for different minimum momentum $p_1$. We use here an optical depth of $\tau =0.001$.[]{data-label="nonthermalsze"}](szeNp10 "fig:"){width="80mm" height="75mm"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- We notice that the frequency at which the maxima of the SZ$_{NP}$ occurs is exactly $x_p$ and is independent of the cluster temperature. This spectral feature offers therefore a direct experimental probe of the plasma frequency in the early universe as measured by the SZ$_{NP}$ in structures of the local universe. We stress that this is a unique characteristic of the SZ$_{NP}$ and allows to use observations of the local universe to infer directly the physics of the early universe.\ We also show that the amplitude of the peak of SZ$_{NP}$ increases with the temperature and density (i.e. with the total Compton parameter $Y\propto \int d \ell P_e$, where $P_e$ is the total electron pressure) of the electron gas. The spectral shape of the thermal SZ$_{NP}$ is very peculiar and it is peaked in a quite narrow frequency range of the order of $\Delta \nu \approx 20$ MHz for $kT= 5$ keV and $\Delta \nu \approx 50$ MHz for $kT= 20$ keV, reflecting hence the relativistic effects of the electron distribution in the photon re-distribution function $P_1(s)$. At frequencies larger than 1 GHz, the SZ$_{NP}$ becomes close to the standard Planckian one, i.e. the standard SZ effect calculated with the non-distorted CMB Planck spectrum [@Colafrancescoetal2003] as an input. The SZ$_{NP}$ can appear in the form of both thermal and non-thermal effects depending on the nature of the electron population producing the ICS of the CMB photons (i.e. thermal electrons in clusters for the thermal SZ$_{NP}$ and non-thermal or relativistic electrons for the non-thermal SZ$_{NP}$). For completeness, therefore, we also compute the effect of non-planckian CMB distribution on the non-thermal SZ$_{NP}$ effect which is shown in Fig \[nonthermalsze\] for a power law electron spectrum with spectral index $\alpha=2.5$ and different values of the minimum momentum $p_1$ of the non-thermal electron distribution. In this case, the peak of the SZ$_{NP}$ decreases with increasing value of $p_1$ because the high-E electrons scatter photons to very high frequency. In this case the peak of the SZ$_{NP}$ is less pronounced (see Fig.\[nonthermalsze\]) but it is still located at the plasma frequency value $x_p$. The width of the SZ$_{NP}$ spectrum (see Fig.\[nonthermalsze\]) is larger due to the enhanced impact of the relativistic effects of the high-E electron population and its shape reflects therefore the different nature of the scattering plasma. Possible observations of the SZ$_{NP}$ can therefore address the question of the intrinsic nature of the plasma in the target cosmic structure (e.g., galaxy clusters vs. radio galaxies). In order to study the possible detectability of NP effect on the CMB, we have compared the SZ$_{NP}$ spectrum integrated over an angle of $5^{\prime}$ radius of a Bullet-like cluster with temperature of $kT = 15$ keV. Fig \[skasze\] shows the difference between the SZ spectrum $\Delta \tilde{I}$ and $\Delta I$ compared to the sensitivity of the SKA1 and SKA50% for 1000 hrs of observations and to the eVLA sensitivity for 12 hrs of observation. We found that the values on the plasma frequency $\nu_p$ = 206, 346 and 418 MHz at 1, 2 and 3 $\sigma$ c.I, derived from the analysis of the CMB spectrum, can be detected with both SKA-LOW and SKA-MID band1 (350-1050 MHz). ![The difference between the SZ$_{NP}$ and the Planckian SZ for a Bullet-like cluster with temperature of 15 keV (see [@bullettemp]) subtending an angle of $5^{\prime}$ to its $R_{500}$. An optical depth of $\tau =0.001$ is assumed. The SKA1 (red) and SKA-50% (orange) sensitivity is calculated for 1000 hrs integration while the eVLA sensitivity (green) is calculated for 12 hrs integration.[]{data-label="skasze"}](specialska){width="140mm" height="100mm"} Modifications to the cosmological 21-cm background ================================================== The frequency range where NP effects can be observed (see Sects.2 and 3) is the same where the distortion of the CMB spectrum induced by the 21-cm background is more relevant [@Furlanetto2006]. For this reason we explore in this Section how NP effects on the CMB spectrum as in eq.(\[nonplanckian\]) impact on the cosmological 21-cm background spectrum. To this aim, we discuss the results of our spectral analysis in term of the brightness temperature change given by $$\delta T (\nu)= \frac{c^2}{2k_B \nu^2} \delta I(\nu). \label{brigthness.temp}$$ For a Planckian CMB spectrum, the 21-cm background appears as a perturbation on the CMB background, so we have: $$\delta T(\nu) = T_{21}(\nu) - T_0 (\nu) ,$$ where $T_{21}$ is the brightness temperature of the 21-cm background and $T_0$ is the brightness temperature of the CMB. In the case of a non-planckian CMB spectrum, the previous relation can be generalized as follows $$\delta \tilde{T}(\nu;\nu_p) = \tilde{T}_{21}(\nu;\nu_p) -\tilde{T_0} (\nu;\nu_p).$$ Once we fixed the value of $x_p$, we can distinguish two frequencies regimes:\ i) $\nu<\nu_p$: in this case $\tilde{T_0}=0$, and therefore $\delta \tilde{T}=\tilde{T}_{21}$. If we assume that $\tilde{T}_{21} \approx T_{21}$, i.e. the 21-cm background is not sensibly changed by the NP spectral distortion, as it is expected because the main physical processes affecting the 21-cm background depend on the global temperature of the system which is not heavily affected by a distortion at small frequencies, then we can write: $$\delta \tilde{T} = \delta T + T_0 ,$$ so the frequency change of the brightness temperature is the same as in the Planckian case, but its amplitude is shifted by a value $T_0$ (that in the RJ region has a constant value equal to the CMB temperature);\ ii) $\nu>\nu_p$: in this case we can write the NP modified spectrum in the RJ region as: $$\tilde{I}_0 \propto x \sqrt{x^2-x_p^2} = x^2 \sqrt{1-\frac{x_p^2}{x^2}},$$ and the corresponding brightness temperature is: $$\tilde{T}_0= T_0 \sqrt{1-\frac{x_p^2}{x^2}}.$$ By assuming again $\tilde{T}_{21} \approx T_{21}$, the change in the temperature brightness is given by: $$\delta \tilde{T} = \delta T + T_0 - \tilde{T}_0 = \delta T + T_0 \left(1-\sqrt{1-\frac{x_p^2}{x^2}}\right).$$ In Fig.\[dt21cm\] we show the spectral change of $\delta \tilde{T}$ for different values of $\nu_p$ from 0.2 to 100 MHz, by assuming a standard benchmark model for $\delta T$ (as discussed in[@Cooray2006; @ColafrancescoMarchegiani2014b]). For $\nu<\nu_p$ the frequency change of $\delta \tilde{T}$ is the same $\delta T$ obtained in the standard Planckian case but its amplitude is increased by the value $T_0$; for $\nu \simgt \nu_p$ the quantity $\delta \tilde{T}$ decreases very rapidly with increasing frequency, and for $\nu \gg \nu_p$ it then becomes equal to the standard value $\delta T$ in the Planckian case. ![The change in brigthness temperature $\delta \tilde{T}$ for $\nu_p=0$ (solid line), $\nu_p=0.2$ (dotted), 1 (dashed), 10 (dot-dashed), 20 (three dots-dashed) and 100 MHz (long dashed) as a function of the frequency.[]{data-label="dt21cm"}](21cm) Fig.\[dt21cm\] shows that even low values of the plasma frequency $\nu_p \simlt 206$ MHz (i.e., the 1 $\sigma$ upper limit set by the CMB spectrum analysis in Sect.2) can largely change the 21-cm radiation background spectrum and hence our ability to recover the history of the EoR and of the DA. NP effects do not change the physical mechanisms that are working during these epochs but change the brightness temperature contrast $\delta \tilde{T}$ relative to the CMB that we can use to prove these remote cosmic epochs. Interestingly, the brightness temperature contrast $\delta \tilde{T}$ in the case in which NP effects are present on the CMB is increased w.r.t. the standard assumption of a pure Planckian CMB spectrum. This fact on one hand yields larger signals that could be detected more easily even with precursors of the SKA and HERA, and on the other hand provides an effective possibility to set limits or even prove the existence of NP effects in the early universe by looking at the global spectrum of the 21-cm background. Available limits from the PAPER experiment [@Parson2014] in the 100–200 MHz provide limits only on the average temperature brightness of $\langle T_b \rangle < 275$ and $291$ mK for values of the ionization fraction $x_i = 0.5$ and $x_i=0.75$ of the ionization power spectra, respectively, which cannot be unfortunately directly compared to the predictions for the quantity $\delta T$ used in our paper. The next generation SKA and HERA low-frequency interferometers will have the possibility to improve the knowledge on the brightness temperature contrast $\delta T$ down to levels of $\sim 3$ mK ($\sim 0.3$ mK) with SKA phase-1 (SKA phase-2) at $3 \sigma$ c.l. in about 1000 hrs at 150 MHz [@Koopmans2010] and possibly cover a wider frequency range, thus allowing to set much stronger limits on $\nu_p$ from the spectrum of $\delta T (\nu)$. The SKA1-LOW band 50–350 MHz will be able to set limits only on plasma frequency values $\nu_p \simgt 10 $ MHz because for lower values of $\nu_p$ of order of a few MHz the NP modified 21-cm background spectrum is very close to the Planckian case at the lower end of the SKA-LOW band (see Fig.\[dt21cm\]). Therefore, the SKA-LOW will have a sensitivity to the NP effects at the redshift (frequency) of the EoR. We speculate here that other sources of more recent NP effects can arise during the EoR due to the increasing ionization fraction of the cosmic plasma at this epoch caused by Ly-$\alpha$ interactions, heating due to UV/X-ray primordial sources and DM annihilation. The treatment of these secondary NP effects is well beyond the purposes of this initial paper and will be addressed more specifically elsewhere. Discussion and conclusions ========================== We have performed for the first time in this paper an analysis of NP effects on the cosmological radio background and we derived predictions for their amplitude on three different observables: the CMB spectrum, the SZ effect and the 21-cm temperature brightness change.\ We have shown that NP effect can manifest on the CMB spectrum at $\nu \simlt 400$ MHz as a drastic cut-off in the intensity of the CMB (see Fig.\[CMBlow\]). Using the CMB data of COBE and lower frequency measurements, we derived for the first time upper limits on the plasma frequency $\nu_p$ = 206, 346 and 418 MHz at 1, 2, 3 $\sigma$ c.l., respectively. The difference between the pure BB Planck spectrum and the one modified by NP effects at these low frequencies is of the order of mJy/arcmin$^2$ and this difference becomes smaller at higher frequencies ($\nu \approx 150$ GHz) where it is of the order of 0.1 mJy/arcmin$^2$, thus indicating that the experimental route to probe NP effects in the early universe is to observe the cosmological radio background at very low frequencies. We have also computed the SZ$_{NP}$ effect using the upper limits on $\nu_p$ allowed by the present data of the CMB (Sect.2) and it has been found that the NP effect produces a unique imprint on the SZ$_{NP}$ effect spectrum at $\nu \simlt 1$ GHz, i.e., a peak located exactly at the plasma frequency $\nu_p$ and that this is independent of the cluster parameters (such as its temperature or optical depth). This offers a way to measure directly and unambiguously the plasma frequency in the early universe at the epoch of recombination by using (in principle only one) cluster in the local universe, thus opening a unique window for the experimental exploration of plasma effects in the early universe. We have shown that the SKA-LOW and SKA-MID has the potential to observe such a signal integrating over the central regions ($\approx 5^{\prime}$ radius) of high-temperature ($k_B T \sim 15$ keV) clusters.\ Observing the SZ$_{NP}$ also benefits from its differential nature thus being less affected by the large impact of large-scale foreground emission that is one of the main systematic biases that limits the study of the intensity spectrum of the cosmological background radiation. The SZ$_{NP}$ has also the appealing property that we can study the presence of NP effects in the early universe by looking at very local cosmic structures for which the structural parameters are known with high accuracy. Finally, we mention that studies of NP effects through the SZ$_{NP}$ can be done by intensive observations of only one galaxy cluster, or with a stacked spectrum of a few well known clusters, thus avoiding the need of large statistical studies of source populations or wide area surveys. Finally, we have shown that future low-frequency observations of the cosmological 21-cm brightness temperature spectral changes have the possibility to set global constraints on NP effects by constraining the spectral variations of $\delta \tilde{T}$ induced by the plasma frequency value at the epoch of recombination. We discussed, in this context, that even moderate limits on the average brightness temperature of the 21-cm background obtained with SKA precursors, like e.g. the PAPER experiment, are able to start limiting the possible values of $\nu_p$ in its high-frequency domain (of order of 100 to a few hundreds MHz). In conclusion, we have demonstrated that the study of low-frequency cosmological radio background has a strong and unique potential for proving the physics of the early universe. This paper aligns such the idea of proving NP effects with our previous studies of the photon decay effects in the early universe [@ColafrancescoMarchegiani2014a] and on the study of the DA and EoR through the SZE-21cm [@ColafrancescoMarchegiani2014b; @Colafrancescoetal2015], and indicates that this area of investigation of the fundamental physics of the universe will receive a boost with the next generation high-sensitivity radio telescopes like the SKA. Acknowledgments {#acknowledgments .unnumbered} =============== S.C. acknowledges support by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation and by the Square Kilometre Array (SKA). P.M. and M.S.E. acknowledge support from the DST/NRF SKA post-graduate bursary initiative. [9]{} Ade, P.A.R., et al., 2015, arXiv:1502.01589 Backer, D.C.et al. 2010, Submitted for consideration by the Astro2010 Decadal Survey Program Panel RMS: Radio (Meter/Centimeter) and Millimeter/Submillimeter Birkinshaw, M. 1999, Phys. Rep., 310, 97 Burigana C., Danese L., de Zotti G., 1991a, ApJ, 379, 1 Burigana C., Danese L., de Zotti G., 1991b, A&A, 246, 49 Chen, X. & Kamionkowski, M., 2004, Phys.Rev. 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--- abstract: 'Many state-of-the-art signal decomposition techniques rely on a low-rank factorization of a time-frequency (t-f) transform. In particular, nonnegative matrix factorization (NMF) of the spectrogram has been considered in many audio applications. This is an [analysis]{} approach in the sense that the factorization is applied to the squared magnitude of the analysis coefficients returned by the t-f transform. In this paper we instead propose a [synthesis]{} approach, where low-rankness is imposed to the synthesis coefficients of the data signal over a given t-f dictionary (such as a Gabor frame). As such we offer a novel modeling paradigm that bridges t-f synthesis modeling and traditional analysis-based NMF approaches. The proposed generative model allows in turn to design more sophisticated multi-layer representations that can efficiently capture diverse forms of structure. Additionally, the generative modeling allows to exploit t-f low-rankness for compressive sensing. We present efficient iterative shrinkage algorithms to perform estimation in the proposed models and illustrate the capabilities of the new modeling paradigm over audio signal processing examples.' author: - 'Cédric Févotte, and Matthieu Kowalski [^1] [^2]' title: 'Estimation with Low-Rank Time-Frequency Synthesis Models' --- =1 Introduction ============ factorization methods currently enjoy a large popularity in machine learning and signal processing. In signal processing, the input data is usually a time-frequency (t-f) transform of some original time series $x(t)$. For example, in the audio setting, nonnegative matrix factorization (NMF) is commonly used to decompose magnitude or power spectrograms into elementary components [@Smaragdis2014]; the spectrogram $\ve{P}$ is approximately factorized into $\W \H$, where $\W$ is the dictionary matrix collecting spectral patterns in its columns and $\H$ is the activation matrix. The approximate $\W \H$ is generally of lower rank than $\ve{P}$, unless additional constraints are imposed on the factors. NMF is at the core of [classical]{} source separation systems such as [@vir07; @ieee_asl10]. The spectrogram $\ve{P}$ is usually obtained from the short-time Fourier transform $\ve{Y}$. The coefficients $y_{fn}$ of $\ve{Y}$ are the inner products of $x(t)$ with t-f atoms $\phi_{fn}(t)$, where $f$ and $n$ index frequencies and time frames, respectively, and a common choice is $\ve{P} = \| \ve{Y} \|^{2}$. The STFT coefficients are so-called [analysis]{} coefficients. As such, spectral decomposition by NMF can be viewed as a low-rank time-frequency [analysis]{} procedure. Leveraging on the potential of [synthesis]{} models as opposed to analytical ones (see, e.g., [@elad2007analysis; @balazs2013adapted; @sprechmann2013supervised; @nam2013cosparse]), we propose to explore a dual view of the usual NMF approach and present a new paradigm that we name [low-rank time-frequency synthesis]{} (LRTFS). In this new paradigm, the signal is decomposed as where the synthesis coefficients $\{\alpha_{fn}\}$ are endowed with a low-rank structure such that $\|\alpha_{fn}\|^{2} \approx [\W \H]_{fn}$. Formulation provides a generative representation of the raw data $x(t)$ and extends the modeling capacities of standard NMF-based signal decomposition towards more advanced multi-layer hybrid decompositions. Having a generative model of the raw data (instead of its transform) is also useful for some inverse problems such as compressive sampling, an application that will be considered in the paper. The low-rankness of the synthesis coefficients $\{ \alpha_{fn} \}$ is induced through a probabilistic model named [Gaussian Composite Model]{} (GCM) [@neco09]. The GCM underlies Itakura-Saito NMF, a baseline method that will be recalled in Section \[sec:isnmf\]. Section \[sec:lrtfs-comp\] presents our new paradigm LRTFS in the general case of complex-valued signals. It also describes an alternate minimization algorithm for maximum joint likelihood estimation of the parameters. Section \[sec:lrtfs-real\] shows how the methodology for complex signals can be adapted to real-valued signals. [Section \[sec:related\] discusses how LRTFS relates to other temporal models with low-rank spectrograms and/or structured variance [@Kameoka2015; @Kameoka2017; @Liutkus2011; @Yoshii2013; @Turner2014; @Liutkus2017; @Yoshii2018].]{} Section \[sec:hybrid\] describes how LRTFS can accommodate more advanced multi-layer decompositions in which every layer can have its own t-f resolution or structure (e.g., a sparse instead of low-rank time-frequency structure). Section \[sec:compress\] describes a new approach to compressive sampling, which exploits latent low-rank time-frequency structure instead of sparsity, with superior results for the considered type of data. The article is illustrated throughout with experiments on audio signals (the presented methodology is however not limited to audio signals). [In particular, we use a running piano toy example to illustrate every stage of our contributions.]{} This article unifies and continues our work presented in the conference papers [@nips14; @Fevotte2015a]. In particular, it provides more detailed experiments and presents the following novel methodological contributions: the case of real-valued signals (which require to properly handle the Hermitian symmetry of their synthesis coefficients) is now rigorously treated in Section \[sec:lrtfs-real\], algorithm accelerations are presented in Section \[sec:mjle\], and the concept of compressive LRTFS presented in Section \[sec:compress\] is entirely novel. A baseline: Itakura-Saito NMF and the Gaussian Composite Model (GCM) \[sec:isnmf\] ================================================================================== NMF was originally designed in a deterministic setting [@lee99]: a measure of fit between $\ve{P}$ and $\W \H$ is minimized with respect to (w.r.t) $\W$ and $\H$. Choosing the “right” measure for a specific type of data and task is not straightforward. Furthermore, NMF-based spectral decompositions often arbitrarily discard phase information: only the magnitude of the complex-valued short-time Fourier transform (STFT) is considered. To remedy these limitations, a generative probabilistic latent factor model of the STFT, the GCM, was proposed in [@neco09]. It is defined by where $N_{c}$ refers to the circular complex-valued normal distribution.[^3] As shown by Eq. , in the GCM the STFT is assumed centered and its variance has a low-rank structure. Many temporal waveforms (such as audio signals) can be assumed centered and this remains true for their Fourier coefficients by linearity of the transformation. This explains the zero-mean assumption in the GCM on the one hand. The low-rank variance structure on the other hand underlies a composite signal structure that makes the model relevant for decomposition task. Indeed, introducing the latent complex-valued components $y_{kfn}$, Eq. is equivalent to The latent component $\ve{Y}_{k}$ with coefficients $\{y_{kfn}\}_{fn}$ reflects the contribution of the spectral pattern $\ve{w}_{k}$, the $k^{th}$ column of $\W$, amplitude-modulated in time by the activation coefficients of the $k^{th}$ row of $\H$. Under these assumptions, the negative log-likelihood $-\log p(\ve{Y}\|\W,\H)$ is equal, up to a constant, to the Itakura-Saito (IS) divergence $D_{\IS}(\ve{P}\|\W \H)$ between the power spectrogram $\ve{P} = \|\ve{Y}\|^{2}$ and $\W \H$. The IS divergence between nonnegative matrices $\ve{A}$ and $\ve{B}$ is defined by The GCM is a step forward from traditional NMF approaches that fail to provide a valid generative model of the STFT itself – other approaches have only considered probabilistic models of the magnitude spectrogram under Poisson or multinomial assumptions, see [@Smaragdis2014] for a review. Still, the GCM is not yet a generative model of the raw signal $x(t)$ itself, but of its STFT. LRTFS fills in this ultimate gap. Low-rank time-frequency synthesis (LRTFS) \[sec:lrtfs\] ======================================================= In this section we first present LRTFS for complex-valued signals, closely following [@nips14]. Then we rigorously address the case of real-valued signals represented as a complex-valued linear combination of complex-valued t-f atoms (such as Gabor atoms) with Hermitian symmetry. Finally, we discuss relevant connections with the state-of-the-art and illustrate the potential of LRTFS on an audio example. Complex-valued signals \[sec:lrtfs-comp\] ----------------------------------------- ### Model {#sec:lrtfs-mod} Let ${x}(t)$ denote a complex-valued signal of length $T$ and $\{\phi_{fn}(t)\}_{f=1..F,n=1..N}$ denote a dictionary of complex-valued t-f atoms of length $T$. LRTFS is defined as follows. For $t = 1, \ldots, T$, $f = 1,\ldots,F$, $n = 1,\ldots,N$: $$\begin{aligned} x(t) &= \sum_{fn} \alpha_{fn} \phi_{fn}(t) + e(t), \label{eqn:sig} \\ \alpha_{fn} &\sim N_{c}(0,[\W \H]_{fn}), \label{eqn:alpha} \\ e(t) &\sim N_{c}(0,\lambda), \label{eqn:res} \end{aligned}$$ where $\{ \alpha_{fn} \}$ are the complex-valued synthesis coefficients, $\W$ and $\H$ are nonnegative matrices of sizes $F \times K$ and $K \times N$, respectively, and $e(t)$ is an additive complex-valued residual term with Gaussian distribution $N_{c}(0,\lambda)$. The synthesis coefficients $\{ \alpha_{fn} \}$ are furthermore assumed independent given $\W$ and $\H$. The synthesis coefficients are dual of the analysis coefficients, defined by $y_{fn} = \sum_{t} x(t) \phi_{fn}^{}(t)$, where $\cdot^{}$ denotes conjugation. IS-NMF assumes that the [analysis]{} coefficients follow a GCM, see Eq. . In contrast, LRTFS assumes that the [synthesis]{} coefficients follow a GCM, as given by Eq. . As announced, LRTFS provides a generative model of the raw data $x(t)$, where IS-NMF only provides a generative model of the transformed data $\ve{Y}$. Let us denote by $\ve{x}$ and $\ve{e}$ the column vectors of size $T$ with coefficients $x(t)$ and $e(t)$, respectively. [Let ${\cal I}$ be an arbitrary one-to-one “vectorizing” mapping from $(f,n) \in \{1,\ldots,F\} \times \{1,\ldots,N\} $ to $m = {\cal I}(f,n) \in \{1,\ldots, M\} $, where $M=FN$. We denote by $\balpha$ the column vector of dimension $M$ with coefficients $\alpha_{m} = \alpha_{ {\cal I}(f,n)} = \alpha_{fn} $. We are abusing the notations by indexing the synthesis coefficients by either $m$ (unstructured vectorized form) or $(f,n)$ (matrix form where $f$ indexes frequencies and $n$ indexes time frames). It should be understood that $m$ and $(f,n)$ are in one-to-one correspondence and the meaning should be clear from the context. The notation $\alpha_{m}$ discards the inherent t-f structure of the coefficients while the notation $\alpha_{fn}$ makes it explicit. Despite abusing, this convention allows to significantly reduce cluttering in the following. Similarly, we denote by $\bPhi$ the matrix of size $T\times M$ with columns $\bphi_{m} = \bphi_{fn}$, where $\bphi_{fn}$ is the column vector of size $T$ with coefficients $\phi_{fn}(t)$. We denote by $\ve{v}$ the column vector of dimension $M$ with coefficients $v_{m} = v_{fn} \defequal [\W \H]_{fn}$. We will sometimes write $\ve{v} = \text{vect}[\W \H]$, where $\text{vect}[\cdot]$ refers to the vectorizing operator induced by ${\cal I}$.]{} Equipped with these notations, we may write Eq. and as Ignoring the low-rank structure of $\ve{v}$, Eqs. - resemble sparse Bayesian learning (SBL), as introduced in [@tip01; @Wipf2004], where it is shown that marginal likelihood estimation of the variance induces sparse solutions of $\ve{v}$ (and as a consequence, of $\balpha$). The essential difference between our model and SBL is that the coefficients are no longer unstructured in LRTFS. Indeed, in SBL, each coefficient $\alpha_{m}$ has a free variance parameter $v_{m}$. This property is fundamental to the sparsity-inducing effect of SBL [@tip01]. In contrast, in LRTFS, the variances are now tied together and such that $v_{m} = v_{fn} = [\W \H]_{fn}\ $. ### Maximum joint likelihood estimation \[sec:mjle\] We now address the estimation of $\W$, $\H$ and $\balpha$ and possibly $\lambda$ in LRTFS. We consider maximum joint likelihood estimation (MJLE), also referred to as type-I maximum likelihood estimation in [@Wipf2004]. MJLE relies on the minimization of the following objective function: where $cst = T \log\lambda + (T+M) \log \pi $ and we recall that $\ve{v} = \text{vect}[\W \H]$. Another possible estimation procedure for LRTFS is maximum marginal likelihood estimation (MMLE), also referred to as type-II maximum likelihood estimation in [@Wipf2004]. It relies on the minimization of $- \log p(\ve{x}\|\W,\H,\lambda)$, i.e., involves the marginalization of $\balpha$ from the joint likelihood, following the principle of SBL. We considered MMLE for LRTFS in [@nips14] and presented a valid EM algorithm. However our implementation does not scale with the dimensions involved in signal processing, [as it requires the estimation of the diagonal elements of the inverse of a $M\times M$ matrix.]{} Large-scale algorithms for MMLE are left as future work. ### Alternate minimization algorithm for MJLE \[sec:algo\] We now describe an alternate minimization algorithm that returns stationary points of $C_{\JL}(\btheta)$, where $\btheta = \{\balpha,\W,\H,\lambda\}$. The optimization of $\balpha$ given the other parameters reduces to which defines a convex ridge regression problem. The problem has the closed-form solution We used in our implementation a complex-valued version of ISA, similar to the complex-valued cases treated in [@Chaari2011; @Florescu2014], and using the acceleration described in [@chambolle2015convergence]. This leads to a simple and parameter-free implementation with satisfactory speed of convergence. This in particular results in a faster algorithm than the original EM algorithm presented in our initial contribution [@nips14]. [Moreover, ISA leads in our case to a simpler algorithm than GCD, with same theoretical and practical speed of convergence [@Ben-Tal2004]]{}. The resulting updates are given in Algorithm \[alg:LRTFS\]. The value of the inverse step-size $L$ should be set to the maximum eigenvalue of $\bPhi{^{\mathsf{H}}}\bPhi$, i.e., the squared spectral norm of $\bPhi$. If this value is not available in closed form or difficult to compute, a larger value $L \ge \\| \bPhi \\|_{2}^{2}$ is also permissible but will result in smaller step sizes. In Algorithm \[alg:LRTFS\], the operations $\ve{A} \circ \ve{B}$, ${\ve{A}}^{\circ p }$ and $\frac{\ve{A}}{\ve{B}}$ denote entry-wise multiplication, exponentiation and division, respectively. The optimization of $\W$ and $\H$ given $\balpha$ reduces to which defines a IS-NMF problem with input matrix $\ve{S} = [\|\alpha_{fn}\|^{2}]_{fn}$. This a non-convex problem that is generally approached with alternating updates of $\W$ and $\H$ and majorization-minimization (MM) [@betanmf]. This results in the multiplicative updates given in Algorithm \[alg:LRTFS\]. Finally, the optimization of $\lambda$ given $\balpha$ is trivially given by $\hat{\lambda} = \\| \ve{x} - \bPhi \balpha \\|/T$. However, the MJLE setting is known to be inefficient for the estimation of both the variance parameters of $\balpha$ and of $\ve{e}$, with either $\bPhi \hat{\balpha}$ or $\hat{\ve{e}}$ capturing most of the signal variance. As such, though the estimation of $\lambda$ is possible in principle, we will consider $\lambda$ to be a fixed hyper-parameter in the following. The objective function $C_{\JL}$ being non-convex and because we are using an alternate minimization algorithm, the output of Algorithm \[alg:LRTFS\] depends on the initialization. In all simulations we initialized the synthesis coefficients $\balpha$ with the [analysis]{} coefficients $\bPhi{^{\mathsf{H}}}\ve{x}$. The matrices $\W$ and $\H$ are initialized using the absolute values of the complex SVD of the synthesis coefficients [@becker2015complex]. Finally, a tempering strategy with warm restart is used to speed up convergence for small target values of $\lambda$. The hyper-parameter $\lambda$ is set to an arbitrarily large value in the first iterations and is then gradually decreased to its target value, as proposed in [@hale2008fixed]. [Convergence of the main and inner loops is monitored using the relative difference in norm between successive parameter iterates, see Section \[sec:exlrtfs\].]{} \[alg:LRTFS\] Set $L = \\| \bPhi \\|_{2}^{2}$ (or a larger value)\ Compute the synthesis coefficients $\ve{y} = \bPhi{^{\mathsf{H}}}\ve{x}$ (with matrix form $\ve{Y}$)\ Set $\balpha^{(0)} = \ve{y}$\ Initialize $\W^{(0)}$ and $\H^{(0)}$ with the absolute values of the complex SVD of $\ve{Y}$\ Set $i=0$ ### Reconstruction of the latent components [Algorithm \[alg:LRTFS\] outputs an estimate of $\balpha$, $\W$, $\H$. The approximate signal can directly be recovered from the estimated synthesis coefficients as $\hat{\ve{x}} = \bPhi \hat{\balpha}$.]{} LRTFS further assumes that the synthesis coefficients follow a GCM, see Eq. . As such, $\alpha_{fn}$ may be written as a sum of Gaussian latent components, such that $\alpha_{fn} = \sum_{k} \alpha_{kfn}$, with $\alpha_{kfn} \sim N_{c}(0,w_{fk} h_{kn})$. Denoting by $\balpha_{k}$ the column vector of dimension $M$ with coefficients $\{\alpha_{kfn} \}_{fn}$, Eq. may be written as where $\ve{c}_{k} = \bPhi \balpha_{k} $. The component $\ve{c}_{k}$ is the “temporal expression” of spectral pattern $\ve{w}_{k}$, the $k^{th}$ column of $\ve{W}$. Given estimates of $\balpha$, $\W$ and $\H$, the components may be reconstructed [a posteriori]{} in various ways. A natural choice is $\hat{\ve{c}}_{k}^{\text{MMSE}}= \bPhi \hat{\balpha}_{k}^{\text{MMSE}}$ with The coefficients of $\hat{\balpha}_{k}^{\text{MMSE}}$ are given by Using this estimate, the latent components are reconstructed by applying a t-f dependent “Wiener mask” to the synthesis coefficients. This procedure and the expression of $\hat{\alpha}_{fkn}^{\text{MMSE}}$ is analog to the standard Wiener estimate of the latent components in IS-NMF applied to $\| \ve{Y}\|^{2}$ [@neco09] and given by The estimate $\hat{\balpha}$ is used as an intermediate variable in the expression of $\hat{\alpha}_{kfn}^{\text{MMSE}}$ given by Eq. . Another possible estimate, which marginalizes $\balpha$, is where $\hat{\ve{v}}_{k}$ is the vector of dimension $M$ with coefficients $\{\hat{w}_{fk}\hat{h}_{kn}\}_{fn}$. The input of the estimator is now the raw data $\ve{x}$ which may be more sensible. [The expression of this alternative estimate can be derived in closed-form but the resulting expression involves the large-scale inversion of $T \times T$ matrices, which is hardly feasible in practice.]{} Real-valued signals \[sec:lrtfs-real\] -------------------------------------- ### Model {#sec:rLRTFS} In many signal processing settings the data is a real-valued signal $x(t)$ expressed as a linear combination of complex-valued t-f atoms with Hermitian symmetry. More specifically, the dictionary and synthesis coefficients are such that $\bphi_{fn} = \bphi_{(F-f)n}^{}$ and $\alpha_{fn} = \alpha_{(F-f)n}^{}$ for $f=1,\ldots, F/2$ (assuming $F$ to be even-valued for simplicity), where $\cdot^{}$ denotes conjugation. Under this particular structure, we have and we define real-valued LRTFS (rLFTS) as follows. For $t = 1, \ldots, T$, $f = 1,\ldots,F/2$, $n = 1,\ldots,N$: $$\begin{aligned} x(t) &= \sum_{f=1}^{F/2} \sum_{n=1}^{N} 2 \Re[\alpha_{fn} \phi_{fn}(t)] + e(t) \label{eqn:sigR}, \\ \alpha_{fn} &\sim N_{c}(0,[\W \H]_{fn}) \label{eqn:alphaR}, \\ e(t) &\sim N(0,\lambda) \label{eqn:resR}. \end{aligned}$$ Note how $F$ now runs from $1$ to $F/2$ instead of 1 to $F$. The synthesis coefficients $\alpha_{fn}$ remain complex-valued and the residual $e(t)$ becomes real-valued. $\W$ and $\H$ are nonnegative matrices of sizes $F/2 \times K$ and $K \times N$, respectively. Let us now denote by $\underline{\balpha}$ and ${\ve{v}}$ the vectors of dimension $M/2$ with coefficients $\alpha_{fn}$ and $v_{fn} = [\W \H]_{fn}$, respectively, and by $\underline{\bPhi}$ the matrix of dimension $T \times M/2$ with columns $\bphi_{fn}$, for $f=1,\ldots,F/2$ and $n=1,\ldots,N$. With these notations we have $\balpha = [\underline{\balpha}^{{\mathsf{T}}}, \underline{\balpha}{^{\mathsf{H}}}]^{{\mathsf{T}}}$, $\bPhi = [\underline{\bPhi}, \underline{\bPhi}^{}]$ and $\bPhi \balpha = 2\Re[\underline{\bPhi} \underline{\balpha}]$. Consequently, we may write Eq. - as ### Estimation {#sec:rLFTSestim} The MJLE objective function for rLRTFS writes where $cst = \frac{T}{2} \log (2\pi\lambda) + \frac{M}{2} \log \pi$. Using an alternate minimization setting like in Section \[sec:algo\], the updates of $\W$ and $\H$ are virtually unchanged. They amount to IS-NMF of the matrix form of the synthesis spectrogram $\|\bualpha\|^{2}$ (of size $F/2 \times N$). The update of $\lambda$ is easily given by $\hat{\lambda} = \\|\ve{x} - 2\Re[\buPhi \bualpha]\\|_2^2 /T$, but here again we prefer to treat $\lambda$ as an hyper-parameter. The update of $\bualpha$ involves the following minimization problem: The problem defined by Eq. has a closed-form solution, with a less simpler expression than Eq. . The solution is still computationally demanding and the following numerical procedure is preferable. \[thm\] Let $L = \\| \bPhi \\|_{2}^{2}$ (with $\bPhi = [\underline{\bPhi}, \underline{\bPhi}^{}]$) and $\bualpha^{(0)}$ be an initial estimate. The following sequence of updates converge to the global solution of problem : The proof consists in reformulating Eq. as a quadratic optimization problem over the real and imaginary parts of $\bualpha$ and applying ISA. Let $\ve{A} = 2 [\Re[\buPhi], -\Im[\buPhi]] $, $\ve{b} = [\Re[\bualpha]^{{\mathsf{T}}}\Im[\bualpha]^{{\mathsf{T}}}]^{{\mathsf{T}}}$ and $\ve{c} = \frac{1}{2} [\ve{v}^{{\mathsf{T}}}\ve{v}^{{\mathsf{T}}}]^{{\mathsf{T}}}$. Then we may write Denoting $L_{\ve{A}} = \\| \ve{A} \\|_{2}^{2}$, the ISA update for problem writes [@Figueiredo2003; @Beck2009] Using the identities $\ve{A} \ve{b} = 2 \Re[\buPhi \bualpha]$ and Eqs. - can be rearranged in complex form as To complete the proof we only need to show that $L = 2 L_{\ve{A}} $. Let $\balpha = [\bualpha_{1}^{{\mathsf{T}}}, \bualpha_{2}^{{\mathsf{T}}}]^{{\mathsf{T}}}$ be an eigenvector of $\bPhi{^{\mathsf{H}}}\bPhi$ with maximum eigenvalue $L = \\|\bPhi \\|_{2}^{2}$. By definition we have By taking the conjugate of Eq. , we easily show that $[\bualpha_{2}{^{\mathsf{H}}}, \bualpha_{1}{^{\mathsf{H}}}]^{{\mathsf{T}}}$ is also an eigenvector with eigenvalue $L$. It follows that $[\bualpha_{1} + \bualpha_{2}{^{\mathsf{H}}}, \bualpha_{2} + \bualpha_{1}{^{\mathsf{H}}}]^{{\mathsf{T}}}$ is also an eigenvector, which happens to have a Hermitian structure. We may thus impose $\bualpha_{2} = \bualpha_{1}^{}$ and as such $\balpha = [\bualpha^{{\mathsf{T}}}, \bualpha{^{\mathsf{H}}}]^{{\mathsf{T}}}$. Then we have the following series of equivalences: As such, the spectra of $\buPhi{^{\mathsf{H}}}\buPhi$ and $\ve{A}^{{\mathsf{T}}}\ve{A}$ coincide up to a factor 2 and we have $L = 2 L_{\ve{A}}$, which concludes the proof. ### Comments about implementation \[alg:rLRTFS\] Set $L = \\| \bPhi \\|_{2}^{2}$ (or a larger value)\ Compute the synthesis coefficients ${\ve{y}} = \buPhi{^{\mathsf{H}}}\ve{x}$ (with matrix form ${\ve{Y}}$)\ Set $\bualpha^{(0)} = {\ve{y}}$\ Initialize $\W^{(0)}$ and $\H^{(0)}$ with the absolute values of the complex SVD of ${\ve{Y}}$\ Set $i=0$ Eq. and can be accelerated like before and this results in the general procedure summarized in Algorithm \[alg:rLRTFS\]. As compared to Algorithm \[alg:LRTFS\], $\balpha$ is essentially replaced by $\bualpha$, of size half, and the expression of $\ve{z}^{(i+1)}$ is changed with Eq. . Although the same notations are used for convenience, $\ve{Y}$, $\ve{S}^{(i)}$ and $\ve{W}$ become matrices with $F/2$ rows. Eq. can be read as follows. The operation $2 \Re[\buPhi \bualpha]$ consists of reconstructing an approximation $\hat{\ve{x}}$ of $\ve{x}$ based on the current synthesis coefficients $\bualpha$. The operation $\buPhi{^{\mathsf{H}}}\ve{e}$ then consists in computing the analysis coefficients (restricted to “positive” frequencies, i.e., $f=1,\ldots,F/2$) of the current residual $\ve{e} = \ve{x} - \hat{\ve{x}}$. When $\bPhi$ is a tight Gabor frame, these operations can be efficiently performed with dedicated time-frequency libraries, such as the MATLAB & Python Large Time-Frequency Analysis Toolbox (LTFAT) [@Pruuvsa2014].[^4] When the Gabor frame is tight, i.e., $\bPhi \bPhi{^{\mathsf{H}}}\ve{x} = \ve{x}$, $\bPhi$ has a unit spectral norm and we may set $L=1$. A MATLAB implementation of Algorithm \[alg:rLRTFS\] is available online.[^5] Finally, given estimates of $\bualpha$, $\W$ and $\H$, latent component coefficients $\hat{\bualpha}_{k}$ may be reconstructed like in Eq. , and then $\hat{\ve{c}}_{k}= 2 \Re[\buPhi \hat{\bualpha}_{k}]$. Related work {#sec:related} ------------ The closest to our work are probably the recent papers by Kameoka [@Kameoka2015; @Kameoka2017] which addresses temporal models of the form $\ve{x} = \sum_{k} \ve{c}_{k}$, like Eq. , where the spectrograms of the latent components are approximately rank-one. In essence (and slightly simplifying) these papers address optimization problems of the form where $\|\bPhi{^{\mathsf{H}}}\ve{c}_{k}\|^{2}$ is the spectrogram of $\ve{c}_{k}$, indexed by $f$ and $n$ according to the convention of Section \[sec:lrtfs-mod\], and $D(\cdot \| \cdot)$ is a divergence between nonnegative matrices (either the generalized Kullback-Leibler divergence or the quadratic cost in [@Kameoka2015; @Kameoka2017]). Though very elegant in our opinion, the approaches of [@Kameoka2015; @Kameoka2017] are still analysis-based and do not yet provide a fully generative synthesis-based model like LRTFS. [Another related trend of work are the approaches of [@Liutkus2011; @Yoshii2013; @Turner2014] which essentially model $x(t)$ as a sum of variance-structured Gaussian processes. Using our notations, the model in [@Turner2014] sets $N=T$ and assumes $x(t) = \sum_{f} \sqrt{[\W \H]_{ft}} \, \Re[s_{f}(t) e^{-j 2\pi ft/F}]$ where $s_{f}(t)$ is a complex-valued Gaussian autoregressive sequence. In [@Liutkus2011; @Yoshii2013], $x(t)$ is modeled as a short-time stationary process. The signal is segmented into overlapping temporal frames $\ve{x}_{n}$ of size $P$, like in the first stage of a STFT. Each temporal frame $\ve{x}_{n}$ is then assumed to follow a multivariate Gaussian distribution with covariance $\ve{R}_{n} = \sum_{k} h_{kn} \bSigma_{k}$, where $\bSigma_{k}$ is a full covariance matrix of size $P \times P$ (real-valued and symmetric). Estimation then consists in estimating the set of parameters $\{ \bSigma_{k} \}_{k}$ and $\H$ from the entire set of temporal frames (the whole approach is coined PSDTF in [@Yoshii2013]). When it is further assumed that $\bSigma_{k}$ is the covariance matrix of a real-valued latent stationary process, $\bSigma_{k}$ becomes circulant and is diagonalized by the discrete Fourier transform (DFT). In that case, it can be shown that PSDTF specializes to IS-NMF [@Liutkus2011; @Yoshii2013]. PSDTF remains close to the analysis view of IS-NMF: the raw data is segmented into overlapping frames and each frame is individually assigned a covariance model. In contrast, LRTFS provides a generative model of the entire signal $x(t)$, assumed to be a linear combination of elementary t-f bricks endowed with a low-rank variance structure. In LRTFS, we assumed the synthesis coefficients to be conditionally mutually independent, i.e., $p(\balpha \| \W, \H) = \prod_{fn}(\alpha_{fn} \| [\W \H]_{fn})$. We could very well consider more sophisticated models similar to PSDTF or extensions [@Liutkus2017; @Yoshii2018] which assume some correlation across frequencies or frames. Papers [@Liutkus2011; @Yoshii2013; @Turner2014] describe NMF-related generative probabilistic models rooted in time series analysis while LRTFS offers a different perspective, rooted in the sparse approximation literature. In particular, LRTFS can be used with any time-frequency dictionary $\bPhi$, can easily accommodate multi-layer variants (see Section \[sec:hybrid\]) or be considered for inverse problems (see Section \[sec:compress\]).]{} Example \[sec:exlrtfs\] {#sec:piano1} ----------------------- \(a) Input data\ \ (b) IS-NMF decomposition (c) LRTFS decomposition\ \ \(a) Analysis coefficients of input data\ \ (b) Synthesis coefficients estimated by LRTFS We illustrate the performance of LRTFS compared to standard IS-NMF using the piano example used in [@neco09]. The sequence has a simple structure: four notes are played together at once in the first measure and are then played by pairs in all possible combinations in the subsequent measures. The duration is $15.6$ s and the sampling rate $22050$ Hz. In noise-free conditions and with appropriate initialization, [standard IS-NMF is able to extract six identifiable latent components from this musical signal: the temporal expression of the four individual notes in a set of four components, the transient parts produced by the hammer hitting the strings [in a fifth component]{} and the sound produced by the sustain pedal when it is released [in a sixth component]{} [@neco09]]{}. We here consider a noisy example using additive white Gaussian noise with $20$ dB input Signal to Noise Ratio (SNR). [The resulting signal is displayed in Fig. \[fig:piano\_1Layer\] (a).]{} A tight Gabor dictionary (with Hermitian symmetry) built on a Hann window of $1024$ samples ($46$ ms) with $50\%$ overlap is used for $\bPhi$. IS-NMF is applied to the analysis power spectrogram $\|\buPhi \ve{x}\|^{2}$ [displayed in Fig. \[fig:piano\_1Layer\_TF\] (a).]{} The number of latent components is arbitrarily set to $K=10$ for both IS-NMF and rLTFS and the two methods are run from the same initialization (based on the SVD of $\ve{Y}$, see Algorithm \[alg:rLRTFS\]). Iteration of the main and inner loops is stopped when the relative error between two successive parameter iterates falls under $10^{-5}$. rLTFS is run with $30$ different values of $\lambda$ logarithmically equally spaced between $10^{-1}$ and $10^{-6}$. [The initialization described in Algorithm \[alg:rLRTFS\] was used for the first value $\lambda = 10^{-1}$ and warm restart was used for the subsequent experiments. The CPU time for the total 30 experiments is $\sim$5 min using a MATLAB implementation running on a Intel Core i5 processor.]{} We show results corresponding to the value of $\lambda$ that maximizes the output SNR given by Decomposition results are reported in Fig. \[fig:piano\_1Layer\]. LRTFS is able to recover the four notes in the first four components, [like standard IS-NMF in the noise free case,]{} while the fifth component recovers the transient components produced by the hammer and the sustain pedal, [corresponding to the last two components estimated by standard IS-NMF in the noise free case. As expected,]{} the remaining five components are inaudible because of the denoising performed by LRTFS. In this [noisy]{} setting, IS-NMF fails to recover this transient part and splits the first note into two components. The input noise is spread over the five remaining components. Audio files are available online. [Fig. \[fig:piano\_1Layer\_TF\] (b) displays the squared magnitude of the synthesis coefficients estimated by LRTFS. Denoising is clearly illustrated by the recovery of high frequencies.]{} Multi-layer LRTFS \[sec:hybrid\] ================================ Besides the advantage of modeling the raw signal itself, and not its STFT, another major strength of LRTFS is that it offers the possibility of multi-layer modeling. This means we may envisage models of the form where $\ve{x}_{\text{a}} = \bPhi_{\text{a}} \balpha_{\text{a}}$ and $\ve{x}_{\text{b}} = \bPhi_{\text{b}} \balpha_{\text{b}}$ are referred to as [layers]{}. This setting covers a variety of situations. $\bPhi_{\aa}$ and $\bPhi_{\bb}$ may be equal with $\balpha_{\aa}$ and $\balpha_{\bb}$ having a different structure. For example, $\balpha_{\aa}$ may follow a GCM like before and $\balpha_{\bb}$ may be given a sparsity-inducing prior. In such a case, multi-layer LRTFS offers a synthesis perspective to sparse + low-rank spectrogram decompositions, such as those presented in [@Huang2012; @Chen2013; @Sun2014] which propose variants of robust principal component analysis (RPCA) [@Candes2009] for spectral unmixing. Even more interestingly, the time-frequency dictionaries $\bPhi_{\text{a}}$ and $\bPhi_{\text{b}}$ may be chosen with different t-f resolutions. This yields so-called hybrid or morphological decompositions [@Daudet2002; @starck2005morphological], in which each layer may capture specific resolution-dependent structures. A typical audio example is transient + tonal decomposition: transient components are by nature adequately represented by a t-f dictionary with short time resolution while tonal components (such as the sustained parts of musical notes) are better represented by a t-f dictionary with larger time resolution (and as a consequence, finer frequency resolution). A variety of priors can be considered for $\balpha_{\aa}$ and $\balpha_{\bb}$, such as frequency grouping for the transient synthesis coefficients and temporal grouping for the tonal synthesis coefficients [@Kowalski2009a]. Sparse and low-rank time-frequency synthesis -------------------------------------------- We consider a special case of multi-layer LRTFS that illustrates the potential of the synthesis approach. We present the methodology in the complex case for simplicity, but the results can readily be adapted to the real case following the procedure described in Section \[sec:lrtfs\]. ### Model {#model} Let $\bPhi_{\aa}$ and $\bPhi_{\bb}$ be time-frequency dictionaries consisting of atoms $\bphi_{fn}^{\aa}(t)$ and $\bphi_{fn}^{\bb}(t)$ with common dimension $T$ and t-f pavings of size $F_{\aa} \times N_{\aa}$ and $F_{\bb} \times N_{\bb}$, respectively. We consider the following model, for $t=1,\ldots,T$: $$\begin{aligned} x(t) &= \sum_{f=1}^{F_{\aa}} \sum_{n=1}^{N_{\aa}} \alpha_{fn}^{\aa} \phi_{fn}^{\aa}(t) + \sum_{f=1}^{F_{\bb}} \sum_{n=1}^{N_{\bb}} \alpha^{\bb}_{fn} \phi^{\bb}_{fn}(t) + e(t) \label{eqn:sigh} \\ \alpha^{\aa}_{fn} &\sim N_{c}(0,[\W \H]_{fn}), f=1,\ldots,F_{\aa}, n=1,\ldots,N_{\aa} \label{eqn:alphaa} \\ \alpha^{\bb}_{fn} &\sim N_{c}(0,v^{\bb}_{fn}), f=1,\ldots,F_{\bb}, n=1,\ldots,N_{\bb} \label{eqn:alphab} \\ e(t) &\sim N_{c}(0,\lambda) \label{eqn:resh} \end{aligned}$$ where $\{ \alpha^{\aa}_{fn} \}$ and $\{ \alpha^{\bb}_{fn} \}$ are the complex-valued synthesis coefficients, $\W$ and $\H$ are nonnegative matrices of sizes $F_{\aa} \times K$ and $K \times N_{\aa}$, respectively, $\{ v^{\bb}_{fn} \}$ are nonnegative variance parameters and $e(t)$ is an additive complex-valued residual term. Eq. is nothing but the scalar form of Eq. . Eq. defines a GCM, while Eq. defines the sparse-inducing prior that is used in SBL. Like before, we denote by $\ve{v}^{\aa}$ and $\ve{v}^{\bb}$ the column vectors with coefficients $[\W \H]_{fn}$ and $v^{\bb}_{fn}$, respectively. Both $\ve{v}^{\aa}$ and $\ve{v}^{\bb}$ are parameters of a hierarchical variance model. Notice however how $\ve{v}^{\bb}$ is a [free]{} parameter, while $\ve{v}^{\aa}$ is structured through $\W$ and $\H$. Overall, Eq. -, define a multi-layer LRTFS model with latent low-rank t-f structure for layer $\ve{x}_{\aa}$ and latent sparse t-f structure for layer $\ve{x}_{\bb}$. [Note that a wavelet dictionary could alternatively be used to represent the sparse layer: the estimation procedure presented next would apply in the exact same way. LRTFS however has to be supported by a regular t-f lattice and is not compatible with wavelets (in which time resolution decreases with frequency). LRTFS can however accommodate constant-$Q$ t-f representations (constant time resolution, logarithmic frequency resolution) provided it can be inverted (near-accurate synthesis operator are proposed in [@Brown1991; @Fitzgerald2006]).]{} ### Estimation {#estimation} The negative log-likelihood of the data and parameters in model - is given by where $cst = T \log \lambda + (T+ F_{\aa}N_{\aa} + F_{\bb}N_{\bb}) \log \pi$. Unfortunately, and similarly to the difficulty of estimating $\lambda$ raised in Section \[sec:mjle\], MJLE fails to evenly distribute the signal variance onto the two layers, and one of the two layers takes it all in practice. Such a problem can be mitigated using MMLE instead of MJLE, but again, MMLE is too costly in our setting. To solve this issue we introduce an extra hyper-parameter $\mu$ that balances the contributions of each layer and propose to optimize the following objective where $0\le\mu\le1$, $\btheta = \{\balpha_{\aa},\balpha_{\bb}, \W,\H, \ve{v}_{\bb} \} $ is the set of latent variables and parameters and SLR stands for “sparse + low-rank”. We may again find a stationary point of $C_\text{SLR}(\btheta)$ by alternate minimization. The update of $\ve{v}_{\bb}$ is trivially given by $\ve{v}_{\bb} = \|\balpha_{\bb}\|^{2}$. The update of $\W$ and $\H$ amounts to finding an IS-NMF of the synthesis spectrogram $\|\alpha^{\aa}_{fn}\|^{2}$ like in Algorithm \[alg:LRTFS\]. The synthesis coefficients $\balpha_{\aa}$ and $\balpha_{\bb}$ may be updated jointly via ridge regression over the joint dictionary $[ \bPhi_{\aa}, \bPhi_{\bb}]$. This leads to the following updates where the inverse-step size should satisfy $L \ge \\| [ \bPhi_{\aa}, \bPhi_{\bb}] \\|_{2}^{2}$. A convenient choice is $L = \\|\bPhi_{\aa}\\|^2_{2}+\\|\bPhi_{\bb}\\|^2_{2}$. Eq. computes the current residual, Eqs. and produce a step in the descent direction and Eqs. and shrink the resulting iterates. Example {#sec:exhlrtfs} ------- We use exactly the same data and setting as in Section \[sec:exlrtfs\] but we now add a sparse layer $\bPhi_{\bb} \balpha_{\bb} $ to the LRTFS layer. $\bPhi_{\bb}$ is set to be a tight Gabor dictionary built on a Hann window of 128 samples ($6$ ms) with $50\%$ overlap. $\bPhi_{\aa}$ is set as in Section \[sec:exlrtfs\]. The parameter $\mu$ was experimentally fixed to $\mu=0.05$, and $\lambda$ was again chosen among logarithmically spaced vales. Fig. \[fig:multilrtfs\] displays the 10 latent components characterizing the tonal layer and the transient layer. The components of the tonal layer are similar to those obtained from the single-layer LRTFS decomposition of Fig. \[fig:piano\_1Layer\]. The fourth component captures part of the hammer attacks (especially from the first, most energetic note) with the shortest resolution components relegated to the transient layer $x^{\bb}(t)$ as expected. Audio files are available online. \ \ [(b) Transient layer $x^{\bb}(t)$]{}\ Compressive LRTFS \[sec:compress\] ================================== A striking advantage of LRTFS is that it may be used as a source model in inverse problems. For instance, LRTFS has been used in multichannel source separation in [@Leglaive2017]. We here consider compressive sensing (CS) in which a source signal $x(t)$ must be recovered from $S < <T$ random projections. Traditionally, CS exploits the sparsity of the synthesis coefficients of $x(t)$ onto a suitable dictionary. In this section we show that sparsity can be efficiently replaced with low-rankness, [for the class of signals considered]{}. Model {#model-1} ----- Let us denote by $\ve{x} \in \CC^T $ the vector source signal. The source is assumed to be sensed through the given linear operator $\ve{A} \in \CC^{S \times T}$ ($S < T$), with output $\ve{b} \in \CC^{S}$. We assume the following observation model: where $\ve{\bPhi} \in \CC^{T \times M}$ is a given dictionary, $\balpha$ are the synthesis coefficients of $\ve{x}$, and $\ve{e}$ is a residual term that accounts for noise or model errors. Where traditional CS assumes some form of sparsity for $\balpha$, we assume the synthesis coefficients to have the LRTFS low-rank structure described by Eq. . Like in traditional CS settings, we assume $\ve{A}$ to be a random matrix. Finally, we assume $\ve{e}$ to follow a complex Gaussian distribution like in Eq. . Estimation {#estimation-1} ---------- MJLE amounts to minimizing the following objective function: $$\begin{aligned} & C_{\text{\text{CS}}}(\balpha,\W,\H) = - \log p(\ve{b}, \balpha \| \W, \H, \lambda) \\ & = \frac{1}{\lambda} \\|\ve{b} - \bA\bPhi\balpha\\|_2^2 \nonumber + D_{\text{IS}}(\|\balpha\|^2\|\W\H) + \log(\|\balpha\| ^2) + cst\label{eqn:cjl_inv} \end{aligned}$$ where $cst = T \log \lambda + (T+ M) \log \pi$. The problem of optimizing $C_{\text{\text{CS}}}(\balpha,\W,\H)$ is equivalent to the one of optimizing $C_{\text{\text{JL}}}(\balpha,\W,\H)$ given by Eq. . In the complex case, the methodology developed in Section \[sec:algo\] can be readily applied by replacing $\bPhi$ with $\ve{M} = \ve{A} \bPhi$. The spectral norm of $\ve{M}$ may be difficult to derive or compute and we may set $L = \\| \ve{A} \\|_{2}^{2} \\| \ve{\bPhi} \\|_{2}^{2} $ thanks to the inequality In the real case, i.e, when $\ve{x} \in \RR^T$, the methodology developed in Section \[sec:lrtfs-real\] may again be applied by assuming $\ve{A} \in \RR^{S \times T}$ and replacing $\buPhi$ with $\ve{M} = \ve{A} \buPhi$. Posterior to estimation, an estimate of the original source is given by $\hat{\ve{x}} = \bPhi \hat{\balpha}$. Note that we have addressed compressive sampling of real or complex-valued signals by exploiting a latent NMF-type structure, which is different from compressive sampling of non-negative signals, a topic addressed for example in [@OGrady2008]. Example {#example} ------- We evaluate the recovery accuracy of the piano sequence used in Sections \[sec:exlrtfs\] and \[sec:exhlrtfs\] using a number of measurements $S$ varying increasingly from $T/100$ to $T/10$. For this experiment, the length of the sequence remains $15.6$ s but the sampling rate has been fixed at 11025 Hz because of the memory and computational complexities. The Gabor parameters have been adjusted accordingly with a Hann window of length $512$ samples ($46$ ms) with $50\%$ overlap. We compare CS recovery methods based on LRTFS, SBL and $\ell_{1}$ regularization, using a common alternating minimization setting (only the shrinkage or thresholding operators are changed). Note that we here consider type-I SBL (equivalent to MJLE) and not type-II (which again does not scale with the dimensions of our problem). The algorithms are initialized with $ \balpha =\mathbf{0}_{M \times 1}$. The first IS-NMF step of the LRTFS estimation was initialized with the absolute value of the complex-SVD as explained in Section \[sec:algo\]. LRTFS was applied with $K=10$ and the hyper-parameter $\lambda$ was incrementally decreased from $10^3$ to $10^{-2}$. [In addition, we provide the performance results of two oracles. In the first oracle, the vector of variances in Eq. is set to the power spectrogram of the ground-truth uncompressed signal $x(t)$ (updates of $\W$, $\H$ are thus removed). In the second oracle, we set the matrices $\W$, $\H$ to their estimates returned by IS-NMF applied to the power spectrogram of $x(t)$. These two oracles allow one to evaluate the remaining gap between adaptive and optimal CS recovery.]{} Estimation accuracy was measured by means of output SNR. The results are displayed in Fig. \[fig:CS\] and show that LRTFS-based recovery improves accuracy by several dBs as compared to sparsity-based methods. [This means that for this type of signals which are endowed with a strong low-rank t-f structure, there is a significant gain in exploiting low-rankness instead of mere unstructured sparsity for CS]{}. Such a recovery approach is made possible thanks to the generative design of LRTFS. Fig. \[fig:piano\_CS\_comp\] displays the estimated components $\hat{\ve{c}}_{k}$ returned by LRFTS. It is interesting to note that only $4$ components are meaningful. The first two notes are well recovered, like in the experiment of Section \[sec:piano1\], see Fig. \[fig:piano\_1Layer\] (b), while the two other notes are mixed in the third component. The fourth component still captures some transient information. We also run experiments for various values of the rank $K$ in the case $S=5\,T/100$. The recovery results appeared very robust to this parameter. For $K \in \{ 5,8,10,15,20,30 \}$ the largest difference in the output SNRs was less than $0.5$ dB. [This robustness is partially explained by the fact that LRTFS tends to shrink irrelevant components, as explained by Eq. and illustrated by Fig. \[fig:piano\_1Layer\]. We believe that the deterministic initialization provided by SVD is another explanation.]{} Finally, we run the same CS experiment using the first $12$ s of the song [Mamavatu]{} from Susheela Raman. The excerpt contains acoustic guitar and drums. Output SNRs are displayed on Fig. \[fig:mamavatu\_CS\]. Again, LRTFS recovery outperforms $\ell_{1}$ regularization and SBL by several dBs which confirms the potential of the proposed model for audio inverse problems, [where t-f low-rankness is a valid assumption.]{} \ Conclusion {#sec:conclusion} ========== We have presented a new modeling paradigm that bridges t-f synthesis modeling and traditional analysis-based approaches. The proposed generative model allows in turn to design more sophisticated multi-layer representations that can efficiently capture diverse forms of structure. Additionally, the generative modeling allows to exploit NMF-like structure for compressive sensing which, to the best of our knowledge, is entirely new. Maximum joint likelihood estimation in the proposed models can be efficiently addressed using state-of-the-art iterative shrinkage and NMF algorithms. They can be efficiently implemented thanks to dedicated time-frequency analysis/synthesis packages. In this paper, we also addressed the modeling and decomposition of real signals in a rigorous way, which was missing from our preliminary contributions and appeared more tricky than initially expected. The MLJE objective function induced by the proposed generative modeling suggests more general problems of the form where $\ve{v}=\text{vec}[\W \H]$, $D(\cdot\|\cdot)$ is an arbitrary divergence between nonnegative numbers and $p$ is an arbitrary exponent. $D=D_{\text{IS}}$ and $p=2$ follow naturally from the GCM assumptions but other choices could be more suitable for other families of signals or images. Such problems do not seem to have been addressed yet in the literature and offer stimulating optimization problems. The exact reconstruction case $\lambda = 0$ is also very interesting in itself. Another challenging line of research is the design of workable large-scale optimization algorithms for type-II maximum marginal likelihood estimation. As known from [@Wipf2004], such an estimator would be robust to the joint estimation of $\lambda$ and $\ve{v}$, something in which MJLE fails in practice. [The low-rank structure used in Eq. to model the variance of the synthesis coefficients could be changed for more complex structures, such as neural architectures. This has been considered to model STFT synthesis coefficients in audio applications [@Bando2018], using for example variational auto-encoders for training [@Kingma2004]. The framework presented in this paper can readily accommodate such variants, in particular in the multi-layer setting in which a layer can be assigned a pre-trained variance (for a specific class of signals such as speech) and another layer can be endowed with a free adaptive low-rank variance.]{} Acknowledgment {#acknowledgment .unnumbered} ============== We are grateful to the reviewers for their valuable comments. Cédric Févotte acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 681839 (project FACTORY). [10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{} P. Smaragdis, C. F[é]{}votte, G. Mysore, N. Mohammadiha, and M. Hoffman, “Static and dynamic source separation using nonnegative factorizations: [A]{} unified view,” IEEE Signal Processing Magazine, vol. 31, no. 3, pp. 66–75, May 2014. T. 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IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012. Z. Chen and D. P. W. Ellis, “Speech enhancement by sparse, low-rank, and dictionary spectrogram decomposition,” in Proc. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), 2013. C. Sun, Q. Zhu, and M. Wan, “A novel speech enhancement method based on constrained low-rank and sparse matrix decomposition,” Speech Communication, vol. 60, pp. 44–55, 2014. E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?” Journal of ACM, vol. 58, no. 1, pp. 1–37, 2009. L. Daudet and B. Torrésani, “Hybrid representations for audiophonic signal encoding,” Signal Processing, vol. 82, no. 11, pp. 1595 – 1617, 2002. J.-L. Starck, Y. Moudden, J. Bobin, M. Elad, and D. Donoho, “Morphological component analysis,” in Optics & Photonics, 2005. M. Kowalski, “Sparse regression using mixed norms,” Applied and Computational Harmonic Analysis, vol. 27, no. 3, pp. 303–324, 2009. J. C. Brown, “Calculation of a constant [Q]{} spectral transform,” The Journal of the Acoustical Society of America, vol. 89, no. 1, pp. 425–434, 1991. D. Fitzgerald, M. Cranitch, and M. T. Cychowski, “Towards an inverse constant [Q]{} transform,” in Proc. Audio Engineering Society Convention, 2006. S. Leglaive, R. Badeau, and G. Richard, “Separating time-frequency sources from time-domain convolutive mixtures using non-negative matrix factorization,” in Proc. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), 2017. P. D. O’Grady and S. T. Rickard, “Compressive sampling of non-negative signals,” in Proc. IEEE Workshop on Machine Learning for Signal Processing (MLSP), 2008. Y. Bando, M. Mimura, K. Itoyama, K. Yoshii, and T. Kawahara, “Statistical speech enhancement based on probabilistic integration of variational autoencoder and non-negative matrix factorization,” in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2018. D. Kingma and M. Welling, “Auto-encoding variational [B]{}ayes,” in Proc. International Conference on Learning Representations, 2004. [Cédric Févotte]{} is a CNRS senior researcher at Institut de Recherche en Informatique de Toulouse (IRIT). Previously, he has been a CNRS researcher at Laboratoire Lagrange (Nice, 2013-2016) & Télécom ParisTech (2007-2013), a research engineer at Mist-Technologies (the startup that became Audionamix, 2006-2007) and a postdoc at University of Cambridge (2003-2006). He holds MEng and PhD degrees in EECS from École Centrale de Nantes. His research interests concern statistical signal processing and machine learning, for inverse problems and source separation. He was a member of the IEEE Machine Learning for Signal Processing technical committee (2012-2018) and is a member of SPARS steering committee since 2018. He has been a member of the editorial board of the IEEE Transactions on Signal Processing since 2014, first as an associate editor and then as a senior area editor (from 2018). In 2014, he was the co-recipient of an IEEE Signal Processing Society Best Paper Award for his work on audio source separation using multichannel nonnegative matrix factorization. He is the principal investigator of the European Research Council project FACTORY (New paradigms for latent factor estimation, 2016-2021). [Matthieu Kowalski]{} received the engineering degree in computer science from the Universit[é]{} de Technologie de Compi[è]{}gne in 2005, and the master degree in Mathematics Vision and Learning from the Ecole Normale Supérieur, Cachan, the same year. He received the PhD degree in applied mathematics from the University of Provence in 2008. His thesis was axed on sparse time-frequency decompositions. He is now an associate professor at the University of Paris-Sud, in the L2S Lab, and his research focuses on Inverse Problems and structured sparse approximations. He is an elected member of the SPARS Steering committee since 2013. [^1]: C. Févotte is with IRIT, Université de Toulouse, CNRS, France. [^2]: M. Kowalski is with L2S, Université Paris-Sud, Gif-sur-Yvette, France. [^3]: A random variable $x$ has distribution $N_{c}(x\|\mu,\lambda) = (\pi \lambda)^{-1} \exp - (\|x-\mu\|^{2}/\lambda)$ if and only if its real and imaginary parts are independent and with distribution $N( \Re[\mu],\lambda/2)$ and $N(\Im[\mu],\lambda/2)$, respectively. [^4]: The specific commands being of the like $\bualpha = \text{\tt dgtreal}(\ve{x},\text{`parameters'})$ and $\ve{x} = \text{\tt idgtreal}(\bualpha,\text{`parameters'})$, where $\texttt{dgt}$ stands for discrete Gabor transform. [^5]: [<https://www.irit.fr/~Cedric.Fevotte/extras/tsp2018/>]{}. Future references to online material refer to this same url.
--- abstract: 'We consider a class of second order linear nonautonomous parabolic equations in ${{\mathbb R}}^d$ with time periodic unbounded coefficients. We give sufficient conditions for the evolution operator $G(t,s)$ be compact in $C_b({{\mathbb R}}^d)$ for $t>s$, and describe the asymptotic behavior of $G(t,s)f$ as $t-s \to \infty$ in terms of a family of measures $\mu_s$, $s\in {{\mathbb R}}$, solution of the associated Fokker-Planck equation.' address: \| Dipartimento di Matematica\ Viale G.P. Usberti, 53/A\ I-43100 Parma\ Italia author: - Alessandra Lunardi title: 'Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in ${{\mathbb R}}^d$' --- Introduction ============ Linear nonautonomous parabolic equations in ${{\mathbb R}}^d$ are a classical subject in the mathematical literature. Most papers and books about regular solutions are devoted to the case of bounded coefficients (e.g., [@LSU; @friedman], but the list is very long), and recently the interest towards unbounded coefficients grew up. The standard motivations to the study of unbounded coefficients are on one side the well known connections with stochastic ODEs with unbounded nonlinearities, and on the other side the changes of variables that transform bounded into unbounded coefficients, occurring in different mathematical models. However, only for a few equations with unbounded coefficients it is possible to recover the familiar results about the bounded coefficients case. Many of them exhibit very different, and at first glance surprising, aspects. Therefore, a third motivation is the interest in new phenomena in PDEs. This paper deals with one of these new phenomena, giving sufficient conditions in order that the evolution operator $G(t,s)$ associated to a class of second order parabolic equations is a compact contraction in $C_b({{\mathbb R}}^d)$ for $t>s$. Precisely, Cauchy problems such as $$u_t(t,x) = {\mathcal{A}}(t)u(t, \cdot)(x), \quad t>s, \;x\in {{\mathbb R}}^d, \label{equazione}$$ $$u (s,x) =\varphi(x), \quad x\in {{\mathbb R}}^d \label{Cauchy}$$ will be considered, where the elliptic operators ${\mathcal{A}}(t)$ are defined by $$\begin{aligned} \label{A(t)} ({\mathcal{A}}(t)\varphi) (x) &: =\sum_{i,j=1}^d q_{ij}(t,x)D_{ij}\varphi (x) + \sum_{i=1}^d b_i(t,x) D_i\varphi (x)\notag\\ &: = {\mathrm{Tr}}\left( Q(t,x) D^2 \varphi (x)\right) + \langle b(t,x), \nabla \varphi (x) \rangle ,\end{aligned}$$ and the (smooth enough) coefficients $q_{ij}$, $b_i$ are allowed to be unbounded. If $\varphi$ is smooth and it has compact support, a classical bounded solution to – is readily constructed, as the limit as $R\to \infty$ of the solutions $u_R$ of Cauchy-Dirichlet problems in the balls $B(0,R)$. However, classical bounded solutions need not be unique. Under assumptions that guarantee positivity preserving in – (and hence, uniqueness of its bounded classical solution), a basic study of the evolution operator $G(t,s)$ for in $C_b({{\mathbb R}}^d)$ is in the paper [@KLL]. The evolution operator turns out to be markovian, since it has the representation $$G(t,s)\varphi(x) = \int_{{{\mathbb R}}^d} \varphi(y) p_{t,s,x}(dy), \quad t>s, \;x\in {{\mathbb R}}^d, \;\varphi\in C_b({{\mathbb R}}^d),$$ where the probability measures $p_{t,s,x}$ are given by $p_{t,s,x}(dy) = g(t,s,x,y)dy$ for a positive function $g$. It is easy to see that if a markovian $G(t,s)$ is compact in $C_b({{\mathbb R}}^d)$, then it does not preserve $C_0({{\mathbb R}}^d)$, the space of the continuos functions vanishing as $\|x\|\to \infty$, and it cannot be extended to a bounded operator in $L^p({{\mathbb R}}^d, dx)$ for $1\leq p<\infty$. Therefore, much of the theory developed for bounded coefficients fails. When a parabolic problem is not well posed in $L^p$ spaces with respect to the Lebesgue measure, it is natural to look for other measures $\mu$, and in particular to weighted Lebesgue measures, such that $G(t,s)$ acts in $L^p({{\mathbb R}}^d, \mu)$. This is well understood in the autonomous case ${\mathcal{A}}(t)\equiv {\mathcal{A}}$, where the dynamics is held by a semigroup $T(t)$ and $G(t,s) = T(t-s)$. Then, an important role is played by [invariant measures]{}, that are Borel probability measures $\mu$ such that $$\int_{{{\mathbb R}}^d} T(t)\varphi\,d\mu = \int_{{{\mathbb R}}^d}\varphi\,d\mu, \quad \varphi \in C_b({{\mathbb R}}^d).$$ If a Markov semigroup has an invariant measure $\mu$, it can be extended in a standard way to a contraction semigroup in all the spaces $L^p({{\mathbb R}}^d, \mu)$, $1\leq p<\infty$. Under broad assumptions the invariant measure is unique, and it is strongly related with the asymptotic behavior of $T(t)$, since $\lim_{t\to \infty} T(t)\varphi = \int_{{{\mathbb R}}^d}\varphi \,d\mu$, locally uniformly if $\varphi\in C_b({{\mathbb R}}^d)$ and in $L^p({{\mathbb R}}^d, \mu)$ if $\varphi \in L^p({{\mathbb R}}^d, \mu)$, $1<p<\infty$. In the nonautonomous case the role of the invariant measure is played by families of measures $\{ \mu_s:\; s\in {{\mathbb R}}\}$, called [evolution systems of measures]{}, that satisfy $$\int_{{{\mathbb R}}^d} G(t,s)\varphi\,d\mu_t = \int_{{{\mathbb R}}^d}\varphi\,d\mu_s, \quad t>s, \; \varphi \in C_b({{\mathbb R}}^d). \label{invariance}$$ If is satisfied, the function $s\mapsto \mu_s$ satisfies (at least, formally) the Fokker-Planck equation $$D_s \mu_s + {\mathcal{A}}(s)^\mu_s = 0, \quad s\in {{\mathbb R}},$$ which is a parabolic equation for measures without any initial, or final, condition. Therefore it is natural to have infinitely many solutions, and to look for uniqueness of special solutions. For instance, in the autonomous case the unique stationary solution is the invariant measure, in the periodic case ${\mathcal{A}}(t) = {\mathcal{A}}(t+T)$ under reasonable assumptions there is a unique $T$-periodic solution, etc. Arguing as in the autonomous case, it is easy to see that if holds then $G(t,s)$ may be extended to a contraction from $L^p({{\mathbb R}}^d, \mu_s)$ to $L^p({{\mathbb R}}^d, \mu_t)$ for $t>s$. Therefore, it is natural to investigate asymptotic behavior of $G(t,s)$ not only in $C_b({{\mathbb R}}^d)$ but also in these $L^p$ spaces. A basic study of the evolution operator for parabolic equations with (smooth enough) unbounded coefficients is in [@KLL]. In its sequel [@LLZ] we studied asymptotic behavior of $G(t,s)$ in the case of time-periodic coefficients. In this paper sufficient conditions will be given for the evolution operator $G(t,s)$ be compact in $C_b({{\mathbb R}}^d)$. Then, compactness will be used to obtain asymptotic behavior results in the case of time-periodic coefficients. Indeed, compactness implies that there exists a unique $T$-periodic evolution system of measures $\{ \mu_s:\; s\in {{\mathbb R}}\}$, and that denoting by $m_s\varphi $ the mean value $$\label{m_s} m_s\varphi := \int_{{{\mathbb R}}^d} \varphi(x) \mu_s(dx), \quad s\in {{\mathbb R}},$$ there is $\omega <0$ such that for each ${\varepsilon}>0$ we have $$\label{CompAsInfty} \\| G(t,s)\varphi - m_s\varphi\\|_{\infty} \leq M_{{\varepsilon}} e^{(\omega +{\varepsilon}) (t-s)} \\|\varphi\\|_{\infty} , \quad t>s, \; \varphi \in C_b({{\mathbb R}}^d),$$ for some $ M_{{\varepsilon}} >0$. As a consequence, for every $p\in (1, \infty)$ and ${\varepsilon}>0$ we get $$\label{CompAsExp} \\| G(t,s)\varphi - m_s\varphi\\|_{L^p({{\mathbb R}}^d, \mu_t)} \leq Me^{(\omega +{\varepsilon}) (t-s)} \\|\varphi\\|_{L^p({{\mathbb R}}^d, \mu_s)} , \quad t>s, \; \varphi \in L^p({{\mathbb R}}^d, \mu_s),$$ for some $M= M(p, {\varepsilon})>0$. Note that while the constant $M$ may depend on $p$, the exponential rate of decay is independent of $p$. These results complement the asymptotic behavior results of [@LLZ], where was obtained under different assumptions. Preliminaries: the evolution operator $G(t,s)$. =============================================== We use standard notations. $C_b({{\mathbb R}}^d)$ is the space of the bounded continuous functions from ${{\mathbb R}}^d$ to ${{\mathbb R}}$, endowed with the sup norm. $C_0({{\mathbb R}}^d)$ is the space of the continuous functions that vanish at infinity. For $k\in {{\mathbb N}}\cup \{0\}$, $C^k_c({{\mathbb R}}^d)$ is the space of $k$ times differentiable functions with compact support. For $0<\alpha <1$, $a<b$ and $R>0$, $C^{\alpha/2, \alpha}([a,b]\times B(0,R))$ and $C^{1+\alpha/2, 2+ \alpha}([a,b]\times B(0,R))$ are the usual parabolic Hölder spaces in the set $[a,b]\times B(0,R)$; $C^{\alpha/2, \alpha}_{loc}({{\mathbb R}}^{1+d})$ and $C^{1+\alpha/2, 2+ \alpha}_{loc}({{\mathbb R}}^{1+d})$ are the subspaces of $C_b({{\mathbb R}}^{1+d})$ consisting of functions whose restrictions to $[a,b]\times B(0,R)$ belong to $C^{\alpha/2, \alpha}([a,b]\times B(0,R))$ and to $C^{1+\alpha/2, 2+ \alpha}([a,b]\times B(0,R))$, respectively, for every $a<b$ and $R>0$. In this section we recall some results from [@KLL] about the evolution operator for parabolic equations with unbounded coefficients. They were proved under standard regularity and ellipticity assumptions, and nonstandard qualitative assumptions. \[hyp1\] - The coefficients $q_{ij}$, $b_i$ $(i,j=1,\ldots,d)$ belong to $C^{\alpha/2, \alpha}_{loc}({{\mathbb R}}^{1+d})$ for some $\alpha\in (0, 1)$. - For every $(s,x)\in {{\mathbb R}}^{1+d}$, the matrix $Q(s,x)$ is symmetric and there exists a function $\eta: {{\mathbb R}}^{1+d}\to {{\mathbb R}}$ such that $0<\eta_0:=\inf_{{{\mathbb R}}^{1+d}}\eta$ and $$\begin{aligned} \langle Q(s,x)\xi , \xi \rangle \geq \eta(s,x) \|\xi \|^2, \qquad\;\, \xi \in {{\mathbb R}}^d,\;\,(s,x) \in {{\mathbb R}}^{1+d}.\end{aligned}$$ - There exist a positive function $W \in C^2({{\mathbb R}}^d)$ and a number $\lambda \in {{\mathbb R}}$ such that $$\begin{aligned} \qquad\;\, \lim_{\|x\|\to \infty} W (x) = \infty \quad \mbox{ and } \quad \sup_{s\in {{\mathbb R}}, \,x\in {{\mathbb R}}^d}({\mathcal{A}}(s) W)( x) -\lambda W(x)< 0.\end{aligned}$$ Assumptions (i) and (ii) imply that for every $s\in {{\mathbb R}}$ and $\varphi\in C_b({{\mathbb R}}^d)$, the Cauchy problem $$\label{CP} \left\{ \begin{array}{rcll} \displaystyle D_tu(t,x) & = & {\mathcal{A}}(t)u(t,x), & t>s,\; x \in {{\mathbb R}}^{d},\\[2mm] u(s,x) & = & \varphi(x), & x\in {{\mathbb R}}^d, \end{array}\right.$$ has a bounded classical solution. Assumption (iii) implies that the bounded classical solution to is unique (in fact, a maximum principle that yields uniqueness is proved in [@KLL] under a slightly weaker assumption). The evolution operator $G(t,s)$ is defined by $$\label{(t,s)} G(t,s)\varphi = u(t, \cdot), \quad t\geq s \in {{\mathbb R}},$$ where $u $ is the unique bounded solution to . Some of the properties of $G(t,s)$ are in next theorem. \[Th:G(t,s)\] Let Hypothesis \[hyp1\] hold. Define $ \Lambda := \{(t,s,x)\in {{\mathbb R}}^{2+d}: t > s, \;x\in {{\mathbb R}}^{d}\} $. Then: - for every $\varphi \in C_b({{\mathbb R}}^d)$, the function $(t,s,x)\mapsto G(t,s)\varphi(x)$ is continuous in $\overline{\Lambda }$. For each $s\in {{\mathbb R}}$, $(t,x)\mapsto G(t,s)\varphi(x)$ belongs to $C^{1+\alpha/2, 2+\alpha}_{loc}((s, \infty) \times {{\mathbb R}}^d )$; - for every $\varphi\in C^{2}_{c}( {{\mathbb R}}^d )$, the function $(t,s,x)\mapsto G(t,s)\varphi(x)$ is continuously differentiable with respect to $s$ in $\overline{\Lambda }$ and $D_sG(t,s)\varphi(x) = -G(t,s){\mathcal{A}}(s)\varphi(x)$ for any $(t,s,x)\in\overline\Lambda$; - for each $(t,s,x)\in \Lambda$ there exists a Borel probability measure $p_{t,s,x}$ in ${{\mathbb R}}^d$ such that $$\label{representation} G(t,s)\varphi(x) = \int_{{{\mathbb R}}^d}\varphi (y) p_{t,s,x}(dy), \quad \varphi \in C_b({{\mathbb R}}^d).$$ Moreover, $ p_{t,s,x}(dy) = g(t,s,x,y)dy$ for a positive function $g$. - $G(t,s)$ is strong Feller; extending it to $L^{\infty}({{\mathbb R}}^d, dx)$ through formula , it maps $L^{\infty}({{\mathbb R}}^d, dx)$ (and, in particular, $B_b({{\mathbb R}}^d)$) into $C_b({{\mathbb R}}^d)$ for $t>s$, and $$\begin{aligned} \\|G(t,s)\varphi \\|_{\infty} \leq \\|\varphi \\|_{\infty}, \quad \varphi \in L^{\infty}({{\mathbb R}}^d, dx), \;\,t>s. \end{aligned}$$ - If $(\varphi_n)$ is a bounded sequence in $C_b({{\mathbb R}}^d)$ that converges uniformly to $\varphi $ in each compact set $K\subset{{\mathbb R}}^d$, then for each $s\in {{\mathbb R}}$ and $T>0$, $G(\cdot,s)\varphi _n$ converges to $G(\cdot , s )\varphi $ uniformly in $[s,s+T] \times K$, for each compact set $K \subset {{\mathbb R}}^d$. - For every $s\in {{\mathbb R}}$ and $R>0$, $0<{\varepsilon}<T$ there is $C= C(s,{\varepsilon}, T, R)>0$ such that $$\sup_{ s+{\varepsilon}\leq t\leq s+T}\\|G(t,s)\varphi \\|_{C^2(B(0,R))} \leq C\\|\varphi\\|_{\infty} , \quad \varphi \in C_b({{\mathbb R}}^d).$$ Statements (i) to (v) are explicitly mentioned in [@KLL] (Thm. 2.1, Prop. 2.4, Cor. 2.5, Prop. 3.1, Lemma 3.2). Statement (vi) is hidden in the proof of Theorem 2.1, where $G(t,s)\varphi$ is obtained by an approximation procedure, in three steps: first, for $\varphi\in C^{2+\alpha}_{c}({{\mathbb R}}^d)$, then for $\varphi\in C_{0}({{\mathbb R}}^d)$, and then for $\varphi \in C_b({{\mathbb R}}^d)$. At each step, we have interior Schauder estimates for a sequence $u_n$ that approaches $G(t,s)\varphi$, namely for $s\in {{\mathbb R}}$ and $R>0$, $0<{\varepsilon}<T$ there is $C= C(s,{\varepsilon}, T, R)>0$ such that $$\\|u_n\\|_{C^{1+\alpha/2 , 2+\alpha}([s+{\varepsilon}, s+T]\times B(0,R))} \leq C\\|\varphi\\|_{\infty}, \quad n\in {{\mathbb N}},$$ and $u_n$ converges to $G(t,s)\varphi$ locally uniformly. This yields (vi). To get evolution system of measures we have to strenghten assumption \[hyp1\](iii). The following theorem is proved in [@KLL]. \[evsyst\] Under Hypotheses \[hyp1\], assume in addition that there exist a positive function $W \in C^2({{\mathbb R}}^d)$ and numbers $a$, $c>0$ such that $$\label{V} \qquad\;\, \lim_{\|x\|\to \infty} W (x) = +\infty \quad \mbox{ and } \quad ({\mathcal{A}}(s) W)( x) \leq a - c W(x), \quad (s,x)\in {{\mathbb R}}^{1+d}.$$ Then there exists a tight$^($[^1]$^)$ evolution system of measures $\{ \mu_s: s\in {{\mathbb R}}\}$ for $G(t,s)$. Moreover, $$\label{G(t,s)V} G(t,s)W(x) := \int_{{{\mathbb R}}^d}W(y) p_{t,s,x}(dy) \leq W(x) + \frac{a}{c}, \quad t>s, \;x\in {{\mathbb R}}^d,$$ and $$\label{m_tV} \int_{{{\mathbb R}}^d}W(y)\mu_t(dy) \leq \min W + \frac{a}{c}, \quad t\in {{\mathbb R}}.$$ Compactness in $C_b({{\mathbb R}}^d)$ ===================================== A necessary and sufficient condition for $G(t,s)$ be compact in $C_b({{\mathbb R}}^d)$ for $t>s$ is very similar to the corresponding condition in the autonomous case ([@MPW]). \[tight-compattezza\] Under Hypothesis \[hyp1\] the following statements are equivalent: - for any $t>s$, $G(t,s): C_b({{\mathbb R}}^d)\to C_b({{\mathbb R}}^d)$ is compact. - for any $t>s$ the family of measures $\{p_{t,s,x}(dy):\; x\in {{\mathbb R}}^d\}$ is tight, i.e., for every ${\varepsilon}>0$ there exists $R=R(t,s,{\varepsilon})>0$ such that $$\begin{aligned} p_{t,s,x}(B(0,R)) \geq 1-{\varepsilon}, \quad x\in {{\mathbb R}}^d.\end{aligned}$$ We follow the proof given in [@MPW Prop. 3.6] for the autonomous case. Let statement (a) hold. For every $R >0$ let $\varphi_R:{{\mathbb R}}^d\to{{\mathbb R}}$ be a continuous function such that ${1\!\!\!\;\mathrm{l}}_{B(0,R)} \leq \varphi_R\leq {1\!\!\!\;\mathrm{l}}_{B(0,R+1)}$. Since $\\|\varphi_R\\|_{\infty} \leq 1$ and $G(t,s)$ is compact, there is a sequence $G(t,s) \varphi_{R_n}$ that converges uniformly in the whole ${{\mathbb R}}^d$ to a limit function $g$. Since $\varphi_R$ goes to ${1\!\!\!\;\mathrm{l}}$ as $R\to +\infty$, uniformly on each compact set and $\\|\varphi_R\\|_{\infty} \leq 1$ for every $R$, by Theorem \[Th:G(t,s)\](v) $\lim_{R\to \infty} G(t,s) \varphi_R = G(t,s){1\!\!\!\;\mathrm{l}}= {1\!\!\!\;\mathrm{l}}$ uniformly on each compact set. Then, $g\equiv 1$ and $\lim_{R\to +\infty} \\|G(t,s) \varphi_R -{1\!\!\!\;\mathrm{l}}\\|_{\infty} =0$. Therefore, fixed any ${\varepsilon}>0$, we have $$\begin{aligned} p_{t,s,x}(B(0,R) ) = (G(t,s){1\!\!\!\;\mathrm{l}}_{B(0,R)})(x) \geq (G(t,s)\varphi_{R})(x) \geq 1-{\varepsilon}, \quad x\in{{\mathbb R}}^d,\end{aligned}$$ for $R$ large enough. Let now statement (b) hold. For $t>s$ fix $r\in (s,t)$ and recall that (see formula ) $$\begin{aligned} (G(t,s)\varphi)(x) = (G(t,r)G(r,s)\varphi)(x)=\int_{{{\mathbb R}}^d} (G(r,s)\varphi) (y) p_{t,r,x}(dy), \end{aligned}$$ for any $\varphi\in C_b({{\mathbb R}}^d)$ and any $x\in {{\mathbb R}}^d$. For every $R>0$ set $$\begin{aligned} (G_{R}\varphi)(x) = \int_{B(0,R)} (G(r,s)\varphi) (y) p_{t,r,x}(dy), \quad x\in {{\mathbb R}}^d.\end{aligned}$$ Each $G_R: C_b({{\mathbb R}}^d)\to C_b({{\mathbb R}}^d)$ is a compact operator, since it may be written as $G_{R} = {\mathcal S}\circ {\mathcal R} \circ G(r,s)$ where $G(r,s): C_b({{\mathbb R}}^d)\to C_b({{\mathbb R}}^d)$ is continuous, ${\mathcal R}: C_b({{\mathbb R}}^d)\to C( B(0,R))$ is the restriction operator, and $ {\mathcal S}: C(B(0,R)) \to C_b({{\mathbb R}}^d)$ is defined by $$\begin{aligned} {\mathcal S} \psi(x) = \int_{B(0,R)} \psi(y) p_{t,r,x}(dy) = (G(t,r)\widetilde{\psi})(x),\qquad\;\,x\in B(0,R),\end{aligned}$$ where $\widetilde{\psi}(x)$ is the null extension of $\psi$ to the whole ${{\mathbb R}}^d$. Now, ${\mathcal R}\circ G(s,r): C_b({{\mathbb R}}^d)\to C(B(0,R))$ is compact by Theorem \[Th:G(t,s)\](v), and $ {\mathcal S}$ is continuous from $C(B(0,R))$ to $C_b({{\mathbb R}}^d)$ because $G(t,r)$ is strong Feller by Theorem \[Th:G(t,s)\](iv). Moreover, $G_R\to G(t,s)$ in ${\mathcal L}(C_b({{\mathbb R}}^d))$, as $R\to+\infty$. Indeed, for ${\varepsilon}>0$ there is $R_0>0$ such that $ p_{t,r,x}(B(0,R))\geq 1-{\varepsilon}$ for each $x\in {{\mathbb R}}^d$ and $R\geq R_0$, and consequently $$\begin{aligned} \|(G(t,s)\varphi)(x) - (G_R\varphi)(x) \| \leq \\|G(r,s)\varphi\\|_{\infty}\int_{{{\mathbb R}}^d\setminus B(0,R)}p_{t,r,x}(dy) \leq {\varepsilon}\\|\varphi\\|_{\infty},\end{aligned}$$ for $R\geq R_0$ and for each $x\in {{\mathbb R}}^d$. Being limit of compact operators, $G(t,s)$ is compact. \[rem:comp\] Some remarks are in order. - An insight in the proof shows that if $G(t,s)$ is compact for some $t>s$, then the family $\{p_{t,s,x}(dy):\; x\in {{\mathbb R}}^d\}$ is tight; conversely if for some $r>s$ the family $\{p_{r,s,x}(dy):\; x\in {{\mathbb R}}^d\}$ is tight then $G(t,s)$ is compact for each $t>r$. - If for some $r>s$ the family $\{p_{r,s,x}(dy):\; x\in {{\mathbb R}}^d\}$ is tight, then the family $\{p_{t,s,x}(dy):\; t\geq r, \;x\in {{\mathbb R}}^d\}$ is tight. Indeed, for every $R>0$ we have $$\begin{array}{lll} p_{t,s,x}({{\mathbb R}}^d\setminus B(0,R)) &= &(G(t,s){1\!\!\!\;\mathrm{l}}_{{{\mathbb R}}^d\setminus B(0,R)})(x) \\ \\ &=&(G(t,r)G(r,s){1\!\!\!\;\mathrm{l}}_{{{\mathbb R}}^d\setminus B(0,R)})(x) \\ \\ &\leq &\\|G(r,s){1\!\!\!\;\mathrm{l}}_{{{\mathbb R}}^d\setminus B(0,R)}\\|_{\infty}, \end{array}$$ so that, if $p_{r,s,x}({{\mathbb R}}^d\setminus B(0,R)) = G(r,s){1\!\!\!\;\mathrm{l}}_{{{\mathbb R}}^d\setminus B(0,R)}(x)\leq {\varepsilon}$ for every $x$, also $p_{t,s,x}({{\mathbb R}}^d\setminus B(0,R))\leq {\varepsilon}$ for every $x$. - As in the autonomous case ([@MPW]), if $G(t,s)$ is compact in $C_b({{\mathbb R}}^d)$, it does not preserve $L^p({{\mathbb R}}^d,dx)$ for any $p\in [1, +\infty)$ and it does not preserve $C_0({{\mathbb R}}^d)$. Indeed, let $R>0$ be so large that $p_{t,s,x}(B(0,R)) \geq 1/2$ for every $x\in {{\mathbb R}}^d$, and let $\varphi \in C_c({{\mathbb R}}^d)$ be such that $\varphi \geq {1\!\!\!\;\mathrm{l}}_{B(0,R)}$. Then, $$\begin{aligned} (G(t,s)\varphi)(x) \geq (G(t,s){1\!\!\!\;\mathrm{l}}_{B(0,R)})(x) = p_{t,s,x}(B(0,R))\geq \frac{1}{2},\end{aligned}$$ for every $x$, so that $G(t,s)\varphi $ does not belong to any space $L^p({{\mathbb R}}^d,dx)$ and to $C_0({{\mathbb R}}^d)$. - A similar argument shows that $\inf G(t,s)\varphi >0$ for each $t>s$ and $\varphi \in C_b({{\mathbb R}}^d)\setminus \{0\}$, $\varphi \geq 0$. Indeed, if $\varphi(x) >0$ for each $x$, and $R>0$ is as before, then $(G(t,s)\varphi)(x) \geq \delta (G(t,s){1\!\!\!\;\mathrm{l}}_{B(0,R)})(x) \geq \delta/2$, with $\delta = \min_{\|x\|\leq R}\varphi(x) >0$. If $\varphi(x) \geq 0$ for each $x$, it is sufficient to recall that $G(t,s)\varphi = G(t,(s+t)/2) G((s+t)/2, s)\varphi$ and that $G((s+t)/2, s)\varphi (x)>0$ for each $x$ by Theorem \[Th:G(t,s)\] (iii). However, to check the tightness condition of Proposition \[tight-compattezza\] is not obvious, since the measures $p_{t,s,x}$ are not explicit, in general. In the case of time depending Ornstein-Uhlenbeck operators (e.g. [@DPL]), $$({\mathcal{A}}(t)\varphi)(x) = \frac12 {\mathrm{Tr}}\left(Q(t)D_x^2\varphi (x)\right)+ \langle A(t)x+f(t),\nabla \varphi(x)\rangle ,\quad x\in{{\mathbb R}}^d,$$ the measures $p_{t,s,x}$ are explicit Gaussian measures and it is possible to see that the tightness condition does not hold. Alternatively, one can check that $G(t,s)$ maps $L^p({{\mathbb R}}^d, dx)$ into itself for every $p\in (1, \infty)$ and therefore it cannot be compact in $C_b({{\mathbb R}}^d)$. If the assumptions of Theorem \[evsyst\] hold, estimate implies that the family $\{ p_{t,s,x}: t>s, \;x\in B(0,r)\}$ is tight for every $r>0$. However, this is not enough for compactness. To obtain compactness we have to strenghten condition on the auxiliary function $W$. \[Giorgio\] Let Hypotheses \[hyp1\] hold. Assume in addition that there exist a $C^2$ function $W:{{\mathbb R}}^d\mapsto {{\mathbb R}}$, such that $\lim_{\|x\|\to \infty} W(x) = +\infty$, a number $R>0$ and a convex increasing function $g: [0, +\infty)\to {{\mathbb R}}$ such that $1/g$ is in $L^1(a, +\infty)$ for large $a $, and $$({\mathcal{A}}(s) W)(x) \leq -g(W(x)), \qquad\;\,s\in {{\mathbb R}},\;\, \|x\| \geq R. \label{stima-conv}$$ Then, for every $\delta >0$ there is $C=C(\delta) >0$ such that $(G(t,s)W)(x) \leq C$ for every $x\in {{\mathbb R}}^d$ and $s\leq t-\delta $. Consequently, the family of probabilities $\{ p_{t,s,x}(dy): s\leq t-\delta, \,x\in {{\mathbb R}}^d\}$ is tight, and $G(t,s)$ is compact in $C_b({{\mathbb R}}^d)$ for $t>s$. As a first step we show that $$\label{maggV} (G(t,s)W)(x) - (G(t,r)W)(x) \geq - \int_{r}^{s} (G(t, \sigma){\mathcal{A}}(\sigma) W)(x)d\sigma, \quad r<s<t, \;x\in {{\mathbb R}}^d.$$ Let $\varphi \in C^{\infty}({{\mathbb R}})$ be a nonincreasing function such that $\varphi\equiv 1$ in $(-\infty , 0]$, $\varphi \equiv 0$ in $[1, + \infty)$, and define $\psi_n (t) = \int_{0}^{t} \varphi (s-n)ds$ for each $n\in {{\mathbb N}}$. The functions $\psi_n$ are smooth and enjoy the following properties: - $\psi_n(t) = t$ for $t \in [0, n]$, - $\psi_n(t) \equiv \mathrm{const.}$ for $ t \geq n+1$, - $0 \leq \psi_n' \leq 1$ and $\psi_n'' \leq 0$, - for every $t\geq 0$, the sequence $(\psi_n'(t))$ is increasing. Then, the function $W_n := \psi_n\circ W$ belongs to $C^2_b({{\mathbb R}}^d)$ and it is constant outside a compact set. By Theorem \[Th:G(t,s)\](ii), applied to $W_n - c$, we have $$\begin{aligned} (G(t,s)W_n)(x) -&(G(t,r)W_n)(x) = - \int_r^s (G(t,\sigma){\mathcal{A}}(\sigma )W_n)(x)\, d\sigma\notag\\ &= - \int_r^s G(t,\sigma) \left \{\psi_n'(W) ({\mathcal{A}}(\sigma) W) + \psi_n''(W) \langle Q \nabla W , \nabla W \rangle\right\} (x) d\sigma\notag\\ &\ge - \int_r^s G(t,\sigma)( \psi_n'(W) {\mathcal{A}}(\sigma )W)(x)d\sigma \notag\\ & = - \int_r^sd\sigma\int_{E_{\sigma}}\psi_n'(W(y))({\mathcal{A}}(\sigma )W)(y)p_{t,\sigma,x}(dy) \\ &- \int_r^sd\sigma\int_{{{\mathbb R}}^d\setminus E_{\sigma}}\psi_n'(W(y))({\mathcal{A}}(\sigma )W)(y)p_{t,\sigma,x}(dy),\end{aligned}$$ where $E_{\sigma}=\{x\in{{\mathbb R}}^d: ({\mathcal{A}}(\sigma )W)(x)>0\}$. Letting $n\to +\infty$, the left-hand side goes to $(G(t,s)W)(x) -(G(t,r)W)(x)$. Concerning the right-hand side, both integrals converge by monotone convergence. We have to prove that their limits are finite. The first term converges to $-\int_r^sd\sigma\int_{E_{\sigma}}({\mathcal{A}}(\sigma )W)(y)p_{t,\sigma,x}(dy)$, which is finite since the sets $E_{\sigma}$ are equibounded in ${{\mathbb R}}^d$ (recall that the function ${\mathcal A}(\sigma)W$ tends to $-\infty$ as $\|x\|\to +\infty$, uniformly with respect to $\sigma\in [r,s]$). The second term may be estimated by $$\begin{aligned} &-\int_r^sd\sigma\int_{{{\mathbb R}}^d\setminus E_{\sigma}}\psi_n'(W(y))({\mathcal{A}}(\sigma )W)(y)p_{t,\sigma,x}(dy)\\ \le &\int_r^sd\sigma\int_{E_{\sigma}}\psi_n'(W(y))({\mathcal{A}}(\sigma )W)(y)p_{t,\sigma,x}(dy)+ (G(t,s)W_n)(x) -(G(t,r)W_n)(x).\end{aligned}$$ Letting $n\to +\infty$, we obtain that $\int_r^sd\sigma\int_{{{\mathbb R}}^d\setminus E_{\sigma}}({\mathcal{A}}(\sigma )W)(y)p_{t,\sigma,x}(dy)$ is finite. Summing up, the function $\sigma\mapsto (G(t,\sigma)({\mathcal{A}}(\sigma)W))(x)$ is in $L^1(r,s)$ and follows. Possibly replacing $g$ by $\widetilde{g} = g-C$ for a suitable constant $C$, we may assume that $({\mathcal{A}}(s)W)(x) \leq -g(W(x))$ for every $s\in {{\mathbb R}}$ and $x\in {{\mathbb R}}^d$. Fix $x\in {{\mathbb R}}^d$, $t\in {{\mathbb R}}$, and set $$\begin{aligned} \beta(s) := (G(t, t-s)W)(x), \quad s\geq 0.\end{aligned}$$ Then $\beta$ is measurable, since it is the limit of the sequence of continuous functions $s\mapsto G(t,t-s)W_n(x)$. Inequality implies $$\beta(b)-\beta(a) \leq -\int_{t-b}^{t-a} (G(t,\sigma)g(W))(x)d\sigma, \qquad\;\, a<b,$$ and, since $g$ is convex, $$\begin{array}{l} (G(t, \sigma)g(W))(x) = \displaystyle{ \int_{{{\mathbb R}}^d} g(W(y))p_{t,\sigma, x}(dy) } \\ \\ \geq g\bigg( \displaystyle{ \int_{{{\mathbb R}}^d} W(y)p_{t,\sigma, x}(dy) } \bigg) = g((G(t, \sigma)W)(x)) \end{array}$$ so that $$\label{eq:beta} \begin{array}{lll} \beta(b)-\beta(a) & \le & \displaystyle{-\int_{t-b}^{t-a}g((G(t,\sigma)W)(x))d\sigma } \\ \\ & = & \displaystyle{-\int_{t-b}^{t-a}g(\beta(t-\sigma))d\sigma } \\ \\ & = & \displaystyle{ -\int_{a}^{b} g(\beta(\sigma))d\sigma, } \end{array}$$ for any $a<b$. Then, for every $s\geq 0$, $\beta(s)\leq \zeta(s)$, where $\zeta$ is the solution of the Cauchy problem $$\begin{aligned} \left\{ \begin{array}{l} \zeta'(s) = -g(\zeta (s)), \quad s\geq 0, \\[3mm] \zeta(0) = W(x). \end{array}\right.\end{aligned}$$ Indeed, assume by contradiction that there exists $s_0>0$ such that $\beta(s_0)>\zeta(s_0)$, and denote by $I$ the largest interval containing $s_0$ such that $\beta(s )>\zeta(s )$ for each $s\in I$. Inequality implies that $ \beta(b)-\beta(a) \leq -m (b-a)$ for $b>a$, with $m :=\min g$. In other words, the function $s\mapsto \beta(s) + ms $ is decreasing. This implies that $I$ contains some left neighborhood of $s_0$. Indeed, since $s\mapsto \beta(s)+ms $ is decreasing, then $$\lim_{s\to s_0^-} \beta(s)+ms \geq \beta(s_0) + ms_0 > \zeta(s_0) + ms_0 = \lim_{s\to s_0^-} \zeta(s)+ms$$ so that $\lim_{s\to s_0^-} ( \beta(s)- \zeta(s)) >0$, which yields $\beta>\zeta$ in a left neighborhood of $s_0$. Let $a= \inf I $. Then $a< s_0$, and there is a sequence $(s_n)\uparrow a$ such that $\beta(s_n) \leq \zeta(s_n) $, so that $\beta(a) + ma \leq \lim_{n\to \infty}\beta(s_n) + ms_n \leq \zeta(a) + ma$, that is $\beta(a) \leq \zeta(a)$. On the other hand, for each $s\in I$ we have $$\begin{aligned} \beta(s) - \beta(a) \leq \int_{a}^{s} -g(\beta(\sigma))d\sigma, \quad \zeta(s) -\zeta(a) = \int_{a}^{s} -g(\zeta(\sigma))d\sigma ,\end{aligned}$$ so that $$\begin{aligned} \beta(s) - \zeta(s) \leq \int_{a}^{s} [-g(\beta(\sigma))+ g(\zeta (\sigma))] \,d\sigma , \quad s\in I.\end{aligned}$$ Since $\beta(\sigma) > \zeta(\sigma)$ for every $\sigma \in I$ and $g$ is increasing, the integral in the right-hand side is nonpositive, a contradiction. Therefore, $\beta(s)\leq \zeta(s)$ for every $s\geq 0$. By standard arguments about ODE’s, for every $\delta >0$ there is $C=C(\delta )$ independent on the initial datum $W(x)$ such that $\zeta(s)\leq C$ for every $s\geq \delta$. Therefore, $$\beta (s)= (G(t,t-s)W)(x)\leq C, \quad s\geq \delta ,$$ with $C$ independent of $t$. This implies that for every $\delta >0$ the family of probabilities $p_{t,s,x}(dy)$ with $s\leq t-\delta$ and $x\in {{\mathbb R}}^d$ is tight, because for every $R>0$ we have $$\begin{aligned} p_{t,s,x}({{\mathbb R}}^d \setminus B(0,R)) &= \int_{{{\mathbb R}}^d \setminus B(0,R)}p_{t,s,x}(dy)\\ &\le \frac{1}{\inf \{W(y):\;\|y\|\geq R\}} \int_{{{\mathbb R}}^d \setminus B(0,R)}W(y)p_{t,s,x}(dy)\\ &\le \frac{1}{\inf \{W(y):\;\|y\|\geq R\}} (G(t,s)W)(x) \\ &\le \frac{C}{\inf \{W(y):\;\|y\|\geq R\}}\end{aligned}$$ and $\inf \{W(y):\;\|y\|\geq R\} $ goes to $+\infty $ as $r\to +\infty$. So, condition (b) of Proposition \[tight-compattezza\] is satisfied. If there is $R>0$ such that $$\begin{aligned} {\rm Tr}\; Q(s,x) + \langle b(s,x),x\rangle - \frac{2}{\|x\|^2} \langle Q(s,x)x, x\rangle \leq -c \|x\|^2(\log \|x\|)^{\gamma}, \quad s\in {{\mathbb R}},\;\, \|x\|\geq R,\end{aligned}$$ with $c>0$ and $\gamma >1$, then the condition is satisfied by any $W$ such that $W(x) = \log \|x\|$ for $\|x\|\ge R$, with $g(s) = cs^{\gamma}$. If the regularity and ellipticity assumptions \[hyp1\](i)(ii) hold, Theorem \[Giorgio\] implies that the evolution operator $G(t,s)$ is compact in $C_b({{\mathbb R}}^d)$ for $t>s$. Compactness and asymptotic behavior =================================== In this section we derive asymptotic behavior results from compactness of $G(t,s)$ in $C_b({{\mathbb R}}^d)$. Throughout the section we assume that Hypothesis \[hyp1\] holds, and that the coefficients $q_{ij}$ and $b_i$, $i,j=1, \ldots, d$ are periodic in time, with period $T>0$. Then the asymptotic behavior of $G(t,s)$ is driven by the spectral properties of the operators $$V(s) : = G(s+T, s), \quad s\in {{\mathbb R}}.$$ This is well known in the case of evolution operators associated to families $A(t)$ of generators of analytic semigroups, see e.g. [@Henry sect. 7.2], [@L Ch. 6], [@F]. Most of the arguments are independent of analyticity and will be adapted to our situation. To begin with, since each $V(s)$ is a contraction in $C_b({{\mathbb R}}^d)$, its spectrum is contained in the unit circle. Its spectral radius is $1$, since $1$ is an eigenvalue. The nonzero eigenvalues of $V(s)$ are independent of $s$, since the equality $G(t,s)V(s) = V(t)G(t,s)$ implies that for each eigenfunction $\varphi $ of $V(s)$, $G(t,s)\varphi \neq 0$ is an eigenfunction of $V(t)$ with the same eigenvalue, for $t>s$. If $G(t,s)$ is compact in $C_b({{\mathbb R}}^d)$ for $t>s$, then $\sigma (V(s))\setminus \{0\}$ consists of isolated eigenvalues, hence it is independent of $s$. Therefore, $$\label{r} \sup\{\|\lambda\| : \;\lambda\in \sigma(V(s)), \; \|\lambda\|<1, \;s\in {{\mathbb R}}\} := r <1.$$ Denoting by $Q(s)$ the spectral projection $$Q(s) = \frac{1}{2\pi i} \int_{\partial B(0,a)} (\lambda I- V(s))^{-1}d\lambda , \quad s\in {{\mathbb R}},$$ with any $a\in (r, 1)$, it is not difficult to see that for every ${\varepsilon}>0$ there is $M_{{\varepsilon}}>0$ such that $$\label{decexp} \\|G(t,s)Q(s)\varphi \\|_{\infty} \leq M_{{\varepsilon}}e^{(t-s)(\log r(s) +{\varepsilon})/T}\\|\varphi\\|_{\infty}, \quad t>s, \;\varphi \in C_b({{\mathbb R}}^d).$$ (The proof may be obtained from the proof of in Proposition \[comp\_as\_Lp\], replacing the $L^p$ spaces considered there by $C_b({{\mathbb R}}^d)$). In the proof of the next proposition we use an important corollary of the Krein-Rutman Theorem, whose proof may be found in e.g. [@Du Ch. 1]. \[KR\] Let $K$ be a cone with nonempty interior part $\widetilde{K}$ in a Banach space $X$, and let $L:X\mapsto X$ be a linear compact operator such that $L\varphi \in \widetilde{K}$ for each $\varphi \in K\setminus\{0\}$. Then the spectral radius $r$ of $L$ is a simple eigenvalue of $L$, and all the other eigenvalues have modulus $<r$. \[pr:compact\] If $G(t,s)$ is compact in $C_b({{\mathbb R}}^d)$ for $t>s$, then $1$ is a simple eigenvalue of $V(s)$ for each $s$, and it is the unique eigenvalue on the unit circle. The spectral projection $P(s) = I - Q(s)$ is given by $$P(s)\varphi(x) = \int_{{{\mathbb R}}^d} \varphi(y)\mu_s(dy), \quad \varphi \in C_b({{\mathbb R}}^d), \;s\in {{\mathbb R}}, x\in {{\mathbb R}}^d,$$ where $\{ \mu_s :\;s\in {{\mathbb R}}\}$ is a $T$-periodic evolution system of measures. Let $K= \{\varphi \in C_b({{\mathbb R}}^d):\; \forall x\in {{\mathbb R}}^d, \;\varphi(x) \geq 0\}$ be the cone of the nonnegative functions in $C_b({{\mathbb R}}^d)$. By Remark \[rem:comp\](iv), if $\varphi \in C_b({{\mathbb R}}^d)\setminus \{0\}$ is such that $\varphi(x)\geq 0$ for each $x$, then $\inf V(s)\varphi = \inf G(s+T,s)\varphi(x) >0$. In other words, $V(s)$ maps $K\setminus \{0\}$ into the interior part of $K$. Theorem \[KR\] implies that the spectral radius $1$ of $V(s)$ is a simple eigenvalue, and it is the unique eigenvalue of $V(s)$ on the unit circle. The associated spectral projection $P(s) = I- Q(s)$, with $Q(s)$ defined above, may be expressed as $$P(s)\varphi = m_s\varphi \, {1\!\!\!\;\mathrm{l}}, \quad \varphi\in C_b({{\mathbb R}}^d) ,$$ for some $m_s$ in the dual space of $C_b({{\mathbb R}}^d)$. To prove that $m_s\varphi = \int_{{{\mathbb R}}^d} \varphi(y)\mu_s(dy)$ for some measure $\mu_s$ we use the Stone–Daniell Theorem (e.g., [@Dudley Thm. 4.5.2]): it is enough to check that $m_s\varphi \geq 0$ if $\varphi \geq 0$, and that for each sequence $(\varphi_n) \subset C_b({{\mathbb R}}^d)$ such that $(\varphi_n(x))$ is decreasing and converges to $0$ for each $x\in {{\mathbb R}}^d$, we have $\lim_{n \to \infty} m_s\varphi_n = 0$. In this case, $\mu_s$ is a probability measure for every $s$, because $P(s){1\!\!\!\;\mathrm{l}}= {1\!\!\!\;\mathrm{l}}$. By the general spectral theory, $P(s) = \lim_{\lambda \to 1^-}V_{\lambda}$, where $V_{\lambda}:= (\lambda -1)(\lambda I - V(s))^{-1}$. In its turn, $(\lambda I - V(s))^{-1} = \sum_{k=0}^{\infty}V(s)^k/\lambda^{k+1}$ maps nonnegative functions into nonnegative functions because $V(s)$ does. Therefore, $m_s\varphi \, {1\!\!\!\;\mathrm{l}}= P(s)\varphi \geq 0$ for each $\varphi \geq 0$. Let now $\varphi_n \downarrow 0$. We claim that $V(s) \varphi_n$ converges to $0$ uniformly. Indeed, since the measures $\{ p_{s+T,s,x}:\;x\in {{\mathbb R}}^d\}$ are tight, for each $\varepsilon >0$ there is $R>0$ such that $\int_{{{\mathbb R}}^d\setminus B(0,R)} p_{s+T,s,x}(dy) \leq {\varepsilon}$, for each $x\in {{\mathbb R}}^d$. On the other hand, $\varphi_n$ converges to $0$ uniformly on $B(0,R)$ by the Dini Monotone Convergence Theorem, so that for $n$ large, say $n\geq n_0$, we have $ \varphi_n(y) \leq {\varepsilon}$, for $\|y\|\leq R$. Therefore, for $n\geq n_0$ we have $$\begin{array}{lll} 0\leq V(s) \varphi_n(x) & = & \displaystyle{\int_{B(0,R)} \varphi_n(y)p_{s+T,s,x}(dy) + \int_{{{\mathbb R}}^d\setminus B(0,R)} \varphi_n(y)p_{s+T,s,x}(dy) } \\ \\ & \leq &{\varepsilon}+ \\| \varphi_n\\|_{\infty} {\varepsilon}\end{array}$$ for each $x\in {{\mathbb R}}^d$. Since $V(s) \varphi_n$ converges to $0$ uniformly, then $P(s)V(s) \varphi_n = V(s) P(s) \varphi_n$ converges to $0$ uniformly. But $V(s)$ is the identity on the range of $P(s)$. Then, $P(s) \varphi_n$ converges uniformly to $0$, which implies that $\lim_{n \to \infty} m_s\varphi_n = 0$. Let us prove that $\{ \mu_s :\;s\in {{\mathbb R}}\}$ is a $T$-periodic evolution system of measures. Since $s\mapsto P(s)$ is $T$-periodic, then $\mu_s = \mu_{s+T}$ for each $s\in {{\mathbb R}}$. Moreover, since $V(t)G(t,s) = G(t,s)V(s)$, then $P(t)G(t,s)\varphi =G(t,s)P(s)\varphi$, for each $\varphi\in C_b({{\mathbb R}}^d)$. This means $$\int_{{{\mathbb R}}^d} G(t,s)\varphi \,d\mu_t {1\!\!\!\;\mathrm{l}}= G(t,s) \bigg( \int_{{{\mathbb R}}^d} \varphi \, d\mu_s \,{1\!\!\!\;\mathrm{l}}\bigg) , \quad \varphi\in C_b({{\mathbb R}}^d),$$ and since $G(t,s){1\!\!\!\;\mathrm{l}}= {1\!\!\!\;\mathrm{l}}$, then $$\int_{{{\mathbb R}}^d} G(t,s)\varphi \,d\mu_t = \int_{{{\mathbb R}}^d} \varphi \, d\mu_s, \quad \varphi\in C_b({{\mathbb R}}^d),$$ so that $\{ \mu_s :\;s\in {{\mathbb R}}\}$ is an evolution system of measures. \[decay\_infty\] Assume that $G(t,s)$ is compact in $C_b({{\mathbb R}}^d)$ for $t>s$. Then: - There exists a unique $T$-periodic evolution system of measures $\{ \mu_s :\;s\in {{\mathbb R}}\}$; - Setting $\omega_0 = \log r /T $, where $r $ is defined in , for each $\omega >\omega_0$ there exists $M = M(\omega)>0$ such that $$\\| G(t,s)\varphi - \int_{{{\mathbb R}}^d} \varphi \, d\mu_s\\|_{\infty} \leq M e^{\omega (t-s)}\\|\varphi\\|_{\infty}, \quad t>s, \;\varphi\in C_b({{\mathbb R}}^d), \label{decexp1}$$ while for $\omega<\omega_0$ there is no $M$ such that holds. Let $\{ \mu_s :\;s\in {{\mathbb R}}\}$ be the $T$-periodic evolution system of measures given by Proposition \[pr:compact\]. Since $P(s) \varphi =\int_{{{\mathbb R}}^d} \varphi \, d\mu_s {1\!\!\!\;\mathrm{l}}$, estimate implies . Since there exist eigenvalues of $V(s) $ with modulus $r$, cannot hold for $\omega<\omega_0$. Indeed, if $V(s)\varphi = r\varphi $ then $P(s)\varphi =0$ and $G(s+nT,s)\varphi = r^n\varphi = e^{\omega_0nT}\varphi$ for each $n\in {{\mathbb N}}$, so that $\\| G(s+nT, s)\varphi - m_s \varphi \\|_{\infty} = \\| G(s+nT,s)\varphi \\|_{\infty} = e^{\omega_0nT}\\|\varphi\\|_{\infty}$. If $\{ \nu_s :\;s\in {{\mathbb R}}\}$ is another $T$-periodic evolution system of measures, fix $t\in {{\mathbb R}}$ and $\varphi \in C_b({{\mathbb R}}^d)$. Since $G(t,s)\varphi - \int_{{{\mathbb R}}^d} \varphi \, d\mu_s$ goes to zero uniformly as $s\to -\infty$, then $$0 = \lim_{s\to -\infty} \int_{{{\mathbb R}}^d} \bigg( G(t,s)\varphi - \int_{{{\mathbb R}}^d} \varphi \, d\mu_s\bigg) d\nu_t = \lim_{s\to -\infty}\bigg( \int_{{{\mathbb R}}^d} \varphi \, d\nu_s - \int_{{{\mathbb R}}^d} \varphi \, d\mu_s\bigg).$$ Since $s\mapsto \int_{{{\mathbb R}}^d} \varphi \, d\nu_s - \int_{{{\mathbb R}}^d} \varphi \, d\mu_s$ is $T$-periodic and goes to $0$ as $s\to -\infty$, it vanishes in ${{\mathbb R}}$. By the arbitrariness of $\varphi$, $\nu_s = \mu_s$ for every $s\in {{\mathbb R}}$. Once we have an evolution system of measures $\{ \mu_s :\;s\in {{\mathbb R}}\}$, $G(t,s)$ is extendable to a contraction (still denoted by $G(t,s)$) from $L^p({{\mathbb R}}^d, \mu_s)$ to $L^p({{\mathbb R}}^d, \mu_t)$ for $t>s$. Compactness and asymptotic behavior results in $C_b({{\mathbb R}}^d)$ imply compactness and asymptotic behavior results in such $L^p$ spaces, as the next proposition shows. \[comp\_as\_Lp\] Let $G(t,s)$ be compact in $C_b({{\mathbb R}}^d)$ for $t>s$. Then for every $p\in (1, \infty)$, $G(t,s): L^p({{\mathbb R}}^d, \mu_s)\mapsto L^p({{\mathbb R}}^d, \mu_t)$ is compact for $t>s$. Moreover, for every $\omega \in (\omega_0, 0)$ and $p\in (1, \infty)$ there exist $M = M(\omega, p)>0$ such that $$\\| G(t,s)\varphi - \int_{{{\mathbb R}}^d} \varphi \, d\mu_s\\|_{L^p({{\mathbb R}}^d, \mu_t)} \leq M e^{\omega (t-s)}\\|\varphi\\|_{L^p({{\mathbb R}}^d, \mu_s)}, \quad t>s, \;\varphi\in L^p({{\mathbb R}}^d, \mu_s), \label{decexp2}$$ and for every $\omega<\omega_0$ there is no $M$ such that holds. Here $\omega_0 $ is given by Corollary \[decay\_infty\](ii). Let us prove that $G(t,s): L^p({{\mathbb R}}^d, \mu_s)\mapsto L^p({{\mathbb R}}^d, \mu_t)$ is compact for $t>s$. We have $G(t,s) = G(t, (t+s)/2) G((t+s)/2, s)$ where $G((t+s)/2, s)$ is bounded from $L^{\infty}({{\mathbb R}}^d)$ to $C_b({{\mathbb R}}^d)$ by Theorem \[Th:G(t,s)\](iv), and $G(t, (t+s)/2)$ is compact in $C_b({{\mathbb R}}^d)$. Therefore, $G(t,s)$ is compact in $L^{\infty}({{\mathbb R}}^d)$. Now, if $\mu_1$ and $\mu_2$ are probability measures and a linear operator is bounded from $L^1({{\mathbb R}}^d, \mu_1)$ to $L^1({{\mathbb R}}^d, \mu_2)$ and compact from $L^{\infty}({{\mathbb R}}^d, \mu_1)$ to $L^{\infty}({{\mathbb R}}^d, \mu_2)$, then it is compact from $L^p({{\mathbb R}}^d, \mu_1)$ to $L^p({{\mathbb R}}^d, \mu_2)$, for every $p\in (1, \infty)$ (the proof is the same as in [@MPW Prop. 4.6], where only one probability measure was considered). Let us prove . Since $L^{\infty}({{\mathbb R}}^d, \mu_t) = L^{\infty}({{\mathbb R}}^d, \mu_s) = L^{\infty}({{\mathbb R}}^d, dx)$ by [@KLL Prop. 5.2], interpolating and $\\| G(t,s)\varphi - m_s\varphi\\|_{L^1({{\mathbb R}}^d, \mu_t)} \leq 2\\|\varphi\\|_{L^1({{\mathbb R}}^d , \mu_s)} $ we obtain that $\\| G(t,s) - m_s \\|_{{\mathcal L}(L^p({{\mathbb R}}^d, \mu_s), L^p({{\mathbb R}}^d , \mu_t))} $ decays exponentially as $(t-s)\to \infty$. However, the decay rate that we obtain by interpolation depends on $p$. To prove it is enough to show that the spectrum of the operators $V(s)$ in $L^p({{\mathbb R}}^d, \mu_s)$ does not depend on $s$, and coincides with the spectrum in $C_b({{\mathbb R}}^d)$. Since $V(s) = G(s+T,s)$, then $V(s)$ is compact in $L^p({{\mathbb R}}^d, \mu_s)$. Therefore, its $L^p$ spectrum (except zero) consists of eigenvalues, that are independent of $s$. They are independent of $p$ too, as well as the associated spectral projections, by [@davies Cor. 1.6.2]. The statement follows now as in the case of evolution operators in a fixed Banach space as in the mentioned references [@Henry; @L; @F]. Note however that our Banach spaces $L^p({{\mathbb R}}^d, \mu_s)$ vary with $s$, so that the classical theory cannot be used verbatim. For the reader’s convenience we give the proof below. Let $t-s = \sigma + kT$, with $k\in {{\mathbb N}}$ and $\sigma \in [0, T)$. We have $$\begin{array}{l} \\|G(t,s)\varphi - m_s \varphi \\|_{L^p({{\mathbb R}}^d, \mu_t)} = \\|G(t, t-\sigma) V(s)^k (I-P(s))\varphi \\|_{L^p({{\mathbb R}}^d, \mu_t)} \\ \\ \leq \\| V(s)^k (I-P(s))\varphi \\|_{L^p({{\mathbb R}}^d, \mu_s)} = \\| [V(s)(I-P(s))]^k \varphi \\|_{L^p({{\mathbb R}}^d, \mu_s)}. \end{array}$$ For $\omega>\omega_0$ let ${\varepsilon}>0$ be such that $\log (r+{\varepsilon})\leq \omega $, and let $k(s)\in {{\mathbb N}}$ be such that $ \\| [V(s) (I-P(s))]^k\varphi \\|_{L^p({{\mathbb R}}^d, \mu_s)}\leq (r+{\varepsilon})^k$ for each $k > k(s)$. Therefore, if the integer part $[(t-s)/T]$ is larger than $k(s)$ we have $$\\|G(t,s)\varphi - m_s\varphi \\|_{L^p({{\mathbb R}}^d, \mu_t)} \leq (r+{\varepsilon})^k\\|\varphi \\|_{L^p({{\mathbb R}}^d, \mu_s)} \leq e^{(t-s) \omega} \\|\varphi \\|_{L^p({{\mathbb R}}^d, \mu_s)},$$ for each $ \varphi \in L^p({{\mathbb R}}^d, \mu_s)$. Using the obvious inequality $\\|G(t,s)\varphi - m_s\varphi \\|_{L^p({{\mathbb R}}^d, \mu_t)} \leq 2 \\|\varphi \\|_{L^p({{\mathbb R}}^d, \mu_s)}$ for $[(t-s)/T]\leq k(s)$ we arrive at $$\\|G(t,s)\varphi - m_s\varphi \\|_{L^p({{\mathbb R}}^d, \mu_t)} \leq M_s e^{(t-s) \omega} \\|\varphi \\|_{L^p({{\mathbb R}}^d, \mu_s)}, \quad \varphi \in L^p({{\mathbb R}}^d, \mu_s)$$ for some $M_s>0$. It remains to show that $M_s$ can be taken independent of $s$. Since $V$ is $T$-periodic, we may take $k(s)= k(s+T)$ and hence $M_s = M_{s+T}$ for every $s\in {{\mathbb R}}$. Therefore it is enough to show that $M_s$ can be taken independent of $s$ for $s\in [0,T)$. For $0\leq s<T$ and $t\geq T$ we have $m_T G(T,s)\varphi = \int_{{{\mathbb R}}^d} G(T,s)\varphi\,d\mu_T = m_s\varphi$, hence $$\begin{array}{l} \\|G(t,s)\varphi - m_s\varphi \\|_{L^p({{\mathbb R}}^d, \mu_t)} = \\| (G(t, T)-m_T) G(T,s)\varphi \\|_{L^p({{\mathbb R}}^d, \mu_t)} \\ \\ \leq M_Te^{\omega(t-T)} \\| G(T,s)\varphi \\|_{L^p({{\mathbb R}}^d, \mu_T)} \leq M_Te^{\|\omega\| T}e^{\omega(t-s)} \\| \varphi \\|_{L^p({{\mathbb R}}^d, \mu_s)} \end{array}$$ So, we can take $M_s = M_Te^{\|\omega\| T}$ for $0\leq s<T$. follows. [999]{} G. Da Prato, A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients, J. Evol. Equ. 7* (2007), no. 4, 587–614. E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989. Y. Du, Order structure and topological methods in nonlinear partial differential equations. Vol. 1. Maximum principles and applications. Series in Partial Differential Equations and Applications, 2. World Scientific Publishing Co. Pte. 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--- abstract: 'The coupling of spin-orbit materials to high energy barrier ($\sim$40-60 $k_BT$) nano-magnets has attracted growing interest for exciting new physics and various spintronic applications. We predict that a coupling between the spin-momentum locking (SML) observed in spin-orbit materials and low-energy barrier magnets (LBM) should exhibit a unique multi-terminal rectification for arbitrarily small amplitude channel currents. The basic idea is to measure the charge current induced spin accumulation in the SML channel in the form of a magnetization dependent voltage using an LBM, either with an in-plane or perpendicular anisotropy (IMA or PMA). The LBM feels an instantaneous spin-orbit torque due to the accumulated spins in the channel which causes the average magnetization to follow the current, leading to the non-linear rectification. We discuss the frequency band of this multi-terminal rectification which can be understood in terms of the angular momentum conservation in the LBM. For a fixed spin-current from the SML channel, the frequency band is same for LBMs with IMA and PMA, as long as they have the same total magnetic moment in a given volume. The proposed all-metallic structure could find application as highly sensitive passive rf detectors and as energy harvesters from weak ambient sources where standard technologies may not operate.' author: - Shehrin Sayed - 'Kerem Y. Camsari' - Rafatul Faria - Supriyo Datta bibliography: - 'TI\_rect.bib' - 'Ref.bib' - 'Ref1.bib' - 'Ref2.bib' title: 'Rectification in Spin-Orbit Materials Using Low Energy Barrier Magnets' --- Introduction ============ The interplay between spin-orbit materials and nano-magnetism has attracted much attention for interesting phenomena e.g. spin-orbit torque switching [@Ralph_Science_2012; @Ralph_PRL_2012], probing the spin-momentum locking [@JonkerNatNano2014; @ChenPRL2017; @SamarthPRB2015; @Dash_NanoLett_2015; @Pham_NanoLett_2016; @Koo_New], spin amplification [@AGhosh_PRL_2015], spin battery [@ChenScience2017], skyrmion dynamics [@Woo2017; @Wang_NanoLett2016], among other examples. In this paper, we predict that the spin-momentum locking (SML) observed in spin-orbit materials when coupled to a nano-magnet with low-energy barrier, will rectify the channel current in the form of a voltage in a multi-terminal structure. We start our arguments with the spin-potentiometric measurements well-established in diverse classes of spin-orbit materials (see, for example, [@JonkerNatNano2014; @ChenPRL2017; @SamarthPRB2015; @Dash_NanoLett_2015; @Pham_NanoLett_2016; @Koo_New]) where a high-energy-barrier stable ferromagnet (FM) is used to measure the charge current induced spin potential in the SML channel. We show that such spin-potential measurement on a metallic SML channel using a low-energy barrier magnet (LBM) will result in a rectified voltage, even for arbitrarily small channel current. The discussions on the multi-terminal rectification is limited in the linear response regime of transport in the SML channel and the non-linearity occurs due to the spin-orbit torque (SOT) driven magnetization dynamics of the LBM. We further show that the multi-terminal rectification is limited by a characteristic frequency of the LBM that can be understood in terms of angular momentum conservation between the spins injected from spin-orbit materials and the spins absorbed by the LBM. We argue that, for a fixed spin-current from the SML channel, the characteristic frequency is the same for LBMs with in-plane and perpendicular magnetic anisotropies (IMA and PMA), as long as they have the same total magnetic moment in a given volume. We analyze the rectification in the proposed all-metallic structure (see Fig. \[1\](a)) considering both IMA and PMA LBMs and provide simple models to understand the underlying mechanisms of (i) the spin-orbit torque (SOT) induced magnetization pinning and (ii) the frequency band of the rectification. We compare the simple models with detailed numerical simulations using an experimentally benchmarked multi-physics framework [@Camsari_SciRep_2015]. The simulations are carried out using a transmission line model for the SML channel [@Sayed_arXiv_2017] and a stochastic Landau-Lifshitz-Gilbert (s-LLG) model for LBM [@PhysRevX.7.031014], considering thermal noise within the magnet. We consider the spin-polarization axis to be in-plane of the SML channel and orthogonal to the current flow direction. Hence, in the present discussion, pinning for a IMA or a PMA magnet occurs along the easy-axis or hard-axis, respectively. We argue that such wideband rectification in an all-metallic structure (Fig. \[1\](a)) could be used for ‘passive’ radio frequency (rf) detection. Recently, Magnetic Tunnel Junction (MTJ) diodes with stable magnet as free layer and under an external dc current bias have demonstrated orders of magnitude higher sensitivity compared to the state-of-the-art Schottky diodes [@Miwa2013; @Fang2016; @doi:10.1063/1.5047547]. However, the reported no-bias sensitivity is lower or comparable to that of semiconductor diodes. The low-barrier nature of the magnet in the proposed structure should exhibit no-bias sensitivity as high as those observed using state-of-the-art technologies under external biases [@Miwa2013; @Fang2016; @doi:10.1063/1.5047547]. Furthermore, we discuss the possibility to harvest energy from weak ambient sources where standard technologies may not operate. The paper is organized as follows. In Section \[sec2\], we establish the concept of the multi-terminal rectification in the SML channel using LBM, starting from the well-established spin-potentiometric measurements typically done with high-energy-barrier stable magnets. In Section \[sec3\], we discuss the frequency bandwidth of the rectification and provide a simple model that applies to LBMs with both IMA and PMA. We argue using detailed simulaiton results that such bandwidth arises due to the principles of angular momentum conservation between the spins injected from the SML channel and the spins absorbed by the LBM. In Section IV, we discuss possible applications of the proposed all-metallic structure in ‘passive’ rf detection and energy harvesting. We argue that the no-bias sensitivity of the proposed rectifier can be as high as those observed in state-of-the-art technologies under external bias. Finally, in Section \[sec5\], we end with a brief conclusion. Multi-Terminal Rectification {#sec2} ============================ We start our arguments with the well-established spin-potentiometric measurements [@JonkerNatNano2014; @ChenPRL2017; @SamarthPRB2015; @Dash_NanoLett_2015; @Pham_NanoLett_2016; @Koo_New] where the charge current induced spin potential in the SML channel is measured in the form of a magnetization dependent voltage using a stable ferromagnet (FM). The voltage at the FM with respect to a reference normal metal (NM) contact, placed at the same position along the current path as the FM (see Fig. \[1\](a)), is given by [@Hong_PRB_2012; @SayedEDL2017] $$\label{spin_vol} {V_{34}\left(\vec m\right)} = {\left(\hat{s}\cdot \vec{m} \right)}\,\dfrac{{\alpha \xi {p_0}{p_f}R_B}}{{2}}{I_{12}},$$ which shows opposite signs for the two magnetic states of the FM under a fixed channel current $I_{12}$ flowing along $\hat{n}$-direction (see Fig. \[1\](a)). Here $\hat{s}$ is the spin polarization axis in the SML channel defined by $\hat{y}\times \hat{n}$ with $\hat{y}$ being the out-of-plane direction [@Sayed_arXiv_2017], $\vec{m}$ is the magnetization vector, $p_f$ is the FM polarization, $0\leq\xi\leq1$ is the current shunting factor [@Sayed_SciRep_2016] of the contact with 0 and 1 indicating very high and very low shunting respectively, $p_0$ is the degree of SML in the channel [@Sayed_arXiv_2017], $\alpha\approx 2/\pi$ is an angular averaging factor [@Sayed_arXiv_2017], and $R_B=(h/q^2)(1/M_t)$ is the ballistic resistance of the channel with total number of modes $M_t$ ($q:$ electron charge, $h:$ Planck’s constant). Note that Eq. is valid all the way from ballistic to diffusive regime of operation [@Hong_PRB_2012; @Sayed_arXiv_2017]. We restrict our discussion to linear response where ${V_{34}\left(\vec m\right)}$ in Eq. scales linearly with $I_{12}$ and satisfies the Onsager reciprocity relation [@Buttiker_PRB_2012; @Sayed_SciRep_2016] $$R_{ij,kl}\left(\vec m\right) = R_{kl,ij}\left(-\vec m\right),$$ with $R_{ij,kl}=V_{kl}/I_{ij}$. The Onsager reciprocity does not require any specific relation between $R_{ij,kl}\left(\vec m\right)$ and $R_{ij,kl}\left(-\vec m\right)$ in linear response and the phenomenon described by Eq. has been observed on diverse spin-orbit materials e.g. topological insulator (TI) [@JonkerNatNano2014; @ChenPRL2017; @SamarthPRB2015; @Dash_NanoLett_2015], Kondo insulators [@Hong_KIST_arXiv_2018], transition metals [@Pham_NanoLett_2016], semimetals [@Li_NatComm_2018], and semiconductors [@Koo_New]. To measure Eq. from a highly resistive SML channel (e.g. TI [@JonkerNatNano2014; @ChenPRL2017; @SamarthPRB2015; @Dash_NanoLett_2015], semiconductor [@Koo_New], etc.) using a metallic FM, usually a thin tunnel barrier is inserted at the interface. This tunnel barrier effectively enhances $V_{34}$ by improving $\xi$ [@Sayed_SciRep_2016], however, degrades the spin injection into the FM from the SML channel. It has been recently demonstrated [@Pham_NanoLett_2016] that $V_{34}$ can be measured with metallic FM in direct contact with metallic SML channels (e.g. Pt, Ta, W, etc.), which indicates the possibility of spin-voltage reading (e.g. [@Pham_NanoLett_2016]) and spin-orbit torque (SOT) writing (e.g. [@Ralph_Science_2012; @Ralph_PRL_2012]) of the nano-magnet within same setup with different current magnitudes [@SayedEDL2017]. The energy barrier of a mono-domain magnet is given by $\Delta_B=\frac{1}{2}H_k M_s \Omega$ [@SunPRB2000] where $H_k$ is the anisotropy field, $M_s$ is the saturation magnetization, and $\Omega$ is the FM volume. For a stable FM, $\Delta_B \approx 40\sim60\;k_BT$ and exhibit very long retention time $\tau \propto \exp\left(\Delta_B/k_BT\right)$ of the magnetization state ($k_B:$ Boltzmann constant, $T:$ temperature). LBMs have very small $\tau$ and the $\hat{s}\cdot\vec{m}$ component becomes random within the range $\left\{+1,-1\right\}$ driven by the thermal noise. Experimentally, LBMs have been achieved by lowering the total moment ($M_s\Omega$) [@PhysRevApplied.8.054045] or by lowering the anisotropy field ($H_k$) either by increasing the thickness of a PMA [@8421599], or by making a circular IMA with no shape anisotropy [@7838539]. At equilibrium ($I_{12}=0$), the time-averaged $\langle \hat{s}\cdot\vec{m} \rangle = 0$ for an LBM. For $I_{12}\neq 0$, induced non-equilibrium spins in the channel apply SOT on the LBM and $\langle \hat{s}\cdot\vec{m} \rangle$ follows the accumulated spins, which can be calculated using $$\label{av_fpe1} \langle \hat{s}\cdot\vec{m} \rangle = \dfrac{\displaystyle \int_{\phi=-\pi}^{\phi=\pi}\displaystyle \int_{\theta=0}^{\theta=\pi} \left(\hat{s}\cdot\vec{m}\right) \;\rho\;\sin\theta\;d\theta d\phi}{\displaystyle \int_{\phi=-\pi}^{\phi=\pi}\displaystyle \int_{\theta=0}^{\theta=\pi}\rho\;\sin\theta\;d\theta d\phi}.$$ where $\rho$ is the probability distribution function of the magnetization of the LBM under a particular $I_{12}$, which can be obtained from the Fokker-Planck equation [@PhysRev.130.1677; @ButlerFPE2012]. The dependence of $\langle \hat{s}\cdot\vec{m} \rangle$ on $I_{12}$ deduced from Eq. for a particular LBM, can in-principle be any saturating odd-functions e.g. Langevin function for low-barrier PMA (see Appendix \[App\_I\_0\]). We approximate $\langle \hat{s}\cdot\vec{m} \rangle$ in Eq. with a $\tanh$ functional dependence on $I_{12}$, given by $$\label{avg_Vfm} \langle \hat{s}\cdot\vec{m} \rangle\approx \tanh\left(\dfrac{I_{12}}{I_0}\right),$$ which is in good agreement with the detailed numerical simulations for both IMA and PMA, as shown in Fig. \[1\](b). The simulations are carried out within a multi-physics framework [@Camsari_SciRep_2015] using our experimentally benchmarked transmission line model for SML [@Sayed_arXiv_2017] and stochastic Landau-Lifshitz-Gilbert (s-LLG) model for LBM [@PhysRevX.7.031014] which considers thermal noise. The details of the simulation setup is discussed in Appendix \[App\_Sim\]. Here, $I_0$ is a parameter that determines the SOT induced magnetization pinning of the LBM. $I_0$ depends on the temperature, geometry, and material parameters and much larger for an IMA as compared to a PMA due to the demagnetization field. $I_{12}$ along $\mp\hat{n}$-direction causes the magnetization pinning along $\pm\hat{s}$-direction. In the present discussion, easy axis for PMA is along $\hat y$-direction, hence, the pinning occurs along the hard axis. The easy axis of IMA, in principle, can be in any direction on the plane spanned by $\hat{n}$ and $\hat{s}$ and the magnetization pinning along $\hat{s}$-direction should be described by Eq. with a modified $I_0$. However, we set the easy axis along $\pm\hat{s}$-direction in our IMA simulations for simplicity. For a given structure, $I_0$ can be determined directly from experiments using a characteristic curve similar to that in Figs. \[1\](b) or (c). We provide a simple expression using Eq. and considering easy-axis pinning of a PMA magnet (see Appendix \[App\_I\_0\] for the derivation), as given by $$\label{I0_LBM} {I_0} \approx \frac{{6q}}{\hbar }\frac{{{k_B}T{\alpha _g}}}{\beta},$$ where $\alpha_g$ is the Gilbert damping and $\beta$ is the charge to spin current conversion ratio. Eq. is reasonably valid up to $\Delta_B\approx k_BT$ and provides the correct order of magnitude up to several $k_BT$ (see Appendix \[App\_I\_0\] for details). In this discussion, we consider very low energy barrier ($\leq 1k_BT$) nano-magnets that do not have bistable states. A higher barrier magnet that exhibits bistable states, in principle could exhibit effects like stochastic resonance [@PhysRevLett.105.047202], which is not the subject of the present discussion. For $\|I_{12}\|\gg I_0$ in Eq. , we have $\tanh\left(I_{12}/I_0\right)\approx$ $+1$ or $-1$ when $I_{12}>0$ or $I_{12}<0$ respectively. Hence, $I_{12}\times\tanh\left(I_{12}/I_0\right)\approx \|I_{12}\|$. On the other hand, for $\|I_{12}\|\ll I_0$ in Eq. , we have $I_{12}\times\tanh\left(I_{12}/I_0\right)\approx I_{12}^2/I_0$. Thus, the time-average of the voltage in Eq. is given by $$\label{avg_spin_vol} \langle V_{34} \rangle= \begin{cases} \left(\dfrac{{\alpha \xi {p_0}{p_f}R_B}}{{2I_0}}\right)\,{I_{12}^2}, & \text{for $\|I_{12}\| \ll I_0$} \\ \left(\dfrac{{\alpha \xi {p_0}{p_f}R_B}}{{2}}\right)\,{\|I_{12}\|}. & \text{for $\|I_{12}\| \gg I_0$} \end{cases}$$ Note that $\langle V_{34} \rangle$ represents the steady-state voltage of the capacitor $C_L$ placed between contacts 3 and 4. The relative position between contacts 3 and 4 along $\hat{s}$-direction do not affect Eq. , however, a shift in the $\hat{n}$-direction creates an offset due to Ohmic drop [@SayedEDL2017], that should cancel out over averaging in Eq. when an ac $I_{12}$ is applied. For arbitrary $I_{12}$, $\langle V_{34} \rangle$ is always of the same sign leading to a multi-terminal rectification. This observation agrees well with simulation results for IMA and PMA, as shown in Fig. \[1\](c). For $\|I_{12}\|\ll I_0$, $\langle V_{34} \rangle$ exhibits a parabolic nature (see zoomed inset of Fig. \[1\](c)), as suggested by Eq. . All simulation results presented in this paper are normalized by $I_0$, $V_0=I_0R_B$, and $f_0=I_0/q$ for current, voltage, and frequency, respectively. In all simulations, $M_t=100$ which yields $R_B=259\Omega$. Frequency Bandwidth {#sec3} =================== The frequency bandwidth of the the multi-terminal rectification is limited by a characteristic frequency $f_c$ that is determined by the angular momentum conservation between the spins injected from the SML channel and the spins absorbed by the LBM. We plot the $\langle V_{34} \rangle$ as a function of the frequency $f$ of the ac $I_{12}=i_{c0}\,\sin\left(2\pi f t \right)$ (see Fig. \[sup2\](a)) while other parameters are kept constant in our simulations. Note that $\langle V_{34} \rangle$ is relatively constant in the low frequency region and degrades significantly for $f>f_c$. We have defined $f_c$ as the frequency where $\langle V_{34} \rangle$ degrades by an order of magnitude compared to the region where $\langle V_{34} \rangle$ vs. $f$ is relatively flat. We observe the time-dynamics of $\hat{s}\cdot \vec{m}$ and $V_{34}$ for the two cases indicated with red-dots in Fig. \[sup2\](a): (I) $f<f_c$ and (II) $f>f_c$. For the first case, $I_{12}$ is slow enough that the injected spins from SML channel into the LBM satisfies the angular momentum conservation and $\hat{s}\cdot \vec{m}$ follows the $I_{12}$ at the same frequency, as shown in Fig. \[sup2\](b). This leads to a rectified voltage $V_{34}$ that charges up the capacitor $C_L$ to the steady-state value $\langle V_{34} \rangle$. The ripples observed in $V_{34}$ is similar to those in conventional rectifiers and gets attenuated for increased $C_L$. For the latter case, $\hat{s}\cdot \vec{m}$ struggles to follow $I_{12}$ (see Fig. \[sup2\](c)) since the spins injected from the SML channel to the LBM is fast enough that they do not satisfy the angular momentum conservation. $\hat{s}\cdot \vec{m}$ has no correlation with $I_{12}$, as a result, there is no rectification that charges up $C_L$ to a steady dc voltage. We obtain an empirical expression for $f_c$ from the detailed s-LLG simulations using a broad range of parameter values, given by $$\label{cut_off} 2\pi {f_c} = \dfrac{i_{s0}}{{2q{N_s}}},$$ where the injected spin current amplitude $i_{s0}=\beta i_{c0}$, $N_s=M_s \Omega/\mu_B$ is the total number of spins in LBM and $\mu_B$ is the Bohr magneton. The functional dependence of $f_c$ on $i_{s0}$ and $N_s$ is very similar to the switching delay for stable magnets [@Behtash2011] that also arises from the principles of angular momentum conservation. Note that Eq. is valid for both IMA and PMA. We show comparison between Eq. and simulation results in Fig. \[4\]. The simulation data points shown on Fig. \[4\] are extracted from a plot similar to Fig. \[sup2\](a). Eq. shows good agreement with the simulation for LBMs having IMA with easy-axis pinning and PMA with hard-axis pinning (see Figs. \[4\](a)-(b)). Fig. \[4\](a) shows that $f_c$ scales linearly with $i_{s0}$. A similar scenario has been reported [@PhysRevApplied.2.034009] for a stochastic MTJ oscillator made with relatively lower-barrier free magnetic layer. According to Eq. and detailed simulations, the conclusion that $f_c\propto i_{s0}$ seems valid even if $i_{s0}$ changes by orders of magnitude. Moreover, Fig. \[4\](b) shows that $f_c$ scales inversely proportional to the $N_s$ which depends only on $M_s\Omega$ of the magnet and independent of the magnetic anisotropy. Eq. could be useful for recent interest on LBM based applications e.g. stochastic oscillators [@PhysRevApplied.2.034009], random number generators [@doi:10.1063/1.5006422; @PhysRevApplied.8.054045], probabilistic spin logic [@PhysRevX.7.031014; @Mizrahi2018], etc. Applications: RF Detection and Energy Harvesting {#sec4} ================================================ The proposed all-metallic structure can find useful applications like rf detection and energy harvesting. In this section, we show that the low-barrier nature of the magnet can lead to very high rf detection sensitivity without any external bias, comparable to those observed in state-of-the-art technologies under an external bias. We provide a simple model for no-bias sensitivity which provides insight into the design of a high-sensitivity device. This could be of interest for rf detection from weak sources typically proposed to sense with quantum sensors (see, e.g., [@House2015; @Gely1072]).We further argue that the proposed structure can extract useful energy from the ambient rf energy, especially from the weak sources where standard technologies may not operate. It can be seen from Eq. that $\langle V_{34} \rangle$ scales $\propto \sqrt{P_{in}}$ when $\max\left(I_{12}\right) \gg I_0$, where $P_{in} =\frac{1}{T}\int_{0}^{T}I_{12}^2\left(t\right) R_{12}\,dt$, $T=1/f$, and $R_{12}$ is the channel resistance. However, for $\max\left(I_{12}\right) \ll I_0$, $\langle V_{34} \rangle$ scales $\propto P_{in}$ with a constant slope given by $$\label{sensitivity} \dfrac{d\langle V_{34} \rangle}{dP_{in}} = \dfrac{{\alpha \xi {p_0}{p_f}}}{{2}}\dfrac{{R_B}}{{{I_0}{R_{12}}}}.$$ The derivation is given in Appendix \[App\_ACtoDC\]. The quantity in Eq. is often considered as the sensitivity of rf detectors [@Miwa2013; @Fang2016; @doi:10.1063/1.5047547]. Recently, Magnetic Tunnel Junction (MTJ) diodes with stable magnet as free layer and under an external dc current bias have demonstrated orders of magnitude higher sensitivity compared to the state-of-the-art Schottky diodes [@Miwa2013; @Fang2016; @doi:10.1063/1.5047547]. However, the reported no-bias sensitivity is lower or comparable to that of semiconductor diodes. Eq. indicates that the no-external-bias sensitivity can be very high within the all-metallic structure in Fig. \[1\](a) when designed to have very low $I_0$, enabling highly sensitive ‘passive’ rf detection. With $\alpha_g=0.01$ and $T=300$ K we have $I_0 \approx 0.37\mu$A$/\beta$ from Eq. . For a Py LBM of dimension of 49 nm $\times$ 61 nm $\times$ 5 nm (see Ref. [@8421599]) and 2 nm thick Pt channel, $\beta$ can be $\sim 2$ as estimated from the charge to spin conversion ratio reported in Ref. [@Ralph_PRL_2012], yielding $I_0\approx0.18$ $\mu$A. Note that $\beta$ can be much higher based on the geometry and the choice of the SML material. For Bi$_2$Se$_3$ and Pt, we roughly estimate the sensitivity as 21,000 and 860 mV/mW respectively, assuming 2D SML channel of width $w=$ 210 nm and length $L=$ 500 nm. These estimations were done based on Eq. using: (i) $R_B=$ 259 $\Omega$ (Bi$_2$Se$_3$) and 58 $\Omega$ (Pt), (ii) $R_{ch}\approx$ $6.5$ k$\Omega$ (Bi$_2$Se$_3$) and $\sim3$ k$\Omega$ (Pt), (iii) $p_0\approx$ 0.6 (Bi$_2$Se$_3$) and 0.05 (Pt) (see Ref. [@SayedEDL2017]), and (iv) $p_f\approx0.5$ [@Koo_New]. We have assumed $\xi\approx 1$ and the quoted estimations will be lower for higher shunting. $R_B$ has been estimated using $M_t=k_F w/\pi$, where $k_F=1.5$ nm$^{-1}$ (Bi$_2$Se$_3$) and $6.7$ nm$^{-1}$ (Pt) [@SayedEDL2017]. The channel resistance has been estimated using $R_{ch}=R_B(L+\lambda)/\lambda$ with mean free path $\lambda$ of 20 nm (Bi$_2$Se$_3$ [@Wang2016]) and 10 nm (Pt [@Johann_PRB_1980]), respectively. More detailed analysis and performance evaluation considering signal-to-noise ratio we leave for future work. With proper materials and geometry, it may be possible to extract usable energy from such rectification of rf signals, especially from weak ambient sources. The dc power $P_L=V_LI_L$ extracted by an arbitrary load $R_L$ is limited by the equivalent resistance $R_{34}$ between contacts 3 and 4. The maximum efficiency of such rf to dc power conversion occurs for $\max\left(I_{12}\right) \gg I_0$, given by $$\label{efficiency} \eta_{\max}=\dfrac{P_{L,\max}}{P_{in}}=\left(\dfrac{{\alpha \xi {p_0}{p_f}}}{\sqrt{2}{\pi}}\right)^2 \dfrac{R_B}{{R_{12}}}\dfrac{R_B}{{R_{34}}}.$$ The derivation is given in Appendix \[App\_Efficiency\]. Note that the maximum efficiency is independent of $I_0$. Assuming $R_{eq}=10R_B$ for enhanced $\xi$, we estimate the maximum efficiency to be 0.001% for Bi$_2$Se$_3$ and 3$\times$10$^{-6}$% for Pt even with $P_{in}$ in the $\sim$pW range given $I_0\leq0.18\mu$A. MTJ diodes recently demonstrated rf energy harvesting with similar efficiency [@WANG_arXiv_2018], however, the input power was in the $\mu$W range. Such MTJs should achieve reasonable efficiency at lower input power if the stable free layer is replaced with an LBM. Conclusion {#sec5} ========== In conclusion, we predict multi-terminal rectification in an all-metallic structure that comprises a spin-orbit material exhibiting spin-momentum locking (SML) and a low-energy barrier magnet (LBM) having either in-plane or perpendicular anisotropy (IMA or PMA). The discussion of such multi-terminal rectification was limited in the linear response regime of transport and the non-linearity occurs due to the spin-orbit torque driven magnetization dynamics of the LBM. We draw attention to a frequency band of the rectification which can be understood in terms of angular momentum conservation within the LBM. For a fixed spin-current from the SML channel, the frequency band is same for LBMs with IMA and PMA, as long as they have the same total magnetic moment for a given volume. We further discuss possible applications of the wideband rectification as highly sensitive passive rf detectors and as energy harvesters from ambient sources. This work was supported by ASCENT, one of six centers in JUMP, a SRC program sponsored by DARPA. Average Magnetization of Low-Barrier Magnets and Magnetization Pinning Current {#App_I_0} ============================================================================== This section discusses the pinning current of a LBM and derives Eqs. -, starting from the steady-state solution of the Fokker-Planck Equation.\ We start from the steady-state solution of probability distribution from Fokker-Planck Equation assuming a magnet with perpendicular magnetic anisotropy (PMA) (see Eq. (4.3) in Ref. [@ButlerFPE2012]), given by $$\label{FPE} \rho\left(m_z\right)=\dfrac{1}{{\rm Z}}\exp\left[ -\dfrac{\Delta_B}{k_B T}\left(1-m_z^2+2\left(\dfrac{H_{ext}}{H_k}+\dfrac{i_s}{I_{s0}}\right)m_z\right) \right],$$ where $\rm Z$ is a normalizing factor, $m_z$ is the magnetization along easy-axis ($\hat{s}\cdot \vec{m}$ in the present discussion), $\Delta_B=H_kM_s\Omega/2$ is the energy barrier of a magnet with anisotropy field $H_k$, saturation magnetization $M_s$, and volume $\Omega$, $k_B$ is the Boltzmann constant, $T$ is the temperature, $H_{ext}$ is the external magnetic field along the easy-axis, $i_s$ is the $z$-polarized spin current injected into the magnet, and $I_{s0}$ is the critical spin current for magnetization switching [@SunPRB2000; @SayedEDL2017] for a magnet with PMA, given by $$\label{SunIs} I_{s0}=\dfrac{4q}{\hbar}\Delta_B\alpha_g,$$ where $\hbar=h/(2\pi)$ and $\alpha_g$ is the Gilbert damping constant. We consider the case with no external field i.e. $H_{ext}=0$ which from Eqs. and gives $$\label{FPE1} \rho\left(m_z\right)=\dfrac{1}{{\rm Z}}\exp\left(-\dfrac{\Delta_B}{k_B T}\left(1-m_z^2\right)-\left(\dfrac{i_s}{\dfrac{2q}{\hbar}k_B T\alpha_g}\right)m_z\right).$$ We consider very low energy barrier magnet i.e. $\frac{\Delta_B}{k_BT}\rightarrow0$, which in Eq. yields $$\label{FPE2} \rho\left(m_z\right)=\dfrac{1}{{\rm Z}}\exp\left(-\dfrac{i_s}{\dfrac{2q}{\hbar}k_B T\alpha_g}m_z\right).$$ The steady-state average $\langle m_z \rangle$ is defined as (see Eq. ) $$\label{av_fpe} \langle m_z \rangle = \dfrac{\int_{\phi=-\pi}^{\phi=\pi}\int_{\theta=0}^{\theta=\pi}m_z \;\rho(m_z)\;\sin\theta \;d\theta d\phi}{\int_{\phi=-\pi}^{\phi=\pi}\int_{\theta=0}^{\theta=\pi}\rho(m_z)\;\sin\theta \;d\theta d\phi},$$ with $\left(m_z,m_x,m_y\right)\equiv\left(\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi\right)$. Combining Eq. with Eq. we get the long time averaged magnetization $\langle m_z \rangle$ for a very low barrier PMA without external magnetic field as $$\label{langevin} \langle m_z \rangle = \coth\left(\dfrac{i_s}{\dfrac{2q}{\hbar}k_B T\alpha_g}\right) - \dfrac{\dfrac{2q}{\hbar}k_B T\alpha_g}{i_s},$$ which is a Langevin function $L(x)$ of $x\equiv{i_s}/\left({\frac{2q}{\hbar}k_B T\alpha_g}\right)$. Note that Eq. was derived assuming $\frac{\Delta_B}{k_BT}\rightarrow0$, however, the expression remains reasonably valid up to ${\Delta_B}\approx{k_BT}$. We have compared Eq. with numerical calculations directly from Eqs. and for $\Delta_B=0.1k_BT$ (see Fig. \[sup1\](a)) and $k_BT$ (see Fig. \[sup1\](b)) respectively, which shows reasonably good agreement. For $\Delta_B>{k_BT}$, the simple expression in Eq. deviates from Eqs. and . For an estimation of the pinning spin current we can approximate the Langevin function $L(x)\approx \tanh\dfrac{x}{3}$, hence $$\label{lang_tan} \langle m_z \rangle \approx \tanh\left(\dfrac{i_s}{\dfrac{6q}{\hbar}k_B T\alpha_g}\right).$$ Note that $i_s$ from SML materials are related to input charge current $i_c$ with a conversion factor $\beta$ given by $$\label{ic_is} i_s = \beta i_c.$$ Combining Eq. with Eq. yields $$\label{lang_tan_ic} \langle m_z \rangle \approx \tanh\left(\dfrac{i_c}{\dfrac{6qk_B T\alpha_g}{\hbar \beta}}\right).$$ Comparing Eq. with Eq. yields $$I_0\approx \dfrac{6qk_B T\alpha_g}{\hbar \beta},$$ which gives the expression in Eq. . Simulation Setup {#App_Sim} ================ This section provides the details of the simulation setup in SPICE that was used to analyze the proposed rectifier.\ We have discretized the structure in Fig. [1]{}(a) into 100 small sections and represented each of the small sections with the corresponding circuit model. Note that each of the nodes in Fig. \[sup3\] are two component: charge ($c$) and $z$-component of spin ($s$). We have connected the charge and spin terminals of the models for all the small sections in a modular fashion using standard circuit rules as shown in Fig. \[sup3\]. The models are connected in a series to reconstruct the structure along length direction. We have two of such parallel chains to take into account the structure along width direction and the two chains represent the area under the LBM and the reference NM respectively. The SML block with LBM is connected to a s-LLG block which takes the spin current from the SML block as input and self-consistently solves for $m_z$ and feeds back to the SML block. The contacts (1, 2, 3, and 4) in this discussion are point contacts. The polarization of contacts 1, 2, and 4 are $p_f=0$ since they represent normal metals. Polarization of contact 3 is 0.8 which represents an LBM. We set the total number of modes $M+N$ in the channel to be 100. We have assumed that the reflection with spin-flip scattering mechanism is dominant in the channel i.e. $r_{s1,2}\gg r,t_s$. The scattering rate per unit mode was set to 0.04 per lattice point. We apply the charge open and spin ground boundary condition at the two boundaries given by $${\left\{ {\begin{array}{{20}{c}} {{i_c}}\\ {{v_s}} \end{array}} \right\}_L} = \left\{ {\begin{array}{{20}{c}} 0\\ 0 \end{array}} \right\},\,\,\,\,\text{and}\,\,{\left\{ {\begin{array}{{20}{c}} {{i_c}}\\ {{v_s}} \end{array}} \right\}_R} = \left\{ {\begin{array}{{20}{c}} 0\\ 0 \end{array}} \right\}.$$ Here, $i_c$ and $v_s$ indicates boundary charge current and boundary spin voltage. Indices $L$ and $R$ indicate left and right boundaries respectively. Both charge and spin terminals of contact 1 and 2 and the two boundaries of the two parallel model chains are connected together. We apply a current $i_c$ at the charge terminal of contact 1 and make the spin terminal grounded to take into account the spin relaxation process in the contact. We ground both charge and spin terminals of contact 2. The boundary conditions of contacts 1 and 2 are given by $${\left\{ {\begin{array}{{20}{c}} {{i_c}}\\ {{v_s}} \end{array}} \right\}_1} = \left\{ {\begin{array}{{20}{c}} i_c\\ 0 \end{array}} \right\},\,\,\,\,\text{and}\,\,{\left\{ {\begin{array}{{20}{c}} {{v_c}}\\ {{v_s}} \end{array}} \right\}_2} = \left\{ {\begin{array}{{20}{c}} 0\\ 0 \end{array}} \right\}.$$ We place a capacitor $C_L$ and load $R_L$ across the charge terminals of contacts 3 and 4. The spin terminals of contacts 3 and 4 are grounded. The boundary conditions of the contacts 3 and 4 are given by $${\left\{ {\begin{array}{{20}{c}} {{i_c}}\\ {{v_s}} \end{array}} \right\}_3} = \left\{ {\begin{array}{{20}{c}} 0\\ 0 \end{array}} \right\},\,\,\,\,\text{and}\,\,{\left\{ {\begin{array}{{20}{c}} {{i_c}}\\ {{v_s}} \end{array}} \right\}_4} = \left\{ {\begin{array}{{20}{c}} 0\\ 0 \end{array}} \right\}.$$ Sensitivity {#App_ACtoDC} =========== This section discusses the detailed derivation of the sensitivity model in Eq. .\ We start from Eq. with $\vec{m}(t)$ being the instantaneous magnetization of the LBM. and calculate the average as $$\label{averaging_fm} \begin{aligned} \langle V_{34} \rangle &= \dfrac{1}{T} \int_{0}^{T} V_{34}\,dt\\ &=\dfrac{1}{T}\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}}} \int_{0}^{T} \tanh\left(\dfrac{I_{12}(t)}{I_0}\right)I_{12}(t)\,dt. \end{aligned}$$ Note that the timed average of the random fluctuation in LBM is zero. Here, $G_B=1/R_B$. We apply an alternating current as input, given by $$I_{12}(t)=i_{c0}\sin\left(\dfrac{2\pi t}{T} \right),$$ The average ac input power applied to the channel with resistance $R_{12}$ is given by $$\begin{aligned} P_{in}&=\frac{1}{T}\int_{0}^{T}I_{12}^2(t) R_{12}\,dt\\ &=\frac{i_{c0}^2 R_{12}}{T}\int_{0}^{T}\sin^2 \left(\dfrac{2\pi t}{T}\right)\,dt\\ &=\left(\frac{i_{c0}}{\sqrt{2}}\right)^2R_{12}. \end{aligned}$$ Case I: $i_{c0}\gg I_0$ ----------------------- For $i_{c0}\gg I_0$, we get $\tanh\left(I_{12}(t)/I_0\right)\approx +1$ when $I_{12}(t)>0$ and $\tanh\left(I_{12}(t)/I_0\right)\approx -1$ when $I_{12}(t)<0$. Thus we have $$\tanh\left(\dfrac{I_{12}(t)}{I_0}\right) \times I_{12}(t)\approx \|I_{12}(t)\|,$$ and from Eq. , we get $$\label{averaging_fm2} \begin{aligned} \langle V_{34} \rangle &=\dfrac{1}{T}\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}}} \int_{0}^{T} \|I_{12}(t)\|\,dt\\ &=\dfrac{1}{T}\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}}} {i}_{c0} \int_{0}^{T} \left\|\sin \left(\dfrac{2\pi t}{T}\right)\right\|\,dt\\ &=\dfrac{1}{T}\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}}} {i}_{c0} \int_{0}^{\frac{T}{2}} \sin \left(\dfrac{2\pi t}{T}\right)\,dt\\ &+\dfrac{1}{T}\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}}} {i}_{c0} \int_{\frac{T}{2}}^{T} -\sin \left(\dfrac{2\pi t}{T}\right)\,dt\\ &=\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}}} \times \dfrac{2}{\pi}{i}_{c0}.\\ \end{aligned}$$ We write Eq. as $$\label{max_vfm2} \begin{aligned} \langle V_{34} \rangle=\dfrac{{\alpha \xi {p_0}{p_f}}}{{\pi{G_B}}} \times \dfrac{\sqrt{2}}{\sqrt{R_{12}}}\times \sqrt{P_{in}}, \end{aligned}$$ and the sensitivity is given by $$\begin{aligned} \dfrac{d\langle V_{34} \rangle}{dP_{in}} = \dfrac{{\alpha \xi {p_0}{p_f}}}{{\pi{G_B}}} \times \dfrac{1}{\sqrt{2R_{12}}}\times \dfrac{1}{\sqrt{P_{in}}}. \end{aligned}$$ The sensitivity for $i_{c0}\gg I_0$ decreases inversely proportional to $\sqrt{P_{in}}$. Sensitivity increases for decreasing $P_{in}$ and eventually saturates to a maximum value for $i_{c0}\ll I_0$. Case II: $i_{c0}\ll I_0$ ------------------------ For $i_{c0}\ll I_0$, we get $\tanh\left(I_{12}(t)/I_0\right)\approx I_{12}(t)/I_0$. Thus from Eq. , we get $$\label{averaging_fm1} \begin{aligned} \langle V_{34} \rangle &=\dfrac{1}{T}\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}I_0}} \int_{0}^{T} I_{12}^2(t)\,dt\\ &=\dfrac{1}{T}\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}}} \dfrac{{i}_{c0}^2}{I_0} \int_{0}^{T} \sin^2 \left(\dfrac{2\pi t}{T}\right)\,dt\\ &=\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}}} \times \dfrac{\left(i_{c0}/\sqrt{2}\right)^2}{I_0} . \end{aligned}$$ We write Eq. as $$\label{max_vfm} \langle V_{34} \rangle =\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}R_{12}I_0}} P_{in},$$ and the sensitivity is given as $$\dfrac{d\langle V_{34} \rangle}{dP_{in}}=\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}R_{12}I_0}},$$ which gives the maximum sensitivity in Eq. . Power Conversion Efficiency {#App_Efficiency} =========================== This section discusses the ac to dc power conversion efficiency and provides the details of derivation of Eq. .\ Under the no load condition ($R_L\rightarrow\infty$), we have the open circuit dc voltage from Eq. for $i_{c0}\ll I_0$ $$\langle V_{34} \rangle =\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}R_{12}I_0}} P_{in},$$ and from Eq. we know that for $i_{c0}\gg I_0$ $$\begin{aligned} \langle V_{34} \rangle=\dfrac{{\alpha \xi {p_0}{p_f}}}{{\pi{G_B}}} \times \dfrac{\sqrt{2}}{\sqrt{R_{12}}}\times \sqrt{P_{in}}. \end{aligned}$$ Under the short circuit condition ($R_L\rightarrow0$), we have the short circuit dc current $I_L\|_{R_L\rightarrow0}=\langle V_{34}\rangle/R_{34}$, where $R_{34}$ is the equivalent resistance between the LBM and the reference NM. The maximum power transferred to the load is given by $$P_{L,\max}=\dfrac{1}{4}\times V_L\|_{R_L\rightarrow\infty}\times I_L\|_{R_L\rightarrow0}=\dfrac{\langle V_{34}\rangle^2}{4R_{34}}.$$ which yields $$\begin{aligned} P_{L,\max}&=\left(\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}R_{12}I_0}} \right)^2 \dfrac{P_{in}^2}{4R_{34}}\;\;\;\text{for}\;\;i_{c0}\ll I_0\\ &=\left(\dfrac{{\alpha \xi {p_0}{p_f}}}{{\pi{G_B}}}\right)^2 \dfrac{P_{in}}{{2R_{12}R_{34}}} \;\;\;\text{for}\;\;i_{c0}\gg I_0. \end{aligned}$$ The ac to dc power conversion efficiency is given by $$\begin{aligned} \eta=\dfrac{dP_{L,\max}}{dP_{in}}&=\left(\dfrac{{\alpha \xi {p_0}{p_f}}}{{2{G_B}R_{12}I_0}} \right)^2 \dfrac{P_{in}}{2R_{34}}\;\;\;\text{for}\;\;i_{c0}\ll I_0\\ &=\left(\dfrac{{\alpha \xi {p_0}{p_f}}}{{\pi{G_B}}}\right)^2 \dfrac{1}{{2R_{12}R_{34}}} \;\;\;\text{for}\;\;i_{c0}\gg I_0. \end{aligned}$$ Note that $\eta$ increases with input ac power $P_{in}$ and reaches a maximum when $i_{c0}\gg I_0$ given by $$\begin{aligned} \eta_{\max}=\left(\dfrac{{\alpha \xi {p_0}{p_f}}}{{\pi{G_B}}}\right)^2 \dfrac{1}{{2R_{12}R_{34}}}. \end{aligned}$$
--- abstract: 'In 1961, P. Erdős, A. Ginzburg, and A. Ziv proved a remarkable theorem stating that each set of $2n-1$ integers contains a subset of size $n$, the sum of whose elements is divisible by $n.$ We will prove a similar result for pairs of integers, i.e., planar lattice points, usually referred to as Kemnitz’ conjecture.' address: 'Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany' author: - Christian Reiher title: 'On Kemnitz’ Conjecture Concerning Lattice Points in the Plane' --- Introduction ============ Denoting by $f(n, k)$ the minimal number $f,$ such that any set of $f$ lattice points in the $k$-dimensional Euclidean space contains a subset of cardinality $n,$ the sum of whose elements is divisible by $n,$ it was first proved by P. Erdős, A. Ginzburg, and A. Ziv [@EGZ], that $f(n, 1)=2n-1$. The problem, however, to determine $f(n, 2)$ turned out to be unexpectedly difficult: A. Kemnitz [@Kem83] conjectured it to equal $4n-3$ and knew, (1) that $4n-3$ is a rather straighforward lower bound, (2) that the set of all integers $n$ satisfying $f(n, 2)=4n-3$ is closed under multiplication and that it therefore suffices to prove this equation for prime values of $n$, and (3) that his assertion was correct for $n=2, 3, 5, 7$ and, consequently, also for every $n$ that is expressible as a product of these numbers. Linear upper bounds estimating $f(p, 2)$, where $p$ denotes any prime number, appeared for the first time in an article by N. Alon and M. Dubiner [@AD95] who proved $f(p, 2)\le 6p-5$ for all $p$ and $f(p, 2)\le 5p-2$ for large $p$. Later this was improved to $f(p, 2) \le 4p-2$ by L. Rónyai [@Ro00]. In the third section of this article we prove Kemnitz’ conjecture. Preliminary Results =================== Notational conventions {#notational-conventions .unnumbered} ---------------------- In the sequel the letter $p$ is always assumed to designate an odd prime number and congruence modulo $p$ is simply denoted by “$\equiv$”. Roman capital letters (such as $J, X,\ldots$) will always stand for finite sets of lattice points in the Euclidean plane. The sum of the elements of such a set, taken coordinatewise, will be indicated by a preposed “$\sum$”. Finally the symbol ${(n\,\|\,X)}$ expresses the number of $n$-subsets of $X$, the sum of whose elements is divisible by $p$. All propositions contained in this section are deduced without the use of combinatorial arguments from the following result due to Chevalley and Warning (see e.g., [@Schm76]). Let $P_1, P_2, \ldots, P_m\in F[x_1, \ldots,x_n]$ be some polynomials over a finite field $F$ of characteristic $p$. Provided that the sum of their degrees is less than $n$, the number $\Omega$ of their common zeros in $F^n$ is divisible by $p$. It is easy to see that $$\Omega\equiv\sum_{y_1, \ldots, y_n\in F}\quad\prod_{\mu=1}^{m} \bigl(1-P_{\mu}(y_1,\,\ldots\,y_n)^{q-1}\bigr)\,,$$ where $q=\|F\|$. Expanding the product and taking into account that $$\sum_{y\in F}y^r\equiv 0 \qquad \text{ holds whenever } 1\le r\le q-2\,,$$ we get indeed $\Omega\equiv 0$. \[cor:1\] If $\|J\|=3p-3$, then $1-{(p-1\,\|\,J)}-{(p\,\|\,J)}+{(2p-1\,\|\,J)}+{(2p\,\|\,J)}\equiv 0$. Let $J=\bigl\{(a_n, b_n)\,\big\|\, 1\le n\le 3p-3\bigr\}$ and apply the above theorem to $$\sum_{n=1}^{3p-3}x_n^{\,p-1}+x_{3p-2}^{\,p-1}\,, \sum_{n=1}^{3p-3}a_nx_n^{\,p-1} \qqand \sum_{n=1}^{3p-3}b_nx_n^{\,p-1}$$ considered as polynomials over the field containing $p$ elements. Their common zeros fall into two classes depending on whether $x_{3p-2}=0$ or not. The first class consists of $$1+(p-1)^p{(p\,\|\,J)}+(p-1)^{2p}{(2p\,\|\,J)}$$ solutions, whereas the second class includes $$(p-1)^p{(p-1\,\|\,J)}+(p-1)^{2p}{(2p-1\,\|\,J)}$$ solutions. The first of the following two assertions is proved quite analogously and entails the second one immediately. \[cor:2a\] If $\|J\|=3p-2$ or $\|J\|=3p-1$, then $1-{(p\,\|\,J)}+{(2p\,\|\,J)}\equiv 0$. \[cor:2b\] If $\|J\|=3p-2$ or $\|J\|=3p-1$, then ${(p\,\|\,J)}=0$ implies ${(2p\,\|\,J)}\equiv -1$. Now we come to an important statement due to N. Alon and M. Dubiner [@AD95]. \[cor:3\] If $J$ contains exactly $3p$ elements whose sum is $\equiv(0,0)$, then ${(p\,\|\,J)}>0$. Let $\mathfrak{A}\in J$ be arbitrary. Arguing indirectly we assume that ${(p\,\|\,J)}=0$. This obviously implies ${(p\,\|\,J-\mathfrak{A})}=0$ and owing to $\|J-\mathfrak{A}\|=3p-1$ the above Corollary \[cor:2b\] yields ${(2p, J-\mathfrak{A})\equiv -1}$. So in particular we have ${(2p\,\|\,J-\mathfrak{A})}>0$ and the condition $\sum J\equiv (0,0)$ entails indeed ${(p\,\|\,J)}={(2p\,\|\,J)}\ge {(2p\,\|\,J-\mathfrak{A})}>0$. The next two statements are similar to Corollary \[cor:2a\] and may also be proved in the same manner. \[cor:4\] If $\|X\|=4p-3,$ then 1. \[it:4a\] $-1+{(p\,\|\,X)}-{(2p\,\|\,X)}+{(3p\,\|\,X)}\equiv 0$ 2. \[it:4b\] and ${(p-1\,\|\,X)}-{(2p-1\,\|\,X)}+{(3p-1\,\|\,X)}\equiv 0$. \[cor:5\] If $\|X\|=4p-3$, then $3-2{(p-1\,\|\,X)}-2{(p\,\|\,X)}+{(2p-1\,\|\,X)}+{(2p\,\|\,X)}\equiv 0$. Corollary \[cor:1\] implies $$\sum_I \bigl[ 1- {(p-1\,\|\,I)}-{(p\,\|\,I)}+{(2p-1\,\|\,I)}+{(2p\,\|\,I)}\bigr] \equiv 0\,,$$ where the sum is extended over all $I\subseteq X$ of cardinality $3p-3$. Analysing the number of times each set is counted one obtains $$\begin{aligned} \binom{4p-3}{3p-3}&-\binom{3p-2}{2p-2}{(p-1\,\|\,X)}-\binom{3p-3}{2p-3}{(p\,\|\,X)} \\ &+ \binom{2p-2}{p-2}{(2p-1\,\|\,X)}+\binom{2p-3}{p-3}{(2p\,\|\,X)}\equiv 0\,.\end{aligned}$$ The reduction of the binomial coefficients modulo $p$ leads directly to the claim. Resolution of Kemnitz’ Conjecture ================================= If $\|X\|=4p-3$ and ${(p\,\|\,X)}=0$, then ${(p-1\,\|\,X)}\equiv {(3p-1\,\|\,X)}$. Let $\chi$ denote the number of partitions $X=A\cup B\cup C$ satisfying $$\|A\|=p-1, \qquad \|B\|=p-2, \qquad \|C\|=2p\,,$$ and moreover $$\sum A\equiv (0,0), \qquad \sum B\equiv \sum X, \qquad \sum C\equiv (0,0)\,.$$ To determine $\chi$, at least modulo $p$, we first run through all admissible $A$ and employing Corollary \[cor:2b\] we count for each of them how many possibilities for $B$ are contained in its complement, thus getting $$\chi\equiv\sum_A{(2p\,\|\,X-A)}\equiv\sum_A -1\equiv -{(p-1\,\|\,X)}\,.$$ Working the other way around we infer similarly $$\chi\equiv\sum_B{(2p\,\|\,X-B)}\equiv\sum_{X-B} -1\equiv -{(3p-1\,\|\,X)}\,.$$ Therefore indeed, by counting the same entities twice, ${(p-1\,\|\,X)}\equiv {(3p-1\,\|\,X)}$. Any choice of $4p-3$ lattice–points in the plane contains a subset of cardinality $p$ whose centroid is a lattice-point as well. Adding up the congruences obtained in the Corollaries \[cor:4\]\[it:4a\], \[cor:4\]\[it:4b\], \[cor:5\], and the previous lemma one deduces $2-{(p\,\|\,X)}+{(3p\,\|\,X)}\equiv 0$. Since $p$ is odd, this implies that ${(p\,\|\,X)}$ and ${(3p\,\|\,X)}$ cannot vanish simultaneously which in turn yields our assertion ${(p\,\|\,X)}\ne 0$ via Corollary \[cor:3\] As Kemnitz [@Kem83] remarked, for $p=2$ the above result is an easy consequence of the box-principle. Since according to fact (1) mentioned in the introduction the general statement $f(n, 2)=4n-3$ (for every positive integer $n$) follows immediately from the special case where $n$ is a prime number, we have thereby proved Kemnitz’ conjecture.
--- abstract: 'For a large class of linear neutral type systems the problem of eigenvalues and eigenvectors assignment is investigated, i.e. finding the system which has the given spectrum and almost all, in some sense, eigenvectors.' address: - \| Institute of Mathematics, University of Szczecin\ Wielkopolska 15, 70-451 Szczecin, Poland - \| IRCCyN/École des Mines de Nantes\ 4 rue Alfred Kastler, BP 20722, 44307 Nantes, France - \| Institute of Mathematics, University of Szczecin\ Wielkopolska 15, 70-451 Szczecin, Poland author: - 'K. V. Sklyar' - 'R. Rabah' - 'G. M. Sklyar' title: On spectral assignment for neutral type systems --- [^1] [^2] [^3] Introduction ============ One of central problems in control theory is the spectral assignment problem. This question is well investigated for linear finite dimensional systems. It is important to emphasize that the assignment of eigenvalues is not sufficient in several cases. One needs also the assignment of eigenvectors or of the geometric eigenstructure. For infinite dimensional problems (delay systems, partial derivative equations) the problem is much more complicated. Our purpose is to investigate this kind of problems for a large class of neutral type systems given by the equation $$\label{rownanie1.1} \dot{z}(t)=A_{-1}\dot{z}(t-1) +\int^{0}_{-1}A_2(\theta)\dot{z}(t+\theta){\mathrm{d}}\theta +\int^{0}_{-1}A_3(\theta)z(t+\theta){\mathrm{d}}\theta,$$ where $z(t) \in {\mathbb{R}}^n$ and $A_{-1}, A_2, A_3$ are $n\times n$ matrices. The elements of $A_2$ and $A_3$ taking values in $L_2(-1,0)$. The neutral type term $A_{-1}\dot{z}(t-1)$ consists on a simple delay, while the other include as multiple as distributed delays. The behavior of such systems can be described mainly by the algebraic and geometric properties of the spectrum of the matrix $A_{-1}$ (cf. [@Rabah_Sklyar_Rezounenko_2003; @Rabah_Sklyar_Rezounenko_2005]). It is well known that then spectral properties of this system are described by the characteristic matrix $\Delta(\lambda)$ given by $$\Delta(\lambda)= \lambda I-\lambda {\mathrm{e}}^{-\lambda}A_{-1}-\int^{0}_{-1}\lambda {\mathrm{e}}^{{\lambda}\theta}A_2(\theta){\mathrm{d}}\theta- \int^{0}_{-1}{\mathrm{e}}^{{\lambda}\theta}A_3(\theta){\mathrm{d}}\theta.$$ The eigenvalues are roots of the equation $\det\Delta(\lambda)=0$. The eigenvectors of the system (more precisely of the functional operator model of the system) are expressed through the matrix $\Delta(\lambda_k)$, where $\lambda_k$ is an eigenvalue. In fact the problem of an assignment of an infinite number of eigenvalues and eigenvectors is reduced to a problem of assignment of singular values and degenerating vectors of an entire matrix value function $\Delta(\lambda)$. It is remarkable [@Rabah_Sklyar_Rezounenko_2005] that the roots of $\det\Delta(\lambda)=0$ are quadratically close to a fixed set of complex number which are the logarithm of eigenvalues of the matrix $A_{-1}$. Moreover, the degenerating vectors of $\Delta(\lambda_k)$ are also quadratically close to the eigenvectors of the matrix $A_{-1}$. In this paper we investigate an inverse problem:\ What conditions must satisfy a sequence of complex numbers $\{\lambda\}$ and a sequence of vectors $\{v\}$ in order to be a sequence of roots of the characteristic equation $\det\Delta (\lambda)=0$ and a sequence of degenerating vectors of the characteristic matrix $\Delta(\lambda)$ of equation respectively for some choice of matrices $A_{-1},A_{2}(\theta), A_{3}(\theta)$ ?\ One of the possible application of this problem is to investigate a vector moment problem via the solution of the exact controllability property for a corresponding neutral type system by a relation devlopped in [@Rabah_Sklyar_2007].\ The present paper is a detailed version of the short note published in Comptes Rendus Mathematiques [@Sklyar_Rabah_Sklyar_2013]. Operator form of perturbation ============================= We consider neutral type systems of the form $$\label{eq:1} \dot{z}(t)=A_{-1}\dot{z}(t-1) +\int^{0}_{-1}A_2(\theta)\dot{z}(t+\theta){\mathrm{d}}\theta +\int^{0}_{-1}A_3(\theta)z(t+\theta){\mathrm{d}}\theta,$$ where $A_{-1}$ is a constant $n\times n$ matrix, $A_{2},A_{3}$ are $n\times n$ matrices whose elements belong to $L_{2}[-1,0].$ As it is shown in [@Rabah_Sklyar_Rezounenko_2003], [@Rabah_Sklyar_Rezounenko_2005] this system can be rewritten in the operator form $$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\binom{y}{z_t(\cdot)}=\mathcal{A}\binom{y}{z_t(\cdot)},$$ where $\mathcal{A} : D(\mathcal{A})\to M_{2}=\mathbb{C}^n\times L_{2}([-1,0],\mathbb{C}^n)$, $$D(\mathcal{A})=\left\{\binom{y}{\varphi(\cdot)}\mid \varphi(\cdot)\in H^1([-1,0],\mathbb{C}^n),\ y=\varphi(0)-A_{-1}\varphi(-1)\right\}\subset M_2,$$ and the operator $\mathcal{A}$ is given by formula $$\mathcal{A}\binom{y}{\varphi(\cdot)}=\binom{\int^{0}_{-1}A_2(\theta)\dot{\varphi}(\theta){\mathrm{d}}\theta +\int^{0}_{-1}A_3(\theta)\varphi(\theta){\mathrm{d}}\theta}{\frac{{\mathrm{d}}\varphi}{{\mathrm{d}}\theta}(\cdot)}.$$ This operator is noted $\widetilde{\mathcal{A}}$ instead of $\mathcal{A}$ if $A_2(\theta)=A_3(\theta)\equiv0$. The operator $\widetilde{\mathcal{A}}$ is defined on the same domain $D({\mathcal{A}})$. One can consider that the state operator ${\mathcal{A}}$ is a perturbation of the operator $\widetilde{\mathcal{A}}$, namely $$\mathcal{A}\binom{y}{\varphi(\cdot)}=\widetilde{\mathcal{A}}\binom{y}{\varphi(\cdot)}+ \binom{\int^{0}_{-1}A_2(\theta)\dot{\varphi}(\theta){\mathrm{d}}\theta +\int^{0}_{-1}A_3(\theta)\varphi(\theta){\mathrm{d}}\theta}{0}.$$ Let $\mathcal{B}_0:\mathbb{C}^n\to M_2$ be given by $$\mathcal{B}_0y=\binom{y}{0},$$ and $\mathcal{P}_0:D(\mathcal{A})\to\mathbb{C}^n$ by $$\label{rownanie1.2} \mathcal{P}_0\binom{y}{\varphi(\cdot)}=\int^{0}_{-1}A_2(\theta)\dot{\varphi}(\theta){\mathrm{d}}\theta +\int^{0}_{-1}A_3(\theta)\varphi(\theta){\mathrm{d}}\theta.$$ Then $\mathcal{A}=\widetilde{\mathcal{A}}+\mathcal{B}_0\mathcal{P}_0$. Denote by $X_{\mathcal{A}}$ the set $D(\mathcal{A})$ endowed with the graph norm. Let us show that $\mathcal{P}_0$ browses the set of all linear bounded operators $\mathcal{L}(X_{\mathcal{A}},\mathbb{C}^n)$ as $A_2(\cdot),A_3(\cdot)$ run over the set of $n\times n$ matrices with components from $L_2[-1,0].$ Indeed, an arbitrary linear operator $Q$ from $\mathcal{L}(X_{\mathcal{A}},\mathbb{C}^n)$ can be presented as $$\begin{aligned} Q\binom{y}{\varphi(\cdot)}& = &Q_1y +Q_2{\varphi(\cdot)}\\ &=&Q_1(\varphi(0)-A_{-1}\varphi(-1))+ \int^{0}_{-1}\widehat{A}_2(\theta)\dot{\varphi}(\theta){\mathrm{d}}\theta +\int^{0}_{-1}\widehat{A}_3(\theta)\varphi(\theta){\mathrm{d}}\theta,\end{aligned}$$ where $\widehat{A}_2(\cdot),\widehat{A}_3(\cdot)$ are $(n\times n)$-matrices with component from $L_2[-1,0]$ and $Q_1$ is a $(n\times n)$ matrix. Let us observe that $$\begin{aligned} \varphi(-1)&=&\int^{0}_{-1}\theta\dot{\varphi}(\theta){\mathrm{d}}\theta+\int^{0}_{-1}\varphi(\theta){\mathrm{d}}\theta,\cr \varphi(0)&=&\int^{0}_{-1}(\theta+1)\dot{\varphi}(\theta){\mathrm{d}}\theta+\int^{0}_{-1}\varphi(\theta){\mathrm{d}}\theta \end{aligned}$$ and denote $$\begin{aligned} A_2(\theta)&=&\widehat{A}_2(\theta)+(\theta+1)Q_1-\theta Q_1A_{-1},\cr A_3(\theta)&=&\widehat{A}_3(\theta)+Q_1-Q_1A_{-1}.\end{aligned}$$ Then, with these notations, the operator $Q$ may be written as $$Q\binom{y}{\varphi(\cdot)}=\int^{0}_{-1}A_2(\theta)\dot{\varphi}(\theta){\mathrm{d}}\theta+ \int^{0}_{-1}A_3(\theta){\varphi}(\theta){\mathrm{d}}\theta.$$ Hence formula describes all the operators from $\mathcal{L}(X_{\mathcal{A}},\mathbb{C}^n).$ An equation for eigenvalues and eigenvectors of the characteristic matrix (spectral equation) ============================================================================================= Consider the operator $\mathcal{A}=\widetilde{\mathcal{A}}+\mathcal{B}_0\mathcal{P}_0$ and assume that $\lambda_0$ is an eigenvalue of $\mathcal{A}$ and $x_0$ is a corresponding eigenvector, i.e. $$\label{rownanie2.1} (\widetilde{\mathcal{A}}+\mathcal{B}_0\mathcal{P}_0)x_0=\lambda_0 x_0.$$ Let us assume further that $\lambda_0$ does not belong to spectrum of $\widetilde{\mathcal{A}}$ and denote by $R(\widetilde{\mathcal{A}},\lambda_{0})=(\widetilde{\mathcal{A}}-\lambda_{0}I)^{-1},$ with this notation reads as $$\label{rownanie 2.2} x_0+R(\widetilde{\mathcal{A}},\lambda_0)\mathcal{B}_0\mathcal{P}_0 x_0=0.$$ Let us notice that $v_0=\mathcal{P}_0 x_0\neq 0,$ because $\lambda_0\notin\sigma(\widetilde{\mathcal{A}}).$ Then applying operator $\mathcal{P}_0$ to the left hand side of we get $$v_0+\mathcal{P}_0R(\widetilde{\mathcal{A}},\lambda_0)\mathcal{B}_0 v_0=0$$ This equality means that $\lambda_0$ is a point of singularity of the matrix-valued function $F(\lambda)=I+\mathcal{P}_0R(\widetilde{\mathcal{A}},\lambda)\mathcal{B}_0$ and $v_0$ is a vector degenerating $F(\lambda_0)$ from the right. Let $w_0^\ast$ be a non-zero row such that $$\label{rownanie2.3} w_0^\ast F(\lambda_0)=0.$$ In order to describe the vector $w_0$, let us find first another form for the matrix $F(\lambda)$. For any $v\in\mathbb{C}^n$ we denote $$R(\widetilde{\mathcal{A}},\lambda)\mathcal{B}_0v=\binom{y}{\varphi(\cdot)}, \quad\lambda\notin\sigma(\widetilde{\mathcal{A}}),$$ Then $$(\widetilde{\mathcal{A}}-\lambda I)\binom{y}{\varphi(\cdot)}=\mathcal{B}_0v=\binom{v}{0}.$$ This gives $$v=-\lambda y,\qquad \frac{{\mathrm{d}}\varphi(\theta)}{{\mathrm{d}}\theta}-\lambda\varphi(\theta)=0,$$ and, as $\binom{y}{\varphi(\cdot)}\in D(\widetilde{\mathcal{A}})$, we obtain $y=\varphi(0)-A_{-1}\varphi(-1).$ Therefore $$\varphi(\theta)=D{\mathrm{e}}^{\lambda\theta},\quad D\in\mathbb{C}^n,$$ and $$v=-\lambda(I-{\mathrm{e}}^{-\lambda}A_{-1})D.$$ Since the matrix $(I-{\mathrm{e}}^{-\lambda}A_{-1})$ is invertible $(\lambda\notin\sigma(\widetilde{\mathcal{A}})),$ then we get $$\varphi(\theta)=-\frac{{\mathrm{e}}^{\lambda\theta}}{\lambda}(I-{\mathrm{e}}^{-\lambda}A_{-1})^{-1}v,$$ and hence $$R(\widetilde{\mathcal{A}},\lambda)\mathcal{B}_0v= \begin{pmatrix}{y}\cr {-\dfrac{{\mathrm{e}}^{\lambda\theta}}{\lambda}(I-{\mathrm{e}}^{-\lambda}A_{-1})^{-1}v} \end{pmatrix}.$$ This formula and implies $$\begin{aligned} \lefteqn{v+\mathcal{P}_{0}R(\widetilde{\mathcal{A}},\lambda)\mathcal{B}_{0}v=}\\ && =v-\left(\int^{0}_{-1}{\mathrm{e}}^{{\lambda}\theta}A_2(\theta){\mathrm{d}}\theta- \int^{0}_{-1}\frac{{\mathrm{e}}^{{\lambda}\theta}}{\lambda}A_3(\theta){\mathrm{d}}\theta\right)(I-{\mathrm{e}}^{-\lambda}A_{-1})^{-1}v\end{aligned}$$ and hence $$F(\lambda)v=\Delta(\lambda)\frac{(I-{\mathrm{e}}^{-\lambda}A_{-1})^{-1}}{\lambda}v,$$ where $$\Delta(\lambda)=\lambda I-\lambda {\mathrm{e}}^{-\lambda}A_{-1}-\int^{0}_{-1}\lambda {\mathrm{e}}^{{\lambda}\theta}A_2(\theta){\mathrm{d}}\theta- \int^{0}_{-1}{\mathrm{e}}^{{\lambda}\theta}A_3(\theta){\mathrm{d}}\theta$$ is the characteristic matrix of equation . Thus, the equality for $\lambda_{0}\notin\sigma(\widetilde{\mathcal{A}})$ is equivalent to $${w_{0}}^{}\Delta(\lambda_{0})=0.$$ Summarizing we obtain the following \[statement1\] Let $\lambda_{0}$ do not belong to $\sigma(\widetilde{\mathcal{A}}).$ Then the pair $(\lambda_{0},w_{0}),\:w_{0}\in\mathbb{C}^n,\:w_{0}\neq {0},$ satisfies equation if and only if $\lambda_{0}$ is a root of the characteristic equation $$\det\Delta(\lambda)=0$$ and ${w_{0}}^{}$ is a row-vector degenerating $\Delta(\lambda_{0})$ from the left, i.e. ${w_{0}}^{}\Delta(\lambda_{0})=0.$ Thus, one can consider the equation $w^F(\lambda)=0$ as an equation whose roots $(\lambda,w)=(\lambda_{0},w_0)$ describe all eigenvalues and (right) eigenvectors of the characteristic matrix $\Delta(\lambda).$ A component-wise representation of spectral equation ==================================================== We recall spectral properties of operators $\widetilde{\mathcal{A}}$ and $\widetilde{\mathcal{A}}^\ast$ obtained in [@Rabah_Sklyar_Rezounenko_2003; @Rabah_Sklyar_Rezounenko_2005]. We will assume that the matrix $A_{-1}$ has a simple non-zero eigenvalues $\mu_{1},\mu_{2},\mu_{3},\ldots,\mu_{n}.$ In this case the spectrum $\sigma(\widetilde{\mathcal{A}})$ consists of simple eigenvalues which we denote by $$\widetilde\lambda^{m}_{k}=\ln\left\|\mu_{m}\right\|+{\mathrm{i}}(\operatorname{Arg}\mu_{m}+2\pi k),\quad m=1,\ldots,n,\ \!k\in\mathbb{Z},$$ and of eigenvalue $\widetilde{\lambda}_{0}=0.$ First assume $\mu_{m}\neq 1,\ m=1,\ldots,n.$ Then the corresponding to ${\widetilde{\lambda}^{m}_{k}}$ eigenvectors are of the form $$\label{rownanie3.1} \widetilde\varphi^{m}_{k}=\binom{0}{{{\mathrm{e}}^{{\widetilde{\lambda}^{m}_{k}}\theta}}y_{m}}, \quad k\in\mathbb{Z},\:m=1,\ldots,n,$$ where $y_{1},\ldots,y_{n}$ are eigenvectors of $A_{-1}$ corresponding to $\mu_{1},\mu_{2},\ldots,\mu_{n}$. The eigenspace corresponding to $\tilde{\lambda}_{0}=0$ is $n$ dimensional and its basis is $$\label{rownanie3.2} \widetilde\varphi^{0}_{j}=\binom{(1-\mu_{j})y_{j}}{y_{j}}, \quad j=1,\ldots,n.$$ If some $\mu_{m}$ say $\mu_{1}$ equals 1, then $\tilde{\lambda}^{1}_{0}=\tilde{\lambda}_{0}=0.$ In that case the eigenspace corresponding to 0 is $(n+1)$-dimensional and its basis consists of $n$ eigenvectors $\widetilde\varphi^{0}_{j},\:j=1,\ldots,n,$ given by , and one rootvector $$\widetilde\varphi^{1}_{0}=\binom{y_{1}}{\theta y_{1}}, \qquad \widetilde{\mathcal{A}}\widetilde\varphi^{1}_{0} =\widetilde\varphi^{0}_{1}.$$ All the vectors $\widetilde\varphi^{m}_{k}$ given by , except of $\widetilde\varphi^{0}_{1},$ are still eigenvectors of $\widetilde{\mathcal{A}}$ and correspond to eigenvalues ${\widetilde \lambda^{m}_{k}}.$ In both cases the family $\Phi=\left\{{\widetilde\varphi^{m}_{k}}\right\}\cup\left\{{\widetilde\varphi^{0}_{j}}\right\}$ forms a Riesz basis in the space $M_2.$ Denote by $\Psi=\left\{{\widetilde\psi^{m}_{k}}\right\}\cup\left\{{\widetilde\psi^{0}_{j}}\right\}$ the bi-orthogonal basis to $\Phi$. Then $$\label{rownanie3.3} {\widetilde\psi^{m}_{k}}= \begin{pmatrix} {z_{m}}/{\overline{\widetilde\lambda_k^m}} \\[1.5ex] {{\mathrm{e}}^{-\overline{\widetilde{\lambda}_{k}^{m}}\theta}z_{m}} \end{pmatrix}, \qquad m=1,\ldots,n, \ k\in\mathbb{Z},\ \widetilde{\lambda}^{1}_{0}\neq 0,$$ where $\overline{\widetilde{\lambda}}$ is the complex conjugate of $\widetilde{\lambda}$ and $${\widetilde\psi^{1}_{0}}=\binom{z_{1}}{0}$$ if $\widetilde{\lambda}^{1}_0=0,$ here $z_{m}$ are eigenvectors of matrix $A_{-1}^{}$ such that $$\left\langle y_{i},z_{j}\right\rangle=\delta_{i j}.$$ It is easy to see that $\widetilde\psi^{m}_{k}$ are eigenvectors of the operator $\mathcal{A}^{}$ corresponding to eigenvalues $\overline{\widetilde\lambda_k^m}$. Our nearest goal is to rewrite the matrix $F(\lambda_{0})$ in the basis $\Phi$. Let us first rewrite the expression $R(\widetilde{\mathcal{A}},\lambda_{0})\mathcal{B}_0$. We present $\mathcal{B}_0$ as a matrix $\mathcal{B}_0=(b_1^0,b_2^0,\ldots,b_n^0)$ with infinite columns $b_i^0$ which are vectors from $M_2$ of the form $$b_i^0=\binom{e_i}{0}, \qquad i=1,\dots, n,$$ where $e_i$ is the canonical basis of $\mathbb{C}^n$. Then $$b_i^0=\sum_{k,m}\left\langle b_i^0,\widetilde{\psi}_{k}^{m}\right\rangle\widetilde{\varphi}_{k}^{m} +\sum_j\left\langle b_i^0,\widetilde{\psi}_{j}^{0}\right\rangle\widetilde{\varphi}_{j}^{0}, \qquad i=1,\dots, n,$$ and in the case $\mu_m\neq1,\ m=1,\ldots,n$, we have $$R(\widetilde{\mathcal{A}},\lambda_0)b_i^0=\sum_{k,m}\frac{1}{\widetilde\lambda_k^m -\lambda_0}\left\langle b_i^0,\widetilde\psi_{k}^{m}\right\rangle\widetilde\varphi_{k}^{m} -\sum_j\frac{1}{\lambda}_0\left\langle b_i^0,\widetilde\psi_{j}^{0}\right\rangle\widetilde\varphi_{j}^{0}, \quad i=1,\dots, n.$$ Taking into account the form of eigenvectors we find that for all $m=1,\dots,n$ $$\left(\left\langle b_1^0,\widetilde\psi_{k}^{m}\right\rangle,\ldots,\left\langle b_n^0, \widetilde\psi_{k}^{m}\right\rangle\right)=\frac{1}{\widetilde{\lambda}_k^m}(\left\langle e_1,z_m\right\rangle,\ldots, \left\langle e_n,z_m\right\rangle)=z_m^{}/\widetilde{\lambda}_k^m.$$ So we obtain $$\label{rownanie3.4} R(\widetilde{\mathcal{A}},\lambda_0)\mathcal{B}_0 =\sum_{k,m}\frac{1}{\widetilde{\lambda}_k^m -\lambda_0}\times \frac{1}{\widetilde{\lambda}_k^m}\widetilde\varphi^{m}_{k}z_m^{} -\sum_{j}\frac{1}{\lambda_0}\widetilde\varphi_{j}^{0} \left(\left\langle b_1^0,\widetilde\psi_{j}^{0}\right\rangle,\ldots,\left\langle b_n^0,\widetilde\psi_{j}^{0} \right\rangle\right),$$ if all the numbers $\widetilde\lambda_k^m\neq 0.$ If $\mu_1=1$ and $\widetilde\lambda_0^1=0$ then in the sum the term corresponding to $k=1,\:m=1,$ is replaced by $$\left(\frac{1}{\widetilde\lambda_0^1-\lambda_0}\widetilde\varphi_{0}^{1} -\frac{1}{({\widetilde\lambda_0^1-\lambda_0})^{2}}\widetilde\varphi_{1}^{0}\right)z_1^.$$ In what follows, we shall use the notation $$\widetilde\beta_k^m= \left\{ \begin{array}{ll} \widetilde\lambda_k^m, & \widetilde\lambda_k^m\neq 0,\\ 1, & \widetilde\lambda_k^m=0, \end{array} \right. \qquad m=1,\dots,n, \ k \in {\mathbb{Z}}.$$ Let us observe that the expression $w_0^\mathcal{P}_0$ is a linear bounded functional on the space $X_{\mathcal{A}}$, i.e. $w_0^\mathcal{P}_0\in\mathcal{L}(X_{\mathcal{A}},\mathbb{C})$. The representation of $w_0^$ in the basis $z_j$ is as follows: $$w_0^=\sum_{j}\alpha_jz_j^.$$ Consider now $n$ functionals $z_j^\mathcal{P}_0\in\mathcal{L}(X_{\mathcal{A}},\mathbb{C}),\:j=1,\ldots,n$. One can decompose them in the basis $\Psi$: $$z_j^\mathcal{P}_0=\sum_{k,m}{p}_{k,m}^j\widetilde{\psi}_k^m+\sum_{i} {\widetilde{p}}_i^j\widetilde{\psi}_i^0,$$ where $$\label{rownanie3.5} \sum_{k}\left\|\frac{p_{k,m}^j}{\widetilde\beta_k^m}\right\|^{2}<\infty, \qquad m=1,\dots,n.$$ In the sequel, we shall assume $\widetilde{p}_i^j=0,\ i=1,\ldots,n$. This means that from now we consider perturbations $\mathcal{P}_0$ satisfying the condition $$\label{rownanie3.6} \int^{0}_{-1}A_3(\theta){\mathrm{d}}\theta=0.$$ Then we have $$w_0^\mathcal{P}_0=\sum_{j}{{\alpha}_j} z_j^\mathcal{P}_0=\sum_{j} {\alpha}_j\sum_{k,m}{p}_{k,m}^j\widetilde{\psi}_k^m.$$ From this relation and expression we obtain the equality $$\begin{aligned} w_0^\mathcal{P}_0\mathcal{R}(\widetilde{\mathcal{A}},\lambda_0)\mathcal{B}_0& =&\sum_m\left\langle\sum_k \frac{1}{\widetilde{\beta}_k^m}\times \frac{1}{\widetilde\lambda_k^m-\lambda_0}\widetilde{\varphi}_k^m, \sum_{j}\overline{{\alpha}_j}\sum_{k_0,m_0}\overline{\beta}_{k_0,m_0}^j\widetilde{\psi}_{k_0}^{m_0}\right\rangle z_m^\\ &=&\sum_{j}\alpha_j\sum_{k_0,m_0} \frac{p_{k,m}^j}{\widetilde{\beta}_k^m}\times\frac{1}{\widetilde\lambda_k^m- \lambda_0}z_m^.\end{aligned}$$ With these notations, the equation reads $$0=w_0^\left(I+\mathcal{P}_0\mathcal{R}(\widetilde{\mathcal{A}}, \lambda_0)\mathcal{B}_0\right)=\sum_{m}\alpha_m z_m^+\sum_{m} \left(\sum_{j,k}\alpha_j\frac{p_{k,m}^j}{\widetilde{\beta}_k^m}\times \frac{1}{\widetilde\lambda_k^m-\lambda_0}\right) z_m^.$$ Thus, the condition for a pair $(\lambda_0,w_0)$ to satisfy the spectral equation can be rewritten in the form of the following system of $n$ equations: $$\label{rownanie3.7} \alpha_m=-\sum_{\stackrel{k\in\mathbb{Z}}{j=1,\ldots,n}}\alpha_j\left(\frac{p_{k,m}^j}{\widetilde{\beta}_k^m} \times \frac{1}{\widetilde\lambda_k^m-\lambda_0}\right), \quad m=1,\ldots,n,$$ where for any fixed couple $m,j$, the needed* $n$-tuple $\left\{p_{k,m}^j\right\}$ satisfies . Conditions for spectral assignment ================================== Now we discuss the following question: What conditions must satisfy a sequence of complex numbers $\{\lambda\}$ and a sequence of vectors $\{v\}$ in order to be a sequence of roots of the characteristic equation $\det\Delta (\lambda)=0$ and a sequence of degenerating vectors of the characteristic matrix $\Delta(\lambda)$ of equation respectively for some choice of matrices $A_{-1},A_{2}(\theta), A_{3}(\theta)$? We will assume that the corresponding operator $\mathcal{A}$ has simple eigenvalues only. Let us remember that we assumed earlier that all eigenvalues of matrix $A_{-1}$ are also simple. Then one can enumerate those eigenvalues as $\left\{\lambda_k^m\right\}\cup\left\{\lambda_j^0\right\},\:m,j=1,\ldots,n;\:k\in\mathbb{Z},$ where (see [@Rabah_Sklyar_Rezounenko_2005 Theorem 1]) the sequence $\left\{\lambda_k^m\right\}$ satisfies $$\label{rownanie4.1} \sum_{k,m}\left\|\lambda_k^m-\widetilde{\lambda}_k^m\right\|^{2}<\infty.$$ Denote by $\left\{\varphi_k^m\right\}\cup\left\{\varphi_j^0\right\},\:m,j=1,\ldots,n;\:k\in\mathbb{Z},$ corresponding eigenvectors of $\mathcal{A}.$ Then (see [@Rabah_Sklyar_Rezounenko_2005 Lemma 13, Theorem 15]) these vectors form a Riesz basis in $M_2$ which is quadratically close to the basis $\left\{\widetilde{\varphi}_k^m\right\}\cup\left\{\widetilde{\varphi}_j^0\right\}$ if we assume that the corresponding elements have the same norm. Eigenvectors $\left\{\varphi\right\}$ has the form $$\mathbf{\varphi}= \left( \begin{array}{ccc} v-{\mathrm{e}}^{-\lambda}A_{-1}v\\ {\mathrm{e}}^{\lambda\theta}v\\ \end{array} \right)$$ with $\Delta(\lambda)v=0.$ Therefore the fact that the basis $\left\{\varphi\right\}$ and $\left\{\widetilde{\varphi}\right\}$ are quadratically close implies the condition $$\label{rownanie4.2} \sum_k{\left\\|v_k^m-y_m\right\\|}^2<\infty,\quad m=1,\ldots,n,$$ where $\Delta(\lambda_k^m)v_k^m=0,\:m=1,\ldots,n;\:k\in\mathbb{Z}$, and $y_m$ are eigenvectors of $A_{-1}$ corresponding to $\mu_{m}$ as in (\[rownanie3.1\]). If we apply the same arguments for the dual system $$\label{rownanie4.3} \dot{z}(t)=A_{-1}^\dot{z}(t-1)+\int^{0}_{-1}A_2^(\theta)\dot{z}(t+\theta){\mathrm{d}}\theta+ \int^{0}_{-1}A_3^(\theta)z(t+\theta){\mathrm{d}}\theta,$$ then we obtain the symmetric condition $$\label{rownanie4.4} \sum_k{\left\\|w_k^m-z_m\right\\|}^2<\infty,\quad m=1,\ldots,n,$$ where $(w_k^m) ^\Delta(\lambda_k^m)=0,\:m=1,\ldots,n;\:k\in\mathbb{Z}.$ Our further goal is to show that conditions , or , are almost sufficient for couples of sequences $\left\{\lambda\right\}, \left\{w\right\}$ or $\left\{\lambda\right\}, \left\{v\right\}$ to be spectral ones for the system . Let us consider a sequence of different complex numbers $\left\{\lambda_{k_0}^{m_0}\right\},\:m_0=1,\ldots,n;$ $k_0\in\mathbb{Z},$ satisfying . We also assume that the index numbering of $\left\{\lambda\right\}$ is such that if $\lambda_{k_0}^{m_0}=\widetilde{\lambda}_k^m$ then $m_0=m,\:k_0=k.$ To begin with, however, we put $\lambda_{k_0}^{m_0}\neq \widetilde{\lambda}_k^m,$ for all $k,k_0\in\mathbb{Z},\:m, m_0=1,\ldots,n.$ Let now $\left\{\lambda_{k_0}^{m_0}\right\}$ be simple eigenvalues of operator $\mathcal{A}=\widetilde{\mathcal{A}}+\mathcal{B}_0\mathcal{P}_0,$ where $\mathcal{P}_0$ is given by , in which matrix $A_3(\theta)$ satisfies . Then $\Delta(\lambda_{k_0}^{m_0})=0$ and let $\left\{w_{k_0}^{m_0}\right\}$ be a sequence of the left degenerating vectors of $\Delta(\lambda_{k_{0}}^{m_{0}}),$ i.e. $$(w_{k_0}^{m_0})^\Delta(\lambda_{k_0}^{m_0})=0.$$ We assume that the sequence $w_{k_{0}}^{m_{0}}$ satisfies . For all indices $m_0=1,\ldots,n;$ $k_0\in\mathbb{Z},$ consider decompositions $$(w_{k_{0}}^{m_{0}})^=\sum_{{j}=1}^n\alpha_{jm_0}^{k_0}z_j^.$$ Then condition is equivalent to $$\label{rownanie4.5} \sum_{k_{0}}{\left\|\alpha_{mm_{0}}^{k_0}\right\|}^2<\infty,\quad m\neq m_0,\ \sum_{k_{0}}{\left\|\alpha_{mm}^{k_0}-1\right\|}^2<\infty, \qquad m, m_0=1,\ldots,n.$$ Let us rewrite now relations for $\lambda_0=\lambda_{k_{0}}^{m_{0}}$ and $w_0=w_{k_{0}}^{m_{0}}.$ We now consider the space $\ell_2$ of infinite sequences (columns) indexed as $\left\{a_k\right\}_{k\in\mathbb{Z}}$ with a scalar product defined by $\left\langle \left\{a_k\right\},\left\{b_k\right\}\right\rangle=\sum_{k}a_k\overline{b_k}.$ From the relation we obtain that vectors $$p_m^j=-\left\{\frac{\overline{p}_{k,m}^j}{\overline{\widetilde{\beta}}_k^m}\right\}_{k\in\mathbb{Z},\ j,\:m=1,\ldots,n}$$ belong to $\ell_2$. One can also easily see that $$\left\{\frac{1}{\widetilde{\lambda}_k^m-\lambda_{k_{0}}^{m_{0}}}\right\}_{k\in\mathbb{Z}}\in \ell_2, \quad m,m_0=1,\ldots,n;\ k_0\in\mathbb{Z}.$$ Then, putting $\lambda_0=\lambda_{k_0}^{m_{0}}$ and $w_0=w_{k_{0}}^{m_{0}}$ in the equations , we obtain $$\label{rownanie4.6} \alpha_{m,m_{0}}^{k_{0}}= \sum^{n}_{j=1} \alpha_{j m_{0}}^{k_{0}}\left\langle \left\{\frac{1}{\widetilde\lambda_{k}^{m} -\lambda_{k_{0}}^{m_{0}}}\right\}, p_{m}^{j}\right\rangle, \qquad m,m_0=1,\ldots,n;\ k_0\in\mathbb{Z}.$$ Now we would like to rewrite relation in a vector-matrix abstract form. In order to do that, we introduce a more convenient notation. Denote $$\alpha_{mm_{0}}=\left\{\alpha_{mm_{0}}^{k_0}\right\}_{{k_0}\in\mathbb{Z}},$$ $m,m_0=1,\ldots,n.$ By $S_{mm_{0}}=\left\{s_{k_{0}k}^{mm_0}\right\}_{k,\:k_0\in\mathbb{Z}},\quad m,m_0=1,\ldots,n,$ we denote infinite matrices with elements $$s_{k_{0}k}^{mm_0}=\frac{1}{\widetilde{\lambda}_k^m-\lambda_{k_{0}}^{m_{0}}},$$ and by $A_{j m_{0}},\:j,m_0=1,\ldots,n,$ infinite diagonal matrices $$A_{j m_{0}}=\operatorname{diag}\left\{\alpha_{j m_{0}}^{k_0}\right\}_{k_{0}\in\mathbb{Z}}.$$ With these notations relations can be rewritten as $$\label{rownanie4.7} \sum_{j=1}^n A_{j m_{0}}S_{m m_{0}}p_m^j=\alpha_{mm_0},$$ $m,m_0=1,\ldots,n.$ Now let us fix index $m$ and consider $n$ equations with this index and $m_0=1,\!2,\ldots,n$. Consider another infinite diagonal matrix $$\Lambda_m={\operatorname{diag}\left\{\widetilde\lambda_k^m-\lambda_k^m\right\}}_{k\in\mathbb{Z}},$$ and multiply both sides of the $m$-th equality (for $m_0=m$) by this matrix from the left. This gives the following system of equalities $$\label{rownanie4.8} \left\{ \begin{array}{lcl} \displaystyle\sum_{j=1}^n A_{j m_{0}}S_{m m_{0}}p_m^j&=&\alpha_{mm_0},\qquad m_0=1,\ldots,n,\ m_0\neq m,\\ \displaystyle\sum_{j=1}^n A_{jm}\Lambda_m S_{mm}p_m^j&=&\Lambda_m\alpha_{mm}, \end{array} \right.$$ where we used the fact that diagonal matrices commute : $\Lambda_mA_{jm}=A_{jm}\Lambda_m$. Finally, we introduce block matrix operators $$D_m=\left[\begin{array}{ccccc}A_{11}S_{m1}&\ldots&A_{m1}S_{m1}&\ldots&A_{n1}S_{m1}\\ \vdots&\ddots&\vdots& &\vdots\\ A_{1m}\Lambda_m S_{mm}&\ldots&A_{mm}\Lambda_m S_{mm}&\ldots&A_{nm}\Lambda_mS_{mm}\\ \vdots&&\vdots&\ddots&\vdots\\ A_{1n}S_{m n}&\ldots&A_{mn}S_{mn}&\ldots&A_{n n}S_{mn}\end{array}\right],\:m=1,\ldots,n,$$ and present in the form $$D_m \begin{bmatrix} p_m^1\\ \vdots\\p_m^m\\ \vdots\\p_m^n\end{bmatrix} =\begin{bmatrix} \alpha_{m1}\\ \vdots\\\Lambda_m\alpha_{mm}\\ \vdots\\ \alpha_{mn}\end{bmatrix}.$$ Let us observe that both vectors $(p_m^1,\ldots,p_m^n)^T$ and $(\alpha_{m1}\ldots\Lambda_m\alpha_{mm}\ldots\alpha_{m n})^T$ belong to $\ell_2^n=\underbrace{_{}\ell_2\times \ell_2\times\ldots\times \ell_2}_{n \,\textrm{times}}$ (see ,). Therefore the system is solvable if and only if the vector $(\alpha_{m1}\ldots\Lambda_m\alpha_{mm}\ldots\alpha_{m n})^T$ belongs to the image of operator $D_m$ as an operator from $\ell_2^n$ to $\ell_2^n.$ In the sequel we show that, for all sequences $\left\{\lambda\right\}$ satisfying, and for almost all sequences $\left\{w\right\}$ satisfying , operators $D_m, m=1,\ldots,n,$ are bounded and with bounded inverse operators from $\ell_2^n$ to $\ell_2^n$. This means that the spectral assignment problem is solvable. In the further argument we use the following \[statement2\] Let $\left\{\lambda_k\right\}_{k\in\mathbb{Z}}$ be a sequence such that $$\sum_{k \in {\mathbb{Z}}}{\left\|\lambda_k-a+{\mathrm{i}}(b+2\pi k)\right\|}^2<\infty,$$ for some $a, b\in \mathbb{R}$. Then the family $\left\{{\mathrm{e}}^{\lambda_k t}\right\}_{k\in\mathbb{Z}}$ forms a Riesz basis in $L_2(0,1)$. There are several ways to prove this classic result (see [@Avdonin_Ivanov_1995]). It may be be obtained, for example, from the Paley-Wiener theorem [@Paley-Wiener] and Lemma II.4.11 [@Avdonin_Ivanov_1995]. Next we prove the following preliminary result. \[Lemma1\] 1. For $m\neq m_o$ operators $S_{mm_{0}}$ are bounded as operators from $\mathcal{L}(\ell_2)$ and have bounded inverses. 2. $\Lambda_mS_{mm}$ is a bounded operator from $\mathcal{L}(\ell_2)$ and has a bounded inverse. Let $\left\{\varphi_k\right\},\left\{\tilde{\varphi}_k\right\},k\in\mathbb{Z}$, be two Riesz basis of a Hilbert space $H$ and let $R$ be a bounded operator with a bounded inverse, such that $R\varphi_k=\tilde{\varphi}_k,\:k\in\mathbb{Z}.$ For $f\in H$ we have $$f=\sum_{j}a_j\varphi_j,\qquad Rf=\sum_{j}a_jR\varphi_j$$ Then $$R\varphi_j =\widetilde{\varphi}_j=\sum_{k}c_{jk}\varphi_k =\sum_{k}\left\langle \widetilde{\varphi}_j,\psi_k\right\rangle\varphi_k,\quad j\in\mathbb{Z},$$ where ${\left\{\psi_k\right\}}_{k\in\mathbb{Z}}$ is the bi-orthogonal with respect basis to ${\left\{\varphi_k\right\}}_{k\in\mathbb{Z}}.$ Hence $$Rf=\sum_{j}a_j\sum_{k}\left\langle \widetilde{\varphi}_j,\psi_k\right\rangle\varphi_k =\sum_{k}\sum_{j}a_j\left\langle \widetilde{\varphi}_j,\psi_k\right\rangle\varphi_k=\sum_{k}b_k\varphi_k,$$ where $b_k=\sum_{j}a_j\left\langle \widetilde{\varphi}_j,\psi_k\right\rangle.$ This means that the infinite matrix $\widehat{R}$ corresponding to $R$ in the basis $\left\{\varphi_k\right\}$ is of the form $$\widehat{R}=\left\{\widehat{r}_{kj}= \left\langle \widetilde{\varphi}_j,\psi_k\right\rangle\right\}_{\stackrel{k\in\mathbb{Z}}{j\in\mathbb{Z}}},$$ where $$\widehat{R}\begin{bmatrix} \vdots \cr a_{-1}\cr a_0\cr a_1\cr \vdots\cr \end{bmatrix} = \begin{bmatrix} \vdots \\ b_{-1}\cr b_0\cr b_1\cr \vdots\cr \end{bmatrix},\;\{a_j\},\:\{b_j\} \in \ell_2.$$ Let now $H=L_2(0,1)$ and $\left\{\widetilde{\varphi}_k\right\}_{k\in\mathbb{Z}}$ be a Riesz basis of the form $\widetilde{\varphi}_k={\mathrm{e}}^{{\widetilde{\lambda}_k^m}t}$ for some $m=1,\ldots,n$. Let now $\left\{\varphi_k\right\}$ be a Riesz basis which is bi-orthogonal to $\left\{\psi_k={\mathrm{e}}^{-\overline{\lambda_k^{m_{0}}}t}\right\}_{k\in\mathbb{Z}}$ for some $m_{0}=1,\ldots,n$ (the fact that $\left\{\psi_k\right\}_{k\in\mathbb{Z}}$ is a Riesz basis of $L_2(0,1)$ follows from Proposition \[statement2\]). One has $$\begin{aligned} \left\langle \widetilde{\varphi}_j,\psi_k\right\rangle&=&\int^{0}_{-1}{\mathrm{e}}^{\widetilde{\lambda}_j^m t}{\mathrm{e}}^{-{\lambda_k^{m_0}}t}dt=\frac{1}{{\widetilde{\lambda}_j^m}-{\lambda}_k^{m_0}}\left( {\mathrm{e}}^{\widetilde{\lambda}_j^m-{\lambda_k^{m_0}}}-1\right)\\ &=&\frac{1}{\widetilde{\lambda}_j^m-{\lambda}_k^{m_0}}\left(\mu_m {\mathrm{e}}^{-{\lambda_k^{m_0}}}-1\right),\end{aligned}$$ i.e. $\hat{R}=\left\{\hat{r}_{kj}\right\}_{\stackrel{k\in\mathbb{Z}}{j\in\mathbb{Z}}}$, where $$\widehat{r}_{k j}=s_{k j}\left(\mu_m {{\mathrm{e}}^{-\lambda_k^{m_{0}}}}-1\right),\:k,\;j\in\mathbb{Z}.$$ Thus $$\widehat{R}=\varepsilon_{m m_{0}}S_{mm_{0}},$$ where $\varepsilon_{m m_{0}}$ is the infinite matrix $$\varepsilon_{m m_{0}}= \left[\begin{array}{ccccc} \ddots & \ldots & \ldots & \ldots & \ldots\\ \vdots & \mu_m {\mathrm{e}}^{-{\lambda_{-1}^{m_0}}}-1 & 0 & 0 & \vdots \\ \vdots & 0 & \mu_m {\mathrm{e}}^{-{\lambda_0^{m_0}}}-1 & 0 & \vdots \\ \vdots & 0 & 0 & \mu_m {\mathrm{e}}^{-{\lambda_1^{m_0}}}-1 & \vdots \\ \ldots & \ldots & \ldots & \ldots & \ddots\\ \end{array}\right].$$ Hence, we have the following alternative:\ 1. If $m_0\neq m$, then the sequence $\left\{\mu_m{\mathrm{e}}^{-\lambda_k^{m_{0}}}-1\right\}_{k\in\mathbb{Z}}$ is bounded and separated from $0,$ i.e. $\varepsilon_{mm_{0}} : \ell_2\rightarrow \ell_2$ is a bounded opeartor with a bounded inverse. Hence, $$\label{rownanie} S_{mm_{0}}=\varepsilon_{mm_{0}}^{-1}\hat{R}.$$ 2. If $m=m_{0}$, then $\mu_m {\mathrm{e}}^{-\lambda_k^m}\rightarrow1,\:k\rightarrow\infty,$ moreover $$\begin{aligned} \mu_m {\mathrm{e}}^{-\lambda_k^m}-1 &=&{\mathrm{e}}^{\widetilde{\lambda}_k^m-\lambda_k^m}-1\\ &=&\left(1+\left(\widetilde{\lambda}_k^m-\lambda_k^m\right)+\ldots+ \frac{\left(\widetilde{\lambda}_k^m-\lambda_k^m\right)^s}{s!}+\ldots\right)-1\\ &=&\left(\widetilde{\lambda}_k^m-\lambda_k^m\right)\left(1+\bar{o} \left(\widetilde{\lambda}_k^m-\lambda_k^m\right)\right).\end{aligned}$$ Therefore, $$\varepsilon_{mm}=\Lambda_m Q_m=Q_m\Lambda_m,$$ where $Q_m=\operatorname{diag}\left(1+\bar{o}(\widetilde{\lambda}_k^m-\lambda_k^m)\right)_{k\in\mathbb{Z}}$ has a bounded inverse, so $$\label{rownanie} \Lambda_mS_{mm}=Q_m^{-1}\hat{R}$$ From , it follows that $S_{m m_{0}}, \ m\neq m_0$ and $\Lambda_mS_{mm}$ are bounded and have bounded inverse. In our previous consideration we assumed implicitly that our sequences $\left\{\lambda_{k_{0}}^{m_{0}} \right\}_{k_{0}\in\mathbb{Z}}$ are different from $\left\{\widetilde\lambda_k^m \right\}_{k\in\mathbb{Z}},$ i.e. $\widetilde{\lambda}_k^m\neq{\lambda}_{k_{0}}^{m_{0}}$ for all $k,k_0\in\mathbb{Z},\ m,m_{0}\in\left\{1,\ldots,\:,n\right\}$ in particular $\lambda_k^m\neq\widetilde{\lambda}_k^m$ for every $k\in\mathbb{Z},\:m\in\left\{1,\ldots,n\right\}.$ Now let us allow $\lambda_k^m=\widetilde{\lambda}_k^m$ for some indices $k\in I\subset\mathbb{Z}.$ Note that in this case the operators $S_{m m_{0}},\;m_{0}\neq m$, are still well-defined and the operator $\Lambda_m S_{mm}$ can be well-defined as well if we define its components as limits of correspondent components when $\lambda_k^m\rightarrow\widetilde{\lambda}_k^m,\ k\in I$. This means that for $k\in I$ all non-diagonal elements of the $k$-th line of $\Lambda_m S_{mm}$ equal $0$ and the diagonal elements equal $1.$ Besides, $S_{m m_{0}},\;m_0\neq m$ and $\Lambda_m S_{mm}$ remains bounded and with a bounded inverse $\ell_2\rightarrow \ell_2$ operators since formulas remain true also when $\lambda_k^m=\widetilde{\lambda}_k^m.$ Finally if we consider the dependence $\Lambda_m S_{mm}$ of the sequence $\left\{\lambda_k^m\right\}_{k\in\mathbb{Z}}$, one can easy prove that $$\Lambda_m S_{mm}\left(\left\{{{\lambda'}_k^m}\right\}\right) \rightarrow\Lambda_m S_{mm}\left(\left\{\lambda_k^m\right\}\right)$$ as $$\sum_{k}{\left\|{\lambda'}_k^m-\lambda_k^m\right\|}^2\rightarrow 0.$$ In other words this means that operators $\Lambda_m S_{mm}$ and, as a consequence also its inverse operators ${\left(\Lambda_m S_{mm}\right)}^{-1},$ depend continuously of sequence ${\left\{\lambda_k^m\right\}}_{k\in\mathbb{Z}}$ on the set $$\left\{\left\{\lambda_k^m\right\}:\sum_{k}{\left\|\lambda_k^m-\widetilde{\lambda}_k^m\right\|}^2<\infty\right\}.$$ Now we are ready to prove our main results on the spectral assignment. \[Theorem1\] Let $\mu_1$, $\mu_2$, $\ldots$, $\mu_n$ be different nonzero complex numbers and $z_1$, $z_2$, $\ldots$, $z_n$ be nonzero n-dimensional linear independent vectors. Denote $$\widetilde{\lambda}_k^m=\ln\left\|\mu_m\right\|+{\mathrm{i}}\left(\operatorname{Arg}\mu_m+2\pi k\right),\quad m=1,\ldots,n,\:k\in \mathbb{Z}.$$ Let us consider an arbitrary sequence of different complex numbers $\left\{\lambda_k^m\right\}_{\stackrel{k\in\mathbb{Z}}{m=1,\ldots,n}}$ such that $$\sum_{k}{\left\|\lambda_k^m-\widetilde{\lambda}_k^m\right\|}^2<\infty,\quad m=1,\ldots,n.$$ Then there is a small enough $\varepsilon>0$ such for any sequence of nonzero vectors $\left\{d_k^m\right\}_{\stackrel{k\in\mathbb{Z}}{m=1,\ldots,n}}$ satisfying $$\sum_{k}{\left\\|d_k^m-z_m\right\\|}^2<\varepsilon, \quad m=1,\ldots,n$$ one can choose matrices $A_{-1},A_{2}(\theta),A_3(\theta)$ such that for the system , with these matrices, the following two conditions hold: 1. all the numbers $\left\{\lambda_k^m\right\}$ are roots of the characteristic equation $\det\Delta(\lambda_k^m)=0$, $k\in\mathbb{Z}, m=1,\dots, n$, 2. ${d_k^m}$ are right degenerating vectors for $\Delta(\lambda_k^m)$ : ${d_k^m}^\Delta(\lambda_k^m)=0,\;m=1,\ldots,n;\ k\in\mathbb{Z}$. Such a choice is unique if we put the following additional condition on matrix $A_3(\theta)$: $$(C) \qquad \qquad \qquad \int^{0}_{-1}A_3(\theta){\mathrm{d}}\theta=0.$$ First we denote by $A_{-1}$ the matrix uniquely defined by the relations: $$z_m^A_{-1} =\mu_m z_m^,\:m=1,\ldots,n,$$ and denote by $\left\{y_j\right\}_{j=1,\ldots,n}$ the bi-orthogonal basis with respect to $\left\{z_j\right\}_{j=1,\ldots,n}$ in $\mathbb{C}^n$. Then the corresponding operator $\widetilde{\mathcal{A}}$, generated by the matrix $A_{-1}$, has eigenvalues $\widetilde{\lambda}_k^m,\ m=1,\ldots,n,\ k\in\mathbb{Z};\:\widetilde{\lambda_0}=0$ and possesses the Riesz basis of eigenvectors of $\left\{\widetilde{\varphi}_k^m\right\}_{k\in\mathbb{Z}}\cup\left\{\widetilde{\varphi}_j^0\right\}_{j=1,\ldots,n}.$ In the case when all $\mu_m\neq1,\:m=1,\ldots,n$, the corresponding eigenvectors of $\widetilde{\mathcal{A}}$ are $$\label{rownanie(a)} \left\{ \begin{array}{l} \displaystyle{\widetilde{\varphi}}_k^m= \begin{pmatrix} 0\cr {\mathrm{e}}^{\widetilde{\lambda}_k^m\theta}y_m \end{pmatrix},\quad m=1,\ldots,n;\ k\in\mathbb{Z},\quad \widetilde{\mathcal{A}}\widetilde{\varphi}_k^m=\widetilde{\lambda}_k^m\widetilde{\varphi}_k^m,\\[2ex] \displaystyle\widetilde{\varphi}_j^0= \begin{pmatrix} y_j-A_{-1}y_j\cr y_j \end{pmatrix}, \quad j=1,\ldots,n;\quad\widetilde{\mathcal{A}}{\widetilde{\varphi}_j^0}=0. \end{array} \right.$$ In the case $\mu_1=1$, the vector $$\varphi_0^1= \begin{pmatrix} A_{-1}y_j \cr \theta y_j \end{pmatrix}, \quad j=1,\ldots,n;$$ is a generalized eigenvector of $\widetilde{\mathcal{A}}$ corresponding to $\widetilde{\lambda}_0$ and the other $\widetilde{\varphi}_k^m,\;\widetilde{\varphi}_j^0$ are given by formula . Let us show that there is a choice of a bounded operator $\mathcal{P}_0:X_{\widetilde{\mathcal{A}}}\rightarrow\mathbb{C}^n$ (or equivalently a choice of matrices $A_2(\theta),A_3(\theta))$ such that $$\label{rownanie(a1)} \lambda_{k_{0}}^{m_{0}}\in\sigma(\widetilde{\mathcal{A}}+\mathcal{B}_0\mathcal{P}_0),$$ or equivalently $\det\Delta(\lambda_{k_{0}}^{m_{0}})=0,$ and $$\label{rownanie(a2)} d_{k_{0}}^{{m_{0}}}\Delta(\lambda_{k_{0}}^{m_{0}})=0,\quad m=1,\ldots,n;\ k\in\mathbb{Z}.$$ We represent vectors $d_{k_{0}}^{m_{0}}$ in the basis $\left\{{z_j}\right\}_{j=1,\ldots,n},$ namely $$d_{k_{0}}^{m_{0}}=\sum_{m=1}^n \alpha_{m m_{0}}^{k_{0}}z_m,\quad k_{0}\in\mathbb{Z};\ m,m_{0}=1,\ldots,n.$$ With these notations the condition $$\sum_{k_{0}}{\\|d_{k_{0}}^m-z_m\\|}^2<\infty,\quad m=1,\ldots,n$$ implies $$\sum_{k_{0}}\|\alpha_{mm_{0}}^{k_{0}}\|^2<\infty,\quad m\neq m_{0}; \qquad \sum_{k_{0}}{\|\alpha_{mm}^{k_{0}}-1\|}^2<\infty,\;m=1,\ldots,n.$$ and these sums tend to zero as $$\label{rownanie(b)} \sum_{k_{0}}{\\|d_{k_{0}}^m-z_m\\|}^2\rightarrow 0.$$ Therefore, under condition operators $D_m,\;m=1,\ldots,n$ tend to block diagonal operators $$\widehat{D}_m=\begin{bmatrix} S_{m1} & 0 & \ldots & 0 & \ldots & 0 \cr \vdots & \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & \ldots & \Lambda_mS_{mm} & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \ldots & 0 & \ldots & S_{mn} \end{bmatrix}$$ which have a bounded inverse due to Lemma \[Lemma1\]. This means that for a small enough $\varepsilon >0$, the inequality $\sum\limits_{k}{\\|d_k^m-z_m\\|}^2<\varepsilon$ implies that operators $D_m,\;m=1,\ldots,n$ have bounded inverse. If numbers $\{{\lambda}_k^m\}$ are different from $\{{{\widetilde{\lambda}}_k^m}\}$ the later fact yields the existence of a bounded operator [$\mathcal{P}_0:X_{\mathcal{A}}\rightarrow{\mathbb{C}^{n}}$]{} for which the relations , are satisfied. Besides, this operator is unique if we require additionally: $\mathcal{P}_0\widetilde{\varphi}_j^0=0,\;j=1,\ldots,n$, which is equivalent to condition (C). If we allow coincidence $\lambda_k^m=\widetilde{\lambda}_k^m$ for some indices $\{k,m\}\in\mathbb{I}$ one needs to use continuous dependence of operators $\mathcal{D}$ on the sequence $\{\lambda_k^m\}$ (see Remark to Lemma \[Lemma1\]). We approximate $\{{\lambda'}_k^m\}$ by $\{{\lambda'}_k^m\}\;(\sum\limits_k{\|{\lambda'}_k^m-\lambda_k^m\|}^2\rightarrow 0)$ such that $\{{\lambda'}_k^m\}\neq\widetilde{\lambda}_k^m$. Since the conditions , are satisfied for operator $\mathcal{P}_0(\{{\lambda'}_k^m\})$ they are also satisfied for $\mathcal{P}_0(\{{\lambda}_k^m\}).$ This completes the proof \[Lemma2\] Let $\{\lambda_k^m\}$ be a given sequence such that $\sum\limits_k{\|\lambda_k^m-\widetilde{{\lambda}}_k^m\|}^2<\infty$ and $\{\widehat{\alpha}_{mm_{0}}^{k_{0}}\}$ be an arbitrary sequence satisfying $$\sum\limits_{k_{0}}{\|\widehat{\alpha}_{mm_{0}}^{k_{0}}\|}^2<\infty,\quad m\neq m_{0}; \qquad \sum\limits_{k_{0}}{\|\widehat{\alpha}_{mm}^{k_{0}}-1\|}^2<\infty.$$ Then for any $\varepsilon>0$ and $m=1,\ldots,n$ there is a sequence $\{{\alpha}_{mm_{0}}^{k_{0}}\}$ satisfying $$\sum\limits_{k_{0}}{\|{\alpha}_{mm_{0}}^{k_{0}}-\widehat{\alpha}_{mm_{0}}^{k_{0}}\|}^2<\varepsilon$$ and such that the operator $D_m$ has a bounded inverse. First, for given $\{\lambda_k^m\}$, let us choose $\varepsilon_{0}>0$ such that $D_m$ will be invertible for $$\sum\limits_{k_{0}}{\|{\alpha}_{mm_{0}}^{k_{0}}\|}^2<\varepsilon_{0}, \quad m\neq m_{0},\qquad \sum\limits_{k_{0}}{\|\alpha_{mm}^{k_{0}}-1\|}^2<\varepsilon_{0}.$$ Then, one can find a great enough $N$ such that $$\sum_{\|k_{0}\|>N}{\|\widehat{\alpha}_{mm_{0}}^{k_{0}}\|}^2<\varepsilon_{0},\quad \sum_{\|k_{0}\|>N}{\|\widehat{\alpha}_{mm}^{k_{0}}-1\|}^2<\varepsilon_{0}.$$ Next we consider the sequences $\{{\alpha}_{mm_{0}}^{k_{0}}\}$ for which $$\label{rownanie(c)} {\alpha}_{mm_{0}}^{k_{0}}=\widehat{\alpha}_{mm_{0}}^{k_{0}},\;as\; \|k_{0}\|>N.$$ Our goal is to choose the remaining components ${\alpha}_{mm_{0}}^{k_{0}},\;\|k_{0}\|\leq N$ in order to satisfy the requirement of Lemma. Denote rows of matrix $D_m$ by $({\ell}_{k_{0}}^{m_{0}})^,\;\ m_{0}=1,\ldots,n;\ k_{0}\in\mathbb{Z}$ and let $q_{k_{0}}^{m_{0}}$ be the correspondent components of the vector $q=D_m p$, i.e. $q_{k_{0}}^{m_{0}}=({\ell}_{k_{0}}^{m_{0}})^p,\;p\in{\ell}_2^n\; m_{0}=1,\ldots,n;\;k_{0}\in\mathbb{Z}.$ The space $L=\underbrace{\ell_2\times\ldots\times \ell_2}_{n \textrm{ times}}$ may be written as $L=L^{1}\oplus L^{2},$ where $$L^{1}=\{q:{q_{k_{0}}^{m_{0}}=0},\ m_0=1,\ldots,n,\ \|k_{0}\|>N\},$$ and $$L^2=\{q:{q_{k_{0}}^{m_{0}}=0},\ \|k_{0}\|\leq N,\ m_0=1,\ldots,n\}.$$ Let $P$ be the orthogonal projector on $L_2$. Let us observe that the lines $({\ell}_{k_{0}}^{m_{0}})^$ for $\|k_{0}\|>N$ do not depend on chosen components $\alpha_{mm_{0}}^{k_{0}},\ \left\|k_{0}\right\|\leq N$ and that if we put ${\alpha}_{mm_{0}}^{k_{0}}=\delta_{mm_{0}}, \|k_{0}\|\leq N,\ m,\:m_0=1,\ldots,n$, then the operator ${D}_m: L\rightarrow L$ has a bounded inverse. This means that for all sequences $\{\alpha_{mm_0}^{k_{0}}\}$ satisfying we have $PD_{m}L=L^2$ and the invertibility of $D_m$ occurs if and only if $$\label{rownanie(!)} D_{m}L\supset L_1.$$ Let $L^{1'}$ be the subspace $$\left\{p\in L:({\ell}_{k_0}^{m_0})^p=0 ,\ \|k_{0}\|>N,\;m_0=1,\ldots,n\right\}$$ of dimension $(2N+1)n$. Denote by ${\ell_k^j}', j=1,\ldots,n;\ \left\|k\right\|\leq N$ a basis of $L^{1'}$. With this notations one can see that is equivalent to the invertibility of $(2N+1)n\times(2N+1)n$ matrix $$M=\left\{({\ell}_{k_{0}}^{m_{0}})^{\ell_k^j}', \ \|k_{0}\|\leq N,\;\|k\|\leq N, \ m_{0}=1,\ldots,n,\ j=1,\ldots,n\right\}$$ i.e. $\det M\neq 0$. The components of $M$ are linear functions of chosen $\alpha_{mm_{0}}^{k_{0}},\ \|k_{0}\|\leq N$. Therefore, its determinant is a polynomial of these coefficients. Besides, $\det M$ is not identical zero because matrix $D_m$ is invertible if we take $\alpha_{mm_{0}}^{k_{0}}=\delta_{mm_{0}}$, i.e. $\det M\neq 0.$ This implies that $M$ is invertible almost everywhere in $\mathbb{C}^{2(N+1)n}$. This fact completes the proof of Lemma because we can choose $\alpha_{mm_{0}}^{k_{0}},\ \|k_{0}\|\leq N$, in such a way that $$\sum\limits_{\|k_{0}\|\leq N}\mid\alpha_{mm_{0}}^{k_{0}}-\widehat{\alpha}_{mm_{0}}^{k_{0}}\mid<\varepsilon$$ and the operator $D_m$ will be invertible. From the proof of Lemma \[Lemma2\] it is easy to see that actually sequence $\alpha_{mm_{0}}^{k_{0}}$ may differ from $\widehat{\alpha}_{mm_{0}}^{k_{0}}$ only for a finite number of components $\|k_{0}\|\leq N.$ Due to Lemma \[Lemma2\] the formulation of Theorem \[Theorem1\] may be generalized in the following way. \[Theorem2\] Let the sequence $\{\lambda_k^m\}_{k\in\mathbb{Z},\;m=1,\ldots,n}$ and vectors $z_m,\;m=1,\ldots,n$ be chosen according the assumptions of Theorem \[Theorem1\]. Then for any sequence of vectors\ $\{\widehat{d}_k^m\}_{k\in\mathbb{Z},\;m=1,\ldots,n}$ satisfying $$\sum\limits_k{\\|\widehat{d}_k^m-z_m\\|}^2<\infty,\; m=1,\ldots, n,$$ and for any $\varepsilon>0,$ there is a sequence $\{d_k^m\}_{k\in\mathbb{Z},\;m=1,\ldots,n}:$ $$\sum\limits_k{\\|d_k^m-\widehat{d}_k^m\\|}^2<\varepsilon$$ such that, for some choice of matrices $A_{-1}, \ A_2(\theta),\ A_3(\theta)$, satisfying $\int^{0}_{-1}A_3(\theta){\mathrm{d}}\theta=0$, the conditions i), ii) of Theorem \[Theorem1\] are verified. Moreover, $\{d_k^m\}$ may be chosen in such a way, that $d_k^m=\widehat{d}_k^m$ for all $\|k\|>N$ and for some $N\in\mathbb{N}.$ And then we obtain the following result. \[Theorem3\] Let the sequences $\{\lambda_k^m\}_{k\in\mathbb{Z},\;m=1,\ldots,n}$ and $\{\widehat{d}_k^m\}_{k\in\mathbb{Z},\;m=1,\ldots,n}$ be from Theorem \[Theorem2\]. Let, in addition, the complex numbers $\lambda_j^0,\;j=1,\ldots,n$ be different from each other and different from $\lambda_k^m$ and let $d_j^0,\;j=1,\ldots,n$ be linear independent vectors. Then, for any $\varepsilon>0$ there exist $N>0,$ a sequence ${\{d_k^m\}}_{k\in\mathbb{Z},\;m=1,\ldots,n}:$ $$\sum\limits_k{\\|d_k^m-\widehat{d}_k^m\\|}^2<\varepsilon,\;d_k^m=\widehat{d}_k^m,\;\mid k\mid>N,\;m=1,\ldots,n$$ and a choice of matrices $A_{-1}, A_2(\theta), A_3(\theta)$ such that: 1. all the numbers $\{\lambda_k^m\}_{k\in\mathbb{Z},\;m=1,\ldots,n}\cup\{\lambda_j^0\}_{j=1.\ldots,n}$ are roots of the characteristic equation $\det \Delta(\lambda)=0$; 2. $d_k^{m^}\Delta(\lambda_k^m)=0,\;m=1,\ldots,n,\;k\in\mathbb{Z}$ and $d_j^{0^}\Delta(\lambda_j^0)=0.$ Denote by $C$ a $(n\times n)$ matrix uniquely defined by the equalities: $$d_j^{0^}C=\lambda_j^0d_j^{0^},\;j=1,\ldots,n,$$ and let us put $$\widehat{f}_k^{m^}=\widehat{d}_k^{m^}(I-\frac{1}{\lambda_k^m}C)^{-1},\;k\in\mathbb{Z} ,\;m=1,\ldots,n$$ if $\lambda_k^m\neq 0$ and $$\widehat{f}_k^{m^}=\widehat{d}_k^{m^}C^{-1}$$ for $\lambda_k^m=0.$ It is easy to see that the sequences $\{\widehat{f}_k^m\}$ are also quadratically closed to $z_m$: $$\sum_k\\|\widehat{f}_k^m-z_m\\|^2<\infty.$$ Therefore, due to Theorem \[Theorem2\], for any $\varepsilon>0$, there exist matrices $A_{-1},\widehat{A}_2(\theta),\widehat{A}_3(\theta) \\(\int^{0}_{-1}\widehat{A}_3(\theta){\mathrm{d}}\theta=0)$, a number $N>0$ and a sequence of vectors $\{f_k^m\}_{k\in\mathbb{Z},\;m=1,\ldots,n}$: $$\sum\limits_k\\|f_k^m-\widehat{f}_k^m\\|^2<\frac{\varepsilon}{M^2},\qquad f_k^m=\widehat{f}_k^m,\;\mid k\mid>N$$ such that $$f_k^{m}\widehat{\Delta}(\lambda_k^m)=0,\; k\in\mathbb{Z},\;m=1,\ldots,n;$$ where $M=\sup\{\\|I-\frac{1}{\lambda_k^m}C\\|,\;\lambda_k^m\neq0;\;\\|C\\|\}$ and $$\widehat{\Delta}(\lambda)= \lambda I-\lambda {\mathrm{e}}^{-\lambda}A_{-1}+\int^{0}_{-1}\lambda {\mathrm{e}}^{\lambda\theta}\widehat{A}_2(\theta){\mathrm{d}}\theta+ \int^{0}_{-1}{\mathrm{e}}^{\lambda\theta}\widehat{A}_3(\theta){\mathrm{d}}\theta.$$ Now, let us put $$\Delta(\lambda)=(I-\frac{1}{\lambda}C)\widehat{\Delta}(\lambda),\;\lambda\neq 0.$$ One can note that $\widehat{\Delta}(0)=0$ and the function $\widehat{\Delta}_1(\lambda)=\frac{1}{\lambda}\widehat{\Delta}(\lambda)$ may be extended to zero by the formula $$\widehat{\Delta}_1(0)=I-A_{-1}+\int^{0}_{-1}\widehat{A}_2(\theta){\mathrm{d}}\theta+ \int^{0}_{-1}\lim_{\lambda\rightarrow 0}{\frac{{\mathrm{e}}^{\lambda\theta}-1}{\lambda}}\widehat{A}_3(\theta){\mathrm{d}}\theta$$ Then, one can define $$\Delta(0)=-C\widehat{\Delta}_1(0).$$ Let us observe that $\Delta(\lambda)$ can be written as $$\Delta(\lambda)=\lambda I-\lambda {\mathrm{e}}^{-\lambda}A_{-1}+ \int^{0}_{-1}\lambda {\mathrm{e}}^{\lambda\theta}\widehat{A}_2(\theta){\mathrm{d}}\theta+ \int^{0}_{-1}{\mathrm{e}}^{\lambda\theta}\widehat{A}_3(\theta){\mathrm{d}}\theta-C+{\mathrm{e}}^{-\lambda}CA_{-1}$$ $$\hskip -15ex -\int^{0}_{-1}{\mathrm{e}}^{\lambda\theta}C\widehat{A}_2(\theta){\mathrm{d}}\theta+\int^{0}_{-1}{\mathrm{e}}^{\lambda\theta} \int^{\theta}_{-1}C\widehat{A}_3(\tau){\mathrm{d}}\tau{\mathrm{d}}\theta,$$ $$\hskip -12ex =\lambda I-\lambda {\mathrm{e}}^{-\lambda}A_{-1}+ \int^{0}_{-1}\lambda {\mathrm{e}}^{\lambda\theta}{A}_2(\theta){\mathrm{d}}\theta+ \int^{0}_{-1}{\mathrm{e}}^{\lambda\theta}{A}_3(\theta){\mathrm{d}}\theta$$ for $A_2(\theta)$ and $A_3(\theta)$ given by $$\begin{aligned} A_2(\theta)&=&\widehat{A}_2(\theta)-(\theta+1)C-\theta CA_{-1},\cr A_3(\theta)&=&\widehat{A}_3(\theta)+\int^{\theta}_{-1}C\widehat{A}_3(\tau){\mathrm{d}}\tau-C-CA_{-1}\end{aligned}$$ It remains to note that, with this choice of matrices, the conditions i), ii) are satisfied. Indeed: $$\det\Delta(\lambda)=\det(I-\frac{1}{\lambda}C)\det\widehat{\Delta}(\lambda),$$ so all the numbers $\{\lambda_k^m\}$ and $\{\lambda\}_j^0$ are roots of the characteristic equation and $$\begin{array}{rcl} d_k^{m}\Delta(\lambda_k^m)&=&f_k^{m}\widehat{\Delta}(\lambda_k^m),\qquad k\in\mathbb{Z},\ m=1,\ldots,n,\\ &=&0,\\ d_j^{0}\Delta(\lambda_j^0)&=&0,\qquad j=1,\ldots, m \end{array}$$ and finally, for all $m=1,\dots,n$, we have $$\sum\limits_k{\\|d_k^m-\widehat{d}_k^m\\|}^2\leq\sum\limits_{k,m}M^2\\|f_k^m-\widehat{f}_k^m\\|<\varepsilon.$$ This completes the proof. Conclusion ========== We give here some conditions on sets of complex numbers $\{\lambda\}$ and $n$-vectors $\{d\}$ such that they form a spectral set for a neutral type systems. This is a first etap for solving vector moment problems using the exact controllability properties of a neutral type system related to the given moment problem. [99]{} Sergei A. Avdonin, Sergei A. Ivanov: Families of Exponentials, Cambridge University press, 1995. R. Paley, N. Wiener:: Fourier Transform in the Complex Domain, AMS Coll.Publ., vol. 19. New York. R. Rabah, G. M. Sklyar: The Analysis of Exact Controllability of Neutral-Type Systems by the Moment Problem Approach, SIAM J. Control Optim. Vol. 46, Issue 6 (2007), 2148–2181. R. Rabah, G. M. Sklyar, A. V. Rezounenko: Generalized Riesz basis property in the analysis of neutral type systems, C. R. Math. Acad. Sci. Paris, 337 (2003), fasc. 1, 19–24. R. Rabah, G. M. Sklyar, A. V. Rezounenko: Stability analysis of neutral type systems in Hilbert space, J. Differential Equations 214 (2005), no. 2, 391-428. R. Rabah, G. M. Sklyar, and P. Yu. Barkhayev: Stability and stabilizability of mixed retarded-neutral type systems, ESAIM: Control, Optimisation and Calculus of Variations, 18* (2012), 656–692. K. Sklyar, R. Rabah, G. Sklyar, Eigenvalues and eigenvectors assignment for neutral type systems, C. R. Math. Acad. Sci. Paris, 351 (2013), fasc. 3-4, 91–95. [^1]: This work was supported in part by the Polish Nat. Sci. Center, grant N N514 238 438 [^2]: [^3]:
--- abstract: 'Emerging wireless communication systems will be characterized by a tight coupling between communication and positioning. This is particularly apparent in millimeter-wave (mm-wave) communications, where devices use a large number of antennas and the propagation is well described by geometric channel models. For mm-wave communications, initial access, consisting in the beam selection and alignment of two devices, is challenging and time-consuming in the absence of location information. Conversely, accurate positioning relies on high-quality communication links with proper beam alignment. This paper studies this interaction and proposes a new position-aided beam selection protocol, which considers the problem of joint communication and positioning in scenarios with direct line-of-sight and scattering. Simulation results show significant reductions in latency with respect to a standard protocol.' author: - \| Gabriel E. Garcia, Gonzalo Seco-Granados, , Eleftherios Karipidis,\ and Henk Wymeersch, [^1] bibliography: - 'references.bib' title: 'Transmitter Beam Selection in Millimeter-wave MIMO with In-Band Position-Aiding' --- Introduction\[sec:Introduction\] ================================ (mm-wave) communications have recently gained attention for the development of high-speed wireless networks. Mm-wave systems operate at frequencies between 30 to 300 GHz with large available bandwidths. Combined with multiple-input-multiple-output (MIMO), using a large number of antennas, mm-wave can provide high data rates to users through dense spatial multiplexing [@Zhouyue; @Rappaport; @Swindlehurst2014; @Sun2014; @Bai2014]. Hence, mm-wave MIMO is considered a key enabler for emerging communication systems, e.g., 5G or IEEE WiGig [@Hansen2011], to deliver throughputs on the order of multi-Gbps for a range of applications from wearables [@Venugopal2016] to automotive [@Choi2016]. However, mm-wave communications face a number of challenges, in particular severe path-loss at these high frequencies. As a solution, system designers improve the link budget through highly directional links involving sophisticated beamforming (BF) at the transmitter and/or receiver [@Wang; @Hur; @Tsang], relying on the knowledge of the mm-wave MIMO propagation channel. Given the quasi-optical propagation of mm-wave, stochastic geometrical channel models have become an attractive approach to characterize the channel with few parameters. These models relate the propagation to the geometry of the operating environment, thus creating an explicit interplay between the communication channel and the positions of the transmitter, receiver, and reflectors [@Lopes1995; @DenSaya; @Li2014; @Leus; @Steinbauer2001]. This interplay becomes apparent during the initial access phase, where two devices, a transmitter and a receiver, here termed D1 and D2, aim to establish a connection by achieving beam alignment. This consists in finding a pair of transmit and receive beams to reach a required signal-to-noise ratio (SNR) for the link. From the communications perspective, this is achieved by a dedicated protocol that searches across the angle-of-arrival (AOA) and angle-of-departure (AOD) space. Both AOA and AOD can be related to the location of D1 and D2, thus presenting an opportunity to exploit location information. From the positioning perspective, obtaining the position information of a device through exchange of mm-wave signals requires the establishment of a communication link. Hence, the communication and positioning problems are coupled, indicating that a joint solution strategy may yield better performance. Conventional beam selection protocols do not consider the positioning aspect explicitly. For instance, the authors in [@Wang; @Lee2011; @Xia2008; @Chen2011; @Xiao2016] designed BF protocols based on discretized iterative beam codebooks, while in [@Tsang] the use of simultaneous beams through beam coding is introduced. In [@Leus; @Kokshoorn2017], the authors developed a hierarchical multi-resolution codebooks: in [@Leus], codebooks are based on hybrid analog/digital precoding and proposed low-overhead channel estimation algorithms, while in [@Kokshoorn2017] the codebook allows for beam overlapping for channel estimation purposes. In [@Barati2016], the initial access problem is tackled by means of scanning and signaling procedures, while in [@Liu2017] the authors propose a strategy for transmitting reference signals using pre-designed codebooks for device discovery, and in [@Kim2014], prioritized beam ordering strategies are presented. These protocols involve a time-consuming search over different AOA/AOD pairs in order to determine directions in which to point the beams. On the other hand, contributions in the area of positioning generally ignore the initial access aspect. For instance, the authors in [@hu2014esprit; @DenSaya; @sanchis2002novel; @Zhu2016; @Peng2016] present direction-of-arrival and location estimation algorithms, but do not provide initial access protocols. Similarly, [@Shamansoori2015] exploits mm-wave and MIMO features along with BF to provide sufficient conditions on the identifiability of the position and orientation for a device in a line-of-sight (LOS) scenario but no protocols for the initial access are included. Works that combine positioning with initial access include [@Alexandropoulos2017; @Aviles2016; @Garcia2016; @Va2015]: [@Alexandropoulos2017] proposes a beam alignment method for fixed-position network nodes in mm-wave backhaul systems aided with position information obtained using high-sensitivity displacement sensors in each node. In [@Aviles2016], beam training is presented exploiting a database linked to the geographical position of the users. In [@Garcia2016] location information is harnessed for fast channel estimation in a vehicular context. In [@Va2015], beam alignment is proposed with the use of position information obtained from the on-board train system. What is common in [@Alexandropoulos2017; @Aviles2016; @Garcia2016; @Va2015] is that position information is obtained out-of-band, not from the mm-wave signal itself. In the context of beam tracking (i.e., once the initial access has been solved) in-band information has been harnessed, in the form of either AOD or/and AOA [@Cao2016; @Bae2017; @Kela2016]: authors in [@Bae2017] propose an estimator for the AOD and channel information under Gaussian AOD dynamics, but no protocol is presented; in [@Cao2016], AOA estimation is introduced based on the geometry of the antenna array and the transmitting beam pattern, not including position information; in [@Kela2016], state-space models for the AOD and AOA are inferred aided with channel-aided information rather than position information. In this paper, a novel in-band positioning-aided transmitter beam selection protocol is proposed, with the aim of reducing the set-up time of the initial access procedure for communication in the presence of a line-of-sight path and unknown scatterer locations. In order to gain insight into the fundamental achievable performance, we determine the evolution of the Fisher information of the D2 position and orientation as new beams are utilized, feeding back this location information to D1 in order to adapt the beams. Both discrete and continuous codebooks are considered, in the presence of LOS communication with scatterers. The new protocol is evaluated through simulations, considering as performance metrics the set-up time, signal-to-noise (SNR) ratio, and the position and orientation error bounds after protocol completion. We observe that the position-aided protocol is significantly faster than a conventional protocol based on discretized beam codebooks, with little or no SNR penalty, and can additionally determine the position or orientation of D2. In addition, we find that standard discrete codebooks achieve similar performance to more complex codebooks, indicating that the proposed protocol can be implemented with standard mm-wave communication technologies. The remainder of the paper is structured as follows. Section \[sec:System-Model\] presents the communication model and performance metrics. In Section \[sec:Conventional-Beam-Selection\], the conventional protocol description, operation and performance are described. Then, in Section \[sec:Joint-PBS\] the joint positioning and beam selection protocol, its operation and performance are introduced. Finally, numerical results are given in Section \[sec:Simulation-Results\], followed by the conclusions in Section \[sec:Conclusions\]. System Model\[sec:System-Model\] ================================ Geometric Model --------------- We consider a MIMO mm-wave system consisting of a transmitting device D1 with $N_{t}$ antennas and beamforming capabilities, and a receiving device D2 with $N_{r}$ antennas. The 2-dimensional locations[^2] of D1 and D2 are denoted by $\mathbf{p}=[p_{\mathrm{x}},p_{\mathrm{y}}]^{\mathrm{T}}\in\mathbb{R}^{2}$ and $\mathbf{q}=[q_{\mathrm{x}},q_{\mathrm{y}}]^{\mathrm{T}}\in\mathbb{R}^{2}$, respectively, and let $\alpha\in[0,2\pi)$ be the angle of rotation of the D2 antenna array with respect to the horizontal axis. These parameters in turn imply an AOD $\theta_{\mathrm{tx},0}$ and an AOA $\theta_{\mathrm{rx},0}$, as depicted in Figure \[fig:system\_model\]. Note that under our definitions, $\cos(\theta_{\mathrm{tx},0})=(p_{\mathrm{x}}-p_{\mathrm{y}})/\left\Vert \mathbf{q}-\mathbf{p}\right\Vert ,$ and $\alpha=\pi+\theta_{\mathrm{tx},0}-\theta_{\mathrm{rx},0}.$ We also introduce the LOS propagation delay as between D2 and D1 as $\tau_{0}=\left\Vert \mathbf{q}-\mathbf{p}\right\Vert /c$, where $c$ is the speed of light. We assume that $\mathbf{q}$ is a known reference point. It is easy to show that the knowledge of $\bm{\beta}=[\mathbf{p},\alpha]^{\mathrm{T}}$ is equivalent to the knowledge of $[\tau_{0},\theta_{\mathrm{tx},0},\theta_{\mathrm{rx},0}]$. The environment can also contain scatterers, here modeled as points, with locations $\mathbf{s}_{k}$, $k\ge1$, for which we introduce $\tau_{k}=\left\Vert \mathbf{q}-\mathbf{s}_{k}\right\Vert /c+\left\Vert \mathbf{s}_{k}-\mathbf{p}\right\Vert /c$, as well as the AOD $\theta_{\mathrm{tx},k}$ and AOA $\theta_{\mathrm{rx},k}$, as shown in Figure \[fig:system\_model\]. Furthermore, we consider that the device D1 transmits signals at a carrier frequency $f_{c}$ (or equivalently wavelength $\lambda=c/f_{c}$, where $c$ is the speed of light) and with bandwidth $B$. We employ a narrowband model[^3] where the $N_{r}\times N_{t}$ channel matrix is given by [@Sayeed2002; @BradySay2] $$\mathbf{H}(t)=\sum_{k=0}^{K-1}\underbrace{\sqrt{N_{t}N_{r}}\,h_{k}~\mathbf{a}_{\mathrm{rx}}(\theta_{\mathrm{rx},k})\mathbf{a}_{\mathrm{tx}}^{\mathrm{H}}(\theta_{\mathrm{tx},k})}_{=\mathbf{H}_{k}}\delta(t-\tau_{k}),\label{eq:channelmodel}$$ in which $h_{k}$ is the complex gain of the $k$-th path, $\mathbf{a}_{\mathrm{tx}}(\theta_{\mathrm{tx},k})\in\mathbb{C}^{N_{t}}$ and $\mathbf{a}_{\mathrm{rx}}(\theta_{\mathrm{rx},k})\in\mathbb{C}^{N_{r}}$ are the normalized antenna steering and response vectors associated with the $k$-th path. Without loss of generality, our focus will be on uniform linear arrays[^4] (ULA), for which $$\begin{aligned} \left[\mathbf{a}_{\mathrm{tx}}(\theta_{\mathrm{tx}})\right]_{l=0}^{N_{t}-1} & = & \frac{1}{\sqrt{N_{t}}}\exp\bigl(j\frac{2\pi ld}{\lambda}\sin\theta_{\mathrm{tx}}\bigr)\label{eq:steeringvector}\\ \left[\mathbf{a}_{\mathrm{rx}}(\theta_{\mathrm{rx}})\right]_{l=0}^{N_{r}-1} & = & \frac{1}{\sqrt{N_{r}}}\exp\bigl(j\frac{2\pi ld}{\lambda}\sin\theta_{\mathrm{rx}}\bigr),\label{eq:responsevector}\end{aligned}$$ where $d$ is the antenna spacing. ![Two-dimensional MIMO system model with a D1 with known position and orientation, and a D2 with unknown position ($\mathbf{p}$) and orientation ($\alpha$). The AODs $\theta_{\mathrm{tx},k}$ and AOAs $\theta_{\mathrm{rx},k}$ are also indicated. \[fig:system\_model\]](modelHW2){width="1\columnwidth"} Training Model\[subsec:Training-Model\] --------------------------------------- ### Transmitter With Analog Beamforming We assume the use of analog beamforming, implemented with phase shifters and combined with antenna selection. The transmitter D1 can sequentially send training sequences (TS) using beams pointed towards in different directions, leading to a signal model $\mathbf{f}_{m}x(t)$ in which $x(t)=\sum_{n=1}^{N}a_{n}p(t-n/B),$ where $p(t)$ is a unit-energy transmit pulse (e.g., root-raised cosine), $N$ denotes the number of symbols, and $a_{n}$ are known training symbols with $\mathbb{E}\{\|a_{n}\|^{2}\}=E_{s}$, and $$\begin{aligned} & \mathbf{f}_{m}=\\ & \frac{1}{\sqrt{N'_{t}}}\left[\mathbf{0}_{\frac{N_{t}-N'_{t}}{2}}\,e^{j\phi_{0}}\ldots e^{j\phi_{N'_{t}-1}}\,\mathbf{0}_{\frac{N_{t}-N'_{t}}{2}}\right]^{\mathrm{T}},\nonumber \end{aligned}$$ where $N'_{t}\le N_{t}$ indicates the number of active contiguous antennas [@Celik2006] used to control the beam widths at the expense of the beam gain. Special cases include $$\phi_{i}\in\{0,\pi,\pi/2,-\pi/2\}\label{eq:simpleCodebook}$$ and $$\phi_{i}=e^{j\frac{2\pi di}{\lambda}\sin\theta_{m}},\label{eq:directionCodebook}$$ where $\theta_{m}$ is the direction of the beam, chosen from a given set $\Theta$. The design parameters of the beam patterns consist of the maximum gain direction $\theta_{\mathrm{max}}$ and the half-power beam-width angle, $\theta_{\mathrm{HPBW}}$, which is the angle where the square magnitude of the radiation pattern decreases by 50% with respect to its maximum value, and depends on the type of antenna and operating frequency, among other parameters. For beams based on (\[eq:simpleCodebook\]) or (\[eq:directionCodebook\]), both $\theta_{\mathrm{max}}$ and $\theta_{\mathrm{HPBW}}$ can be calculated and tabulated [@Orfandis2008]. ### Idealized Receiver For each transmitted beam, the receiver observes the following complex baseband signal: $$\mathbf{y}(t)=\sum_{k=0}^{K-1}\mathbf{H}_{k}\mathbf{f}_{m}x(t-\tau_{k})+\mathbf{n}(t),\label{receivedsignal}$$ where $\mathbf{n}(t)\in\mathbb{C}^{N_{r}}$ is a Gaussian noise vector with zero mean and two-sided power spectral density $N_{0}/2$. We will consider an idealized receiver D2, which samples the entire signal $\mathbf{y}(t)$ and is synchronized to D1 [@Shen2010]. While such a receiver may be impractical, it allows us to understand the ultimate performance of position-aided protocols and can thus serve as a benchmark for different receiver structures with analog beamforming as well as low-complexity algorithms. Performance Metrics\[subsec:Performance-Metrics\] ------------------------------------------------- The beam selection protocol works in an iterative manner, where each iteration $i\ge1$ involves selecting a number of active antennas $N_{t}^{(i)}<N_{t}$ and a number of beams $M_{t}^{(i)}$. The objective of protocol is to quickly determine a beamforming vector $\mathbf{f}_{\mathrm{sel}}$ resulting in high SNR. The relevant performance metrics are thus SNR, number of transactions, and positioning quality. 1. SNR: The selection of $\mathbf{f}_{\mathrm{sel}}$ intends to maximize the SNR, defined as $$\mathrm{SNR}\triangleq\sum_{k=0}^{K-1}\frac{N_{t}N_{r}E_{s}}{N_{0}}\|h_{k}\|^{2}\left\Vert \mathbf{a}_{\mathrm{tx}}^{\mathrm{H}}(\theta_{\mathrm{tx},k})\mathbf{f}_{\mathrm{sel}}\right\Vert .\label{eq:SNR}$$ 2. Number of transactions: Considering an iterative beam selection protocol, total beam selection time can be broken down for each iteration $i\ge1$ as follows: (i) training stage during which D1 sends $M_{t}^{(i)}$ training sequences; (ii) feedback stage, during which D2 reports back to D1; (iii) mapping stage, during which D1 informs D2 about the number of required transmit beam patterns $M_{t}^{(i+1)}$; and a one time (iv) acknowledgment after which high-rate data communication can start. The total number of transactions $N_{\mathrm{trans}}$ can be quantified as $N_{\mathrm{trans}}=\sum_{i=1}^{I}(M_{t}^{(i)}+2)+1$ . Note that when the mapping is agreed a priori and feedback messages are neglected, we find that $$N_{\mathrm{trans}}\approx\sum_{i=1}^{I}M_{t}^{(i)}.\label{eq:timeGeneral}$$ 3. Positioning quality: We consider the expected positioning and orientation errors, given by $$\mathbb{E}\{\Vert\mathbf{p}-\hat{\mathbf{p}}\Vert^{2}\}\label{eq:error_pos}$$ and $$\mathbb{E}\{\Vert\alpha-\hat{\alpha}\Vert^{2}\},\label{eq:error_orient}$$ where $\hat{\mathbf{p}}$ and $\hat{\alpha}$ denote the estimated position and angle of rotation for the D2, respectively, obtained from the sequence of received signals of the form (\[receivedsignal\]). Conventional Beam Selection\[sec:Conventional-Beam-Selection\] ============================================================== A beam selection protocol with the goal of minimizing the beamforming set-up time and mitigate the high path-loss has been adopted by the IEEE 802.15.3c standard as an optional functionality [@Wang]. This iterative protocol relies on a multi-level beam tree search starting from lower resolution beams that cover large angular range per beam moving towards higher resolution beams covering a smaller angular range. . Here, we describe a general beam selection protocol for D1, not exploiting or requiring any position information. General Protocol Operation -------------------------- The iterative protocol selects a number of active antennas and a number of beams at each iteration $i\ge1$. In particular, at iteration $i$, D1 selects $M_{t}^{(i)}$ beams with associated beamforming vectors $$\mathbf{F}^{(i)}=\{\mathbf{f}_{1}^{(i)},\ldots,\mathbf{f}_{M_{t}^{(i)}}^{(i)}\}$$ to be used with $N_{t}^{(i)}\leq N_{t}$ selected active antennas. The protocol makes use of a finite codebook from which beams can be selected for each value of $N_{t}^{(i)}.$ D1 transmits a reference signal $x(t)$ for each of the $M_{t}^{(i)}$ beams. Through suitable signal processing, D2 measures the reference signal received power (RSPS) $P_{m}^{(i)}$ for each of the $m=1,\ldots,M_{t}^{(i)}$ transmitted beams, and gathers them in the vector $\mathbf{P}^{(i)}\in\mathbb{R}^{M_{t}^{(i)}}$ . The selection of the beams at each iteration $i$ is dependent on the previous beam selection $\mathbf{F}^{(i-1)}$ and on the reference signal received powers $\mathbf{P}^{(i-1)}$ transmitted as feedback from D2 to D1. The mapping $\mathbf{F}^{(i)}=f_{\mathrm{map}}(\mathbf{F}^{(i-1)},\mathbf{P}^{(i-1)})$ depends on the specific codebook employed. The protocol is summarized as pseudocode in Algorithm \[wangprotocol\]. Input: $N_{t}^{(1)}$, $M_{t}^{(1)}$ , $f_{\mathrm{map}}$, and $N_{t}$ D1 transmits a TS for each $m$-th beam pattern, $\mathbf{f}_{m}^{(i)}$; D2 measures RSPS for each $m$-th pattern, $P_{m}^{(i)}$; D2 transmits all measured received powers $\mathbf{P}^{(i)}$ to D1; $i=i+1;$ A new selection of beams is obtained at D1: $\mathbf{F}^{(i)}=f_{\mathrm{map}}(\mathbf{F}^{(i-1)},\mathbf{P}^{(i-1)})$ Output: Final beam pattern selection $\mathbf{f}_{\mathrm{sel}}$ Protocol-specific Performance \[subsec:Realizations\_snr\] ---------------------------------------------------------- We now present the evaluation of the performance metrics in Section \[subsec:Performance-Metrics\] for the RSPS protocol. 1. SNR: Upon completion of the protocol, at iteration $I$, corresponding to $N_{t}^{(I)}=N_{t},$ the beam with the highest RSPS is selected $$\mathbf{f}_{\mathrm{sel}}=\arg\max_{\mathbf{f}_{m}^{(I)}}P_{m}^{(I)}(\mathbf{f}_{m}^{(I)}),$$ and the SNR is evaluated according to (\[eq:SNR\]). 2. Number of transactions: Depending on the codebook design, which is both known at D1 and D2, different implementations of the protocol can be designed by means of the mapping $\mathbf{F}^{(i)}=f_{\mathrm{map}}(\mathbf{F}^{(i-1)},\mathbf{P}^{(i-1)})$, e.g, [@Garcia2016; @Wang]. A simple mapping would involve one iteration with $M_{t}=N_{t}$ narrow beams, leading to a number of transactions $N_{\mathrm{trans}}^{\mathrm{conv}}\approx N_{t}$ [@Kim2014]. A reduction in delay can be achieved through a multi-level beam search from broad to directive beams, noting that the HPBW scales roughly as $1/N_{t}^{(i)}$ [@Orfandis2008], so that a beam with $N_{t}^{(i-1)}$ antennas can be covered with $M_{t}^{(i)}\le3$ beams with $N_{t}^{(i)}=2N_{t}^{(i-1)}$ antennas. This leads to $I=\log_{2}(N_{t})$ and thus $N_{\mathrm{trans}}^{\mathrm{conv}}\approx3\log_{2}(N_{t})$. 3. Positioning quality: The protocol does not provide any positioning information. Proposed Joint Positioning and Beam Selection\[sec:Joint-PBS\] ============================================================== In this section, we introduce the proposed iterative position-based beam selection protocol. The protocol aims to minimize the set-up time and mitigate the high path-loss using D2 position information as proxy for the optimal pointing of the transmit beams. Before we describe the proposed protocol, followed by its performance and implementation details, we first briefly detail some properties of mm-wave positioning. Performance of Mm-wave Positioning ---------------------------------- D2 can perform estimation of its position and orientation (represented by $\bm{\beta}$) based on the received waveforms from D1. The quality of such estimation can be assessed through the Fisher information matrix (FIM) [@Kay]. Given the statistics of a waveform $\mathbf{y}(t)$ of the form (\[receivedsignal\]) and an unknown vector parameter $$\bm{\eta}=\left[\tau_{0},\mathbf{\bm{\theta}}_{0}^{\mathrm{T}},\mathbf{h}_{0}^{\mathrm{T}},\ldots,\tau_{K-1},\mathbf{\bm{\theta}}_{K-1}^{\mathrm{T}},\mathbf{h}_{K-1}^{\mathrm{T}}\right]^{\mathrm{T}},$$ where $\bm{\theta}_{k}=\left[\theta_{\mathrm{tx},k},\theta_{\mathrm{rx},k}\right]^{\mathrm{T}}$, $\mathbf{h}_{k}=\left[h_{R,k},h_{I,k}\right]=[\Re\{h_{k}\},\Im\{h_{k}\}]$, the FIM associated with a single beam, $\mathbf{J}_{\bm{\mathbf{\eta}}}^{(\mathrm{beam})}$, is a $5K\times5K$ matrix, whose expression is provided in (\[FIM\_gral\]) in the Appendix. While each $\mathbf{y}(t)$ corresponds to a single transmit beam, the FIM for multiple beams is simply the sum of the corresponding FIMs, due to the additive nature of Fisher information. For each iteration $i$ of a beam selection protocol, we can thus compute the FIM associated with the $m$-th beam, say, $\mathbf{J}_{\bm{\eta}}^{(i,m)}$. The total FIM after $i$ iterations can then be expressed as $$\mathbf{J}_{\bm{\eta}}^{(i)}=\sum_{l=1}^{i}\sum_{m=1}^{M_{t}^{(l)}}\mathbf{J}_{\bm{\eta}}^{(l,m)}.$$ Since there is an injective relation between $\bm{\eta}$ and[^5] $$\bm{\eta}'=\left[\bm{\beta}^{\mathrm{T}},\mathbf{h}_{0}^{\mathrm{T}},\mathbf{s}_{1}^{\mathrm{T}},\mathbf{h}_{1}^{\mathrm{T}},\ldots,\mathbf{s}_{K-1}^{\mathrm{T}},\mathbf{h}_{K-1}^{\mathrm{T}}\right]^{\mathrm{T}},$$ we can also determine the FIM of $\mathbf{J}_{\bm{\eta}'}^{(i)}$ as $\mathbf{J}_{\bm{\eta}'}^{(i)}=\mathbf{T}^{\mathrm{T}}\mathbf{J}_{\bm{\eta}}^{(i)}\mathbf{T}$, where $\mathbf{T}$ is the Jacobian matrix associated with the transformation from $\bm{\eta}$ to $\bm{\eta}'$, that is, $T_{ij}=\partial\eta_{i}/\partial\eta'_{j}$ . Finally, the inverse of the FIM can be related to the mean squared error (MSE) of unbiased estimators of $\bm{\eta}'$ [@Kay]: $$\mathbb{E}_{\mathbf{y}\|\bm{\eta}'}\left[\left(\hat{\bm{\eta}}'-\bm{\text{\ensuremath{\eta}}}\right)\left(\hat{\bm{\eta}}'-\bm{\text{\ensuremath{\eta}}}\right)^{\mathrm{T}}\right]\succeq\left[\mathbf{J}_{\bm{\eta}'}^{(i)}\right]^{-1}.$$ From this relationship, we can immediately derive the so-called position error bound (PEB) and rotation error bound (REB) as $$\begin{aligned} \mathrm{PEB}^{(i)} & =\sqrt{\mathrm{tr}\left\{ \left[\mathbf{J}_{\bm{\eta}'}^{(i)}\right]_{1:2,1:2}^{-1}\right\} }\label{eq:PEBdef}\\ & \le\sqrt{\mathbb{E}\{\Vert\mathbf{p}-\hat{\mathbf{p}}\Vert^{2}\}}\end{aligned}$$ and $$\begin{aligned} \mathrm{REB}^{(i)} & =\sqrt{\left[\mathbf{J}_{\bm{\eta}'}^{(i)}\right]_{3,3}^{-1}}\label{eq:REBdef}\\ & \le\sqrt{\mathbb{E}\{\Vert\alpha-\hat{\alpha}\Vert^{2}\}},\end{aligned}$$ where $\left[\cdot\right]_{1:2,1:2}^{-1}$ denotes the $2\times2$ upper left submatrix of the inverse of the argument, and $\left[\cdot\right]_{3,3}^{-1}$ denotes the third diagonal element of the inverse of the argument. ### Remarks {#remarks .unnumbered} - We note that in contrast to conventional range-based positioning, the use of multiple antennas at both devices allows for the determination of both the position and the orientation of D2 using signals from a single reference device D1. - The FIM and corresponding PEB and REB are valid, irrespective of the subsequent processing at the receiver. Such processing can include analog beamforming as well as low-complexity estimation and detection algorithms. - Given the geometric model, the FIM does not account for path resolvability in time and angle spaces. However, this can be addressed as follows. Consider paths $a$ and $b$, with AOAs $\theta_{r,a}$, and $\theta_{r,b}$; delays $\tau_{a},$ and $\tau_{b}$, respectively. Paths $a$ and $b$ are considered unresolvable in time and angle, when $\left\|\tau_{a}-\tau_{b}\right\|\le1/B$ and $N_{r}\lambda\left\|\sin(\theta_{r,a})-\sin(\theta_{r,b})\right\|\le d$ . When two paths are unresolvable, they are to be combined into a single path by adding the complex channel gains, prior to computation of the Fisher information. General Protocol Operation\[subsec:Positioning-based-algorithm\] ---------------------------------------------------------------- From the above FIM analysis, it is apparent that D2 can not only compute the received powers for each beam, but also harness them to compute its position and orientation $\bm{\beta}=[\mathbf{p},\alpha]^{\mathrm{T}}$. We will denote the aggregated waveforms at iteration $i$ by $\mathbf{y}^{(i)}$, and the collection of $\mathbf{y}^{(i)}$ up to iteration $i$ by $\mathbf{y}^{(1:i)}$ . Our idealized receiver D2 can thus be equipped with an estimator, which can determine an estimate of $\bm{\beta}$ from $\mathbf{y}^{(1:i)}$ at the end of each iteration $i$, operating close to the fundamental performance bounds (\[eq:PEBdef\])(\[eq:REBdef\]). Considering a Gaussian approximation of the position and orientation error, we can describe the estimate by a mean $\hat{\bm{\beta}}^{(i)}$ and a covariance matrix $\bm{\Sigma}_{\bm{\beta}}^{(i)}$. This information can be fed back to D1. The protocol then operates according to Algorithm \[alg:positionprotocol-simplified\]. Since D1 has more information about D2 than in the conventional algorithm, a more intelligent mapping function can be designed, as will be described in Section \[subsec:Realizations\_pos\]. In addition, both D2 and D1 have knowledge of D2’s position and orientation. Input: $N_{t}^{(1)}$, $M_{t}^{(1)}$ , $f_{\mathrm{map}}^{\mathrm{pos}}$ and $N_{t}$ D1 transmits a TS for each $m$-th beam pattern, $\mathbf{f}_{m}^{(i)}$; D2 measures received power for each $m$-th beam pattern, $P_{m}^{(i)}$; D2 determines $[\hat{\bm{\beta}}^{(i)},\bm{\Sigma}_{\bm{\beta}}^{(i)}]$; D2 feeds back $[\mathbf{P}^{(i)},\hat{\bm{\beta}}^{(i)},\bm{\Sigma}_{\bm{\beta}}^{(i)}]$ to D1; $i=i+1;$ A new selection of beams is obtained at D1: $\mathbf{F}^{(i)}=f_{\mathrm{map}}^{\mathrm{pos}}(\mathbf{F}^{(i-1)},\mathbf{P}^{(i)},\hat{\bm{\beta}}^{(i)},\bm{\Sigma}_{\bm{\beta}}^{(i)})$; Output: Final beam pattern selection $\mathbf{f}_{\mathrm{sel}}$, final D2’s position and orientation $\hat{\bm{\beta}}^{(i)}$ and their uncertainties $\bm{\Sigma}_{\bm{\beta}}^{(i)}$. Protocol-specific Performance \[subsec:Realizations\_pos\] ---------------------------------------------------------- We now present the evaluation of the performance metrics in Section \[subsec:Performance-Metrics\] for the protocol described by Algorithm \[alg:positionprotocol-simplified\]. 1. SNR: As for the conventional protocol, the SNR is computed using (\[eq:SNR\]), based on the final selected beam. 2. Number of transactions: Different implementations depending on the mapping function $f_{\mathrm{map}}^{\mathrm{pos}}(\mathbf{F}^{(i-1)},\mathbf{P}^{(i)},\hat{\bm{\beta}}^{(i)},\bm{\Sigma}_{\bm{\beta}}^{(i)})$ can be designed, in order to reduce the number of transactions. In particular, when D1 has knowledge of the AOD, it can select an appropriate number of active antennas and beams. More specifically, D1 can process $\hat{\bm{\beta}}^{(i)},\bm{\Sigma}_{\bm{\beta}}^{(i)}$ to compute an AOD estimate $\hat{\theta}_{\mathrm{tx},0}^{(i)}$ and the AOD standard deviation, denoted as $\sigma_{\mathrm{tx},0}^{(i)}$. A conventional hierarchical protocol with $N_{t}^{(i)}=2N_{t}^{(i-1)}$ and $M_{t}^{(i)}=3$ can be used whenever the AOD uncertainty is large, i.e., $\sigma_{\mathrm{tx},0}^{(i)}\ge3\theta_{\mathrm{HPBW}}(2N_{t}^{(i-1)},\hat{\theta}_{\mathrm{tx},0}^{(i)})$. On the other hand, if $\sigma_{\mathrm{tx},0}^{(i)}$$<$$3\theta_{\mathrm{HPBW}}(2N_{t}^{(i-1)},\hat{\theta}_{\mathrm{tx},0}^{(i)})$, the number of transactions can be reduced by using more than $2N_{t}^{(i)}$ active antennas with $M_{t}^{(i)}=3$. In the latter case, we propose to set $N_{t}^{(i)}$ according to $$\begin{aligned} \mathrm{maximize}\,\,\, & N_{t}^{(i)}\\ \mathrm{subject\,to}\,\,\, & N_{t}^{(i)}\le N_{t}\\ & 3\theta_{\mathrm{HPBW}}(N_{t}^{(i)},\hat{\theta}_{\mathrm{tx},0}^{(i)})\ge\sigma_{\mathrm{tx},0}^{(i)},\label{eq:AODcondition}\end{aligned}$$ and transmit 3 beams covering the AOD region $\hat{\theta}_{\mathrm{tx},0}^{(i)}\pm\sigma_{\mathrm{tx},0}^{(i)}$. Such beams can also be optimized to minimize the future expected uncertainty, as detailed in Appendix \[sec:Optimized-Beam-Directions\]. We thus expect that in cases when $\bm{\beta}$ can be accurately estimated with few antennas, then the number of transactions fulfills $N_{\mathrm{trans}}^{\mathrm{pos}}\ll N_{\mathrm{trans}}^{\mathrm{conv}}$, and $N_{\mathrm{trans}}^{\mathrm{pos}}\approx N_{\mathrm{trans}}^{\mathrm{conv}}$ otherwise. From [@Shamansoori2015; @Dardari], it is known that good estimates of $\bm{\beta}$ are possible when enough beams are transmitted pointing roughly in the direction of D2, and the received SNR associated with those beams is sufficiently high. Consequently, we expect $N_{\mathrm{trans}}^{\mathrm{pos}}\ll N_{\mathrm{trans}}^{\mathrm{conv}}$ for D2 locations close to D1. 3. Positioning quality: The proposed protocol can be assessed in terms of position (\[eq:error\_pos\]) and orientation errors (\[eq:error\_orient\]), which can be predicted through the FIM. ![Average SNR as a function of distance to D2. \[fig:SNRsector\]](SNRavg.tikz){width="1\columnwidth"} Simulation Results\[sec:Simulation-Results\] ============================================ Simulation Setup ---------------- We consider a $40\,\mathrm{m}\times40\,\mathrm{m}$ area where the receiver D2 can be placed, D1 is located at a fixed and known position $\mathbf{q}=[0,0]^{\mathrm{T}}$, and a scatterer is fixed at $\mathbf{s}=[5,5]^{\mathrm{T}}$, hence $K=2$ . We set $f_{c}=60$ GHz, $B=100$ MHz, $N_{0}=-84$ dBm/GHz, $\alpha=0$ rad. For the LOS path, we set $h_{0}=\exp(-j2\pi f_{c}\tau_{0})/\sqrt{\rho_{0}}$, where $\rho_{0}=(2\pi\Vert\mathbf{q}-\mathbf{p}\Vert/\lambda)^{2}$ is the path-loss between D1 and D2. For the NLOS paths, we set $h_{k}=\exp(-j2\pi f_{c}\tau_{k})/\sqrt{\rho_{k}}$, in which $\rho_{k}=(2\pi(\Vert\mathbf{q}-\mathbf{s}_{k}\Vert\times\Vert\mathbf{s}_{k}-\mathbf{p}\Vert)/\lambda)^{2}$ [@Rappaport]. The number of antennas at both D1 and D2 is $N_{t}=N_{r}=64$, and the inter-element spacing is $d=\lambda/2$. The ULAs are located along the vertical axis. We generate a signal $x(t)$ with $N=64$ symbols. We set remaining parameters such that the SNR given by (\[eq:SNR\]) on the horizontal axis at 10 meters from D1 is 0 dB (i.e., the nominal communication range is 10 m). We will evaluate three protocols in terms of SNR, number of transactions, and positioning quality (using $\mathrm{PEB}^{(I)}$ and $\mathrm{REB}^{(I)}$): - A conventional beam selection protocol (termed CBS), similar to [@Wang], which can use only $M_{t}\in\{2,4,8,16,32,64\}$ discrete beams and $N'_{t}=M_{t}$ active contiguous antennas sequentially selected, generated using phase shifters $\phi_{i}=\{\pi,-\pi,\pi/2,-\pi/2\}$. - A discretized joint positioning and beam selection protocol (termed D-JPBS), using the same discrete codebook as the conventional beam selection protocol. - A joint positioning and beam selection protocol (termed C-JPBS), with a continuous codebook of the form (\[eq:directionCodebook\]), where $\theta_{m}\in[-\pi/2,\pi/2]$ and $N'_{t}\in[2,64]$. ![Normalized number of transactions with respect to the conventional beam selection protocol for the discrete joint positioning and beam selection protocol.\[fig:Transactionsdjp\]](Trandjpnp.tikz){width="1\columnwidth"} Results and Discussion ---------------------- We show results in two forms: contour plots over the area showing average performance (over 30 realizations) for the D-JPBS and plots that show the performance as a function of distance, where we averaged the values of the contour plots along concentric circles around D1. ### Final SNR Figure \[fig:SNRsector\] shows the SNR as a function of the D1-D2 distance for the CBS, C-JPBS, and D-JPBS, respectively. Overall, all protocols show a similar performance in terms of SNR. The CBS and D-JPBS protocols show identical performance. We can observe that the C-JPBS protocol achieves a slightly higher SNR with increasing distance between devices. The higher SNR is due to the higher degree of freedom that the C-JPBS protocol has compared to the other two protocols, and which allows the C-JPBS protocol to point the beams directly to the position of D2. In contrast, the CBS and D-JPBS protocols employ a more restricted codebook, and hence the SNR is dependent on the discretization of the beams. We conclude that the positioning-based protocols have no significant negative impact on the final SNR. ### Number of transactions The CBS uses a fixed number of transactions to complete the procedure regardless of D2 location. Hence, we show the contour plot of the normalized number of transactions with respect to the fixed CBS transactions in Figure \[fig:Transactionsdjp\] for the D-JPBS protocol, as an example of the behavior of the protocol in terms of transactions. It can be observed that number of transactions is dependent on the discretization of the beams. We note distinct regions in the figure due to discrete number of antennas that can be used by the protocol, combined with the criterion (\[eq:AODcondition\]). Since the beams are wider at the endfire of the D1 array, more transactions are used in the upper and lower left regions of the areas. Moreover, behind the scatterer we can observe a peculiar behavior caused by the inability of D1 in such locations to estimate both its own location and the scatterer location. In particular, the paths within this region are resolvable in angle but not in delay creating the need for more transactions. Figure \[fig:Transsector\] shows the number of transactions as a function of D1-D2 distance. As distance grows larger, the number of transactions for both the D-JPBS and C-JPBS protocols increases, since we need more information in the FIM to jump to a higher number of contiguous active antennas. We can observe a reduction of $67\%$ in the number of transactions is achieved when D2 is close to D1 (3 meters or less). The reduction grows to $50\%$ at inter-device distances between 3 and 10 meters. Beyond 10 meters we start observing a gap between the D-JPBS and C-JPBS protocols. This is due to the codebook restriction in the D-JPBS. The beam discretization has more influence at larger distances, given the separation between the beams; thus, giving an advantage to the C-JPBS protocol which has no codebook restrictions and beams can be pointed at any direction. We can conclude that position information has an impact in the reduction of latency of the device-to-device beam selection protocol. ![Average number of normalized transactions with respect to the conventional beam selection protocol as a function of distance to D1.\[fig:Transsector\]](Tranavg.tikz){width="1\columnwidth"} ### Positioning performance ![PEB for the discrete joint positioning and beam selection protocol.\[fig:PEBdpos\]](PEBdjpnp.tikz){width="1\columnwidth"} ![Average PEB as a function of distance to D1.\[fig:PEBsector\]](PEBavg.tikz){width="1\columnwidth"} Figure \[fig:PEBdpos\] shows the PEB for the D-JPBS. Note that a PEB of 0 dB corresponds to an uncertainty of 1 m, 10 dB corresponds to 3.2 m, and -10 dB to 30 cm. The achieved PEB values depend on the choice of the number of symbols $N$. As expected, the PEB values also depend on the distance and AOD with respect to D2. Due to high SNR gains, very low PEB values are observed close to D1. Moreover, in the region around the position of the scatterer $\mathbf{s}=[5,5]^{\mathrm{T}}$, we can observe a small decrease in PEB due to the aggregate information to the FIM provided by the scatterer. This behavior is only apparent close to the scatterer given our path loss model for the scattered path. The discretized protocol shows more accurate PEB in the directions of the available beams within the codebook. We note that within the region behind the scatterer there is a decrease in PEB due to the poor resolvability of the paths and thus the inability of D2 to estimate both its own location and the scatterer location. The paths for this region are not resolvable in time, but are resolvable in the angle domain. This translates into poor delay information, which causes a degradation of the estimated parameters and hence of the PEB. Figure \[fig:PEBsector\] shows the PEB as a function the D1-D2 distance. The general trend is that the PEB increases as a function of distance, but we can observe a change around 3 meters, where the PEB abruptly decreases due to the increase in number of transactions, which provide more information to the FIM. Similar behavior is observed for the REB, hence only the PEB figures are presented. We can observe how accurate position information is attainable within a moderate distance between devices. Conclusions\[sec:Conclusions\] ============================== Beam selection is an essential step in establishing a mm-wave communication link. Conventional protocols rely on measuring the received power obtained with a set of beamformers, which are successively made more and more directive. Given the characteristics of mm-wave propagation and the use of standard geometric channel models, we have exploited the ability of devices to determine their location during the beam selection process and thus improve the subsequent selection of beams. We have shown that such in-band position-aided protocols have similar performance as the conventional protocol in terms of achieved final SNR, but they are significantly faster and can additionally provide the position or orientation of the device in an accurate manner. Such information can be used in other procedures or applications. Our analysis indicates that standard codebooks can be used to harness these gains, with similar performance to more complex codebooks. Future work will include the removal of assumptions in the idealized receiver, such as the introduction of receiver beamforming. Derivation of the FIM ===================== $$\mathbf{J}_{(i,j)}=\left[\begin{array}{ccccc} \Phi(\tau_{i},\tau_{j}) & \Phi(\tau_{i},\theta_{\mathrm{tx},j}) & \Phi(\tau_{i},\theta_{\mathrm{rx},j}) & \Phi(\tau_{i},h_{R,j}) & \Phi(\tau_{i},h_{I,j})\\ \Phi(\theta_{\mathrm{tx},i},\tau_{j}) & \Phi(\theta_{\mathrm{tx},i},\theta_{\mathrm{tx},j}) & \Phi(\theta_{\mathrm{tx},i},\theta_{\mathrm{rx},j}) & \Phi(\theta_{\mathrm{tx},i},h_{R,j}) & \Phi(\theta_{\mathrm{tx},i},h_{I,j})\\ \Phi(\theta_{\mathrm{rx},i},\tau_{j}) & \Phi(\theta_{\mathrm{rx},i},\theta_{\mathrm{tx},j}) & \Phi(\theta_{\mathrm{rx},i},\theta_{\mathrm{rx},j}) & \Phi(\theta_{\mathrm{rx},i},h_{R,j}) & \Phi(\theta_{\mathrm{rx},i},h_{I,j})\\ \Phi(h_{R,i},\tau_{j}) & \Phi(h_{R,i},\theta_{\mathrm{tx},j}) & \Phi(h_{R,i},\theta_{\mathrm{rx},j}) & \Phi(h_{R,i},h_{R,j}) & \Phi(h_{R,i},h_{I,j})\\ \Phi(h_{I,i},\tau_{j}) & \Phi(h_{I,i},\theta_{\mathrm{tx},j}) & \Phi(h_{I,i},\theta_{\mathrm{rx},j}) & \Phi(h_{I,i},h_{R,j}) & \Phi(h_{I,i},h_{I,j}) \end{array}\right]\label{eq:FIMsub}$$ We consider the case of multiple paths and a single beam. The general form of the FIM for $k$ paths is given by $$\mathbf{J}_{\bm{\mathbf{\eta}}}^{(\mathrm{beam})}=\left[\begin{array}{ccc} \mathbf{J}_{(1,k)} & \cdots & \mathbf{J}_{(1,k)}\\ \vdots & \ddots & \vdots\\ \mathbf{J}_{(k,1)} & \cdots & \mathbf{J}_{(k,k)} \end{array}\right],\label{FIM_gral}$$ where each of the sub-matrices has the form (\[eq:FIMsub\]), in which $$\Phi(x_{1},x_{2})=\mathbb{E}_{\mathbf{y},\mathbf{a}\|\bm{\eta}}\left\{ \frac{\partial}{\partial x_{1}}\Lambda(\mathbf{y}\|\bm{\eta},\mathbf{a})\left(\frac{\partial}{\partial x_{2}}\Lambda(\mathbf{y}\|\bm{\eta},\mathbf{a})\right)^{}\right\} ,$$ where the log-likelihood function is expressed as $$\Lambda(\mathbf{y}\|\bm{\eta},\mathbf{a})=-\frac{1}{N_{0}}\int\Bigl\Vert\mathbf{y}(t)-\sum_{k=0}^{K-1}\mathbf{H}_{k}\mathbf{f}x(t-\tau_{k})\Bigr\Vert^{2}\mathrm{d}t.$$ Denoting the noise-free signal by $$\mathbf{m}(t)=\sum_{k=0}^{K-1}\mathbf{H}_{k}\mathbf{f}x(t-\tau_{k}),$$ it can be shown that $$\begin{aligned} \Phi(x_{1},x_{2}) & =\frac{2}{N_{0}}\Re\left\{ \mathbb{E}_{\mathbf{a}}\left\{ \int\frac{\partial\mathbf{m}^{\mathrm{H}}(t)}{\partial x_{1}}\frac{\partial\mathbf{m}(t)}{\partial x_{2}}\mathrm{d}t\right\} \right\} .\end{aligned}$$ It is readily verified that for an arbitrary path $i$ $$\begin{aligned} \frac{\partial\mathbf{m}(t)}{\partial\tau_{i}} & =-\sqrt{N_{t}N_{r}}h_{i}\mathbf{a}_{\mathrm{rx}}(\theta_{\mathrm{rx},i})\mathbf{a}_{\mathrm{tx}}^{\mathrm{H}}(\theta_{\mathrm{tx},i})\mathbf{f}\dot{x}(t-\tau_{i})\\ \frac{\partial\mathbf{m}(t)}{\partial\theta_{\mathrm{tx},i}} & =\sqrt{N_{t}N_{r}}h_{i}\mathbf{a}_{\mathrm{rx}}(\theta_{\mathrm{rx},i})\mathbf{\dot{a}}_{\mathrm{tx}}^{\mathrm{H}}(\theta_{\mathrm{tx},i})\mathbf{f}x(t-\tau_{i})\\ \frac{\partial\mathbf{m}(t)}{\partial\theta_{\mathrm{rx},i}} & =\sqrt{N_{t}N_{r}}h_{i}\dot{\mathbf{a}}_{\mathrm{rx}}(\theta_{\mathrm{rx},i})\mathbf{a}_{\mathrm{tx}}^{\mathrm{H}}(\theta_{\mathrm{tx},i})\mathbf{f}x(t-\tau_{i})\\ \frac{\partial\mathbf{m}(t)}{\partial h_{R,i}} & =\sqrt{N_{t}N_{r}}\mathbf{a}_{\mathrm{rx}}(\theta_{\mathrm{rx},i})\mathbf{a}_{\mathrm{tx}}^{\mathrm{H}}(\theta_{\mathrm{tx},i})\mathbf{f}x(t-\tau_{i})\\ \frac{\partial\mathbf{m}(t)}{\partial h_{I,i}} & =\sqrt{N_{t}N_{r}}j\mathbf{a}_{\mathrm{rx}}(\theta_{\mathrm{rx},i})\mathbf{a}_{\mathrm{tx}}^{\mathrm{H}}(\theta_{\mathrm{tx},i})\mathbf{f}x(t-\tau_{i}).\end{aligned}$$ Diagonal elements of the FIM {#diagonal-elements-of-the-fim .unnumbered} ---------------------------- We easily find that the diagonal elements of the submatrices $\mathbf{J}_{(i,j)}$. We first define $1/\sigma^{2}=2NE_{s}N_{r}N_{t}/N_{0}$, $\gamma_{\mathrm{tx},i}=\mathbf{f}^{\mathrm{H}}\mathbf{a}_{\mathrm{tx}}(\theta_{\mathrm{tx},i})$, $\beta_{ij}=\mathbf{a}_{\mathrm{rx}}^{\mathrm{H}}(\theta_{\mathrm{rx},i})\mathbf{a}_{\mathrm{rx}}(\theta_{\mathrm{rx},j})$ as well as $\dot{\mathbf{a}}_{\mathrm{tx}}(\theta_{\mathrm{tx}})=\partial\mathbf{a}_{\mathrm{tx}}(\theta_{\mathrm{tx}})/\partial\theta_{\mathrm{tx}}$ , $\dot{\gamma}_{\mathrm{tx,i}}=\dot{\mathbf{a}}_{\mathrm{tx}}^{\mathrm{H}}(\theta_{\mathrm{tx},i})\mathbf{f}$, and $\ddot{\beta}_{ij}=\dot{\mathbf{a}}_{\mathrm{rx}}^{\mathrm{H}}(\theta_{\mathrm{rx},i})\dot{\mathbf{a}}_{\mathrm{rx}}(\theta_{\mathrm{rx},j})$. We also introduce $$\begin{aligned} A_{0}(\Delta) & =\int p^{}(t-\Delta)p(t)\mathrm{d}t\\ A_{1}(\Delta) & =\int\dot{p}^{}(t-\Delta)p(t)\mathrm{d}t\\ A_{2}(\Delta) & =\int\dot{p}^{}(t-\Delta)\dot{p}(t)\mathrm{d}t.\end{aligned}$$ We then find that $$\begin{aligned} \Phi(\tau_{i},\tau_{j}) & =\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}h_{j}\gamma_{\mathrm{tx},i}\gamma_{\mathrm{tx},j}^{}\beta_{ij}A_{2}(\Delta_{ij})\right\} ,\\ \Phi(\theta_{\mathrm{tx},i},\theta_{\mathrm{tx},j}) & =\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}h_{j}\dot{\gamma}_{\mathrm{tx},i}^{}\beta_{ij}\dot{\gamma}_{\mathrm{tx},j}A_{0}(\Delta_{ij})\right\} \\ \Phi(\theta_{\mathrm{rx},i},\theta_{\mathrm{rx},j}) & =\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}h_{j}\gamma_{\mathrm{tx},i}\ddot{\beta}_{ij}\gamma_{\mathrm{tx},j}^{}A_{0}(\Delta_{ij})\right\} \\ \Phi(h_{R,i},h_{R,j}) & =\Phi(h_{I,i},h_{I,j})=\frac{1}{\sigma^{2}}\Re\left\{ \gamma_{\mathrm{tx},i}\gamma_{\mathrm{tx},j}^{}\beta_{ij}A_{0}(\Delta_{ij})\right\} \end{aligned}$$ where $\Delta_{ij}=\tau_{i}-\tau_{j}$. Off-diagonal elements of the FIM {#off-diagonal-elements-of-the-fim .unnumbered} -------------------------------- The off-diagonal elements are computed in similar fashion. Introducing, $\dot{\beta}_{ij}=\mathbf{a}_{\mathrm{rx}}^{\mathrm{H}}(\theta_{\mathrm{rx},i})\dot{\mathbf{a}}_{\mathrm{rx}}(\theta_{\mathrm{rx},j})$, the final expressions for the upper diagonal elements are computed as: $$\begin{aligned} \Phi(\tau_{i},\theta_{\mathrm{tx},j}) & =-\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}h_{j}\gamma_{\mathrm{tx},i}\beta_{ij}\dot{\gamma}_{\mathrm{tx},j}A_{1}(\Delta_{ij})\right\} \\ \Phi(\tau_{i},\theta_{\mathrm{rx},j}) & =-\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}h_{j}\gamma_{\mathrm{tx},i}\dot{\beta}_{ij}\gamma_{\mathrm{tx},j}^{}A_{1}(\Delta_{ij})\right\} \\ \Phi(\tau_{i},h_{R,j}) & =-\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}\gamma_{\mathrm{tx},i}\beta_{ij}\gamma_{\mathrm{tx},j}^{}A_{1}(\Delta_{ij})\right\} \\ \Phi(\tau_{i},h_{I,j}) & =-\frac{1}{\sigma^{2}}\Re\left\{ jh_{i}^{}\gamma_{\mathrm{tx},i}\beta_{ij}\gamma_{\mathrm{tx},j}^{}A_{1}(\Delta_{ij})\right\} \\ \Phi(\theta_{\mathrm{tx},i},\theta_{\mathrm{rx},j}) & =\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}h_{j}\dot{\gamma}_{\mathrm{tx},i}^{}\dot{\beta}_{ij}\gamma_{\mathrm{tx},j}^{}A_{0}(\Delta_{ij})\right\} \\ \Phi(\theta_{\mathrm{tx},i},h_{R,j}) & =\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}\dot{\gamma}_{\mathrm{tx},i}^{}\beta_{ij}\gamma_{\mathrm{tx},j}^{}A_{0}(\Delta_{ij})\right\} \\ \Phi(\theta_{\mathrm{tx},i},h_{I,j}) & =\frac{1}{\sigma^{2}}\Re\left\{ jh_{i}^{}\dot{\gamma}_{\mathrm{tx},i}^{}\beta_{ij}\gamma_{\mathrm{tx},j}^{}A_{0}(\Delta_{ij})\right\} \\ \Phi(\theta_{\mathrm{rx},i},h_{R,j}) & =\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}\gamma_{\mathrm{tx},i}\dot{\beta}_{ji}^{}\gamma_{\mathrm{tx},j}^{}A_{0}(\Delta_{ij})\right\} \\ \Phi(\theta_{\mathrm{rx},i},h_{I,j}) & =\frac{1}{\sigma^{2}}\Re\left\{ jh_{i}^{}\gamma_{\mathrm{tx},i}\dot{\beta}_{ij}^{}\gamma_{\mathrm{tx},j}^{}A_{0}(\Delta_{ij})\right\} \\ \Phi(h_{R,i},h_{I,i}) & =\frac{1}{\sigma^{2}}\Re\left\{ j\gamma_{\mathrm{tx},i}\beta_{ij}\gamma_{\mathrm{tx},j}^{}A_{0}(\Delta_{ij})\right\} .\end{aligned}$$ The elements of the lower off diagonal are obtained as $$\begin{aligned} \Phi(\theta_{\mathrm{tx},i},\tau_{j}) & =-\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}h_{j}\dot{\gamma}_{\mathrm{tx},i}^{}\beta_{ij}\gamma_{\mathrm{tx},j}^{}A_{1}(\Delta_{ji})\right\} \\ \Phi(\theta_{\mathrm{rx},i},\tau_{j}) & =-\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}h_{j}\gamma_{\mathrm{tx},i}\dot{\beta}_{ji}^{}\gamma_{\mathrm{tx},j}^{}A_{1}(\Delta_{ji})\right\} \\ \Phi(h_{R,i},\tau_{j}) & =-\frac{1}{\sigma^{2}}\Re\left\{ h_{j}\gamma_{\mathrm{tx},i}\beta_{ij}\gamma_{\mathrm{tx},j}^{}A_{1}(\Delta_{ji})\right\} \\ \Phi(h_{I,i},\tau_{j}) & =-\frac{1}{\sigma^{2}}\Re\left\{ jh_{j}\gamma_{\mathrm{tx},i}\beta_{ij}\gamma_{\mathrm{tx},j}^{}A_{1}(\Delta_{ji})\right\} \\ \Phi(\theta_{\mathrm{rx},i},\theta_{\mathrm{tx},j}) & =\frac{1}{\sigma^{2}}\Re\left\{ h_{i}^{}h_{j}\gamma_{\mathrm{tx},i}\dot{\beta}_{ji}^{}\dot{\gamma}_{\mathrm{tx},j}A_{0}(\Delta_{ij})\right\} \\ \Phi(h_{R,i},\theta_{\mathrm{tx},j}) & =\frac{1}{\sigma^{2}}\Re\left\{ h_{j}\gamma_{\mathrm{tx},i}\beta_{ij}\dot{\gamma}_{\mathrm{tx},j}A_{0}(\Delta_{ij})\right\} \\ \Phi(h_{I,i},\theta_{\mathrm{tx},j},) & =\frac{1}{\sigma^{2}}\Re\left\{ jh_{j}\gamma_{\mathrm{tx},i}\beta_{ij}\dot{\gamma}_{\mathrm{tx},j}A_{0}(\Delta_{ij})\right\} \\ \Phi(h_{R,i},\theta_{\mathrm{rx},j}) & =\frac{1}{\sigma^{2}}\Re\left\{ h_{j}\gamma_{\mathrm{tx},i}\dot{\beta}_{ij}\gamma_{\mathrm{tx},j}^{}A_{0}(\Delta_{ij})\right\} \\ \Phi(h_{I,i},\theta_{\mathrm{rx},j}) & =\frac{1}{\sigma^{2}}\Re\left\{ jh_{j}\gamma_{\mathrm{tx},i}\dot{\beta}_{ij}\gamma_{\mathrm{tx},j}^{}A_{0}(\Delta_{ij})\right\} \\ \Phi(h_{I,i},h_{R,j}) & =\frac{1}{\sigma^{2}}\Re\left\{ j\gamma_{\mathrm{tx},i}\beta_{ij}\gamma_{\mathrm{tx},j}^{*}A_{0}(\Delta_{ij})\right\} .\end{aligned}$$ ### Remarks {#remarks-1 .unnumbered} - When $p(t)$ is flat in the frequency domain, then $$\begin{aligned} A_{0}(\Delta) & =\frac{\sin(\pi B\Delta)}{\pi B\Delta}\\ A_{1}(\Delta) & =\frac{-\sin(\pi B\Delta)+\pi B\Delta\cos(\pi B\Delta)}{\pi B\Delta^{2}}.\\ A_{2}(\Delta) & =\frac{((\pi B\Delta)^{2}-2)\sin(\pi B\Delta)}{\pi\Delta^{3}B}\\ & +\frac{2\pi B\Delta\cos(\pi B\Delta)}{\pi\Delta^{3}B}.\end{aligned}$$ We observe that $A_{0}(0)=1$, $A_{1}(0)=0$, and $A_{2}(0)=\pi^{2}B^{2}/3$, so that the entries in $\mathbf{J}_{(i,i)}$ have compact expressions, compared to $\mathbf{J}_{(i,j\neq i)}$. - When $\Delta$ is such that $B\Delta\gg1$, then $A_{0,1,2}(\Delta)\approx0$. Hence, when paths have large relative path lengths, this leads to a block diagonal structure in (\[FIM\_gral\]). Optimized Beam Directions\[sec:Optimized-Beam-Directions\] ========================================================== The directions of the beams in the position-aided protocol can be optimized as follows. We select one beam with maximum gain direction $\theta_{0}$ closest to $\hat{\theta}_{\mathrm{tx},0}^{(i)}$. Then, the two additional beams are set to minimize a measure of expected future uncertainty. Let $\mathbf{J}_{\hat{\bm{\eta}}}^{(i,m)}(\theta)$ be the FIM, evaluated in $\hat{\bm{\eta}}$ (the estimate of $\boldsymbol{\eta}$) for a beam pointing towards $\theta$. The we choose beams $$\begin{aligned} \mathrm{minimize}_{\theta_{1},\theta_{2}}\, & \mathrm{trace}\Bigl\{\Bigl[\mathbf{J}_{\hat{\bm{\eta}}}^{(i,1)}(\theta_{0})\\ & +\mathbf{J}_{\hat{\bm{\eta}}}^{(i,2)}(\theta_{1})+\mathbf{J}_{\hat{\bm{\eta}}}^{(i,3)}(\theta_{2})\Bigr]_{1:2,1:2}^{-1}\Bigr\}\\ \mathrm{subject\,to}\,\, & \theta_{1},\theta_{2}\in\Theta.\end{aligned}$$ In case $\Theta$ is $[-\pi/2,\pi/2]$, we can instead set $\theta_{0}=\hat{\theta}_{\mathrm{tx},0}^{(i)}$ , $\theta_{1}=\hat{\theta}_{\mathrm{tx},0}^{(i)}+\varepsilon$ and $\theta_{2}=\hat{\theta}_{\mathrm{tx},0}^{(i)}-\varepsilon$, and optimize with respect to the scalar parameter $\varepsilon\ge0$. [^1]: G. E. Garcia, and H. Wymeersch are with the Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden, e-mails: {ggarcia,henkw}@chalmers.se. G. Seco-Granados is with the Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, Barcelona, Spain, e-mail: gonzalo.seco@uab.cat. E. Karipidis is with Ericsson Research, Stockholm, Sweden, e-mail: eleftherios.karipidis@ericsson.com. This research was supported, in part, by the European Research Council, under Grant No. 258418 (COOPNET), the EU project HIGHTS (High precision positioning for cooperative ITS applications) MG-3.5a-2014-636537, and the R&D Project of Spanish Ministry of Economy and Competitiveness under Grant TEC2014-53656-R. [^2]: A 2-dimensional model is assumed for simplicity. However, the proposed protocols can be extended to 3-dimensional scenarios with 2-dimensional antenna arrays. [^3]: The narrowband assumption imposes the constraints that (i) there is no beam squint; and (ii) $\mbox{max}(N_{t},N_{r})d\ll c/B$ where $d$ denotes the distance between the antenna elements. [^4]: The underlying idea of the proposed protocol is applicable to any array geometry. [^5]: In a practical implementation a priori information on the number of scatterers is not required [@Shahmansoori2017].
--- abstract: '[With this paper we want to investigate the highly variable afterglow light curve and environment of gamma-ray burst (GRB) 060526 at $z=3.221$.]{} [We present one of the largest photometric datasets ever obtained for a GRB afterglow, consisting of multi-color photometric data from the ultraviolet to the near infrared. The data set contains 412 data points in total to which we add additional data from the literature. Furthermore, we present low-resolution high signal-to-noise spectra of the afterglow. The afterglow light curve is modeled with both an analytical model using broken power law fits and with a broad-band numerical model which includes energy injections. The absorption lines detected in the spectra are used to derive column densities using a multi-ion single-component curve-of-growth analysis from which we derive the metallicity of the host of GRB 060526.]{} [The temporal behaviour of the afterglow follows a double broken power law with breaks at $t=0.090\pm0.005$ and $t=2.401\pm0.061$ days. It shows deviations from the smooth set of power laws that can be modeled by additional energy injections from the central engine, although some significant microvariability remains. The broadband spectral-energy distribution of the afterglow shows no significant extinction along the line of sight. The metallicity derived from and of \[S/H\] = –0.57 $\pm$0.25 and \[Fe/H\] = –1.09$\pm$0.24 is relatively high for a galaxy at that redshift but comparable to the metallicity of other GRB hosts at similar redshifts. At the position of the afterglow, no host is detected to F775W(AB) = 28.5 mag with the HST, implying an absolute magnitude of the host M(1500 Å)$>$–18.3 mag which is fainter than most long-duration hosts, although the GRB may be associated with a faint galaxy at a distance of 11 kpc.]{}' author: - 'C. C. Thöne' - 'D. A. Kann' - 'G. Jóhannesson' - 'J. H. Selj' - 'A. O. Jaunsen' - 'J. P. U. Fynbo' - 'C. W. Akerlof' - 'K. S. Baliyan' - 'C. Bartolini' - 'I. F. Bikmaev' - 'J. S. Bloom' - 'R. A. Burenin' - 'B. E. Cobb' - 'S. Covino' - 'P. A. Curran' - 'H. Dahle' - 'A. Ferrero' - 'S. Foley' - 'J. French' - 'A. S. Fruchter' - 'S. Ganesh' - 'J. F. Graham' - 'G. Greco' - 'A. Guarnieri' - 'L. Hanlon' - 'J. Hjorth' - 'M. Ibrahimov' - 'G. L. Israel' - 'P. Jakobsson' - 'M. Jelínek' - 'B. L. Jensen' - 'U. G. Jørgensen' - 'I. M. Khamitov' - 'T. S. Koch' - 'A. J. Levan' - 'D. Malesani' - 'N. Masetti' - 'S. Meehan' - 'G. Melady' - 'D. Nanni' - 'J. Näränen' - 'E. Pakstiene' - 'M. N. Pavlinsky' - 'D. A. Perley' - 'A. Piccioni' - 'G. Pizzichini' - 'A. Pozanenko' - 'P. W. A. Roming' - 'W. Rujopakarn' - 'V. Rumyantsev' - 'E. S. Rykoff' - 'D. Sharapov' - 'D. Starr' - 'R. A. Sunyaev' - 'H. Swan' - 'N. R. Tanvir' - 'F. Terra' - 'P. M. Vreeswijk' - 'A. C. Wilson' - 'S. A. Yost' - 'F. Yuan' date: 'Received 8 June 2008 / accepted 17 July 2010' title: 'Photometry and spectroscopy of GRB 060526: A detailed study of the afterglow and host galaxy of a z=3.2 gamma-ray burst [^1]' --- Introduction ============ Gamma-ray bursts (GRBs) and their afterglows offer a powerful tool to probe the high-redshift universe, both through photometry and spectroscopy. The standard fireball model of GRB afterglows [see @Zhang07; @Meszaros06; @Gehrels09 for recent reviews on the topic] predicts a smooth temporal evolution, and the resulting afterglow light curve can be empirically described by a joint smoothly-broken power law, the so-called Beuermann equation [@Beuermann1999]. For many GRBs in the pre-[Swift]{} era, when temporally dense afterglow photometry was obtained, the afterglow evolution was found to be smooth as, e.g., in GRB 020813 [@LaursenStanek2003], GRB 030226 [@Klose2004] and GRB 041006 [@Stanek2005] [for the complete pre-Swift sample see @ZKK]. Out of the total sample of 59 afterglows of that paper, only four of the GRBs analysed showed significant deviations from the expected decay (though we note that about half of the 59 afterglows were not sampled well enough to allow any conclusions). One might be explained by microlensing [GRB 000301C, @GLS2000] and another one by assuming an inhomogeneously emitting surface [GRB 011211, @JakobssonNA2004]. The other two GRBs which are the only ones showing long-lasting strong deviations are GRB 021004 [e.g. @deUgarte2005 and references therein] and GRB 030329 [e.g. @Lipkin2004 and references therein], which incidentally also have the densest optical monitoring. With the launch of the Swift satellite and its rapid localization capabilities [@Gehrels2004], the number of highly variable light curves has increased considerably, though there are still examples of very smooth light curves (e.g. that of GRB 080210, which showed a smooth behaviour with a high sampling rate of 1s, A. De Cia, priv. comm.). Some light curves show small, achromatic bumps overlying the smooth power law decay (e.g., GRB 050502A, @Guidorzi05; GRB 061007, @Mundell07; GRBs 090323 and 090328, @McBreenLAT), early bumps with chromatic evolution (e.g., GRB 061126, @Perley08a; GRB 071003, @Perley08b), “steps” due to energy injection episodes (e.g., GRB 070125, @Updike08; GRB 071010A, @Covino08; GRB 080913, @Greiner080913, GRB 090926A, @Rau090926A [@CenkoLAT; @Swenson090926A]) or powerful late-time rebrightenings of up to several magnitudes (e.g., GRB 050721, @Antonelli06; GRB 060206, @Wozniak060206 [@Monfardini060206; @Stanek060206]; GRB 070311, @Guidorzi07; GRB 071003, @Perley08b). The early time domain, which can now be routinely accessed by rapid follow-up in the Swift era, has yielded more types of variability, like rising afterglows (e.g., GRB 060418, @Molinari07; GRB 060605, @Ferrero08; GRB 060607A, @Nysewander07, @Molinari07; GRB 081008, @Yuan10; see @Oates09 and @Rykoff09 for further examples) and short-term variability directly linked to the prompt emission (e.g., GRB 041219A, @Vestrand041219A [@Blake041219A]; GRB 050820A, @Vestrand050820A; GRB 080319B, @Racusin08; GRB 080129, @Greiner080129). In all these cases, dense photometric follow-up during the periods of variability was needed to characterize the phenomena involved. In addition their use in to studying the GRB phenomenon itself, GRB afterglows can be used to study their galactic environment through absorption line spectroscopy of material in the line-of-sight towards the GRB. Above a redshift of around $z\sim0.2$, resonant absorption lines from elements present in the interstellar medium (ISM) such as Mg, Zn, Fe, Si, C, and S are shifted into the optical regime and can be studied with ground-based telescopes. For 10 bursts between a redshift of $z=2$ to 6, the metallicity along the line-of-sight in the galaxy could be obtained so far [e.g. @Savaglio06; @Fynbo06a; @Price07; @Prochaska07; @Ledoux09 and references therein]. The values are usually below solar, but higher than for QSO absorbers at comparable redshifts, some of them even higher than theoretical limits for the formation of collapsars [@Woosley06]. The difference to QSO absorbers can be explained if GRB sightlines probe denser parts of the galaxy, or if GRBs reside in galaxies with higher masses and therefore higher metallicities [@Fynbo08; @Pontzen10]. Both for QSO and GRB absorbers, there seems to be a metallicity evolution with redshift [@Savaglio06; @Fynbo06a; @Price07], although the slope is different for the two samples. GRB hosts seem to show a low extinction along the line-of-sight [@Prochaska07], however, relative abundances of heavier elements indicate that some of the ions must be depleted onto dust grains and the depletion pattern resembles the one found in the warm disc and halo of the Milky Way (MW; @Savaglio06). How this can be explained together with the low extinction as also derived from the spectral energy distribution (SED) of the afterglow [@Kann2006; @Starling07; @Kann07], is still an open question. One possible solution to this problem is destruction of the dust present in the line-of-sight by the GRB and afterglow radiation [e.g., @Waxman00; @Perna02], though no strong evidence has ever been found for this. Furthermore, there is clearly a strong observational bias involved, as those GRB afterglows with successful spectroscopy, especially in the case of high-resolution observations, are those which have only low extinctions and thus relatively bright afterglows [@Fynbo09; @Kann07], although rapid observations with large telescopes can achieve detailed spectroscopy of highly extinguished afterglows, as in the case of GRB 080607 [@Prochaska09; @Sheffer09; @Fynbo09]. GRB 060526 was detected by the Swift satellite on May 26.686458 (16:28:29.95 UT). The satellite slewed immediately to the burst, detecting both the X-ray and the optical afterglow [@CampanaGCN1]. The BAT instrument on Swift measured two emission episodes. The first one lasted 13.8 s and consisted of two FRED (fast rise exponential decay) peaks, followed by a second symmetric peak between 230 and 270 s [@CampanaGCN2]. The second peak was coincident with a giant X-ray flare followed by a softer flare at 310 s [@CampanaGCN3] also detected in the $v$ band by the UVOT telescope on-board Swift [@BrownGCN]. The gamma-ray fluence was $(4.9\pm0.6)\times10^{-7}$ erg cm$^{-2}$ during the first emission episode and $(5.9\pm0.6)\times10^{-7}$ erg cm$^{-2}$ during the second, the peak flux of the second episode was however only half of the peak flux of the first one. The photon index of the two epochs changed from $1.66\pm0.20$ to $2.07\pm0.18$, thus showing the typical hard-to-soft evolution [@MarkwardtGCN]. The Watcher telescope provided the first ground based detection with $R\approx15$ mag [@FrenchGCN] 36.2 s after the trigger. ROTSE observations showed a plateau for several thousand seconds after the GRB onset [@RykoffGCN]. A redshift of $z=3.21$ was determined by [@BergerGCN] with the Magellan/Clay telescope. The brightness of the optical afterglow allowed for a dense monitoring which revealed a complex light curve structure including several flares [@HalpernGCN1; @HalpernGCN2] and a steepening attributed to a jet break [@ThoeneGCN]. In this paper, we approach the analysis of GRB 060526 from two directions: Through modelling of the very detailed optical light curve and late optical imaging of the field to detect the host in Sec. 3, and analysis of low to medium resolution spectroscopic observations of absorption lines along the line of sight (Sec. 4). Throughout the paper, we follow the convention $F_\nu (t)\propto t^{-\alpha}\nu^{-\beta}$, and use WMAP concordant cosmology [@Spergel2003] with $H_0=71$km s$^{-1}$ Mpc$^{-1}$, $\Omega_M=0.27$, and $\Omega_{\Lambda}=0.73$. Uncertainties are given at 68% confidence level for one parameter of interest unless stated otherwise. Observations ============ Photometry ---------- In order to get a good coverage of the light curve, we obtained data using several different telescopes around the world. Our complete data set comprises a total of 412 points from the UV to K-band, one of the largest photometric samples of an optical/NIR afterglow in the Swift era. The earliest dataset was obtained by the Watcher telescope, located at Boyden Observatory, South Africa, starting 36.2 s after the burst, followed by Swift UVOT starting 86 s after the trigger. Early ground based optical data were obtained with the ROTSE-IIIc 0.3m telescope at the H.E.S.S. site at Mt. Gamsberg, Namibia, the 1.5m telescope on Mt. Maidanak/Uzbekistan, the 2.6m Shajn telescope at CrAO (Crimean Astrophysical Observatory/Ukraine), the TNG (Telescopio Nazionale Galileo) on La Palma equipped with DOLoRes, the 1.2m MIRO telescope on Mt. Abu/India, with BFOSC (Bologna Faint Object Spectrograph & Camera) at the G. D. Cassini 152 cm telescope of the Bologna University under poor conditions and with the RTT150 (1.5m Russian-Turkish telescope, Bakirlitepe, Turkey), the RTT150 data are also presented in [@Khamitov08]. The light curve was followed up sparsely every night over nearly a week with DFOSC (Danish Faint Object Spectrograph and Camera) on the Danish 1.54m telescope on La Silla/Chile under partially photometric conditions and with MOSCA (MOSaic CAmera) and ALFOSC (Andalucia Faint Object Spectrograph and Camera) at the Nordic Optical Telescope on La Palma. Furthermore, two epochs were obtained with the Tautenburg 1.34m Schmidt telescope and two sets of images were taken several days after the GRB with Keck/LRIS simultaneously in the Kron-Cousins $R$ and the Sloan $g^\prime$ bands, the second observation was performed at high airmass under bad seeing conditions. Late images were obtained with FORS2 at the VLT on Paranal/Chile on Feb. 23, 2007 and Mar. 30, 2008 in the $R_C$ band with exposure times of 2500 and 7500 s, respectively, to look for the host galaxy. We also took observations with the [Hubble Space Telescope]{} on 9 August 2009, utilizing the Advanced Camera for Surveys with the F775W (roughly SDSS $i^\prime$) filter (see Fig. \[hostfig\]). A total of 7844 s of observations were obtained in six dithered exposures. These were reduced via [multidrizzle]{} in the standard fashion. Near infrared data were collected with ANDICAM and the 1.3 m SMARTS telescope (Small and Moderate Aperture Research Telescope System) at CTIO under non-photometric conditions as well as with the robotic 1.3m PAIRITEL telescope on Mt. Hopkins. The UVOT data were reduced and analysed using the standard UVOT tasks within the [heasoft]{} package. For the photometric calibration of the ground-based data, we determined the calibrated magnitude of six comparison stars in the field using photometric zero points from DFOSC in the $V$, $R_C$ and $I_C$ bands (see Table \[compstars\]). These stars were then used to perform relative Point-Spread Function (PSF) photometry to get the calibrated magnitude of the afterglow. For some of the late NOT images as well as the faint MIRO detections, though, we applied relative aperture photometry using a circle of 20 pixels diameter (and an annulus of 10 pixels for the sky). For the $B$-band, where no DFOSC data were available, we took zero points for only three comparison stars from the SDSS (see Table \[compstars\]), converting them with the equations of [@JesterSDSS]. $g^\prime$ zero points were taken from the SDSS, and they are given as AB magnitudes. For the RTT 150 data, we used one USNO-B1 star as reference that was calibrated using Landolt standard stars. The results are in full agreement with the rest of the data set. The Watcher data were analysed using a dedicated photometry pipeline [@Ferrero2010]. The $J$, $H$ and $K_S$ band data from SMARTS/ANDICAM and PAIRITEL were calibrated using three and ten nearby stars, respectively, from the 2MASS catalogue. For the $H$ and $K_S$ band data, we used smaller apertures and applied aperture corrections to reduce the influence of the highly variable background. All data and upper limits are given in Table \[obslog\], the data are not corrected for Galactic extinction. Note we give the $g^\prime$ and HST $F775W$ magnitudes in AB magnitudes. For the final light curve fitting, we add $Br^\prime R_Ci^\prime$ band data from [@Dai2007]. We shift the $B$ and $R_C$ band data of [@Dai2007] by 0.1 magnitudes to bring it to our zero point. The multi-color light curves are shown in Fig. \[lightcurve\]. Spectroscopy ------------ Spectra were obtained with FORS1/VLT on May 27 from 9 to 12 hours after the burst. Four different grisms cover the wavelength range from 3650 to 9200 [Å]{}. For all four grisms a 1$\farcs$0 slit was used which provides resolutions between 2.4 and 11.1 [Å]{}. Reduction, cosmic ray removal, extraction and wavelength calibration were performed using standard tasks in IRAF[^2]. The final spectra were then normalised as no absolute flux calibration was needed. In order to improve the S/N, we combined the datasets taken with the same grism weighted with their variance. A summary of the spectroscopic observations is given in Table \[speclog\]. ------------ --------- --------- ------- ---------------- ------------ -- Time (May Midtime Exptime Grism Spectral range Resolution 27 \[UT\]) \[d\] \[s\] \[Å\] \[Å\] 01:16:56 0.37042 600 300V 3650 – 8900 11.1 01:29:46 0.38105 900 600I 7000 – 9200 4.4 01:47:28 0.39335 900 1200B 3860 – 4400 2.4 02:05:27 0.40583 900 600V 4080 – 7200 4.5 02:24:00 0.42043 1200 300V 3650 – 8900 11.1 02:57:05 0.44681 1800 1200B 3860 – 4400 2.4 03:30:34 0.47007 1800 600V 4080 – 7200 4.5 04:04:08 0.49339 1800 600I 7000 – 9200 4.4 ------------ --------- --------- ------- ---------------- ------------ -- ---- ------------- ------------- ---------------- ---------------- ---------------- ---------------- \# Coordinates $B$ $V$ $R_C$ $I_C$ RA Dec. \[mag\] \[mag\] \[mag\] \[mag\] 1 15:31:19.6 +00:16:59.5 $19.90\pm0.03$ $19.36\pm0.02$ $19.02\pm0.02$ $18.64\pm0.02$ 2 15:31:22.7 +00:17:21.2 $20.30\pm0.03$ $19.51\pm0.02$ $19.09\pm0.02$ $18.71\pm0.02$ 3 15:31:20.9 +00:16:39.1 $19.97\pm0.03$ $19.11\pm0.02$ $18.55\pm0.02$ $18.12\pm0.02$ 4 15:31:14.4 +00:17:49.9 — $19.78\pm0.02$ $19.43\pm0.02$ $18.81\pm0.03$ 5 15:31:16.5 +00:17:55.5 — $20.34\pm0.03$ $19.96\pm0.03$ $20.59\pm0.03$ 6 15:31:18.3 +00:18:17.7 — $20.49\pm0.03$ $19.45\pm0.02$ $18.23\pm0.02$ ---- ------------- ------------- ---------------- ---------------- ---------------- ---------------- ![image](10340fg1.eps){width="18cm"} Prompt emission =============== Up to two hours after the burst, the light curve features an optical flare contemporaneous to the XRT/BAT flare at $\approx250$ s and a following plateau phase [@BrownGCN; @FrenchGCN; @RykoffGCN]. We reduced the BAT event data using standard procedures within the software provided by [HEASOFT]{} (version 6.1). The XRT observations were reduced using the standard [xrtpipeline]{} (version 0.10.4) for XRT data analysis software using the most recent calibration files. The spectral data of BAT and XRT were analysed with [XSPEC]{} version 11.3 [@Arnaud96]. The X-ray Galactic column density was fixed to $5.02\times10^{20}$ cm$^{-2}$ [@Kalberla05]. We estimated the late-time extragalactic column density by fitting the XRT PC data from 517 s to $1.34\times10^{5}$ s post trigger, where spectral evolution is negligible, using an absorbed power law model. We only find an upper limit of N$_\mathrm{HX} < 9.8\times10^{21}$ cm$^{-2}$ [see also @Campana10]. In Fig. \[060526:flare\] we compare the timescale of the flares at high energies (BAT and XRT) with contemporaneous optical data. The first, stronger flare is seen both by the XRT and BAT whereas the second, softer flare is only visible in the XRT data. Our earliest optical data are also coincident with the two flares. A broad early bump is observed in the optical light curve which precedes the second episode of BAT emission and the major X-ray pulse. The Watcher data only show a plateau during the high-energy flares due to a relatively low time resolution. The higher time-resolution of the UVOT data shows that there are also two significant flares in the optical. The first optical flare is contemporaneous with the peak of the BAT/XRT flare within $2\sigma$ of the temporal error. Then, there may also be a small optical bump (significance only $\approx1.5\sigma$) at the time of the second (XRT-only) flare, whereas the second significant optical flare occurs $\approx90$ s after the XRT-only flare implying that these two events are probably not connected. Power-law fitting of the optical light curve (with respect to the BAT trigger time T$_0$) shows the slopes are very steep. The rising slope of the first flare is $\alpha_{r1}=-12.8\pm3.0$, and the decay slope is $\alpha_{d1}=5.8\pm0.8$, or, if one takes the bump as a second flare, the two decay slopes are $\alpha_{d11}=8.2\pm2.0$, $\alpha_{d12}=8.6\pm2.5$. The rise and decay slopes for the second peak are similarly steep, but with larger uncertainties. Prompt optical flashes attributed to reverse shocks, as seen in the landmark burst GRB 990123 [@Akerlof99], have been observed in only a few cases since the launch of Swift, e.g., GRB 060111B [@Klotz06] and GRB 060117 (@Jelinek06, see @Kann07 for a recent overview of light curves with probable reverse shock flashes/steep decays in the Swift era). Most bursts for which early optical data are available display no evidence of reverse shock emission. In a number of cases, the optical light curve is dominated by forward shock emission from very early times, e.g. GRB 050401 [@Rykoff05], GRB 060418 and GRB 060607A [@Molinari07; @Nysewander07], GRB 060605 [@Ferrero08], GRB 061007 [@Mundell07] and GRB 081008 [@Yuan10]. An optical component of the emission from internal shocks has been invoked to explain the correlation between the optical and high-energy light curves observed in several bursts, e.g. GRB 041219A [@Vestrand041219A; @Blake041219A; @Fan05], GRB 050820A [@Vestrand050820A], and GRB 050904 [@Wei06]. The very rapid variability as measured by the steep slopes and the contemporaneous first flare indicates that GRB 060526 is another case where the early optical light curve is dominated by central engine activity. Such time resolution and coincident optical and high-energy observations are still rare, however, a thorough analysis of the flares and their spectral properties goes beyond the scope of this paper. ![The two early flares seen in the BAT and XRT data and temporally coincident optical observations by UVOT and Watcher. The second flare is much softer at high energies and hence it is only seen in the XRT data. Both flares are reflected in the optical data with the first optical flare being coincident with the first BAT/XRT flare while the second flare is delayed compared to the second X-ray flare. The white data have been shifted upward by 0.3 mag to the $v$ zero point.[]{data-label="060526:flare"}](10340fg2.eps){width="\columnwidth"} Afterglow photometry and light curve modelling ============================================== The multi-color light curve {#MCLC} --------------------------- ![The $R_C$-band light curve of GRB 060526 fitted with a double smoothly-broken power law, using data starting at 0.046 days. The residuals of the fit show an early optical flare and later strong variations of up to 0.3 magnitudes. The dashed vertical lines mark the break times, the dotted lines the $1\sigma$ region of uncertainty. The second break is soft ($n\approx4$) and is thus seen as a smooth rollover.[]{data-label="power-law"}](10340fg3.eps){width="\columnwidth"} To analyse the evolution of the light curve, we use the $R_C/CR$ band which has the densest sampling, a total of 319 data points in all, from 50 s to more than 7 days after the GRB. For all fits, we fix the host magnitude to $m_h=29$ (see [[§]{} \[Host\]]{}). A fit with a single power law to these data is very strongly rejected, with ${\chi^2{\textnormal{/d.o.f.}}=}31.89$ (with 319 d.o.f.). Even if we remove the earliest data which are affected by optical flares, starting only at 400 s after the GRB, the fit is still rejected (${\chi^2{\textnormal{/d.o.f.}}=}29.46$ with 308 d.o.f.). A double smoothly-broken power law gives a much better fit, but even so, the fit is formally rejected, with ${\chi^2{\textnormal{/d.o.f.}}=}3.12$ with 302 d.o.f. This is due to the strong variability in the light curve which was first found by [@HalpernGCN1] and is also discussed in [@Dai2007]. This fit, along with the residuals showing the strong variability, is shown in Fig. \[power-law\]. The parameters we find for this fit ($\alpha_{\rm plateau}=0.288\pm0.026$, $\alpha_1=0.971\pm0.008$, $\alpha_2=2.524\pm0.052$, $t_{b1}=0.090\pm0.005$ days, $t_b=2.216\pm0.049$ days) are concurrent with those of [@Dai2007] (who find $\alpha_1\approx1.0$, $\alpha_2\approx2.9$ and $t_b\approx2.55$ days). Due to the high data density, we are able to let the break smoothness parameter $n$ vary for the second break (the first break had to be fixed to $n=10$), and our result ($n=4.3\pm0.7$) is in agreement with the tentative $\alpha_1-n$ correlation found by [@ZKK]. Still, the significant improvement shows that the light curve is basically a double smoothly-broken power law, and the steep late decay indicates that this break is a jet break, as first noted by [@ThoeneGCN] and also found by [@Dai2007]. The rest frame jet break time of 0.52 days is typical for the optical afterglows of the pre-Swift GRB sample [@ZKK]. Jet breaks, a common feature in well-monitored pre-Swift optical afterglows [e.g., @ZKK], have not been found in many Swift afterglows[^3], especially in the X-rays [e.g., @Mangano2007; @Grupe2007; @Sato2007; @Racusin09], and if there are breaks, then the comparison between the optical and X-ray light curves show them to often be chromatic [@Panaitescu2006; @Oates2007]. Analysing both the optical and X-ray data of GRB 060526, [@Dai2007] suggest that the (jet) break is achromatic. ![image](10340fg4.eps){width="17cm"} If we take the X-ray data from [@Dai2007] and fit it with the parameters of our double smoothly-broken power law fit using X- ray data from 0.06 days onward, we concur that the fit is marginally acceptable, with ${\chi^2{\textnormal{/d.o.f.}}=}1.6$ for 40 degrees of freedom. Using our own X-ray reduction, we obtain a worse result, with ${\chi^2{\textnormal{/d.o.f.}}=}1.7$ for 22 degrees of freedom. Similarly, with X-ray data from the Swift XRT repository [@Evans07; @Evans09], the fit is rejected (${\chi^2{\textnormal{/d.o.f.}}=}2.3$ for 33 degrees of freedom). The reason is that there are less data points, but with smaller error bars, so the outliers are weighted more strongly. A fit to the X-ray data alone results in a much earlier break time and slopes that are less steep [again in agreement with @Dai2007], but we caution that the late X-ray afterglow is only sparsely sampled and shows large scatter. For the three different reductions, we derive the following values using a smoothly-broken power law (a double-broken power law yields no statistical improvement), with $n=10$ fixed and no host galaxy: $\alpha_1=0.92\pm0.05$, $\alpha_2=3.02\pm0.63$, $t_b=1.38\pm0.25$ (XRT repository light curve); $\alpha_1=0.90\pm0.05$, $\alpha_2=2.78\pm0.61$, $t_b=1.34\pm0.28$ (our XRT reduction); $\alpha_1=- 0.29\pm0.56$, $\alpha_2=1.69\pm0.17$, $t_b=0.16\pm0.05$ (@Dai2007 data). As can be seen, the latter fit is very different from the other two (which agree fully within error bars). Even using data from 0.01 days onward (end of the X-ray flare), the fit is still significantly different: $\alpha_1=0.43\pm0.06$, $\alpha_2=1.69\pm0.17$, $t_b=0.25\pm0.06$. Modelling the light curve with energy injections {#EI} ------------------------------------------------ Motivated by the similarities to such highly variable light curves as that of the afterglow of GRB 021004, which was successfully modelled by multiple energy injections (“refreshed shocks”) [@deUgarte2005], we used the code of [@Johannesson06] to model the afterglow light curve. The code numerically solves the kinematic equations of an expanding shock front and calculates the resulting synchrotron emission. Relativistic effects are fully taken into account and the code supports delayed energy injection episodes. Several energy injection episodes are applied as a possible scenario to explain the rebrightenings and shallow decay of the afterglow. Preliminary results on a smaller data set were presented in [@Johannesson09]. The number and a time range for the energy injections have to be inserted as initial guess for the fit and the fit then adjusts them to the best possible time within that range and determines the magnitude of the injection. As the very early data likely contain some signature of the prompt emission, we exclude all data before 400 s as well as the very bright X-ray flare. Due to Lyman forest blanketing, data in $B$ and $g^\prime$ bands were excluded from the fit. The $V$ band is also affected by the blanketing, but was corrected with the model from [@Madau1995] (brightened by 0.17 magnitudes) and included in the fit to have a better spectral coverage at early times. Figure \[Energy\] shows the best fit (${\chi^2{\textnormal{/d.o.f.}}=}2.0$) found using a model with a total of five energy injections: at 0.01, 0.04, 0.07, 0.35 and 0.60 days. Each energy injection episode adds 2 free parametres to the fit in addition to the 6 parametres needed for the standard afterglow model discussed below. The initial energy injected into the outflow is[^4] $E_0=5^{+2}_{-3}\times10^{49}$ ergs. The energy injections then add 1.8, 4.1, 4.2, 8, and finally 14 times the initial energy release $E_0$ to the afterglow, for a total energy release in the afterglow of $1.6^{+0.8}_{-1.1}\times10^{51}$ ergs. The first three injections are responsible for the shallow afterglow decay between 0.008 and 0.25 days. The quality of the data does not allow to discriminate between this three-injection scenario and a continuous injection. Using a lower number of injections does not yield a satisfactory fit to the observed data. Since there are no direct indications of injections in the light curve, the time of each of the three injections is not well determined. A direct consequence of this is that the energy of each individual injection in this phase is not well determined, while the total energy released is fairly consistent. The time and energy of the last two injections are, however, better constrained by the data. Further results of the modelling are a high density of the circumburst medium of $n_0=600^{+3000}_{-500}$ cm$^{-3}$ [a high value, comparable to the high-$z$ GRB 050904, @Frail2006], a rather low opening angle of $\theta_0 = 2^\circ\!\!.2^{+1^\circ\!\!.5}_{-1^\circ\!\!.8}$ and an electron index of $p=2.1^{+0.2}_{-0.3}$ (host galaxy extinction is assumed to be negligible, see [[§]{} \[SEDchap\]]{}). The peak frequency $\nu_m$ passes through the optical/NIR at very early times, while the data are most likely affected by the prompt emission. The cooling break $\nu_c$ is between the optical and X-rays up to 6 days after the burst, i.e., over the whole data span. Furthermore, we find $\varepsilon_e=2^{+5}_{-1.5}\times10^{-2}$ and $\varepsilon_B=7^{+40}_{-6}\times10^{-5}$, with $\varepsilon_e$ being the fraction of the energy in the electron population and $\varepsilon_B$ the fraction of the energy in the magnetic field. Note that the definition of $\varepsilon_e$ has been changed in the model from [@Johannesson06] to the definition of [@PK2001] to allow for $p<2$ in the model. This model reproduces the global properties of the optical/NIR light curves well, with the rather high $\chi^2$/d.o.f. $=2.0$ resulting from rapid variability in the late time afterglow. This is incompatible with the homogeneous shock front assumed in the numerical code. The X-ray light curve is also fitted reasonably well. The spectral energy distribution and host extinction {#SEDchap} ---------------------------------------------------- ![Spectral energy distribution of the afterglow of GRB 060526 in $uvw2\,uvm2\,uvw1\,uBg^\prime Vr^\prime R_Ci^\prime I_CJHK_S$, and fits with no extinction (straight black line), MW extinction (dotted line), LMC extinction (dashed line) and SMC extinction (thick dash-dotted line). The fit with MW dust finds (unphysical) negative extinction, causing an “emissive component” instead of a 2175 [Å]{} absorption bump. Data beyond $2.2\times10^{15}$ Hz ($Vg^\prime B u\,uvw1\,uvm2\,uvw2$) were not included in the fit due to Lyman forest blanketing, the grey curves represent extrapolations. The extinction curves of [@Pei1992] are not correctly defined beyond $3.2\times10^{15}$ Hz. The UVOT UV filters are upper limits only, showing the strong flux decrease beyond the Lyman cutoff. The flux density scale is measured at the break time.[]{data-label="SED"}](10340fg5.eps){width="\columnwidth"} Dust $\chi^2$/d.o.f. $\beta$ $A_V$ ------ ----------------- ----------------- ------------------ none 0.425 $0.695\pm0.035$ ... MW 0.309 $0.833\pm0.149$ $-0.126\pm0.132$ LMC 0.466 $0.494\pm0.397$ $0.113\pm0.223$ SMC 0.396 $0.552\pm0.198$ $0.055\pm0.075$ : Fits to the spectral energy distribution of GRB 060526. Columns are the dust model, the goodness of the fit, the spectral slope and the derived extinction in the rest-frame $V$ band.[]{data-label="SEDtab"} Following the procedures outlined in [@Kann2006], we derive the optical spectral energy distribution (SED) of the GRB 060526 afterglow and fit it with several dust models [Milky Way, MW; Large Magellanic Cloud, LMC; and Small Magellanic Cloud, SMC; @Pei1992] to derive the line-of-sight extinction in the host galaxy. Due to the strongly variable light curve, we choose the approach [@Kann2006] used for the SED of GRB 030329, and shift the other bands to the $R_C$-band zero point to derive the colours. With this method, we can also look for colour changes. There may be marginal variations in $B-R$, but this colour remains constant within conservative errors, and is more sensitive to Lyman forest blanketing, as different filters will suffer a different amount of blanketing. The SED clearly shows the decreasing flux in the $B$, $g^\prime$ and $V$ bands, and especially in the $uvw2\;uvm2\;uvw1\;u$ bands, where only upper limits are found, due to the Lyman forest blanketing as well as the Lyman cutoff (see Fig. \[SED\]), and we thus do not include these filters in our fit. Given the size of the errors, all fits, even with no extinction, are acceptable (see Table \[SEDtab\]) and we are thus unable to prefer one dust model over another. The lack of $z$ band data does not allow us to constrain the existence of a 2175 [Å]{} bump, therefore we have no evidence in favor of or ruling out MW and LMC dust. We do note, though, that MW dust leads to slightly negative extinction, while the LMC dust fit yields large errors. Thus, we henceforth use the SMC dust fit, as this is the most common dust type found in GRB host galaxies, both in pre-Swift [@Kann2006; @Starling07] and Swift-era [@Schady07; @Kann07] results. Note that for SMC dust, the extinction is 0 within errors as well. Assuming the cooling break $\nu_c$ to lie blueward of the optical bands [highly likely considering the X-ray spectral slope $\beta_X\approx1$, @Dai2007 and the result of the numerical modelling], the standard fireball model gives for the electron power law index $p=2\beta+1=2.10\pm0.40$. This result (albeit with large errors) is in agreement with the result from the broadband modelling ([[§]{} \[EI\]]{}) and with the canonical $p=2.2$, and very similar to many other GRB afterglows [e.g. @Kann2006; @Starling07]. Our results are in contrast to [@Dai2007], who derive a very steep slope $\beta_0=1.69^{+0.53}_{-0.49}$ from $Br^\prime i^\prime $ data only (correcting for Lyman absorption) and conclude that the optical and the X-ray data lie on the same slope. Host search {#Host} ----------- ![image](10340fg6.eps){width="\textwidth"} In the late-time VLT images, we do not detect any source at the position of the afterglow down to a limit of $R_C>27.1$ mag (which transforms into an absolute magnitude limit of M$_R>-20.1$). There is one source present at [$\sim$15]{} South-East of the afterglow position with $R_C=26.4\pm0.2$, which at a redshift of $z=3.221$ would transform into a physical offset between GRB and host galaxy of $\sim11.5$ kpc. A long-slit spectrum covering the afterglow position and this galaxy does not show any trace at these positions. If this galaxy was associated with the GRB, it would be one of the largest offsets between a long-duration GRB and its host galaxy known [@Bloom2002]. The strong absorption lines from the ISM seen in the afterglow spectrum (see Sect. \[spectrum\]) as well as the high circumburst density inferred from the numerical modeling (see Sect. \[EI\]) do in fact not favour a large distance from the host galaxy. A position outside of their host galaxies had been inferred for GRB 070125 [@Cenko070125] and GRB 071003 [@Perley08b], however, those spectra showed very weak absorption lines, contrary to what we observe for GRB 060526. Concluding, we have no direct spectroscopic evidence for or against an association of the GRB with the galaxy. We also observed the field with ACS on [HST]{}. Astrometry was performed relative to our TNG image, yielding a position accurate to $\sim0.1$. At the location of the burst we do not find any evidence for an underlying host galaxy. To estimate the limits we place 50 apertures randomly on the sky and measure the standard deviation in their count rates. This implies that the underlying host galaxy of GRB 060526 is fainter than F775W(AB)$\,>\,$28.5 ($3 \sigma$) or an absolute magnitude of M(1500 Å) $>$ –18.3mag which is fainter than 0.5 L\* according to [@Gabash04]. The flux density at the host position is $0.004\pm0.005\;\mu$Jy. Host galaxies of long-duration GRBs have often been found to be faint irregular galaxies [@Fruchter06; @Christensen04] which are difficult to detect at higher redshifts. So far, there are only eight bursts with $z>3$ where the detection of a host galaxy has been published, namely GRB 971214 ($z=3.418$, @Kulkarni98), GRB 000131 ($z=4.500$, @Andersen000131 [@Fruchter06]), GRB 030323 ($z=3.3718$, @Vreeswijk04), GRB 060206 ($z=4.04795$, @Fynbo06a [@Thoene060206; @Chen09]), GRB 060210 ($z=3.9133$, @Fynbo09 [@Perley09]), GRB 060605 ($z=3.773$, @Ferrero08), GRB 090205 ($z=4.6503$, @D'Avanzo090205), and GRB 090323 ($z=3.568$, @CenkoLAT [@McBreenLAT]). The host galaxies of GRB 020124 [@Berger2002; @Chen09], GRB 050730, GRB 050908, GRB 060607A, GRB 070721B [all @Chen09], GRB 050904 [@Berger2007] and GRB 060510B [@Perley09], on the other hand, were not detected to very deep limits in the optical and the NIR. The distribution of pre-Swift $R_C$ band host galaxy magnitudes peaks at $R_C(AB)=25$ [@Fruchter06] but extends out to 29 with a typical redshift $z\approx1.4$. Swift GRBs (and thus their hosts) however have a higher mean redshift of $z=2.8$ [@Jakobsson06a][^5], so the distribution will be shifted out to even fainter magnitudes. [@Ovaldsen07] also find a higher magnitude for Swift hosts than for pre-Swift bursts by comparing the expected detection rate from pre-Swift hosts with detections and upper limits derived from imaging the fields of 24 Swift and HETE II bursts from 2005 – 2006. Spectroscopy results {#spectrum} ==================== Line identification ------------------- We detect a range of metal absorption lines as well as a Lyman limit system (LLS) originating in the host galaxy of GRB 060526. A redshift of $z=3.221$ was determined in [@Jakobsson06b] from a number of these absorption lines using the spectra taken with the 600V grism presented in this article. ![image](10340fg7.eps){width="17cm"} Most of the lines were fitted from the combined 600V spectrum which covers all metal absorption lines detected longwards of Ly-$\alpha$, but provides a higher resolution than the 300V spectra. The 600I spectra only covers the AlII $\lambda$ 1670 line at the same resolution as the 600V grism. The range of 1200B is entirely within the Lyman-$\alpha$ forest but does not have a high enough resolution to deblend metal transitions from absorption caused by the Ly-$\alpha$ forest lines. We do, however, detect Ly-$\beta$ and Ly-$\gamma$ absorption in the 1200B grism. In the blue end of the 300V spectrum, one can clearly see the 915 Å Lyman break at the redshift of the host galaxy. In Fig. \[spec\], we show the combined spectrum of the 300V grism with the identified lines indicated. We note that we do not detect any intervening system in the sightline towards GRB 060526, which is rather unusual for a GRB sightline, in particular at that redshift [see, e.g., @Prochter06]. Taking the strong $\lambda$ 2796, 2803 doublet, the redshift path probed for any intervening system is between $z\sim0.8$ and 2.2. The $3\sigma$ limits on the non-detection of the doublet vary between 0.17 [Å]{} ($z=0.8$) and 0.82 [Å]{} ($z=2.2$) (rest frame). In order to determine the equivalent width (EW) of the strong absorption lines, we fitted the continuum around the lines in regions that were free of absorption and summed over the absorption contained within two times the full-width at half-maximum (FWHM) of the lines. For weak lines, we obtained better results due to the low S/N by fitting Gaussians. For this fit, we used a modified version of the gaussfit procedure provided in IDL[^6] which is more reliable in determining the continuum and fitting the actual line even if it is slightly blended with a neighbouring line. The upper limits on the EWs for a range of ions noted in Table \[lines\] was determined from the spectra taken with the 300V grism due to the better S/N of those spectra. Between the individual spectra, we do not find any variability in the EW of the individual absorption lines. $\lambda$ 1259, $\lambda$ 1260 and $\lambda$ 1260 are blended and cannot be fitted separately. We therefore cannot consider them for the derivation of the column density from the curve of growth fit as described below and only give the total EW in Table \[lines\]. In contrast to what is noted in [@Jakobsson06b], we cannot reliably detect any fine-structure lines and only give an upper limit for . Fine-structure lines would be a clear indication that the detected absorption lines indeed originate in the host galaxy of the GRB, as they are assumed to be produced by UV pumping from the afterglow [e.g. @Prochaska051111; @Vreeswijk07; @Delia09]. The redshift derived is therefore to be strictly taken as a lower limit only, the detection of the Lyman $\alpha$ forest redward of the proposed redshift, however, excludes a significantly higher redshift for the burst. --------------------- ---------------------- ------------ --------- -------------------- ----------------- $\lambda_{\rm obs}$ $\lambda_{\rm rest}$ ID $z$ EW$_\mathrm{rest}$ log N/cm$^{-2}$ $[$Å$]$ $[$Å$]$ $[$Å$]$ 4102.26 972.54 Ly$\gamma$ 3.218 $2.39\pm0.10$ — 4329.70 1025.72 Ly$\beta$ 3.221 $2.92\pm0.06$ — 5131 1215.67 Ly$\alpha$ 3.221 — $20.00\pm0.15$ (5254) 1242.80 (3.221) $<0.09$ $<13.66$ (5276) 1250.58 (3.221) $<0.08$ $<14.71$ 5291.01 1253.81 3.220 $0.05\pm0.02$ $14.58\pm0.25$ 5320.00 1259.52 3.221 ($1.32\pm0.07$) — 1260.42 3.221 [blended]{} — 1260.53 3.221 — (5335) 1264.74 (3.221) $<0.10$ $<12.76$ 5497.30 1302.17 3.222 $1.07\pm0.05$ $>15.15$ 5506.82 1304.37 3.222 $0.70\pm0.21$ $15.87\pm0.16$ (5559) 1317.22 (3.221) $<0.11$ $<13.59$ 5634.24 1334.53 3.222 $1.51\pm0.10$ $>15.55$ 5884.87 1393.76 3.222 $0.20\pm0.08$ $13.50\pm0.14$ 5923.05 1402.77 3.222 $0.12\pm0.05$ $13.50\pm0.14$ 6444.50 1526.71 3.221 $0.86\pm0.04$ $15.87\pm0.16$ 6539.58 1548.20 3.223 $0.26\pm0.08$ $14.10\pm0.13$ 6546.39 1550.78 3.221 $0.22\pm0.09$ $14.10\pm0.13$ 6790.24 1608.45 3.221 $0.17\pm0.03$ $14.28\pm0.24$ 6800.38 1611.20 3.221 $<0.01$ — 7052.62 1670.79 3.221 $1.02\pm0.05$ $15.12\pm0.18$ (7863) 1862.79 (3.221) $<0.35$ $<12.71$ (8552) 2026.13 (3.221) $<0.45$ $<12.73$ (8679) 2056.25 (3.221) $<0.54$ $<13.47$ --------------------- ---------------------- ------------ --------- -------------------- ----------------- Column densities from curve of growth analysis ---------------------------------------------- Some of the strong absorption lines are saturated, which is a problem in low resolution spectra as the damping wings are not resolved and Voigt profile (VP) fitting cannot be adopted to derive a reliable column density. Furthermore, high resolution spectra of GRBs [@ProchaskaCoG] have shown that the strong metal absorption lines unresolved in low resolution spectra usually consist of a number of narrow, unsaturated components that would allow an accurate determination of the column density by fitting the different components separately. ![Multi-ion single-component curve of growth fit for the absorption systems in GRB 060526 using 6 different ions. The CoG is expressed in units of the column density “N”, the oscillator strength of the transition “f” and the wavelength of the transition. As comparison, $b = \infty$ is plotted, the dotted line marks the 1 $\sigma$ deviation from the fit.[]{data-label="CoG"}](10340fg8.eps){width="\columnwidth"} If only low resolution spectra are available, one has to adopt a curve of growth (CoG) analysis [@Spitzer78; @SavageSembach96] which directly relates the EW to the column density on the linear part of the CoG where the lines are not saturated (optical depth $\tau_0<1$), but depends on the Doppler $b$ parameter of the medium on the flat part of the curve which applies for mildly saturated lines. For GRBs, one usually has to do a multiple-ion single-component CoG (MISC-CoG) analysis, adopting the same effective Doppler parameter for all ions. Here, we used all unblended ionic lines that were not heavily saturated, namely , , , as well as and and calculate the $\chi^2$ minimum going through the parameter space for the column densities of each ion and a range of $b$ parameters. We then find the best fit for $b=39\pm3$ km s$^{-1}$ (see Fig. \[CoG\]) and the column densities as noted in Table \[lines\]. Most of the ions lie near the linear part of the CoG and are therefore rather independent of $b$, and lie on the flat part of the CoG and the column densities are only noted as lower limits as derived from the linear part of the CoG. We excluded the saturated and transitions from the fit and only list lower limits for these two ions for which we take the column density resulting from the linear part of the CoG. We also discarded the blended absorption lines of $\lambda$ 1260, and even though is the dominant contribution to the absorption since we detect two other unblended transitions at $\lambda$ 1304 and $\lambda$ 1526 Å. There are several problems connected with the use of the CoG that have to be considered. Different ionisation levels should actually be treated in separate analyses as they might occur in different regions in the absorbing system. However, we assume that the absorption takes place in a relatively small region of the host galaxy and the resolution of the spectrum does resolve different components of the absorbing material in the host galaxy. Therefore, we also fit the higher ionisation levels of and in the same CoG. Furthermore, they lie close to the linear part of the CoG and excluding them from the common fit would not change the derived $b$ parameter very much. Another problem with doing multiple-ion CoG analysis using strong lines in low-resolution spectra has been noted by [@ProchaskaCoG], who compared column densities derived from low resolution spectra and CoG with high resolution data and VP fitting from the same bursts. He found that when including saturated lines in the fit, column densities are generally underestimated. Indications for that are if an effective $b$ parameter of $\gg$ 20 km s$^{-1}$ is found, since strong lines actually consist of a range of components with $b<20$ km s$^{-1}$. [@Savaglio06] however performed a similar analysis using CoG and the apparent optical depth (AOD) method as described in [@Pettini02], which can be applied for medium resolution high S/N spectra, and found a good agreement between the two methods. The column densities of the saturated lines and are infact rather sensitive to the adopted $b$ parameter. These lines very likely consist of a number of unresolved weaker components which would lie on the linear part of the CoG, the real errors should therefore be larger but are difficult to estimate. Metallicity and relative abundances ----------------------------------- Absorption lines that are likely not affected by dust depletion can be used to derive a metallicity of the medium in the line of sight to the GRB. The least dust-depleted element is Zn, which is, however, undetected in our spectra. We then use the relatively weak lines and derive relative metallicities compared to the hydrogen density with \[M/H\]=log(N$_M$/N$_H$)–log(N$_{M}$/N$_{H}$)$_\odot$ using solar abundances from [@Asplund05]. Here we derive metallicities of $[{\rm S}/{\rm H}]=-0.57\pm0.25$ and $[{\rm Fe}/{\rm H}]=-1.22\pm0.24$. Fe is usually affected by dust depletion [@SavageSembach96] and corrections have to be adopted. Using the relation between the Zn and Fe abundance in [@Savaglio06], we find a metallicity of $[{\rm Fe}/{\rm H}]=-1.09\pm0.24$ which marginally agrees within the errors with the value derived from sulphur. The only detected and unblended line at $\lambda$ 1253, however, is only marginally detected and therefore the EW has large errors. The doublet taken for the CoG fit, in contrast, is also slightly blended, but the fit of the stronger component can be considered as reliable. Despite the dust depletion, the metallicity derived from Fe might be the most reliable one in this case and we therefore assume a metallicity of \[Fe/H\] = –1.09 for the host of GRB060526. Independent of the ion used, the metallicity is rather high compared to other galaxies at redshift z $\approx$ 3, but among the typical metallicities derived for other GRB hosts [@Fynbo06a] at that redshift. For those measurements, different ions have been used depending on the quality and the wavelength coverage of the spectra. Our results show that caution is required when comparing the metallicities derived from different elements, as they might be differently affected by dust depletion and/or evolution. This is especially true when saturated lines in low resolution spectra are used to derive the column densities and hence the abundances as it is the case for, e.g., GRB 000926 [@Savaglio03] and GRB 011211 [@Vreeswijk06]. For GRB050401 [@Watson04] and GRB 050505 [@Berger05], the authors themselves note that due to saturation the reported metallicity is indeed a lower limit. Our spectra do not allow us to determine a dust depletion pattern from the relative abundances of heavier elements, since at least four elements out of Zn, Si, Mn, Cr, Fe, Ni or S are necessary to do such a fit. The difference in the relative abundances between S, Si and Fe might suggest the presence of some dust in the line-of-sight towards the GRB, however, large extinction is excluded from the afterglow SED. On the other hand, this difference could also be due to an enhancement in $\alpha$ element production[^7] which is likely to happen in the young star-forming host of a GRB. Generally, GRB hosts have higher $\alpha$/Fe ratios than QSO-DLAs [@Prochaska07], which can either be interpreted as large dust depletion consistent with the higher metallicities of GRB sightlines or as $\alpha$ element enhancement. There are indications that most of the $\alpha$/Fe ratio is due to dust depletion traced by a large \[Zn/Fe\] or $[{\rm Ti}/{\rm Fe}]<0$. The latter is assumed to be a clear indicator of dust depletion ($[{\rm Ti}/{\rm Fe}]<0$) vs. $\alpha$ enhancement ($[{\rm Ti}/{\rm Fe}]>0$) [@Dessauges02]. Both elements are, however, not detected and the limit on Zn does not lead to a strong constraint on the \[Zn/Fe\] ratio to settle this issue in the case of GRB 060526. The ratio between high and low ionisation species in the spectrum clearly shows that most of the material in the line of sight is in a low ionization state. and have rather low column densities and lie close to the linear part of the CoG in contrast to their low ionization species and which are both saturated, whereas is not even detected in our spectra. We then derive column density ratios of log(/) $=-2.37$, log(/) $<-1.45$ and log(/) $<-2.41$. From the large sample of Swift long GRB afterglow spectra [@Fynbo09], this seems fairly normal for an average GRB sightline. However, the few GRBs occurring in LLSs ($\log N_{\rm HI}/{\rm cm}^{-2} < 20.3$) usually show a higher fraction of ionized material compared to GRB-DLA sightlines. This might either be due to decreased shielding of the highly ionizing afterglow flux by the lower hydrogen column density, by a rather special arrangement of the GRB inside the host galaxy or simply by a small host galaxy. Since our spectra show a low fraction of highly ionized material this implies that the absorbing material has a relatively large distance from the GRB. We do not detect in our spectra but provide an upper limit of $\log N < 13.66$. was detected in only four sightlines towards GRBs [@Prochaska08] and likely traces the immediate environment of the GRB as it has a high ionization potential and requires a strong radiation field. Our upper limit is lower than the column densities for those GRB sightlines where could be detected which again implies that the absorbing gas probed by our spectra are most likely not close to the GRB itself. Furthermore, we do not detect any fine structure lines and only derive an upper limit on of $\log N/{\rm cm}^{-2} < 12.76$. Fine structure lines are assumed to be pumped by the UV radiation field of the GRB [@Vreeswijk07] which also indicates that the gas is likely very far from the GRB itself. requires a less strong radiation field and has been detected in Lyman break galaxies [@Pettini02] where the UV radiation from young stars provides the necessary radiation field. Discussion and conclusions ========================== GRB 060526 had a relatively bright afterglow that allowed us to obtain a solid dataset, both photometrically and spectroscopically. We achieved a dense light curve coverage over several days which allowed a detailed study of the afterglow properties, and obtained a series of low resolution but high signal-to-noise spectra to study the host environment. The optical light curve can be fitted with a double smoothly-broken power law with a breaks at $t_{b1}=0.090\pm0.005$ and $t_b=2.216\pm0.049$ days, and decay slopes of $\alpha_{plateau}=0.288\pm0.026$, $\alpha_1=0.971\pm0.008$, and $\alpha_2=2.524\pm0.052$. The dense sampling of especially the $R_C$-band light curve also reveals additional variability on top of the power laws. These features could be explained either by extended activity of the central engine or through interactions of the shock with the interstellar medium. For the case that the variability arises from external shocks, several mechanisms have been considered. In GRB 021004, both density variations of the external medium into which the GRB jet plows and angular inhomogenities of the jet surface were considered [@NPG03]. However, [@Nakar07] show that density variations would cause much smaller fluctuations than those observed in GRB afterglows and can therefore be ruled out. Another possibility is the injection of additional energy into the shock by slower shells that catch up with the shocked region as it decelerates, this model was used successfully to describe GRB 021004 [@deUgarte2005] and also works better than two other models (double jet and density fluctuations) to describe the highly complex light curve of GRB 030329 [@Huang2006]. Thus, variability can give either information on the medium surrounding the GRB or on the activity of the central engine. A more intriguing possibility is that the flares may be emitted from another region closer to the central engine, resulting from late internal shocks. Powerful X-ray flares that are attributed to late central engine activity have been observed in about 50% of all Swift GRBs [e.g. @Burrows2005; @Chincarini2007; @Krimm2007], and strong optical/NIR flaring contemporaneous with the GRB prompt emission may also occur [@Vestrand041219A; @Blake041219A; @Vestrand050820A; @Racusin08], thus making optical flares from late central engine activity an interesting prospect [@KannGCN; @Malesani2007]. Indeed, [@Dai2007] have suggested that the optical variability of the afterglow of GRB 060526 is due to flares from late internal shocks (the very early rapid optical variability we present here is very probably due to central engine activity, as it is seen contemporaneously in gamma and X-rays). [@Khamitov08], on the other hand, conclude that the short timescale of the variabilities requires the jet to be non-relativistic already at $\sim1$ day and could then be explained by external density fluctuations. Our analysis lends tentative support to the notion of flares from internal shocks, finding decay slopes for two flares that exceed what should be possible from external shocks. But we caution that the errors of these fits are large due to a low amount of data in the decaying parts. Furthermore, globally, a model using refreshed external shocks is able to account for the light curve variations, although microvariability remains. This creates the intriguing possibility of reverberation effects [see @Vestrand050820A for a case of reverberation between gamma-rays and optical emission]. Short flares in the X-ray or optical bands signal internal shocks from long-term central engine activity, and when these shells catch up with the forward shock front, they re-energise the external forward shock. The detection of such behaviour would probably require dense multi-band observations of a bright afterglow to search for SED changes at high time resolution combined with detailed modelling of the data. This way, one could discern between internal shocks (which are expected to have a different spectral index from the forward shock afterglow) and refreshed, external shocks (which are achromatic). Our data set of the afterglow of GRB 060526 does not allow us such a detailed decomposition. From the analysis of our low resolution spectra with different resolutions, we detect a LLS and a number of metal absorption lines that all lie at a redshift of $z=3.221$. The low resolution only allows us to derive column densities from measuring the EWs of the absorption lines and adopt a MISC-CoG analysis where we exclude the most saturated as well as blended transitions. We find a best fit for the Doppler parameter of $b=39\pm3$ km s$^{-1}$ and most of the ions used for the fit lie on the linear part of the CoG which allows a relatively reliable determination of the column densities. The relative abundances of different metals in the spectra indicate some dust extinction, but an intrinsic difference due to enhancement of the production of certain elements cannot be excluded. The very low amount of dust detected in the afterglow SED may indicate that the latter might be the favored possibility. The column density of neutral hydrogen is rather low compared to other GRBs. We derive a metallicity for the host of $[{\rm Fe}/{\rm H}]=-1.09$ which is slightly higher than metallicities determined from other GRB afterglow spectra. According to the definition of QSO absorbers, the host of GRB 060526 is classified as a LLS ($19 < \log N_{\rm HI}/{\rm cm}^{-2} < 20.3$), which seem to have on average higher metallicities than damped Lyman $\alpha$ systems (DLA; @Peroux07) and a steeper evolution towards lower redshifts. Around redshift 3, however, the metallicities of both samples are within the same range. Also, GRB hosts show a trend towards increasing metallicity with lower redshifts [@Fynbo06a; @Savaglio06]. Taking into account that most of the sample used only low-resolution spectra to derive the metallicity (which only gives lower limits for the column densities and the metallicity) this evolution might, however, not be as pronounced as for DLAs and LLS. This might imply that the enrichment of the ISM in the early universe had taken place at earlier times than assumed. The absorbing material along the line-of-sight is mostly in the neutral state, as usually observed for long GRB-DLAs, while sightlines with lower $\log N_{\rm HI}$ often contain more ionized material. This might either imply that we have a very small host galaxy or that the GRB is placed somewhere in the outskirts of its host. In general, the low ionization points to a relatively large distance of the absorbing material from the GRB itself. There is no underlying host galaxy of GRB060526 detected down to a deep limit of 28.5 mag (in F775W AB) in HST/ACS data. At that redshift, this means the host has an absolute magnitude M(1500Å) $>-18.3$ mag, fainter than an 0.5 L\* galaxy at that redshift. Long GRBs have been found to occur in actively star-forming galaxies and star formation is assumed to shift towards smaller and fainter galaxies over time [e.g. @Cowie96] while massive galaxies prove to be rather unchanged throughout the history of the universe [e.g. @Abraham99; @Heavens04]. One would therefore expect that GRB hosts should also have higher luminosities towards higher redshifts. [@Fynbo08] concluded that the observed metallicity distribution of GRB hosts (as well as QSO absorbers) at $z\approx3$ can be explained by the luminosity function of galaxies at that redshift and assuming a luminosity-metallicity relation as derived for other high-redshift samples [@Ledoux06; @Erb06]. The non-detection of the host of GRB060526 down to deep limits, however, would not support this suggested evolution. The data neither strongly support nor allow us to rule out that the GRB is associated with the nearby galaxy. While the offset would be very large compared to typical long GRB offsets, it is possible the burst occurred in a locally dense star-forming region which is not detected even in our very deep imaging. We thank the referees for constructive comments that improved the paper. CCT wants to thank Cédric Ledoux for comments on the curve-of-growth analysis and S. Campana for comments on the text. We thank the support astronomer and staff at the Nordic Optical Telescope for obtaining the observations.\ The Dark Cosmology Centre is funded by the DNRF. HD acknowledges support from the Research Council of Norway. LH acknowledges support from Science Foundation Ireland grant 07/RFP/PHYF295. PJ acknowledges support by a Marie Curie European Reintegration Grant within the 7th European Community Framework Program and a Grant of Excellence from the Icelandic Research Fund. IFB acknowledges partial support by the RFFI grant ”09-02-97013-povolzhe”. PAC gratefully acknowledges the support of NWO under grant 639.043.302. CB, AG, and GG acknowledge the University of Bologna for the funds Progetti di Ricerca Pluriennali. SF acknowledges the support of the Irish Research Council for Science, Engineering and Technology, cofunded by Marie Curie Actions under FP7.\ Based on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, with the William Herschel Telescope and the Telescopio Nazionale Galileo in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias and at the Astronomical Observatory of Bologna in Loiano (Italy). Collection of SMARTS data is supported by NSF-AST 0707627. MIRO is supported by the Department of Space, Govt. of India. The Peters Automated Infrared Imaging Telescope (PAIRITEL) is operated by the Smithsonian Astrophysical Observatory (SAO) and was made possible by a grant from the Harvard University Milton Fund, the camera loan from the University of Virginia, and the continued support of the SAO and UC Berkeley. 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[^2]: http://iraf.noao.edu [^3]: Note, though, that [@DaiDepp] argue that optical follow-up in the Swift era, especially with UVOT only, often does not persist long enough (and to enough depth) to find jet breaks. [^4]: The error limits presented here are $1\sigma$ estimates, but due to degeneracy in the model parameters, the actual range of parameter values can be much larger. [^5]: An updated version of the redshift distribution can be found under http://raunvis.hi.is/$\sim$pja/GRBsample.html - as of July 1, 2010, the mean redshift has become lower, $z=2.19$, see also [@Fynbo09], who find a mean and median of $z=2.2$. [^6]: Available at http://www.pa.iasf.cnr.it/$\sim$nicastro/IDL/Lib/gfit.pro [^7]: $\alpha$ elements are produced in massive, metal-poor stars through the $\alpha$ process and include elements with integer multiples of the He nucleus mass such as O, Si, S, Ca, Mg and Ti.
A Three-Dimensional Laguerre Geometry and Its Visualization Hans Havlicek and Klaus List, Institut f[ü]{}r Geometrie, TU Wien [We describe and visualize the chains of the $3$-dimensional chain geometry over the ring ${{\mathbb R}}({{\varepsilon}})$, ${{\varepsilon}}^3=0$.\ MSC 2000: 51C05, 53A20.\ Keywords: chain geometry, Laguerre geometry, affine space, twisted cubic. ]{} Introduction {#sect:intro} ============ The aim of the present paper is to discuss in some detail the Laguerre geometry (cf. [@benz-73], [@herz-95]) which arises from the $3$-dimensional real algebra ${{\mathbb L}}:={{\mathbb R}}({{\varepsilon}})$, where ${{\varepsilon}}^3=0$. This algebra generalizes the algebra of real dual numbers ${{\mathbb D}}={{\mathbb R}}({{\varepsilon}})$, where ${{\varepsilon}}^2=0$. The Laguerre geometry over ${{\mathbb D}}$ is the geometry on the so-called Blaschke cylinder (Figure \[abb:1\]); the non-degenerate conics on this cylinder are called chains (or cycles, circles). If one generator of the cylinder is removed then the remaining points of the cylinder are in one-one correspondence (via a stereographic projection) with the points of the plane of dual numbers, which is an isotropic plane; the chains go over to circles and non-isotropic lines. So the point space of the chain geometry over the real dual numbers can be considered as an affine plane with an extra “improper line”. The Laguerre geometry based on ${{\mathbb L}}$ has as point set the projective line ${{\mathbb P}}({{\mathbb L}})$ over ${{\mathbb L}}$. It can be seen as the real affine $3$-space on ${{\mathbb L}}$ together with an “improper affine plane”. There is a point model for this geometry, like the Blaschke cylinder, but it is more complicated, and belongs to a $7$-dimensional projective space ([@herz-95 p. 812]). We are not going to use it. Instead, we describe ${{\mathbb P}}({{\mathbb L}})$ as an extension of the affine space on ${{\mathbb L}}$ by “improper points” which will be described via lines, parabolas, and cubic parabolas. Higher order contact of twisted cubics {#sect:kubiken} ====================================== Here we present some results which will be needed in Section \[sect:laguerre\]. We refer to [@bol-50], [@brau-76-2], and [@semp+k-98] for the basic properties of twisted cubics in the real projective space ${{\mathbb P}}_3({{\mathbb R}})$. \[thm:kubik\] Let $C$ and ${{\widetilde{C}}}$ be twisted cubics of ${{\mathbb P}}_3({{\mathbb R}})$ with a common point $f$, a common tangent line $F$ at $f$, and a common osculating plane $\Phi$ at $f$. Then a collineation of ${{\mathbb P}}_3({{\mathbb R}})$ taking $C$ to ${{\widetilde{C}}}$ is uniquely determined by each of the following conditions:\ --------- ----------------------------------------------------------------- (I) All lines of the pencil ${{\mathcal L}}(f,\Phi)$ are invariant. (II) All points of the line $F$ are invariant. (III) All planes of the pencil with axis $F$ are invariant. --------- ----------------------------------------------------------------- \(I) We recall that distinct tangent lines of a twisted cubic are skew. The tangent surface of $C$ intersects the osculating plane $\Phi$ in a curve which is the union of $F$ and a conic $K$ through $f$; the line $F$ is tangent to $K$. Likewise the tangent surface of ${{\widetilde{C}}}$ yields a conic ${{\widetilde{K}}}$. Let $r_i$, $i\in\{1,2\}$, be distinct points of $C\setminus\{f\}$. The tangent lines of $C$ at these points meet the plane $\Phi$ at points $k_i\in K\setminus\{f\}$, whence the lines $L_i:=f\vee k_i$ are distinct. These lines meet ${{\widetilde{K}}}$ residually at points ${{\widetilde{k}}}_i$ which in turn are incident with tangent lines of ${{\widetilde{C}}}$ at distinct points ${{\widetilde{r}}}_i$. These points ${{\widetilde{r}}}_i$ are determined uniquely. So, every collineation of type (I) takes $r_i$ to ${{\widetilde{r}}}_i$, and $f$ to $f$. Conversely, there is a unique collineation $\kappa$ of ${{\mathbb P}}_3({{\mathbb R}})$ with $C^\kappa={{\widetilde{C}}}$, $r_i^\kappa={{\widetilde{r}}}_i$, and $f^\kappa=f$. Since $F$, $f\vee k_1$, and $f\vee k_2$ remain invariant under $\kappa$, all lines of the pencil ${{\mathcal L}}(f,\Phi)$ remain fixed. So this $\kappa$ is the only collineation with the required properties. \(II) The proof runs in a similar manner. The osculating planes at $r_i$ meet $F$ at points $k_i\neq f$. Now ${{\widetilde{r}}}_i\in {{\widetilde{C}}}\setminus\{f\}$ can be chosen such that their osculating planes meet the line $F$ at $k_i$. \(III) Each of the planes $F\vee r_i$ meets the twisted cubic ${{\widetilde{C}}}$ residually at a point ${{\widetilde{r}}}_i$. Now we can proceed as above. Let $p_0$, $p_3$, and $p$ be three distinct points of $C$. Define the point $p_1$ as the intersection of the tangent line at $p_0$ with the osculating plane at $p_3$. Likewise, by changing the role of $p_0$ and $p_3$, a point $p_2$ is obtained. Then $(p_0,p_1,p_2,p_3,p)$ is a frame of reference such that $$\label{eq:kubik-homogen} C = \{{{\mathbb R}}(s^3,s^2t,st^2,t^3) \mid (0,0)\neq (s,t)\in{{\mathbb R}}^2\}.$$ We assume that $f=p_3={{\mathbb R}}(0,0,0,1)$, whence $F$ is given by $x_0=x_1=0$ and $\Phi$ has an equation $x_0=0$. A collineation of ${{\mathbb P}}_3({{\mathbb R}})$ is of type (I), (II) or (III) if, and only if, it has a regular matrix with one of the following forms: $$\label{eq:matrizen} \mbox{(I): } {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & a_{01} &a_{02} & a_{03}\\ 0 & a_{11} &0 & a_{13}\\ 0 & 0 &a_{11} & a_{23}\\ 0 & 0 &0 & a_{33} } ,\mbox{ (II): } {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & a_{01} &a_{02} & a_{03}\\ 0 & a_{11} &a_{12} & a_{13}\\ 0 & 0 &a_{22} & 0 \\ 0 & 0 &0 & a_{22} },\mbox{ (III): } {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &a_{02} & a_{03}\\ 0 & 1 &a_{12} & a_{13}\\ 0 & 0 &a_{22} & a_{23}\\ 0 & 0 &0 & a_{33} }$$ Next we describe higher-order contact of twisted cubics; cf. also [@bol-50 pp. 211–219]. \[thm:2\] Let $C$ be the twisted cubic (\[eq:kubik-homogen\]) and let $\kappa$ be a collineation of ${{\mathbb P}}_3({{\mathbb R}})$ given by one of the matrices (\[eq:matrizen\]). Then the conditions stated in the first row, in the first and the second row, and in all rows of the table below are necessary and sufficient for the twisted cubics $C$ and $C^\kappa$ to have contact at the point $f={{\mathbb R}}(0,0,0,1)$ of order $2$, $3$, and $4$, respectively. $$\label{eq:tabelle} \begin{array}{\|c\|ll\|\|ll\|\|ll\|} \hline &\multicolumn{2}{c\|\|}{\mbox{\emph{(I)}}} &\multicolumn{2}{c\|\|}{\mbox{\emph{(II)}}} &\multicolumn{2}{c\|}{\mbox{\emph{(III)}}}\\ \hline 1&a_{33}=a_{11}& &a_{22}=a_{11}& & a_{33}=a_{22}^2& \\ 2&a_{11}=1,& a_{23}=-a_{01} &a_{11}=1,& a_{01}=2a_{12}& a_{22}=1,& a_{23}=2a_{12}\\ 3&a_{01}=0,& a_{13}=2a_{02} &a_{12}=0,& a_{13}=2a_{02}& a_{12}=0,& a_{13}=2a_{02}\\ \hline \end{array}$$ The quadratic forms [0.5mm$$\begin{aligned} G_1:{{\mathbb R}}^4\to{{\mathbb R}}&:& (x_0,x_1,x_2,x_3)\mapsto x_0x_3-x_1x_2, \\ G_2:{{\mathbb R}}^4\to{{\mathbb R}}&:& (x_0,x_1,x_2,x_3)\mapsto x_1x_3-x_2^2,\end{aligned}$$]{}define a hyperbolic quadric $x_0x_3-x_1x_2=0$ and a quadratic cone $x_1x_3-x_2^2=0$ with vertex $p_0$. Their intersection is the twisted cubic $C$ and the line $x_2=x_3=0$. The tangent planes of the two surfaces at $f$ are different. Let $\kappa$ be given by a matrix $A$ of type (I). The mapping $$g:{{\mathbb R}}\to{{\mathbb R}}^4 : s\mapsto (s^3,s^2,s,1)\cdot A$$ gives an arc of $C^\kappa$ containing the point $f$, which has the parameter $s=0$. The products of $g$ with $G_i$ are functions [0.5mm$$\begin{aligned} s&\mapsto& (-a_{11}^2+a_{33})s^3 + (-a_{01}a_{11}+a_{23})s^4+(), \\ s&\mapsto& (-a_{11}^2+a_{11}a_{33})s^2+ (a_{01}a_{33}+a_{11}a_{23})s^3+ (a_{01}a_{23}-2a_{11}a_{02}+a_{11}a_{13})s^4+(),\end{aligned}$$]{}where $()$ denotes terms of higher order in $s$. The twisted cubics $C$ and $C^\kappa$ have contact of order $m$ at $f$ if, and only if, in both functions the coefficients at $s^0,s^1,\ldots,s^m$ vanish [@bol-50 p. 147]. Now the assertions follow easily, taking into account that $a_{11}\neq 0$ and $a_{33}\neq 0$. Similarly, if the matrix $A$ is of type (II) then the functions [0.5mm$$\begin{aligned} s&\mapsto& (-a_{11}a_{22}+a_{22})s^3+ (-a_{01}a_{22}-a_{11}a_{12})s^4+(), \\ s&\mapsto& (a_{11}a_{22}-a_{22}^2)s^2+ (a_{01}a_{22}-2a_{12}a_{22})s^3+ (-2a_{02}a_{22}+a_{11}a_{13}-a_{12}^2)s^4+(),\end{aligned}$$]{}are obtained, whereas for an $A$ of type (III) we get [0.5mm$$\begin{aligned} s&\mapsto& (-a_{22}+a_{33})s^3+ (-a_{12}+a_{23})s^4 + (), \\ s&\mapsto& (-a_{22}^2+a_{33})s^2+(-2a_{12}a_{22}+a_{23})s^3 +(-2a_{02}a_{22}-a_{12}^2+a_{13})s^4 + ().\end{aligned}$$]{}As above, the results are immediate. Let us now consider $\Phi$ as plane at infinity. Then our projective frame of reference determines an affine coordinate system in the usual way; a point ${{\mathbb R}}(1,x_1,x_2,x_3)\in{{\mathbb P}}_3({{\mathbb R}})$ has affine coordinates $(x_1,x_2,x_3)$. It is our aim to describe the results of Theorem \[thm:kubik\] and Theorem \[thm:2\] in affine terms. From the affine point of view the twisted cubics $C$ and $C^\kappa$ are cubic parabolas, projectively extended by the point $f=p_3$. So this point of higher order contact is outside the affine space. In what follows an affine transformation is understood to be a collineation fixing the plane $\Phi$. We restrict ourselves to the description of higher order contact via regular matrices of type (I). Such a matrix $A$ admits the following factorization: $$\label{eq:zerlegung} {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &0 &0 \\ 0 & 1 &0 & 0\\ 0 & 0 &1 & 0\\ 0 & 0 &0 & \frac{a_{33}}{\raisebox{0.8mm}{$\scriptstyle a_{11}$}} }\cdot {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &0 & 0\\ 0 & 1 &0 & \frac{a_{13}}{\raisebox{0.8mm}{$\scriptstyle a_{11}$}}\\ 0 & 0 &1 & \frac{a_{23}}{\raisebox{0.8mm}{$\scriptstyle a_{11}$}}\\ 0 & 0 &0 & 1 }\cdot {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &0 & 0\\ 0 & a_{11} &0 & 0\\ 0 & 0 &a_{11} & 0\\ 0 & 0 &0 & a_{11} }\cdot {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & a_{01} &a_{02} & a_{03}\\ 0 & 1 &0 & 0\\ 0 & 0 &1 & 0\\ 0 & 0 &0 & 1 }$$ Conversely, if the entries $a_{ij}$ in (\[eq:zerlegung\]) are chosen arbitrarily, except for $a_{11},a_{33}\neq 0$, then a regular matrix of type (I) is obtained. Formula (\[eq:zerlegung\]) corresponds to a decomposition of $\kappa$ into a perspective affinity with axis $x_3=0$ in the direction of $p_3$, a shear with an axis through the line $x_1=x_2=0$ in the direction of $p_3$, a stretching fixing the origin $p_0$ with scale factor $a_{11}$, and a translation through the vector $(a_{01}, a_{02} , a_{03})$, respectively; this decomposition is uniquely determined. The matrix $A$ is of type (I.1) if, and only if, $a_{11}=a_{33}$, i.e., the first matrix in (\[eq:zerlegung\]) is the unit matrix. The ultimate and the penultimate matrix in (\[eq:zerlegung\]) together yield a dilatation and every dilatation arises in this way. Hence, up to dilatations, we obtain all twisted cubics which have second order contact with $C$ at $f$ by applying to $C$ all shears with the properties mentioned above. Figure \[abb:2\] shows the twisted cubic $C$ and some of its images under a group $\Sigma$ of shears in the direction of $p_3$ with the common axis $x_1+x_2=0$. All these twisted cubics are on a parabolic cylinder $\Psi$ ($x_1^2-x_2=0$) which is invariant under the group $\Sigma$. (3.81,5) (0.0,0.0)[![image](./oskulieren.png){height="5cm"}]{} (2.5,4.5)[$C$]{} (3.5,2.2)[$x_1$]{} (1.5,4.7)[$x_3$]{} (3.1,3.6)[$\Psi$]{} \[abb:2\] Figure \[abb:2\] (4.09,5) (0.0 ,0.0)[![image](./hyperoskulieren1.png){height="5cm"}]{} (2.15,3.55)[$C$]{} (3.75,2.7)[$x_1$]{} (1.3,3.0)[$x_2$]{} (2.1,4.4)[$x_3$]{} (2.4,2.7)[$\Psi$]{} \[abb:3\] Figure \[abb:3\] (4.38,5.0) (0.0 ,0.0)[![image](./hyperoskulieren2.png){height="5cm"}]{} (2.6,3.2)[$C$]{} (4.0,2.3)[$x_1$]{} (2.4,4.3)[$x_3$]{} \[abb:4\] Figure \[abb:4\] \ The matrix $A$ is of type (I.1.2) if, and only if, it can be written as $$\label{eq:zerlegung12} {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &0 & 0\\ 0 & 1 &0 & a_{13}\\ 0 & 0 &1 & 0\\ 0 & 0 &0 & 1 }\cdot {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &0 & 0\\ 0 & 1 &0 & 0\\ 0 & 0 &1 & -a_{01}\\ 0 & 0 &0 & 1 }\cdot {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & a_{01} &0 & 0\\ 0 & 1 &0 & 0\\ 0 & 0 &1 & 0\\ 0 & 0 &0 & 1 }\cdot {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &a_{02} & a_{03}\\ 0 & 1 &0 & 0\\ 0 & 0 &1 & 0\\ 0 & 0 &0 & 1 }.$$ As before, this factorization is unique and the coefficients can be chosen freely. The first (second) matrix gives a shear with axis $x_1=0$ ($x_2=0$) in the direction of $p_3$, whereas the remaining matrices yield a translation in the direction of $p_1$ and a translation parallel to the plane $x_1=0$. However, the second and the third matrix are linked via the common parameter $a_{01}$. As $a_{01}$ varies in ${{\mathbb R}}$, their products comprise a one-parameter group $\Gamma_1$ of affine transformations. (See [@brau-66-67 I, p. 130], III 3, “Nichtisotrope Cliffordschiebungen”: All points of the line $x_0=x_2=0$ are invariant under $\Gamma_1$. All other point orbits are lines of a parabolic linear congruence with axis $x_0=x_2=0$.) Hence, up to translations parallel to the plane $x_1=0$, we obtain all twisted cubics which have third order contact with $C$ at $f$ by applying to $C$ all shears with axis $x_1=0$ and then all transformations of the group $\Gamma_1$. In Figure \[abb:3\] the twisted cubic $C$ and some of its images under affinities of $\Gamma_1$ are displayed. These curves lie on parabolic cylinders which are translates of $\Psi$. Figure \[abb:4\] shows the ruled surface which arises by applying $\Gamma_1$ to the curve $C$. The illustrated lines are point orbits with respect to $\Gamma_1$. In particular, the $x_1$-axis of the coordinate system is the orbit of the origin; this line is an edge of regression of the surface. The matrix $A$ is of type (I.1.2.3) if, and only if, it can be written as $$\label{eq:zerlegung123} {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &0 & 0\\ 0 & 1 &0 &2a_{02}\\ 0 & 0 &1 & 0\\ 0 & 0 &0 & 1 }\cdot {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &a_{02} & 0\\ 0 & 1 &0 & 0\\ 0 & 0 &1 & 0\\ 0 & 0 &0 & 1 }\cdot {{\left(\begin{array}{{}}{c}}4\end{array}\right)}{1 & 0 &0 & a_{03}\\ 0 & 1 &0 & 0\\ 0 & 0 &1 & 0\\ 0 & 0 &0 & 1 }.$$ Again, this decomposition is unique and the coefficients can be chosen arbitrarily. The products of the first and the second matrix in (\[eq:zerlegung123\]) comprise a one-parameter subgroup $\Gamma_2$; cf. the remarks above. Hence, up to translations parallel to the line $x_1=x_2=0$, we obtain all twisted cubics which have fourth order contact with $C$ at $f$ as the orbit of $C$ under $\Gamma_2$. Figure \[abb:5\] illustrates the twisted cubic $C$ and the cylinder $\Psi$, together with some of their images under affinities of $\Gamma_2$. Figure \[abb:6\] shows the ruled surface which is generated by applying $\Gamma_2$ to the curve $C$. This surface is a proper subset (only the points of $F\setminus\{f\}$ are missing) of the (ruled) Cayley surface* with equation $2x_0x_1x_2-x_1^3=x_0^2x_3$. (3.64,4.6) (0.0 ,0.0)[![image](./superoskulieren1.png){height="5cm"}]{} (1.8,2.35)[$C$]{} (1.8,0.4)[$\Psi$]{} (3.3,2.75)[$x_2$]{} (1.41,1.05)[$x$]{} (1.58,1.05)[${}_1$]{} (1.4,4.4)[$x_3$]{} \[abb:5\] Figure \[abb:5\] (3.78,4.5) (0.0 ,0.0)[![image](./superoskulieren2.png){height="5cm"}]{} (3.35,2.75)[$x_2$]{} (1.45,1.05)[$x_1$]{} (1.45,4.45)[$x_3$]{} \[abb:6\] Figure \[abb:6\] The Laguerre Geometry $\Sigma({{\mathbb R}},{{\mathbb L}})$ {#sect:laguerre} =========================================================== Let ${{\mathbb R}}[X]$ be the polynomial ring over the reals. The factor ring ${{\mathbb R}}[X]/\langle X^3\rangle=:{{\mathbb L}}$ is a $3$-dimensional real commutative local algebra with an ${{\mathbb R}}$-basis $1_{{\mathbb L}},{{\varepsilon}},{{\varepsilon}}^2$, where ${{\varepsilon}}:=X+\langle X^3\rangle$. Its non-invertible elements form the only maximal ideal $N:={{\mathbb R}}{{\varepsilon}}+{{\mathbb R}}{{\varepsilon}}^2$. We consider ${{\mathbb R}}$ as a subring of ${{\mathbb L}}$ by identifying $x\in{{\mathbb R}}$ with $x\cdot 1_{{\mathbb L}}\in {{\mathbb L}}$. (Our ring ${{\mathbb L}}$ is the ring $\mathfrak L_4$ in [@benz-73 p. 306].) Let us recall the definition of the projective line over ${{\mathbb L}}$, in symbols ${{\mathbb P}}({{\mathbb L}})$: We consider the free left ${{\mathbb L}}$-module ${{\mathbb L}}^2$. A cyclic submodule ${{\mathbb L}}(u,v)\subset {{\mathbb L}}^2 $ is a point of ${{\mathbb P}}({{\mathbb L}})$ if, and only if, $u$ or $v$ is a unit in ${{\mathbb L}}$. Two such pairs $(u,v)$ and $(u',v')$ in ${{\mathbb L}}^2$ determine the same point precisely when they are proportional by a unit in ${{\mathbb L}}$. (Cf. [@herz-95 p. 785] for a definition of the projective line over an arbitrary ring with a unit element.). We embed the real projective line ${{\mathbb P}}({{\mathbb R}})$ in ${{\mathbb P}}({{\mathbb L}})$ by ${{\mathbb R}}(x,y)\mapsto {{\mathbb L}}(x,y)$. The point set of the chain geometry $\Sigma({{\mathbb R}},{{\mathbb L}})$ is the projective line over ${{\mathbb L}}$, the chains are the images of ${{\mathbb P}}({{\mathbb R}})\subset{{\mathbb P}}({{\mathbb L}})$ under the natural right action of ${{\mathrm{GL}}}_2({{\mathbb L}})$ on ${{\mathbb L}}^2$; cf. [@herz-95 p. 790]. Since ${{\mathbb L}}$ is a local ring, our chain geometry is a Laguerre geometry [@herz-95 p. 793]. If two distinct points of ${{\mathbb P}}({{\mathbb L}})$ can be joined by a chain then they are said to be distant. Non-distant points are also said to be parallel ($\parallel$). Letting $p={{\mathbb L}}(a,b)$ and $q= {{\mathbb L}}(c,d)$ gives $$\label{eq:parallel} p\,\parallel q\,\Leftrightarrow\, \det{{\left(\begin{array}{{}}{c}}2\end{array}\right)}{a&b\\c&d}\in N.$$ This parallelism is an equivalence relation. There is a unique chain through any three mutually distant points. We fix the point ${{\mathbb L}}(1,0)=:\infty\in{{\mathbb P}}({{\mathbb L}})$. Then the point set of ${{\mathbb P}}({{\mathbb L}})$ can be split into two classes: A proper point* has the form ${{\mathbb L}}(z,1)$, and we identify such a point with the element $z\in {{\mathbb L}}$. The proper points are precisely the points which are distant (non-parallel) to $\infty$. Every other point of ${{\mathbb P}}({{\mathbb L}})$ has the form ${{\mathbb L}}(1,z)$ with $z\in N$. Such points are said to be improper. Hence we can regard ${{\mathbb P}}({{\mathbb L}})$ as the real affine $3$-space on ${{\mathbb L}}$ together with an extra “improper plane” which is just a copy of the maximal ideal $N$. The algebra ${{\mathbb L}}$ has two distinguished ideals, namely the maximal ideal $N$ and its annihilator $\{z\in {{\mathbb L}}\mid zN=0\}={{\mathbb R}}{{\varepsilon}}^2$. Accordingly, there are three types of lines: A line ${{\mathbb R}}u+v\subset {{\mathbb L}}$, where $u\in {{\mathbb L}}\setminus\{0\}$, $v\in {{\mathbb L}}$ is called singular if $u\in N$, and regular otherwise. A singular line of the form ${{\mathbb R}}{{\varepsilon}}^2+v$ is said to be vertical. We say that a plane is regular provided that it contains at least one regular line. A singular plane is just a non-regular plane. By (\[eq:parallel\]), the singular planes are the classes of proper parallel points. For each subset ${{\mathcal S}}\subset{{\mathbb P}}({{\mathbb L}})$ let ${{\mathcal S}}^\circ$ be its proper part, i.e. the set of all its proper points. The following is taken from <span style="font-variant:small-caps;">H.-J. Samaga</span> [@sama-76 Satz 4]; cf. also [@benz+s+s-81]: A subset ${{\mathcal C}}$ of ${{\mathbb P}}({{\mathbb L}})$ is a chain of $\Sigma({{\mathbb R}},{{\mathbb L}})$ precisely when one of the following conditions holds: $$\label{eq:gerade} {{\mathcal C}}=\{t+(a_{02}+a_{12} t){{\varepsilon}}+(a_{03}+a_{13} t){{\varepsilon}}^2\mid t\in{{\mathbb R}}\}\cup\{\infty\},$$ whence ${{\mathcal C}}^\circ$ is an affine line; $$\label{eq:parabel} {{\mathcal C}}=\{t+(a_{02}+a_{12} t){{\varepsilon}}+(a_{03}+a_{13} t+a_{33} t^2){{\varepsilon}}^2\mid t\in{{\mathbb R}}, a_{33}\neq 0\} \cup \{ {{\mathbb L}}(1,-a_{33}{{\varepsilon}}^2) \},$$ whence ${{\mathcal C}}^\circ$ is a parabola; [0.5mm$$\begin{aligned} {{\mathcal C}}&=&\{t+(a_{02}+a_{12} t+a_{22} t^2){{\varepsilon}}+(a_{03}+a_{13} t+a_{23} t^2+a_{33} t^3){{\varepsilon}}^2 \mid t\in{{\mathbb R}}, a_{33}=a_{22}^2\neq 0\}\nonumber\\ &&{} \cup\{ {{\mathbb L}}(1,-a_{22}{{\varepsilon}}+(-a_{23}+2a_{12}a_{22}){{\varepsilon}}^2)\},\label{eq:kub_parabel}\end{aligned}$$]{}whence ${{\mathcal C}}^\circ$ is a cubic parabola. In either case the $a_{ij}$’s are real constants subject to the conditions stated above. Obviously, the lines given by (\[eq:gerade\]) are precisely the regular ones. So, all regular lines are representatives for the point $\infty$. We say that a (cubic) parabola in ${{\mathbb L}}$ is admissible if it is the proper part of a chain. By (\[eq:parabel\]), a parabola is admissible if, and only if, its diameters are vertical lines and its plane is regular. Each admissible parabola is a representative of a unique improper point. We describe admissible parabolas which determine the same improper point: \[thm:parabeln\] Let ${{\mathcal C}}^\circ$ and ${{\widetilde{{{\mathcal C}}}}}^\circ$ be admissible parabolas of ${{\mathbb L}}$. Then the chains ${{\mathcal C}}$ and ${{\widetilde{{{\mathcal C}}}}}$ have the same improper point if, and only if, the parallel projection of ${{\widetilde{{{\mathcal C}}}}}^\circ$ onto the plane of ${{\mathcal C}}^\circ$, in the direction of an arbitrary non-vertical singular line, is a translate of ${{\mathcal C}}^\circ$. Let ${{\mathcal C}}$ and ${{\widetilde{{{\mathcal C}}}}}$ be given according to (\[eq:parabel\]) with coefficients $a_{ij}$ and ${{\widetilde{a}}}_{ij}$, respectively. The parallel projection of ${{\widetilde{{{\mathcal C}}}}}^\circ$ onto the plane of ${{\mathcal C}}^\circ$ is a parabola $$\{t+(a_{02}+a_{12} t){{\varepsilon}}+(a_{03}^+a_{13}^ t+a_{33}^* t^2){{\varepsilon}}^2 \mid t\in{{\mathbb R}}\} \mbox{ with } a_{33}^={{\widetilde{a}}}_{33}.$$ An easy calculation shows that the projected parabola is a translate of ${{\mathcal C}}^\circ$ if, and only if, ${{\widetilde{a}}}_{33}=a_{33}$. By (\[eq:parabel\]), this is necessary and sufficient for ${{\mathcal C}}$ and ${{\widetilde{{{\mathcal C}}}}}$ to have the same improper point. Let us consider the projective closure* ${{\mathbb P}}_3({{\mathbb R}})$ of the affine space on ${{\mathbb L}}$, where we do not distinguish between ${{\mathbb R}}(1,x_1,x_2,x_3)\in{{\mathbb P}}_3({{\mathbb R}})$ and $x_1+x_2{{\varepsilon}}+x_3{{\varepsilon}}^2\in {{\mathbb L}}$. Since we are going to work with two different extensions of the affine space on ${{\mathbb L}}$, we reserve the phrases “at infinity” and “improper” for the projective closure and for the chain-geometric closure, respectively. If ${{\mathcal C}}$ is a chain of $\Sigma({{\mathbb R}},{{\mathbb L}})$ then ${{\mathcal C}}^+\subset{{\mathbb P}}_3({{\mathbb R}})$ denotes that unique projective line or conic or twisted cubic which contains ${{\mathcal C}}^\circ$. We denote by $f$, $F$, and $\Phi$ the point at infinity of the vertical line ${{\mathbb R}}{{\varepsilon}}$, the line at infinity of the singular plane $N$, and the plane at infinity, respectively. Let ${{\mathcal C}}$ be a chain. If ${{\mathcal C}}^\circ$ is a line then ${{\mathcal C}}^+\not\subset\Phi$ is a projective line with a point at infinity not on $F$ and vice versa. If ${{\mathcal C}}^\circ$ is a parabola then ${{\mathcal C}}^+\not\subset\Phi$ is a conic through $f$ touching a line at infinity other than $F$. As before, all such conics arise from chains. We note that when ${{\mathcal C}}^\circ$ and ${{\widetilde{{{\mathcal C}}}}}^\circ$ are parabolas in the same plane then the existence of a translation taking ${{\mathcal C}}^\circ$ to ${{\widetilde{{{\mathcal C}}}}}^\circ$ just means that the projective conics ${{\mathcal C}}^+$ and ${{\widetilde{{{\mathcal C}}}}}^+$ have contact of second order at the point $f$. See, e.g., [@pauk-79]. But, since admissible parabolas in different planes may represent the same improper point, we cannot always describe improper points in terms of conics with second order contact at infinity. Now we turn to the case when ${{\mathcal C}}^\circ$ is a cubic parabola: \[thm:3\] The cubic parabola $$\label{eq:standard_kub_parabel} \{t+t^2{{\varepsilon}}+ t^3{{\varepsilon}}^2\mid t\in{{\mathbb R}}\}$$ is admissible. A cubic parabola of ${{\mathbb L}}$ is admissible if, and only if, its projective extension and the projective extension of (\[eq:standard\_kub\_parabel\]) have contact of second order at the point $f={{\mathbb R}}(0,0,0,1)$. By (\[eq:kub\_parabel\]), there is a unique chain ${{\mathcal D}}$, say, such that ${{\mathcal D}}^\circ$ coincides with the cubic parabola (\[eq:standard\_kub\_parabel\]). Its projective extension ${{\mathcal D}}^+$ is given by (\[eq:kubik-homogen\]), whence it is a twisted cubic through $f$, with tangent line $F$, and osculating plane $\Phi$. Now we apply those collineations of ${{\mathbb P}}_3({{\mathbb R}})$ which are given by regular matrices of type (III.1). So we get all the twisted cubics which have second order contact with ${{\mathcal D}}^+$ at $f$ and, by (\[eq:kub\_parabel\]), these are precisely the projectively extended admissible cubic parabolas. \[thm:4\] Let ${{\mathcal C}}^\circ$ and ${{\widetilde{{{\mathcal C}}}}}^\circ$ be admissible cubic parabolas of ${{\mathbb L}}$. Then the chains ${{\mathcal C}}$ and ${{\widetilde{{{\mathcal C}}}}}$ have the same improper point if, and only if, the extended curves ${{\mathcal C}}^+$ and ${{\widetilde{{{\mathcal C}}}}}^+$ have contact of third order at $f={{\mathbb R}}(0,0,0,1)$. \(a) First, we consider that chain ${{\mathcal D}}$ which yields the cubic parabola (\[eq:standard\_kub\_parabel\]). The improper point of ${{\mathcal D}}$ is ${{\mathbb L}}(1,-{{\varepsilon}})$. Now we apply those collineations of ${{\mathbb P}}_3({{\mathbb R}})$ which are given by regular matrices of type (III.1.2). This gives, by Theorem \[thm:2\], precisely those twisted cubics which have third order contact with ${{\mathcal D}}^+$ at $f$ and, by (\[eq:kub\_parabel\]), we get all the projectively extended cubic parabolas that arise from the chains through ${{\mathbb L}}(1,-{{\varepsilon}})$. Since contact of third order is a transitive notion, the assertion follows for all chains ${{\mathcal C}}$ through ${{\mathbb L}}(1,-{{\varepsilon}})$. \(b) Next, let ${{\mathcal C}}$ be any chain whose proper part is a cubic parabola, so that its improper point can be written as ${{\mathbb L}}(1,-a{{\varepsilon}}-b{{\varepsilon}}^2)$, where $a,b\in{{\mathbb R}}$ and $a\neq 0$. The matrix $$\alpha:=\left(\begin{array}{rr} a & 0 \\ -\frac{b}{\raisebox{0.8mm}{$\scriptstyle a$}} & 1 \end{array}\right) \in{{\mathrm{GL}}}_2({{\mathbb R}})\subset{{\mathrm{GL}}}_2({{\mathbb L}})$$ induces a projectivity of ${{\mathbb P}}({{\mathbb L}})$ taking the improper point ${{\mathbb L}}(1,-a{{\varepsilon}}-b{{\varepsilon}}^2)$ to $${{\mathbb L}}\Big(\Big(\frac 1 a-\frac b{a^2}\,{{\varepsilon}}\Big)\Big(a+b{{\varepsilon}}+\frac {b^2}{a}\,{{\varepsilon}}^2,-a{{\varepsilon}}-b{{\varepsilon}}^2\Big)\Big) = {{\mathbb L}}(1,-{{\varepsilon}}).$$ The action of $\alpha$ on the proper points is the affine transformation ${{\mathbb L}}\to {{\mathbb L}}: z\mapsto za-\frac{b}{a}$ which in turn can be extended to a collineation of ${{\mathbb P}}_3({{\mathbb R}})$. Since contact of any order is preserved under collineations, we can apply the results from (a) in order to complete the proof. From the affine point of view, the previous results are not satisfying, because they are formulated in projective terms. However, in Section \[sect:kubiken\] we have explained how one can “see” contact of higher order at $f$ via an affine transformation taking ${{\widetilde{{{\mathcal C}}}}}^\circ$ to ${{\mathcal C}}^\circ$. Another basic topic is to characterize chains ${{\mathcal C}}$ and ${{\widetilde{{{\mathcal C}}}}}$ which touch at a common improper point. If ${{\mathcal C}}^\circ$ is an affine line then this means, by definition, that ${{\mathcal C}}^\circ$ and ${{\widetilde{{{\mathcal C}}}}}$ are parallel lines. If ${{\mathcal C}}^\circ$ is a parabola then a characterization as in Theorem \[thm:2\] can be given, but now the parallel projection of ${{\widetilde{{{\mathcal C}}}}}^\circ$ has to arise from ${{\mathcal C}}^\circ$ under a translation in the direction of ${{\varepsilon}}^2$. (This means contact of third order at $f$.) Likewise, Theorem \[thm:4\] can be modified as to describe touching chains, by replacing “third order contact” with “fourth order contact”. The proofs are left to the reader. The affine space on ${{\mathbb L}}$ is closely related with the flag space (two-fold isotropic space), as the triple $(f,F,\Phi)$ can be considered as its absolute flag. Cf., among others, the papers [@brau-66-67] by <span style="font-variant:small-caps;">H. Brauner</span>. Due to lack of space we have to refrain from presenting here the interesting connections between these two geometries. [10]{} -3pt [ Hans Havlicek and Klaus List\ Institut für Geometrie\ Technische Universität\ Wiedner Hauptstra[ß]{}e 8–10\ A-1040 Wien\ Austria ]{}
--- author: - 'Bernard Le Stum & Adolfo Quirós[^1]' bibliography: - 'BiblioBLS.bib' date: Version of title: Formal confluence of quantum differential operators --- Abstract We prove that a usual differential operator is formally the limit of quantum differential operators. For this purpose, we introduce the notion of twisted differential operator of infinite level and prove that, formally, such an object is independent of the choice of the twist. Our method provides explicit formulas. Introduction {#introduction .unnumbered} ============ The confluence process consists in replacing differential by finite difference (or $q$-difference) in a differential equation and try to derive solutions of the original equation from solutions of the finite difference (or $q$-difference) equation (see [@LevyLessman92] for example in the finite difference case and [@Sauloy00] in the $q$-difference case). Actually, differential equations and finite difference equations (or $q$-difference equations) may be seen as particular cases of twisted differential equations as explained in [@LeStumQuiros15a]. In this more general setting, we are faced with comparing twisted differential equations for different twists. We will introduce various kinds of twisted differential rings and compare them. Note that our methods will apply as well to the positive characteristic situation and also to quantum differential equations (with respect to a root of unity). Let us be more precise. If $R$ is a commutative ring, a twisted $R$-algebra is a commutative $R$-algebra $A$ endowed with an $R$-linear ring endomorphism $\sigma$. Actually, in [@LeStumQuiros15a] we allowed more generally families of endomorphisms satisfying some relations but we will concentrate here on the one dimensional case and we will use either one morphism or a system of roots of one morphism. One may then consider the $\sigma$-derivations $D$ of $A$, or more generally, the $\sigma$-derivations $D$ of an $A$-module $M$: $R$-linear maps that satisfy the twisted Leibnitz rule $$D(xs) = D(x)s + \sigma(x)D(s).$$ A $\sigma$-differential module is an $A$-module $M$ endowed with an action by $\sigma$-derivations of the $\sigma$-derivations of $A$. Assume that $x \in A$ is a $\sigma$-coordinate in the sense for example that there exists a a basis $\partial_{\sigma}$ for the $\sigma$-derivations of $A$ such that $\partial_{\sigma}(x) = 1$ and define the twisted Weyl algebra $\mathrm D_{\sigma}$ as the Ore extension of $A$ by $\sigma$ and $\partial_{\sigma}$ (as in [@Bourbaki12], proposition 1.4). A $\sigma$-differential module is then the same thing as a $\mathrm D_{\sigma}$-module. So now, what we want to understand is the relation between $\mathrm D_{\sigma}$ and $\mathrm D_{\tau}$ when $\tau$ is another $R$-linear endomorphism of $A$ with the same twisted coordinate $x$. In the end, we will be mostly essentially interested in $\tau (x) = x$ (usual case) and $\sigma(x) = qx+h$ (quantum case). One may usually see a ring of differential operators as some dual of a ring of formal functions. Doing this directly for $\mathrm D_{\sigma}$ would require to understand the notion of $\sigma$-divided powers on the function side. This is a very interesting question that we postpone to a forthcoming article. Here, we will actually replace the twisted Weyl Algebra $\mathrm D_{\sigma}$ with a Grothendieck ring of differential operators $\mathrm D^{(\infty)}_{\sigma}$ (so that the $\sigma$-divided powers live naturally on the differential operator side). It so happens that the classical construction works incredibly well for this particular generalization. It is actually sufficient to replace the usual powers of an ideal with the twisted powers introduced in [@LeStumQuiros15]. But now we are faced with two questions: the comparison between $\mathrm D^{(\infty)}_{\sigma}$ and $\mathrm D^{(\infty)}_{\tau}$ on one hand and the comparison of $\mathrm D_{\sigma}$ with $\mathrm D^{(\infty)}_{\sigma}$ on the other. We will show that, if we denote by $\widehat{\mathrm D}^{(\infty)}_{\sigma}$ the completion along the divided powers of $\partial_{\sigma}$, then there always exists a natural isomorphism $\widehat{\mathrm D}^{(\infty)}_{\sigma} \simeq \widehat{\mathrm D}^{(\infty)}_{\tau}$ (the formal deformation of proposition \[dfthm\]) and we will be able to give very explicit formulas. For example, in the case $\tau (x) = x$ and $\sigma(x) = qx$, we can write (over a field of characteristic zero when $q$ is not a root of unity): $$\partial_{q} = \sum_{k=1}^{\infty} \frac 1{k!}(q-1)^{k-1}x^{k-1} \partial^{k} \quad \mathrm{and} \quad \partial = \sum_{k=1}^{\infty} \frac {(1-q)^{k}}{1 -q^k} x^{k-1}\partial_{q}^{k}.$$ In the quantum situation, when $\sigma(x) = qx +h$ with $q,h \in R$, then there exists a canonical map $\mathrm D_{\sigma} \to \mathrm D^{(\infty)}_{\sigma}$ whose image is exactly the ring of small (or naive) differential operators. When the $q$-characteristic of $A$, that was introduced in [@LeStumQuiros15], is zero, then all three rings are actually equal and we easily obtain our first confluence theorem (theorem \[confalg\]) when $R$ is a $\mathbb Q$-algebra: there exists a map $\mathrm D_{\sigma} \to \widehat{\mathrm D}$ with dense image. But we are actually mainly interested in the positive $q$-characteristic case, which means that $q$ is a primitive $p$-th root of unity. It is then necessary to use a complete family $\underline q$ of $p^n$-th roots of $q$. Then we can define what we call a rooted Weyl algebra $\mathrm D_{\underline \sigma}$ by taking the limit on all $\mathrm D_{\sigma_{n}}$ and build a map $\mathrm D_{\underline \sigma} \to \widehat{\mathrm D}$ with dense image. This is the second confluence theorem (theorem \[mainth\]). It is interesting to notice that this theorem puts together the ring $D$ who supports the conjecture of Jacques Dixmier [@Dixmier68] (whose higher version is equivalent to the Jacobian conjecture [@BelovKanelKonsevich07]) and an Azumaya algebra where the Morita equivalence could be applied. We may also notice that a $p$-adic version of one of our results, the derivation of the formal deformation theorem (proposition \[dfthm\]) from the density lemma (lemma \[density\]), would recover Andrea Pulita’s realization of a formal solution of a differential module as formal solution of some $\sigma$-module (results in this direction had already obtained by Yves André and Lucia Di Vizio in [@AndreDiVizio04]). We expect that the quantum refinement could clarify many things and we intend to work this out in the future. Both authors thank Michel Gros for all the fruitful conversations that we had all together. For us, a ring has an associative multiplication (not commutative in general) and a two-sided unit. Morphisms of rings are always assumed to preserve the unit. A module always means a left module. We will essentially consider $1$-twisted rings from [@LeStumQuiros15a] and simply call them twisted rings. Throughout the paper, $R$ will denote a commutative ring and $A$ a twisted commutative $R$-algebra: a commutative $R$-algebra endowed with an $R$-linear ring endomorphism $\sigma_{A}$. Twisted principal parts {#tpp} ======================= We will introduce here the notion of twisted principal part (functions on twisted infinitesimal neighborhoods of the diagonal). We will begin by ignoring the endomorphism $\sigma_{A}$ and concentrating on the commutative $R$-algebra $A$. The tensor product $P_{A/R} := A \otimes_{R} A$ has two $A$-algebra structures, one coming from the action on the left and the other one coming from the action on the right. Unless otherwise specified, we will use the left structure when we see $P_{A/R}$ as an $A$-module. However, when $M$ is an $A$-module, the notation $P \otimes_{A} M$ will implicitly mean that we use the action of $A$ on the right to build the tensor product and that the resulting object will be seen as an $A$-module using the action of $A$ on the left. In practice, we will write $x := x \otimes 1 \in P_{A/R}$ and $\tilde x := 1 \otimes x \in P_{A/R}$. In other words, with these notations, the action on the left is multiplication by $x$ and the action on the right is multiplication by $\tilde x$. Any element of $P_{A/R}$ can be written as a finite sum $\sum x_{i} \tilde y_{i}$. At some point, we will call Taylor map the embedding on the right and denote it by $$\xymatrix@R0cm{\theta_{A/R} :& A \ar[r] & P_{A/R}. \\ & x \ar@{\|->}[r] & \tilde x }$$ We will then call $\tilde x = \theta_{A/R}(x)$ the Taylor expansion of $x \in A$ (more on this vocabulary later on). We will also consider the canonical map corresponding to the projection that forgets the middle term: $$\xymatrix@R0cm{ A \otimes_{R} A \ar[r]^-{\delta_{A/R}} & A \otimes_{R} A \otimes_{R} A \\ x \otimes y \ar@{\|->}[r] & x \otimes 1 \otimes y }.$$ It is a morphism of $R$-algebras that may also be seen as a map $$\xymatrix@R0cm{ P_{A/R} \ar[r]^-{\delta_{A/R}} &P_{A/R} \otimes_{A} P_{A/R} \\ x \tilde y \ar@{\|->}[r] & x \otimes \tilde y },$$ where $A$ acts on the right on the first factor and on the left on the second one in the tensor product. Let $I_{A/R}$ be the kernel of the multiplication morphism $$\xymatrix@R0cm{P_{A/R} \ar[r] & A \\ x \tilde y \ar@{\|->}[r] & xy}$$ (that corresponds to the diagonal embedding). The ideal $I_{A/R}$ is generated by the image of the linear map $$\xymatrix@R0cm{A \ar[r]^-{d_{A/R}} & P_{A/R} \\ x \ar@{\|->}[r] & \xi := \tilde x - x}$$ (that corresponds to the difference between the projections). In practice, we will usually drop the index $A/R$ and simply write $P$, $I$, $\theta$, $\delta$ and $d$. Since we will make regular use of linearization, we introduce it formally now: Let $M, N$ be two $A$-modules and $u : M \to N$ an $R$-linear map. Then, the $A$-linearization of $u$ is the $A$-linear map $$\xymatrix@R0cm{ P \otimes_{A} M \ar@/^.5cm/[rr] ^-{\tilde u} \ar[r]^{\simeq} &A \otimes_{R} M \ar[r] & N \\ & x \otimes s \ar@{\|->}[r] & xu(s).}$$ Recall that, in the tensor product, we use the action of the right on $P$. Therefore, we have $$\forall x,y \in A, \forall s \in M, \quad \tilde u (x\tilde y \otimes s) = xu(ys).$$ As an example, we see that the multiplication map $P \to A$ is the $A$-linearization of the identity of $A$, seen as an $R$-linear map. For further use, we also mention the following result, that follows immediately from the definitions: \[compolin\] If we are given two $R$-linear maps $\varphi : M \to N$ and $\psi : L \to M$, then the linearization of $\varphi \circ \psi$ factors as $$\label{facdif} \xymatrix@R0cm{ P \otimes_{A} L \ar[r]^-{\delta \otimes \mathrm{Id}} & P \otimes_{A} P \otimes_A L \ar[r]^-{\mathrm{Id} \otimes_{A} \tilde \psi} & P \otimes_{A} M \ar[r]^-{\tilde \varphi} & N.} \quad\Box$$ Now we make the endomorphism $\sigma_{A}$ enter the game. We will consider $P := A \otimes_{R} A$ as a twisted $R$-algebra by using $\sigma_{A}$ on the left and the identity the right: in other words, we set $ \sigma_{P} := \sigma_{A} \otimes_{R} \mathrm{Id}_{A}$. Alternatively, this is the unique structure of twisted $R$-algebra on $P$ such that $$\forall x \in A,\quad \sigma_{P}(x) = \sigma_{A}(x) \quad \mathrm{and} \quad \sigma_{P}(\tilde x) = \tilde x.$$ In particular, we may also consider $P$ as a twisted $A$-algebra, in the sense that $P$ is endowed with a $\sigma_{A}$-linear ring endomorphism $\sigma_{P}$. We will often drop the indexes $A$ and $P$ and simply write $\sigma$ for both maps (so that $\sigma(\tilde x) = \tilde x$) when there is no ambiguity. Before we do anything else, let us prove the following result, which is quite elementary, but very useful: \[transl\] If $x \in A$ and $\xi :=\tilde x - x \in P_{A/R}$, then, $$\sigma_{P}(\xi) = \xi + y \quad \mathrm{with} \quad y:= x - \sigma_{A}(x).$$ We have $$\sigma(\xi) = \sigma(\tilde x - x) = \sigma(\tilde x) - \sigma(x) = \tilde x - \sigma(x) = \tilde x - x + x - \sigma(x) = \xi + y. \quad \Box$$ \[kersig\] The kernel of the $A$-linearization $$\xymatrix@R0cm{ P_{A/R} \ar[r]^-{\tilde \sigma_{A}} & A \\ x \tilde y \ar@{\|->}[r] & x\sigma_{A}(y)}$$ of $\sigma_{A}$ is $\sigma_{P}(I_{A/R})$. Recall that $\sigma(I)$ denotes the ideal generated by the image of $I$. As a consequence, since $I$ is generated by the image of $d$, we see that $\ker \tilde \sigma$ will be generated by the image of $$\xymatrix@R0cm{A \ar[r]^-{\sigma \circ d} & P \\ x \ar@{\|->}[r] & \tilde x - \sigma(x)}.$$ First of all, we have $$\forall x \in A, \quad \tilde \sigma( \tilde x - \sigma(x)) = \sigma(x) - \sigma(x) = 0,$$ and it follows that $\sigma(I) \subset \ker \tilde \sigma$. Conversely, by definition, we have $$\forall x \in A, \quad \sigma(x) \equiv \tilde x \mod \sigma(I).$$ Therefore, if $f := \sum x_{i} \tilde y_{i} \in \ker \tilde \sigma$, we have $$f \equiv \sum x_{i}\sigma(y_{i}) = \tilde \sigma (f) = 0 \mod \sigma(I),$$ and we see that $\ker \tilde \sigma \subset \sigma(I)$. $\quad \Box$ 1. It is sometimes convenient to use the bimodule language. An $A$-sesquimodule $M$ is an $A$-bimodule such that $$\forall x \in A, s \in M, \quad \sigma_{A}(x) \cdot s = s \cdot x.$$ Note that we are using the reverse convention from André’s in [@Andre01] so that forgetting the right action induces an equivalence (an isomorphism) between $A$-sesquimodules and left $A$-modules (we will use this identification). 2. The $R$-algebra $P$ has a canonical $A$-bimodule structure which is completely independent of the choice of $\sigma_{A}$. If we endow $A$ with its sesquimodule structure, then the linearization $\tilde \sigma$ of $\sigma_{A}$ is a morphism of $A$-bimodules: we always have $$\forall x, y \in A, f \in P, \quad \tilde\sigma(x\cdot f\cdot y)=\tilde \sigma(xf\tilde y) = x\tilde \sigma(f)\sigma(y)=x\cdot \tilde\sigma(f) \cdot y.$$ It follows that $\ker \tilde \sigma$ has a natural structure of $A$-bimodule. Actually, since $\tilde \sigma$ is a ring homomorphism, then $\ker \tilde \sigma$ is an ideal and therefore automatically an $A$-bimodule. Recall from [@LeStumQuiros15] that the twisted powers of $I$ are $$I^{(0)} = P, \quad I^{(1)} = I , \quad I^{(2)} := I\sigma(I), \quad \ldots, \quad I^{(n)} := I\sigma(I) \cdots \sigma^{n-1}(I),$$ where images and products of ideals are meant as ideals. We will write $I^{(n)_{\sigma}}_{A/R}$ when we want to make clear the dependence on $\sigma$ and $A/R$. The $A$-module of twisted principal parts of order $n \in \mathbb N$ (and infinite level) of $A$ is $$P_{A/R,(n)_{\sigma}} : = P_{A/R}/I_{A/R}^{(n+1)_{\sigma}}.$$ The $A$-module of twisted principal parts of infinite order (and infinite level) of $A$ is the twisted completion: $$\widehat P_{A/R,\sigma} := \varprojlim P_{A/R,(n)_{\sigma}}.$$ Note that these $A$-modules all have a natural structure of $R$-algebra and that the definition also makes sense for $n = -1$ so that $P_{A/R,(-1)_{\sigma}} = 0$. Again, we will often drop the indexes $A/R$ when we believe that there is no risk of confusion and simply write $P_{(n)_{\sigma}}$ and $\widehat P_{\sigma}$. We will also drop the index $\sigma$ when $\sigma_{A} = \mathrm{Id}_{A}$. 1. When $A$ is trivially twisted, which means that $\sigma_{A} := \mathrm{Id}_{A}$, this notion of principal parts coincides with the usual one (definition 16.3.1 of [@EGA44]), and therefore many of the basic objects we will construct are twisted versions of those in [@EGA44]. 2. When $A = R[x]$, we have $P = R[x, \xi]$ with $\xi = \tilde x - x$ and $I = (\xi)$. Moreover, $\sigma(\xi) = \xi + y$ with $y = x - \sigma(x)$. It follows that $$\sigma^n(\xi) = \xi + (n)_{\sigma}(y) = \xi + x - \sigma^{n}(x),$$ with $(n)_{\sigma} := 1 + \cdots + \sigma^{n-1}$. Therefore, we have $$P_{(n)} = R[x, \xi]/\prod_{i=0}^n \left(\xi + (i)_{\sigma}(y)\right) = R[x, \xi]/\prod_{i=0}^n \left(\xi + x - \sigma^{n}(x)\right).$$ 1. In the case $\sigma(x) = x $, we get $P_{(n)} = R[x, \xi]/\xi^{n+1}$ as expected. 2. More generally, if we assume that $\sigma(x) = x + h$ with $h \in R$, we obtain $$P_{(n)} = R[x, \xi]/\prod_{i=0}^n \left(\xi -ih\right).$$ 3. On the other hand, if we let $\sigma(x) = qx$ with $q \in R$, we find $$P_{(n)} = R[x, \xi]/\prod_{i=0}^n \left( \xi + (1-q^i)x\right).$$ When $R \to R'$ is a homomorphism of commutative rings, we endow $A' := R' \otimes_R A$ with $\sigma_{A'} := \mathrm{Id}_{R'} \otimes_{R} \sigma_{A}$. \[extn\] Let $R \to R'$ be a homomorphism of commutative rings and $A' := R' \otimes_R A$. Then, we have for all $n \in \mathbb N$, $$A' \otimes_A P_{A/R,(n)_{\sigma}} \simeq P_{A'/R',(n)_{\sigma}}.$$ If we let $P' := P_{A'/R'}$, then there exists an canonical isomorphism $A' \otimes_{A} P \simeq R' \otimes_{R} P \simeq P'$. Moreover, if we denote by $I'$ the kernel of multiplication on $P'$, we have $A' \otimes_{A} I \simeq I'$. And finally, $\mathrm{Id}_{A} \otimes_{A} \sigma_{P}$ corresponds to $\sigma_{P'}$ under this isomorphism. Our assertion is therefore an immediate consequence of right exactness of tensor product. $\quad \Box$ Recall from [@LeStumQuiros15] that a twisted $A$-algebra is an $A$-algebra $B$ endowed with a $\sigma_{A}$-linear ring endomorphism $\sigma_{B}$. \[locord\] If $B$ is a twisted commutative $A$-algebra, then there exists a canonical morphism of $B$-algebras $$B \otimes_A P_{A/R,(n)_{\sigma}} \to P_{B/R,(n)_{\sigma}}.$$ When $B$ is a quotient (resp. a localization) of $A$ this map is surjective (resp. bijective). Recall from Definition 1.7 of [@LeStumQuiros15a] that we call such a $B$ a twisted quotient (resp. twisted localization) of $A$. The morphism of twisted $R$-algebras $A \to B$ extends naturally to a morphism of twisted $R$-algebras $P_{A} \to P_{B}$. Since $I_{A}$ is sent into $I_{B}$, we see that, for all $n \in \mathbb N$, $\sigma^n(I_{A})$ is sent into $\sigma^n(I_{B})$ and the first assertion formally follows. In the case of a quotient map, all the maps involved are surjective. Now, if $B := S^{-1}A$ is a twisted localization of $A$, then $P_{B}$ is the localization of $P_{A}$ with respect to the monoid $S'$ generated by $S$ and $\tilde S$, and we have $I_{B} = P_{B} \otimes_{P_{A}} I_{A}$. It immediately follows that for all $n \in \mathbb N$, we have $\sigma^n(I_{B}) = P_{B} \otimes_{P_{A}} \sigma^n(I_{A})$, and therefore $I_{B}^{(n)} = P_{B} \otimes_{P_{A}} I_{A}^{(n)}$. Thus we see that $$P_{B,(n)_{\sigma}} = P_{B}/(P_{B} \otimes_{P_{A}} I_{A}^{(n+1)}) = P_{B} \otimes_{P_{A}} P_{A,(n)} = B \otimes_A P_{A,(n)} \otimes_{A} B.$$ We need to remove the $B$ on the right hand side and it is sufficient to show that $\tilde x$ is invertible in $B \otimes_A P_{A,(n)}$ whenever $x \in S$. But we have $$\prod_{i=0}^n (\tilde x - \sigma^i(x)) = \prod_{i=0}^n \sigma^i(\tilde x - x) \in I_{A}^{(n+1)},$$ from which we derive that there exists $f \in P_{A}$ such that $$f\tilde x \equiv \prod_{i=0}^n \sigma^i(x) \mod I_{A}^{(n+1)}.$$ Since $B = S^{-1} A$, we must have $\sigma(S) \subset B^\times$ and it follows that $\prod_{i=0}^n \sigma^i(x) \in B^\times$. Thus, we see that $f\tilde x$ is invertible in $B \otimes_A P_{A,(n)}$ and it follows that $\tilde x$ is invertible too. $\quad \Box$ As an illustration, we can give explicit formulas in the quantum situation. Recall that we introduced in [@LeStumQuiros15] the notion of twisted powers of an element in a twisted ring. In particular, for $f \in P$, we will have $$f^{(0)} := 1, \quad f^{(1)} := f, \quad f^{(2)} = f\sigma(f), \quad \ldots, \quad f^{(n+1)} = f\sigma(f) \cdots \sigma^n(f).$$ Recall also that the quantum binomial coefficients are defined by induction (see [@LeStumQuiros15] for example) as $${n \choose k}_{q} := {n-1 \choose k-1}_{q} + q^k{n-1 \choose k}_{q}.$$ Assume $\sigma(x) = qx$ with $q \in R$ and let $\xi = \tilde x -x$. Then, we have $$\label{binxi} \forall n \in \mathbb N, \quad \xi^{(n)} = \sum_{j=0}^n (-1)^{j} {n \choose j}_q q^{\frac {j(j-1)}2}x^{j}\tilde x^{n-j}$$ and $$\forall n \in \mathbb N, \quad \tilde x^n = \sum_{i=0}^n {n \choose i}_{q} x^i\xi^{(n-i)}.$$ The first equality is essentially the quantum binomial formula (see proposition 2.14 in [@LeStumQuiros15]): $$(\tilde x - x)^{(n)} = \sum_{j=0}^n {n \choose j}_q (-x)^{(j)}\tilde x^{(n-j)}.$$ For the second one, we compute the right hand side with the help of formula : $$\begin{aligned} \sum_{i=0}^n {n \choose i}_{q} x^i\xi^{(n-i)} &=& \sum_{i=0}^n {n \choose i}_{q} x^i \left(\sum_{j=0}^{n-i} (-1)^j {n-i \choose j}_{q} q^{\frac {j(j-1)}2} x^j\tilde x^{n-i-j}\right) \\ &=& \sum_{k=0}^{n} \left( \sum_{i=0}^k {n \choose i}_{q} {n-i \choose k-i}_{q} (-1)^{k-i} q^{\frac {(k-i)(k-i-1)}2} \right) x^{k}\tilde x^{n-k}\end{aligned}$$ after rewriting $i+j=k$. Now, we have (using corollaries 2.7 and 2.8 in [@LeStumQuiros15], for example) $$\sum_{i=0}^k {n \choose i}_{q} {n-i \choose k-i}_{q} (-1)^{k-i} q^{\frac {(k-i)(k-i-1)}2} = {n \choose k}_{q} \sum_{i=0}^k {k \choose i}_{q} (-1)^{k-i} q^{\frac {(k-i)(k-i-1)}2}.$$ But the quantum binomial formula again implies that for $k > 0$, we have $$\sum_{i=0}^k {k \choose i}_{q} (-1)^{k-i} q^{\frac {(k-i)(k-i-1)}2} = \prod_{i=0}^{k-1} (1-q^i) = 0.$$ And it follows that $$\sum_{i=0}^n {n \choose i}_{q} x^i\xi^{(n-i)} = \tilde x^n,$$ as asserted. $\quad \Box$ We will need a slightly stronger notion of coordinate than the one used in [@LeStumQuiros15a]: \[twistdef\] Let $x \in A$ and $\xi := \tilde x - x \in P_{A/R}$. 1. Then, $x$ is a twisted coordinate for $A$ over $R$ if for all $n \in \mathbb N$, $P_{A/R,(n)_{\sigma}}$ is freely generated as an $A$-module by the images of $1, \xi, \xi^{2}, \ldots, \xi^{n}$. 2. If $x$ is a twisted coordinate such that $\sigma(x) = qx + h$ with $q, h \in R$, we will call it a quantum coordinate or $q$-coordinate and call $A$ a quantum $R$-algebra or $q$-$R$-algebra. <!-- --> 1. When $A/R$ is smooth (of pure relative dimension one) and $\sigma_{A} = \mathrm{Id}_{A}$, then a twisted coordinate is nothing but an étale coordinate: it means that the map $$\xymatrix@R0cm{R[T] \ar[r] & A \\ T \ar@{\|->}[r] & x}$$ is étale. 2. If $A = R[x]$, then $x$ is always a twisted coordinate, whatever $\sigma$ is. \[locco\] 1. If $R \to R'$ is a homomorphism of commutative rings and $x$ is a twisted coordinate on $A$, then $x$ becomes a twisted coordinate on $A' := R' \otimes_{R} A$ (relatively to $R'$). 2. If $B$ is a twisted localization of $A$ and $x$ is a twisted coordinate on $A$, then $x$ becomes a twisted coordinate on $B$. Follows from propositions \[extn\] and \[locord\]. $\quad \Box$ Assume $A$ is a quantum $R$-algebra so that there exists a twisted coordinate $x$ on $A$ and $q,h \in R$ such that $\sigma(x) = qx+h$. Let us still denote by the same letter $x$ an indeterminate over $R$ and by $\sigma$ again the endomorphism of $R[x]$ given by the same formula. Then, $A$ becomes an $R[x]$-twisted algebra and we have a canonical isomorphism (compare basis on both sides): $$A \otimes_{R[x]} P_{R[x]/R,(n)_{\sigma}} \simeq P_{A/R,(n)_{\sigma}}.$$ In the next statement, we use the letter $\xi$ as an indeterminate over $A$ so that $A[\xi]$ denotes the polynomial ring and $A[\xi]_{\leq n}$ the submodule of polynomial of degree at most $n$. Ultimately, this should not create any confusion due to corollary \[insid\] below. \[sigis\] Let $x \in A$ and $y := x - \sigma(x)$. We endow $A[\xi]$ with the unique $\sigma_{A}$-linear endomorphism such that $\sigma(\xi) = \xi + y$. Then, $x$ is a twisted coordinate on $A$ if and only if the morphism of twisted algebras $$\xymatrix@R0cm{ \phi\,:\, A[\xi] \ar[r]& P_{A/R} \\ \xi \ar@{\|->}[r] & \tilde x - x}$$ induces for all $n \in \mathbb N$, an isomorphism of $A$-algebras $A[\xi]/\xi^{(n+1)} \simeq P_{A/R,(n)_{\sigma}}$. First of all, it follows from lemma \[transl\] that there exists such a morphism for all $n \in \mathbb N$. On the other hand, $$\xi^{(n+1)}= \prod_0^{n} (\xi + (x - \sigma^i(x)))$$ is a monic polynomial of degree $n+1$. Then, euclidean division tells us that the composite map $$\xymatrix@R0cm{ A[\xi]_{\leq n} \ar[r] & A[\xi] \ar[r] & A[\xi]/ \xi^{(n+1)}}$$ is an isomorphism of $A$-modules. Therefore the condition on $\phi$ is equivalent to the fact that the map $$\xymatrix@R0cm{ A[\xi]_{\leq n} \ar[r] & P_{(n)_{\sigma}} \\ \xi \ar@{\|->}[r] & \overline{\tilde x - x}}$$ is bijective. And this exactly means that $P_{(n)_{\sigma}}$ is freely generated by the $n+1$ first powers of the images of $\tilde x - x$. $\quad \Box$ When the polynomial ring $A[\xi]$ is endowed with a structure of $\sigma_{A}$-algebra, we will set $$A[[\xi]]_{\sigma} := \varprojlim A[\xi]/ \xi^{(n+1)}.$$ \[insid\] With the same hypothesis, $x$ is a twisted coordinate on $A$ if and only if there exists an isomorphism of $A$-algebras $$\xymatrix@R0cm{ A[[\xi]]_{\sigma} \ar[r]^-{\simeq} &\widehat P_{A/R,\sigma}\,. \\ \xi \ar@{\|->}[r] & \tilde x - x & \Box}$$ \[free\] Let $x \in A$ and $\xi := \tilde x - x \in P$. Then, the following conditions are equivalent: 1. $x$ is a twisted coordinate on $A$ 2. for all $n \in \mathbb N$, the $A$-module $P_{(n)_{\sigma}}$ is freely generated by the images of $1, \xi, \xi^{(2)}, \ldots, \xi^{(n)}$ 3. for all $n \in \mathbb N$, the $A$-module $I^{(n)}/I^{(n+1)}$ is free of rank one on the image of $\xi^{(n)}$. $\quad \Box$ If $A$ is a twisted localization of $R[x]$, then $x$ is a twisted coordinate on $A$. Using the second part of proposition \[locco\], we may assume that $A = R[x]$ in which case, this is a trivial consequence of proposition \[sigis\]. $\quad \Box$ Twisted differential forms ========================== In this section, we study the module of twisted differential forms (of degree one) and make the link with twisted derivations. We use same notations as before. The $A$-module of twisted differential forms on $A/R$ is $$\Omega_{A/R, \sigma}^1 := I_{A/R}/I_{A/R}^{(2)_{\sigma}}.$$ Again, we will often drop the index $A/R$. Since we implicitly endow $P$ with the action of $A$ on the left, we will also always see $\Omega^1_{\sigma}$ as an $A$-module through the action on the left. 1. When $A$ is trivially twisted, then $\Omega^1_{\sigma} = I/I^2$ is the usual module of differential forms of $A$ over $R$. 2. If $A = R[x]$ is endowed with any $R$-algebra endomorphism $\sigma$, then $\Omega^1_{\sigma}$ is free of rank $1$: with the notations of lemma \[transl\], we have $$\Omega^1_{\sigma} \simeq \xi R[x,\xi]/\xi(\xi + y) \simeq R[x,\xi]/(\xi + y) \simeq R[x].$$ <!-- --> 1. Clearly, $\Omega^1_{\sigma}$ has a natural structure of $A$-bimodule as a quotient of two ideals of $P$. It happens that this is identical to its $A$-sesquimodule structure: by definition, if $x \in A$, then $\tilde x \equiv \sigma(x) \mod \sigma(I)$ and it follows that $$\label{modisig} \forall f \in I, \quad f\tilde x \equiv f\sigma(x) \mod I\sigma(I).$$ 2. The $\Omega^1_{\sigma}$ that appears in proposition 1.4.2.1 of [@Andre01] is exactly the same as ours (André calls $k$ what we call $R$). 3. Formula \[modisig\] is exactly the first step of the braiding described by Max Karoubi and Mariano Suárez-Álvarez in [@KaroubiSuarez03]. 4. One can define more generally the twisted de Rham complex $\Omega^\bullet_{\sigma}$ of $A$ as the quotient of the non commutative tensor algebra of $I$ by the graded differential ideal generated by $I\sigma(I)$. We will not consider this complex here. \[splitex\] There exists a split exact sequence $$\xymatrix{0 \ar[r] & \Omega^1_{A/R,\sigma} \ar[r] & P_{A/R,(1)_{\sigma}} \ar[r]^-{\tilde \sigma_{A}} & A \ar[r] & 0}.$$ There exists such an exact sequence by definition of $P_{(1)_{\sigma}}$ and $\Omega^1_{\sigma}$. The $A$-module structure of $P$ provides a section of $\widetilde \sigma$. $\quad \Box$ \[omegfonc1\] \[omegfonc2\] 1. If $R \to R'$ is a homomorphism of commutative rings and $A' := R' \otimes_R A$, then there exists an isomorphism $$A' \otimes_A \Omega_{A/R, \sigma}^1 \simeq \Omega_{A'/R', \sigma}^1.$$ 2. If $B$ is a twisted commutative $A$-algebra, then there exists a canonical $B$-linear map $$\label{compom} B \otimes_{A} \Omega_{A/R, \sigma}^1 \to \Omega_{B/R, \sigma}^1.$$ When $B$ is a quotient (resp. a localization) of $A$ this map is surjective (resp. bijective). Using proposition \[splitex\], this follows from propositions \[extn\] and \[locord\]. This last result does not hold however if we only require $A \to B$ to be an étale map (and not a localization map) as the following example shows. Let $R$ be any field of characteristic different from 2, $A := R [x]$ with $\sigma_{A} := \mathrm{Id}_{A}$ and $B := R [x,x^{-1}]$ with $\sigma_{B}(x) := -x$. Then the morphism $x \mapsto x^2$ is an étale twisted morphism but the morphism is the zero map. More precisely, if $\xi = \tilde x - x$, we have $$\label{compom2} B \otimes_A \Omega_{A,\sigma}^1 = (\xi)/(\xi^2) \quad \mathrm{and} \quad \Omega_{B,\sigma}^1 = (\xi)/(\xi^2 +2x\xi) ,$$ where the ideals are taken inside $R[x,x^{-1},\xi]$, and morphism is induced by $\xi \mapsto \xi^2 + 2x\xi$. Recall from [@LeStumQuiros15a] that a twisted derivation of $A$ is an $R$-linear map into an $A$-module $M$ that satisfies the twisted Leibnitz rule: $$\forall x,y \in A, \quad D(xy) = x D(y) + \sigma(y)D(x).$$ They form an $A$-module $\mathrm {Der}_{R,\sigma}(A,M)$. \[deriv\] The canonical map $A \to \Omega^1_{A/R,\sigma}$ induced by $d$ is a twisted derivation. It provides us with a natural isomorphism $$\label{stan} \xymatrix@R0cm{ \mathrm {Hom}_{A}(\Omega^1_{A/R,\sigma}, M) \ar[r]^-\simeq & \mathrm {Der}_{R,\sigma}(A,M) \\ u \ar@{\|->}[r] & D := u \circ d}$$ whenever $M$ is an $A$-module. In the future, we will also denote by $d : A \to \Omega^1_{\sigma}$ this universal twisted derivation when there is no risk of confusion. Using formula , we see that, inside $P$, we have $$\begin{aligned} yd(x) + \sigma(x)d(y) &=& y(\tilde x - x) + \sigma(x)(\tilde y -y) \\ &\equiv& y(\tilde x - x) + \tilde x(\tilde y -y) = \widetilde {xy} - xy = d(xy) \mod I\sigma(I).\end{aligned}$$ It follows that the induced map $d : A \to \Omega^1_{\sigma}$ is indeed a twisted derivation. This also implies that the map in is well defined. And it is clearly injective because $I$ is generated by the image of $d$. We now show that it is surjective. If $D$ is a twisted derivation of $M$, we can consider its linearization $$\xymatrix@R0cm{ P \ar[r]^{\tilde D} & M \\ x \tilde y \ar@{\|->}[r] & xD(y)}.$$ By definition, the ideal $I\sigma(I)$ is generated by elements of the form $$f = (\tilde x - x)(\tilde y - \sigma(y)) = \tilde x \tilde y - x \tilde y - \sigma(y) \tilde x + x\sigma(y),$$ and we have $$\tilde D(f) = D(xy) - xD(y) - \sigma(y)D(x) + x\sigma(y) D(1) = 0$$ because $D$ is a twisted derivation (and in particular $D(1) = 0$). It follows that $\tilde D$ factors through $P/I\sigma(I)$ and we may consider the induced map $u : \Omega^1_{\sigma} \to M$. It only remains to notice that we have $$\forall x \in A, \quad u(d(x)) = \tilde D(d(x)) = \tilde D (\tilde x - x) = D (x) - x D(1) = D(x). \quad \Box$$ There exists a very elegant proof of this last result through the theory of bimodules. It is based on the fact (see proposition 17 in [@Bourbaki70], Chapter III, section 10) that $I$ is universal for bimodule derivations: there exists a natural isomorphism $$\xymatrix@R0cm{ \mathrm {Hom}_{A\mathrm{-Bim}}(I, M) \ar[r]^-\simeq & \mathrm {Der}_{R}(A,M) \\ u \ar@{\|->}[r] & D := u \circ d},$$ where the right-hand side stands for bimodule derivations (see proposition 1.4.2.1 of [@Andre01]). As an immediate consequence of the proposition, if we write $T_{A/R,\sigma} := \mathrm {Der}_{R,\sigma}(A,A)$, that we will often abbreviate to $T_{\sigma}$, we obtain the following: \[dualid\] The $A$-module $T_{A/R,\sigma}$ is the dual of $\Omega^1_{A/R,\sigma}$. $\quad \Box$ \[corfonc\] Assume that $\Omega^1_{A/R,\sigma}$ is projective of finite rank. Then, if $M$ is an $A$-module, we have the following: 1. If $R \to R'$ is a base extension and $A' = R' \otimes_{R} A$, then there exists a canonical isomorphism $$R' \otimes_{R} \mathrm {Der}_{R,\sigma}(A,M) \simeq \mathrm {Der}_{R',\sigma}(A',A' \otimes_{A} M).$$ 2. If $B$ is a twisted $A$-algebra, there exists a canonical map $$B \otimes_{A} \mathrm {Der}_{R,\sigma}(A,M) \leftarrow \mathrm {Der}_{R,\sigma}(B,B \otimes_{A} M).$$ It is injective (resp. bijective) when $B$ is a quotient (resp. a localization) of $A$. Both assertions follow from proposition \[omegfonc1\]. More precisely, in the first case, we have $$R' \otimes_{R} \mathrm {Der}_{R,\sigma}(A,M) \simeq R' \otimes_{R} \mathrm {Hom}_{A}(\Omega^1_{A/R,\sigma}, M) \simeq \mathrm {Hom}_{A}(\Omega^1_{A/R,\sigma}, A' \otimes_{A} M)$$ because $\Omega^1_{A/R,\sigma}$ is projective of finite rank, and then $$\begin{aligned} \mathrm {Hom}_{A}(\Omega^1_{A/R,\sigma}, A' \otimes_{A} M) &\simeq& \mathrm {Hom}_{A'}(A' \otimes_{A} \Omega^1_{A/R,\sigma}, A' \otimes_{A} M) \\ &\simeq& \mathrm {Hom}_{A'}(\Omega^1_{A'/R',\sigma}, A' \otimes_{A} M) \simeq \mathrm {Der}_{R',\sigma}(A',A' \otimes_{A} M).\end{aligned}$$ The proof of the second assertion follows exactly the same lines with the same arguments. We have $$B \otimes_{A} \mathrm {Der}_{R,\sigma}(A,M) \simeq B \otimes_{A} \mathrm {Hom}_{A}(\Omega^1_{A/R,\sigma}, M) \simeq \mathrm {Hom}_{A}(\Omega^1_{A/R,\sigma}, B \otimes_{A} M) \simeq$$ $$\mathrm {Hom}_{B}(B \otimes_{A} \Omega^1_{A/R,\sigma}, B \otimes_{A} M) \leftarrow \mathrm {Hom}_{A}(\Omega^1_{B/R,\sigma}, B \otimes_{A} M) \simeq \mathrm {Der}_{R,\sigma}(B',B \otimes_{A} M).$$ And the only map which is not always an isomorphism will be injective (resp. bijective) when $B$ is a quotient (resp. a localization) of $A$. $\quad \Box$ The following immediate consequence is worth stating: If $\Omega^1_{A/R,\sigma}$ is projective of finite rank, then $\mathrm T_{A/R,\sigma}$ and $\Omega^1_{A/R,\sigma}$ are dual to each other and we have 1. If $R \to R'$ is a base extension and $A' = R' \otimes_{R} A$, then there exists a canonical isomorphism $$R' \otimes_{R} \mathrm T_{A/R,\sigma} \simeq \mathrm T_{A'/R',\sigma'}.$$ 2. If $B$ is a twisted localization of $A$, then there exists a canonical isomorphism $$B \otimes_{A} \mathrm T_{A/R,\sigma} \simeq \mathrm T_{B/R,\sigma}. \quad \Box$$ \[twico\] A twisted connection on an $A$-module $M$ is an $R$-linear map $$\nabla : M \mapsto M \otimes_{A} \Omega^1_{A/R,\sigma}$$ such that $$\forall s \in M, \forall x \in A, \quad \nabla (xs) = s \otimes \mathrm d(x) + \sigma(x)\nabla(s).$$ An $A$-linear map between two $A$-modules with twisted connections is said to be horizontal if it is compatible with the connections. Clearly, $A$-modules endowed with a connection and horizontal maps form a category $\nabla_{\sigma_{A}}\mathrm{-Mod}$. This definition is compatible with definition 2.2.1 by André in [@Andre01]. In particular, all the tannakian formalism applies but this is not what we are interested in. Recall from [@LeStumQuiros15a] that if $D$ is a twisted derivation of $A$, then a twisted $D$-derivation of an $A$-module $M$ is an $R$-linear endomorphism $D_{M}$ of $M$ that satisfies the [twisted Leibnitz rule]{}: $$\forall x \in A, \forall s \in M, \quad D_{M}(xs) = D(x)s + \sigma_{A}(x)D_{M}(s).$$ One may then consider the notion of action by twisted derivations of $T_{A/R,\sigma}$ on $M$: it is an $R$-linear action such that whenever $D \in T_{A/R,\sigma}$, the map $D_{M} : s \mapsto D.s$ is a $D$-derivation. \[dact\] There exists a functor from the category of $A$-modules endowed with a twisted connection to the category of $A$-modules endowed with a linear action of $T_{A/R,\sigma}$ by twisted derivations. It is an equivalence (an isomorphism) when $\Omega^1_{A/R,\sigma}$ is free of finite rank. If $M$ is endowed with a twisted connection $\nabla : M \mapsto M \otimes_{A} \Omega^1_{\sigma}$ and $D$ is a twisted derivation of $A$, we may write uniquely $D = u \circ d$ with $u : \Omega^1_{\sigma} \to A$ and consider the composite map $D_{M} := (\mathrm{Id}_{M} \otimes u) \circ \nabla : M \to M$. Then, we will have $$\begin{aligned} D_{M}(xs) &=& (\mathrm{Id}_{M} \otimes u)(\nabla(xs)) = (\mathrm{Id}_{M} \otimes u)(s \otimes \mathrm d(x) + \sigma(x)\nabla(s)) \\ &=& (\mathrm u \circ d)(x)s +\sigma(x) (\mathrm{Id}_{M} \otimes u)(\nabla(s)) = D(x)s +\sigma(x) D_{M}(s).\end{aligned}$$ Conversely, assume that $M$ is endowed with an action of $T_{\sigma}$ by twisted derivations. Let $D_{1}, \ldots, D_{n}$ be a basis of $\mathrm{T}_{\sigma}$ and $\omega_{1}, \ldots, \omega_{n}$ be the dual basis in $\Omega^1_{\sigma}$. Then, we can define $\nabla(s) = \sum D_{i,M}(s) \otimes \omega_{i}$ and check that $$\begin{aligned} \nabla (xs) &=& \sum D_{i,M}(xs) \otimes \omega_{i} = \left(\sum D_{i}(x)s +\sigma(x) D_{i,M}(s)\right) \otimes \omega_{i} \\ &=& s \otimes \sum D_{i}(x)\omega_{i} +\sigma(x) \sum D_{i,M}(s) \otimes \omega_{i} = s \otimes \mathrm d(x) + \sigma(x)\nabla(s).\end{aligned}$$ Clearly, this is an inverse to the previous functor. $\quad \Box$ \[1coor\] If $x \in A$ is a twisted coordinate on $A$, then $\Omega^1_{A/R,\sigma}$ is free of rank $1$ generated by $\mathrm dx$. Moreover, there exists a unique twisted derivation $\partial_{x,\sigma}$ of $A$ such that $\partial_{x,\sigma}(x) =1$ and we have $$\forall D \in T_{A/R, \sigma}, \quad D = D(x)\partial_{x,\sigma}.$$ The first assertion is a particular case of corollary \[free\]. The second one then follows from corollary \[dualid\]. $\quad \Box$ In particular, we see that a twisted coordinate is also a coordinate in the sense of [@LeStumQuiros15a]. In order to lighten the notations, we will usually drop the index $x$ but we must not forget that $\partial_{\sigma}$ depends on the choice of $x$. Also, we would rather write $\partial_{A,\sigma}$ than $\partial_{\sigma_{A}}$ when we want to make clear the dependence on $A$. If $x$ is a twisted coordinate on $A$, one may consider the twisted Weyl algebra $\mathrm D_{A/R,\sigma,\partial}$ (see [@LeStumQuiros15a] for example), that we will usually denote by $\mathrm D_{A/R,\sigma}$ and sometimes simply by $\mathrm D_{\sigma}$. This is the non commutative polynomial ring in one variable $\partial_{\sigma}$ over $A$ with the commutation rule $$\forall z \in A, \quad \partial_{\sigma} z = \partial_{A,\sigma}(z) + \sigma_{A}(z)\partial_{\sigma}.$$ Moreover, there exists an equivalence (an isomorphism) of categories $\mathrm D_{A/R,\sigma}\mathrm{-Mod} \simeq \partial_{A,\sigma}\mathrm{-Mod}$ where the later denotes the category of $A$-modules $M$ endowed with a $\partial_{A,\sigma}$-derivation. \[eqnabd\] Assume that $x$ is a twisted coordinate on $A$. Then, there exists an equivalence (an isomorphism) of categories $$\nabla_{\sigma_{A}}\mathrm{-Mod} \simeq D_{A/R,\sigma}\mathrm{-Mod}.$$ Follows from proposition \[dact\]. $\quad \Box$ Twisted binomial coefficient theorem for principal parts ======================================================== We prove here the main theorem that will allow us to recover twisted differential operators from principal parts. We use same notations as before. In section \[tpp\] we introduced the canonical map (it is a morphism of $R$-algebras) $$\xymatrix@R0cm{ P_{A/R} \ar[r]^-{\delta_{A/R}} &P_{A/R} \otimes_{A} P_{A/R} \\ x \tilde y \ar@{\|->}[r] & x \otimes \tilde y }.$$ We want to investigate the interaction between $\sigma_{A}$ and $\delta_{A/R}$. Recall that we also considered in section \[tpp\] the maps $$\xymatrix@R0cm{ P_{A/R} \ar[r]^-{\sigma_{P}} &P_{A/R} \\ x \tilde y \ar@{\|->}[r] & \sigma_{A}(x) \tilde y },$$ which is an $R$-linear ring homomorphism, and $$\xymatrix@R0cm{ A \ar[r]^-{d_{A/R}} & P_{A/R} \\ x \ar@{\|->}[r] & \tilde x - x },$$ which is only $R$-linear. As usual, we will drop the subscripts in order to lighten the notations, hoping that the meaning will always be clear from the context. \[decompg\] For all $i = 0, \ldots, n$, we have in $P \otimes_{A} P$: $$\forall x \in A, \quad \delta(\sigma^n(d(x))) = 1 \otimes \sigma^i(d(x)) + \sigma^{n-i}(d(\sigma^i(x)))\otimes 1.$$ We do the computations in $A \otimes_{R} A \otimes_{R} A$. The right hand side is $$\begin{aligned} &1 \otimes (1 \otimes x - \sigma^i(x) \otimes 1) + \sigma^{n-i}(1 \otimes \sigma^i(x) - \sigma^i(x) \otimes 1) \otimes 1 \\ &=1 \otimes 1 \otimes x - 1 \otimes \sigma^i(x) \otimes 1 + 1 \otimes \sigma^i(x) \otimes 1 - \sigma^n(x) \otimes 1 \otimes 1 \\ &= 1 \otimes 1 \otimes x - \sigma^n(x) \otimes 1 \otimes 1.\end{aligned}$$ If we develop the left hand side, we obtain exactly the same thing: $$\delta(\sigma^n(d(x))) = \delta(1 \otimes x - \sigma^n(x) \otimes 1) = 1 \otimes 1 \otimes x - \sigma^n(x) \otimes 1 \otimes 1. \quad \Box$$ We endow $P \otimes_{A} P$ with the endomorphism $\sigma_{P} \otimes_{A} \mathrm{Id}_{P}$ (which is the same thing as $\sigma_{A} \otimes_{R} \mathrm{Id}_{A} \otimes_{R} \mathrm{Id}_{A}$ on $A \otimes_{R} A \otimes_{R} A$). \[sigdel\] The map $\delta : P \to P \otimes_{A} P$ is a morphism of twisted $R$-algebras. This is the case $n = 1$ and $i = 0$ of lemma \[decompg\]. More precisely, if $x \in A$ and $\xi = \tilde x - x$, we have $$\delta(\sigma(\xi)) = 1 \otimes \xi + \sigma(\xi) \otimes 1 = \sigma(\delta(\xi)). \quad \Box$$ We have in $P \otimes_{A} P$: $$\forall n \in \mathbb N, \quad \delta(I^{(n)}) \subset \sum_{i=0}^{n} I^{(i)} \otimes I^{(n-i)}.$$ First of all, since $I$ is generated by the image of $d$, it follows from lemma \[decompg\] that for all $i = 0, \ldots, n$, we have $$\delta(\sigma^n(I)) \subset P \otimes \sigma^i(I) + \sigma^{n-i}(I) \otimes P.$$ Using induction, we obtain $$\begin{aligned} \delta(I^{(n+1)}) &=& \delta(I^{(n)})\delta(\sigma^{n}(I))) \subset \sum_{i=0}^{n} (I^{(i)} \otimes I^{(n-i)})(P \otimes \sigma^{n-i}(I) + \sigma^i(I) \otimes P) \\ &\subset& \sum_{i=0}^{n} (I^{(i)} \otimes I^{(n-i+1)}) + \sum_{i=0}^{n} (I^{(i+1)} \otimes I^{(n-i)}) \subset \sum_{i=0}^{n+1} I^{(i)} \otimes I^{(n+1-i)}. \quad \Box\end{aligned}$$ \[delstab\] For all $m, n \in \mathbb N$, we have in $P \otimes_{A} P$: $$\delta(I^{(n+m+1)}) \subset P \otimes_{A} I^{(m+1)} + I^{(n+1)} \otimes_{A} P.$$ In other words, $\delta$ induces a map $$\xymatrix@R0cm{ P_{(n+m)_{\sigma}} \ar[r]^-{\delta_{n,m}} &P_{(n)_{\sigma}} \otimes_{A} P_{(m)_{\sigma}}. }$$ Proof If $0 \leq i \leq m+n+1$, we have either $i > n$ and then $I^{(i)} \subset I^{(n+1)}$ or else $i \leq n$ so that $m + n + 1 -i > m$ and then $I^{(m+n+1-i)} \subset I^{(m+1)}$. $\quad \Box$ Going to the limit, we obtain a canonical homomorphism of $R$-algebras $$\xymatrix@R0cm{ \widehat P_{\sigma} \ar[r]^-{\widehat \delta} & \widehat P_{\sigma} \widehat \otimes_{A} \widehat P_{\sigma} },$$ where the right hand side is, by definition, the inverse limit of all the $P_{(n)_{\sigma}} \otimes_{A} P_{(m)_{\sigma}}$. In other words, we obtain a comultiplication on $\widehat P_{\sigma}$ that will allow us to turn its “dual” into a ring (more on this later). We finish this section with the quantum binomial theorem for principal parts: \[biconf\] let $A$ be a twisted commutative $R$-algebra and $x \in A$ such that $\sigma(x) = qx + h$ with $q, h \in R$. If we set $\xi = \tilde x - x$, then, we have $$\delta(\xi^{(n)}) := \sum_{i=0}^{n} {n \choose i}_{q} \xi^{(n-i)} \otimes \xi^{(i)}.$$ The formula is proved to be correct by induction on $n$. First of all, since $\delta$ is a ring homomorphism, we have $$\delta(\xi^{(n+1)}) = \delta(\xi^{(n)}\sigma^{n}(\xi)) = \delta(\xi^{(n)})\delta(\sigma^{n}(\xi))).$$ Using induction and lemma \[decomp\] below, we get $$\begin{aligned} \delta(\xi^{(n+1)}) &=& \sum_{i=0}^{n} {n \choose i}_{q} (\xi^{(n-i)} \otimes \xi^{(i)})(1 \otimes \sigma^i(\xi) + q^i\sigma^{n-i}(\xi)\otimes 1) \\ &=& \sum_{i=0}^{n} {n \choose i}_{q} \xi^{(n-i)} \otimes \xi^{(i)}\sigma^{i}(\xi) + \sum_{i=0}^{n} {n \choose i}_{q} q^i\xi^{(n-i)}\sigma^{n-i}(\xi) \otimes \xi^{(i)} \\ &=& \sum_{i=0}^{n} {n \choose i}_{q} \xi^{(n-i)} \otimes \xi^{(i+1)} + \sum_{i=0}^{n} {n \choose i}_{q} q^i\xi^{(n-i+1)} \otimes \xi^{(i)} \\ &=& \sum_{i=1}^{n+1} {n \choose i-1}_{q} \xi^{(n-i+1)} \otimes \xi^{(i)} + \sum_{i=0}^{n} {n \choose i}_{q} q^i\xi^{(n-i+1)} \otimes \xi^{(i)} \\ &=& \sum_{i=0}^{n+1} \left({n \choose i-1}_{q} + q^i {n \choose i}_{q}\right) \xi^{(n+1-i)} \otimes \xi^{(i)} \\ &=& \sum_{i=0}^{n+1} {n +1 \choose i}_{q} \xi^{(n+1-i)} \otimes \xi^{(i)}. \quad \Box\end{aligned}$$ \[decomp\] Under the hypothesis of the proposition, we have for all $i = 0, \ldots n$, $$\delta(\sigma^n(\xi)) = 1 \otimes \sigma^i(\xi) + q^i\sigma^{n-i}(\xi)\otimes 1.$$ We have $$d(\sigma(x)) = \widetilde{\sigma(x)} - \sigma(x) = \widetilde {qx+h} - (qx + h) = q (\tilde x - x) = q \xi.$$ The analogous result holds for the endomorphism $\sigma^i$. It follows that $d(\sigma^i(x)) = q^i \xi$ and we finish with lemma \[decompg\]. $\quad \Box$ Twisted differential operators of infinite level {#tdo} ================================================ We are now able to define the ring of twisted differential operators (of infinite level). We keep the previous notations. \[trudef\] If $M$ and $N$ are two $A$-modules, then a twisted differential operator $\varphi : M \to N$ of order at most $n$ (and infinite level) is an $R$-linear map whose $A$-linearization $$\xymatrix@R0cm{ P \otimes_{A} M \ar@/^.5cm/[rr] ^-{\tilde \varphi} \ar[r]^{\simeq} &A \otimes_{R} M \ar[r] & N \\ & x \otimes s \ar@{\|->}[r] & x\varphi (s)}$$ factors through $P_{(n)_{\sigma}} \otimes_{A} M$. Note that the condition means that the restriction of $\tilde \varphi$ to $I^{(n+1)} \otimes_{A} M$ is zero. We might still write $\tilde \varphi$ for the map induced on $P_{(n)_{\sigma}} \otimes_{A} M$ when there is no risk of confusion. We denote by $\mathrm{Diff}_{n,\sigma}(M, N)$ the set of all twisted differential operators of order at most $n$. Thus, we have a canonical isomorphism $$\mathrm{Diff}_{n,\sigma}(M, N) \simeq \mathrm{Hom}_{A}(P_{(n)_{\sigma}} \otimes_{A} M, N).$$ where $P_{(n)_{\sigma}}$ is seen as an $A$-module for the action on the right with respect to $\otimes_{A}$ and for the action on the left with respect to $\mathrm{Hom}_{A}$. In particular, $\mathrm{Diff}_{n,\sigma}(M, N)$ has a natural structure of $P_{(n)_{\sigma}}$-module given by $x\tilde y \cdot \varphi := x \circ \varphi \circ y$, where multiplication by $y$ takes place in $M$ while we multiply by $x$ in $N$. We will also denote by $\mathrm{Diff}_{\sigma}(M, N)$ the set of all twisted differential operators of any order so that $$\mathrm{Diff}_{\sigma}(M, N) \simeq \varinjlim \mathrm{Hom}_{A}(P_{(n)_{\sigma}} \otimes_{A} M, N).$$ In particular, we see that $\mathrm{Diff}_{\sigma}(M, N)$ has a natural structure of $\widehat P_{\sigma}$-module. In the case $N=M$, we will write $\mathrm{Diff}_{n,\sigma}(M)$ and $\mathrm{Diff}_{\sigma}(M)$. One also sets $\mathrm D^{(\infty)}_{A/R,\sigma} := \mathrm{Diff}_{\sigma}(A)$ and we will often drop the index $A/R$ and simply write $\mathrm D^{(\infty)}_{\sigma}$. \[deka\] Let $x$ be a twisted coordinate on $A$ and $\xi = \tilde x - x$. Then, the standard basis of $\mathrm{Diff}_{n,\sigma}(A)$ is the basis $\partial^{[k]}_{\sigma}$ dual to the images of the $\xi^{(k)}$ in $P_{(n)_{\sigma}}$. We call $\partial^{[k]}_{\sigma}$ the standard twisted divided differential operator of order $k$ associated to $x$. Thus, when $x$ is a twisted coordinate, any $\varphi \in D_{\sigma}^{(\infty)}$ can be uniquely written as a finite sum $\sum z_{k} \partial_{\sigma}^{[k]}$ with $z_{k} \in A$ (and conversely, any such sum is in $D_{\sigma}^{(\infty)}$). The canonical basis is characterized by the property $$\forall k, n \in \mathbb N, \quad \widetilde{\partial^{[k]}_{\sigma}}(\xi^{(n)}) = \left\{\begin{array} {cll} 1 & \mathrm{if} \ k = n \\ 0 & \mathrm{otherwise}\end{array} \right. .$$ We will give explicit examples later on. The next proposition shows that the $A$-module of twisted differential operators could have been also defined by induction on the order $n$ (this is sometimes more convenient and does not require to work out the theory of principal parts). Instead, it uses the notion of twisted bracket $$\forall \varphi \in \mathrm{Hom}_{R}(M, N), \forall x \in A, \quad [\varphi, x]_{\sigma} = \varphi \circ x - \sigma(x) \circ \varphi$$ already used in [@LeStumQuiros15a]. We will need this intermediate result: \[phipsi\] Let $M,N$ be two $A$-modules, $\varphi \in \mathrm{Hom}_{R}(M,N)$ and $x \in A$. If we set $\varphi_{x} := [\varphi, x]_{\sigma}$, then, $$\tilde \varphi_{x} = \tilde \varphi \circ (\sigma(\xi) \otimes_{A} \mathrm{Id}_{M}) : P \otimes_{A} M \to N.$$ We do the computations in $A \otimes_{R} M$. Let $y \in A$ and $s \in M$, we have $$\begin{aligned} \tilde \varphi (\sigma(\xi)(y \otimes s)) &=& \tilde \varphi ((1 \otimes x - \sigma(x) \otimes 1)(y \otimes s))= \tilde \varphi (y \otimes xs) - \tilde \varphi (\sigma(x)y \otimes s) \\ &=& y \varphi(xs) - \sigma(x)y\varphi(s) = y \varphi_{x}(s) = \tilde \varphi_{x}(y \otimes s). \quad \Box\end{aligned}$$ \[carac\] Let $M,N$ be two $A$-modules and $\varphi \in \mathrm{Hom}_{R}(M,N)$. Then, we have $$\forall n \in \mathbb N, \quad \varphi \in \mathrm{Diff}_{n,\sigma}(M, N) \Leftrightarrow \forall x \in A, [\varphi,x]_{\sigma^n} \in \mathrm{Diff}_{n-1,\sigma}(M, N).$$ Note that we can start the induction process with $\mathrm{Diff}_{0,\sigma}(M, N)= \mathrm{Hom}_{A}(M, N)$ or with $\mathrm{Diff}_{-1,\sigma}(M, N) = 0$ if we prefer. For $x \in A$, we set $\varphi_{x} := [\varphi, x]_{\sigma^n}$. Then, we consider the linearizations $$\tilde{\varphi}, \tilde \varphi_{x} : P \otimes_{A} M \to N$$ of $\varphi$ and $\varphi_{x}$, respectively, and apply lemma \[phipsi\] to $\sigma^n$ so that $$\tilde \varphi_{x} = \tilde \varphi \circ (\sigma^n(\xi) \otimes_{A} \mathrm{Id}_{M}).$$ Thus we see that $\tilde \varphi = 0$ on $I^{(n+1)} \otimes_{A} M = I^{(n)}\sigma^n(I) \otimes_{A} M$ if and only if $\tilde \varphi_{x} = 0$ on $I^{(n)} \otimes_{A} M$ for all $x \in A$. In other words, $\tilde \varphi$ factors through $P_{(n)_{\sigma}} \otimes_{A} M$ if and only if all $\tilde \varphi_{x}$ factor through $P_{(n-1)_{\sigma}} \otimes_{A} M$. $\quad \Box$ \[sigmadif\] A twisted differential operator of order at most $n$ (and infinite level) from $M$ to $N$ is an $R$-linear map $\varphi : M \to N$ such that $$\forall x_{0}, \ldots, x_{n}\in A, \quad\ [[\cdots[ [\varphi,x_{n}]_{\sigma^n}, x_{n-1}]_{\sigma^{n-1}} \cdots ]_{\sigma}, x_{0}] = 0. \quad \Box$$ 1. Be careful that, with the notations of Valery Lunts and Alexander L. Rosenberg in [@LuntsRosenberg97], our $\mathrm{Diff}_{\sigma}(M, N)$ is different from their $\mathrm{Diff}(M, N^\sigma)$ which is defined by the condition $$\forall x_{0}, \ldots, x_{n}\in A, \quad [ [\cdots[ [\varphi,x_{n}]_{\sigma}, x_{n-1}]_{\sigma} \cdots ]_{\sigma}, x_{0}] = 0.$$ They only coincide when $n =0, 1$. 2. Our $\mathrm{Diff}_{\sigma}(M, N)$ should however coincide with some flavor of the $\mathrm{Diff}_{\beta}(M, N)$ of Lunts and Rosenberg. More precisely, in order to define this module, they need a $G$-grading on $A$ and a bilinear map $\beta : G \times G \to R^\times$ (they use $k$ and $R$ for our $R$ and $A$). In the simplest non trivial case $A = R[x]$ and $\beta(m, n) = q^{-mn}$, we believe that their $D_{\beta}(A)$ coincides with our $D_{A/R,\sigma}^{(\infty)}$ but their $D_{q}(A)$ is bigger (see [@IyerMcCune02] for example). 3. Charlotte Hardouin introduces in [@Hardouin10], definition 2.4, what she calls an iterative $q$-difference ring or $\mathrm{ID}_{q}$-ring for short. She chooses some non zero $q \in K$ where $K$ is a fixed algebraically closed field and endows the field $A := K(x)$ of rational functions on $K$ with the automorphism $\sigma(x) = qx$. Then, with our notations, an $\mathrm{ID}_{q}$-ring is a finitely generated $A$-algebra $B$ with a structure of $D_{A/K,\sigma}^{(\infty)}$-module, denoted by $(\varphi, y) \mapsto \varphi(y)$, such that the map $y \mapsto \sigma (y)$ is an automorphism of the ring $B$ and $$\label{hardouin} \forall k \in \mathbb N, \forall y, z \in B, \quad \partial^{[k]}_{\sigma}(yz) = \sum_{i+j=k} (\sigma^i\partial^{[j]}_{\sigma})(y)\partial^{[i]}_{\sigma}(z).$$ Note that $B$ becomes an inversive twisted $A$-algebra and that condition is automatic if the $q$-characteristic of $B$ is zero. \[compo\] Composition of twisted differential operators gives a twisted differential operator. Moreover, its order is at most the sum of the order of the components. We let $\varphi : M \to N$ be a twisted differential operator of order $n$ and $\psi : L \to M$ a twisted differential operator of order $m$ and consider the factorization $$\xymatrix@R0cm{ P \otimes_{A} L \ar[r]^-{\delta \otimes \mathrm{Id}} & P \otimes_{A} P \otimes_A L \ar[r]^-{\mathrm{Id} \otimes_{A} \tilde \psi} & P \otimes_{A} M \ar[r]^-{\tilde \varphi} & N}$$ of lemma \[compolin\]. The map $\tilde \varphi$ factors through $P_{(n)_{\sigma}} \otimes_{A} M$ and $\mathrm{Id} \otimes \tilde \psi$ factors through $P \otimes_{A} P_{(m)_{\sigma}} \otimes_{A} L$. Therefore, their composite factors through $P_{(n)_{\sigma}} \otimes_{A} P_{(m)_{\sigma}} \otimes_{A} L$ and it follows from corollary \[delstab\] that the whole thing will factor through $P_{(n+ m)_{\sigma}} \otimes L$. Thus, $\varphi \circ \psi$ is a twisted differential operator of order at most $m + n$. $\quad \Box$ 1. Let $R \to R'$ be any morphism of commutative rings and $A' := R' \otimes_R A$, endowed with $\mathrm{Id}_{R'} \otimes_{R} \sigma_{A}$. Then, we have $$\mathrm{Diff}_{\sigma_{A'}}(A' \otimes_A M, A' \otimes_A N) \simeq A' \otimes_A \mathrm{Diff}_{\sigma_{A}}(M, N).$$ 2. If $A \to B$ is a localization of twisted $R$-algebras, we have $$\mathrm{Diff}_{\sigma_{B}}(B \otimes_A M, B \otimes_A N) \simeq B \otimes_A \mathrm{Diff}_{\sigma_{A}}(M, N).$$ Follows from propositions \[extn\] and \[locord\] and the fact that direct limits commute with tensor product. $\quad \Box$ 1. As a particular case, we will have $$\mathrm D^{(\infty)}_{A'/R',\sigma'} \simeq A' \otimes_A \mathrm D^{(\infty)}_{A/R,\sigma}\quad \mathrm{and} \quad \mathrm D^{(\infty)}_{B/R,\sigma} \simeq B \otimes_A \mathrm D^{(\infty)}_{A/R,\sigma}.$$ 2. When $A$ is a quantum $R$-algebra, we will always have (see the remark following proposition \[locco\]) a natural isomorphism of $A$-modules $$\mathrm D^{(\infty)}_{A/R,\sigma} \simeq A \otimes_{R[x]} \mathrm D^{(\infty)}_{R[x]/R,\sigma}.$$ Note however that the ring structure (or equivalently the action on $A$) plays a fundamental role. We have $\mathrm{Diff}_{0,\sigma}(A) = A$ and $\mathrm{Diff}_{1,\sigma}(A) = A \oplus \mathrm T_{A/R,\sigma}$. The first assertion follows from the fact that $P_{(0)_{\sigma}} = A$ and the second one from proposition \[splitex\]. $\quad \Box$ Recall that we introduced in [@LeStumQuiros15a] the ring $\overline{\mathrm D}_{A/R,\sigma}$ of small twisted differential operators of $A/R$ as the smallest subring of $\mathrm{End}_{R}(A)$ containing both $A$ and $\mathrm T_{A/R,\sigma}$. Again, we will simply write $\overline{\mathrm D}_{\sigma}$ when we believe that there is no risk of confusion. Then, we have the following: \[contin\] The ring of small twisted differential operators is contained inside the ring of twisted differential operators of infinite level: we have $$\overline{\mathrm D}_{A/R,\sigma} \subset \mathrm D^{(\infty)}_{A/R,\sigma}. \quad \Box$$ When there exists a twisted coordinate, we can make the twisted Weyl algebra enter the picture and we have: \[epimon\] If $x$ is a twisted coordinate on $A$, there exists an epi-mono factorization $$\xymatrix@R0cm{ \mathrm D_{A/R,\sigma} \ar@{->>}[r] & \overline{\mathrm D}_{A/R,\sigma} \ar@{^{(}->}[r] & \mathrm D^{(\infty)}_{A/R,\sigma}. }$$ There exists a natural map $\mathrm D_{\sigma} \to \overline{\mathrm D}_{\sigma}$ that sends the parameter $\partial_{\sigma}$ of $\mathrm D_{\sigma,\partial}$ to the corresponding endomorphism $\partial_{A,\sigma}$ of $A$. And it is surjective since $\partial_{A,\sigma}$ is a generator of $\mathrm{T}_{\sigma}$. $\quad \Box$ At some point, we will need to be able to compare twisted differential operators with respect to $\sigma$ and twisted differential operators with respect to the powers (or roots) of $\sigma$. \[pow\] For all $m > 0$, if $M, N$ are two $A$-modules, we have $$\mathrm{Diff}_{\sigma^m}(M, N) \subset \mathrm{Diff}_{\sigma}(M, N).$$ Since $$I^{(mn)_{\sigma}} = I \sigma(I) \cdots \sigma^{mn}(I) \subset I\sigma^m(I) \cdots \sigma^{mn}(I) = I^{(n)_{\sigma^m}},$$ there exists an natural surjective map $$P_{(nm)_{\sigma}} \to P_{(n)_{\sigma^m}},$$ from which we derive an inclusion $\mathrm{Diff}_{n, \sigma^m}(M, N) \subset \mathrm{Diff}_{mn,\sigma}(M, N)$. $\quad \Box$ Recall from [@LeStumQuiros15] that a system of roots of $\sigma$ is a family $\underline \sigma := \{\sigma_{n}\}_{n \in S}$, with $S \subset \mathbb N$, of $R$-linear ring endomorphisms of $A$ such that $\sigma_{n}^m = \sigma_{n'}^{m'}$ whenever $m/n = m'/n'$ and $\sigma_{n}^n = \sigma$. We will always assume that $S$ is filtering for division. We will call the pair $(A, \underline \sigma)$ an $S$-twisted $R$-algebra. Let $\underline \sigma := \{\sigma_{i}\}_{i \in S}$ be a system of roots of $\sigma$. Then the ring of twisted differential operators (of infinite level) $\mathrm D^{(\infty)}_{A/R,\underline \sigma}$ is the $R$-subalgebra of $\mathrm{End}_{R}(A)$ generated by all $\sigma_{n}$-differential operators (of infinite level) for all $n \in S$. Proposition \[pow\] then has the following consequence: If $\underline \sigma$ is a system of roots of $\sigma$, we have $$\mathrm D^{(\infty)}_{A/R,\underline \sigma} =\cup \mathrm D^{(\infty)}_{A/R,\sigma_{n}}. \quad \Box$$ Note that in section 3 of [@LeStumQuiros15a] we defined exactly in the same way the ring of small twisted differential operators $\overline {\mathrm D}_{A/R,\underline \sigma}$ for any family $\underline \sigma$ (not necessarily a root system), but we showed that the analogous statement is not true in general. Twisted taylor series ===================== We will develop here the formalism of twisted Taylor maps which describes the formal solutions of twisted differential modules. Notations are as before. \[disrep\] If $M$ is a $\mathrm D_{A/R,\sigma}^{(\infty)}$-module, then the canonical map $\mathrm D_{A/R,\sigma}^{(\infty)} \to \mathrm{End}_{R}(M)$ induces, for all $n \in \mathbb N$, a $P_{A/R,(n)_{\sigma}}$-linear map $$\mathrm{Diff}_{n,\sigma}(A) \to \mathrm{Diff}_{n,\sigma}(M).$$ Hence, there exists a canonical $\widehat P_{\sigma}$-linear map $$D^{(\infty)}_{A/R} \to \mathrm{Diff}_{\sigma}(M).$$ Since the canonical map $\lambda : D_{\sigma}^{(\infty)} \to \mathrm{End}_{R}(M)$ is a morphism of $A$-algebras, it will commute with the action of $P$. More precisely, for all $x,y \in A$ and $\varphi \in D_{\sigma}^{(\infty)}$, we have (see section \[tdo\] for the definition of the action of $P$ on $D_{\sigma}^{(\infty)}$) $$\lambda(x\tilde y \cdot \varphi) = \lambda(x \circ \varphi \circ y) = x \circ \lambda(\varphi) \circ y = x\tilde y \cdot \lambda(\varphi).$$ In particular, if $\tilde \varphi$ is zero on $I^{(n+1)}$, then $\widetilde {\lambda(\varphi)}$ will be zero on $I^{(n+1)} \otimes_{A} M$. It means that the image of $\mathrm{Diff}_{n,\sigma}(A)$ falls inside $\mathrm{Diff}_{n,\sigma} (M)$. $\quad \Box$ We will usually denote by $\varphi_{M} \in \mathrm{Diff}_{\sigma}(M)$ the image of $\varphi \in D^{(\infty)}_{\sigma}$. In other words, for $\varphi \in D_{\sigma}^{(\infty)}$ and $s \in M$, we will have $\varphi_{M}(s) = \varphi s$. A twisted Taylor structure (of infinite level) on an $A$-module $M$ is a compatible family of $A$-linear maps $\theta_{n} : M \to M \otimes_{A} P_{(n)_{\sigma}}$ (called twisted Taylor maps) with $\theta_{0} = \mathrm{Id}$, making commutative all the diagrams $$\xymatrix{M \ar[rr]^{\theta_{m}} \ar[d]^{\theta_{n+m}} && M \otimes_{A} P_{(m)_{\sigma}} \ar[d]^{\theta_{n} \otimes \mathrm{Id}} \\ M \otimes_{A} P_{(n+m)_{\sigma}} \ar[rr]^{\mathrm{Id} \otimes \delta_{n,m}} && M \otimes_{A} P_{(n)_{\sigma}} \otimes_{A} P_{(m)_{\sigma}}}.$$ For example, the canonical twisted Taylor structure on $A$ is defined by the family of composite maps $$\xymatrix{ A \ar[r]^{\theta} \ar[dr]_{\theta_{n}} & P \ar@{->>}[d] \\ & P_{(n)_{\sigma}}},$$ where the upper map is the Taylor map $x\mapsto \tilde x$ given by the action on the right (see section \[tpp\]). There exists an obvious notion of morphism of $A$-modules endowed with a twisted Taylor structure and they form a category. \[Taylorstruc\] Let $M$ be an $A$-module endowed with a twisted Taylor structure $(\theta_{n})_{n \in \mathbb N}$. Then, there exists a unique structure of $\mathrm D^{(\infty)}_{A/R,\sigma}$-module on $M$ such that $$\label{Taylexp} \forall n \in \mathbb N, \quad \theta_{n}(s) = \sum s_{k} \otimes f_{k} \Rightarrow \forall \varphi \in \mathrm{Diff}_{n,\sigma}(A), \varphi_{M}(s) = \sum \tilde \varphi(f_{k}) s_{k}.$$ This is functorial in $M$ and we obtain an equivalence (an isomorphism) of categories if all $P_{(n)_{\sigma}}$ are finite projective (for the left $A$-module structure). Note that the last condition is satisfied if there exists a twisted coordinate on $A$. First of all, there exists for all $n \in \mathbb N$, a canonical morphism of $A$-modules $$\label{dualm} M \otimes_{A} P_{(n)_{\sigma}} \to \mathrm{Hom}_{A}(\mathrm{Hom}_{A}(P_{(n)_{\sigma}}, A), M).$$ which is automatically $P_{(n)_{\sigma}}$-linear. Now, $A$-linear maps $$\label{nd} \theta_{n} : M \to M \otimes_{A} P_{(n)_{\sigma}}$$ correspond bijectively to $P_{(n)_{\sigma}}$-linear maps $$\epsilon_{n} : P_{(n)_{\sigma}} \otimes_{A} M \to M \otimes_{A} P_{(n)_{\sigma}},$$ and we can compose with the map in order to get $$P_{(n)_{\sigma}} \otimes_{A} M \to \mathrm{Hom}_{A}(\mathrm{Hom}_{A}(P_{(n)_{\sigma}}, A), M),$$ or equivalently, $$\mathrm{Hom}_{A}(P_{(n)_{\sigma}}, A) \to \mathrm{Hom}_{A}(P_{(n)_{\sigma}} \otimes_{A} M, M).$$ In other words, we obtain $P_{(n)_{\sigma}}$-linear maps $$\label{dndn} \xymatrix@R0cm{ \mathrm{Diff}_{n,\sigma}(A) \ar[r] & \mathrm{Diff}_{n,\sigma}(M) \\ \varphi \ar@{\|->}[r] & \varphi_{M}}.$$ Formula follows directly from the construction. Compatibility for various $n$ in follows from compatibility for various $n$ in . We need to show that the corresponding map $\mathrm D_{\sigma}^{(\infty)} \to \mathrm{End}(M)$ is a morphism of rings. To do that, one can use the description of composition of twisted differential operators given in proposition \[compo\]: we need to verify that the maps are compatible with $\delta$ which is exactly the condition in the definition of Taylor structure. This construction is clearly functorial. Moreover, if $P_{(n)_{\sigma}}$ is finite projective (for the left $A$-module structure), then the map is actually an isomorphism. And it follows from lemma \[disrep\] that a $\mathrm D^{(\infty)}_{\sigma}$-module structure on $M$ will provide us with a compatible family of maps as in . $\quad \Box$ \[taylordf\] Let $M$ be an $A$-module endowed with a twisted Taylor structure. Then, the twisted Taylor map of $M$ is the map $$\widehat \theta = \varprojlim \theta_{n} : M \mapsto M \widehat \otimes_{A} \widehat P_{\sigma} := \varprojlim M \otimes_{A} P_{(n)_{\sigma}}.$$ The twisted Taylor series of $s \in M$ is $\widehat \theta(s) \in M \widehat \otimes_{A} \widehat P_{\sigma}$. There exists a commutative diagram $$\xymatrix{M \ar[rr]^{\widehat \theta} \ar[d]^{\widehat\theta} && M \widehat \otimes_{A} \widehat P_{\sigma} \ar[d]^{\widehat \theta \widehat \otimes \mathrm{Id}} \\ M \widehat \otimes_{A} \widehat P_{\sigma} \ar[rr]^{\mathrm{Id} \widehat \otimes \widehat \delta} && M \widehat \otimes_{A} \widehat P_{\sigma} \widehat \otimes_{A} \widehat P_{\sigma}},$$ and the action of $A$ on $\widehat P_{\sigma}$ on the right is given by the Taylor map of $A$. In practice, we will only consider the case of finitely presented $A$-modules $M$, and then, the completed tensor product is the usual tensor product $M \otimes_{A} \widehat P_{\sigma}$ We just showed in proposition \[Taylorstruc\] that any $\mathrm D^{(\infty)}_{\sigma}$-module comes with a canonical twisted Taylor structure. When there exists a twisted coordinate on $A$, we can can describe it explicitly as follows: \[qtayl\] Assume that $x$ is a twisted coordinate on $A$ and let $\xi = \tilde x -x$. If $M$ is a $\mathrm D^{(\infty)}_{\sigma}$-module, we have for all $n \in \mathbb N$, $$\theta_{n}(s) = \sum_{k=0}^{n} \partial_{\sigma}^{[k]}(s) \otimes \xi^{(k)} \in M \otimes_{A} P_{(n)_{\sigma}}$$ and $$\widehat \theta(s) = \sum_{k=0}^{\infty} \partial_{\sigma}^{[k]}(s) \otimes \xi^{(k)} \in M \widehat \otimes_{A} \widehat P_{\sigma}.$$ This follows from equation : if we write $$\theta_{n}(s) = \sum_{k=0}^{n} s_{k} \otimes \xi^{(k)},$$ we will have for all $l \in \mathbb N$, $$\partial_{\sigma}^{[l]}(s) = \sum_{k=0}^{n} \tilde \partial_{\sigma}^{[l]}(\xi^{(k)}) s_{k} = s_{l}.\quad \Box$$ In particular, we see that, if $z \in A$, then the image in $\widehat P_{\sigma}$ of $\tilde z = 1 \otimes z \in P$ is the twisted Taylor series $$\widehat \theta(z) = \sum_{k} \partial_{\sigma}^{[k]}(z)\xi^{(k)}.$$ This explains why it is legitimate to call the map $z \mapsto \tilde z$ the Taylor map. 1. If $x$ is a twisted coordinate on $A$, we have $\widehat \theta(x) = x + \xi$ and $$\widehat \theta(x^2) = x^2 + (x + \sigma(x)) \xi + \xi^2.$$ 2. Assume that $x \in A^\times$ is an invertible twisted coordinate on $A$ and that $\sigma(x) = qx$ with $q \in R^\times$, then we have $$\widehat \theta \left( \frac 1x \right) = \sum_{k=0}^{\infty} (-1)^k \frac {\xi^{(k)}}{ q^{\frac {k(k+1)}2} x^{k+1}} = \frac 1x - \frac \xi{qx^2} + \frac {\xi^{(2)}}{q^3x^3} - \cdots.$$ If $A$ is a twisted localization of $R[x]$, then there exists at most one $R$-algebra homomorphism $\widehat \theta : A \to \widehat P_{\sigma}$ such that $\widehat \theta(x) = \tilde x$. It means that the twisted Taylor map $$\xymatrix@R=0cm{ A \ar[r]^{\widehat \theta } & \widehat P_{\sigma} \\ z \ar@{\|->}[r] & \sum_{k=0}^{\infty} \partial_{\sigma}^{[k]}(z) \xi^{(k)}}$$ is the unique such map. Quantum differential operators ============================== In the quantum situation, we can be a lot more explicit as we shall see now. Thus, we assume in this section that $A$ is a quantum $R$-algebra: we are given a twisted coordinate $x$ such that $\sigma(x) = qx+h$ with $q,h \in R$. We have $$\forall k,l \in \mathbb N, \quad \partial_{\sigma}^{[k]} \circ \partial_{\sigma}^{[l]} = {k+l \choose l}_{q} \partial_{\sigma}^{[k+l]}.$$ It follows from lemma \[compolin\] that $$\widetilde {\partial_{\sigma}^{[k]} \circ \partial_{\sigma}^{[l]}} = \widetilde {\partial_{\sigma}^{[k]}} \circ (\mathrm{Id} \otimes \widetilde {\partial_{\sigma}^{[l]}}) \circ \delta.$$ Thus, using theorem \[biconf\], we see that $$\begin{aligned} (\widetilde {\partial_{\sigma}^{[k]} \circ \partial_{\sigma}^{[l]}})(\xi^{(n)}) &=& \widetilde {\partial_{\sigma}^{[k]}} \left( (\mathrm{Id} \otimes \widetilde {\partial_{\sigma}^{[l]}}) \left( \sum_{i=0}^{n} {n \choose i}_{q} \xi^{(n-i)} \otimes \xi^{(i)}\right)\right) \\ &=& \sum_{i=0}^{n} {n \choose i}_{q} \widetilde {\partial_{\sigma}^{[k]}} \left( \xi^{(n-i)} \otimes \widetilde {\partial_{\sigma}^{[l]}}(\xi^{(i)})\right) \\ &=& {n \choose l}_{q} \widetilde {\partial_{\sigma}^{[k]}} (\xi^{(n-l)}) = \left\{ \begin{array}{cl} {k+l \choose l}_{q} & \mathrm{if} \ n = k + l \\ 0 & \mathrm{otherwise}. \end{array} \right. . \quad \Box\end{aligned}$$ Recall from [@LeStumQuiros15] that if $m \in \mathbb N$, we write $(m)_{q} := {m \choose 1}_{q} = 1 + \cdots + q^{m-1}$ and we define by induction $(m)_{q}! := (m)_{q} (m-1)_{q}!$. \[formd\] We have $$\forall k \in \mathbb N, \forall z \in A, \quad \partial_{\sigma}^{k}(z) = (k)_{q}! \partial_{\sigma}^{[k]}(z).$$ We proceed by induction on $k$ and obtain $$\partial_{\sigma}^{k+1}(z) = \partial_{\sigma}^{k} (\partial_{\sigma}(z)) = (k)_{q}! \partial_{\sigma}^{[k]} (\partial_{\sigma}(z)) = (k)_{q}! (\partial_{\sigma}^{[k]} \partial_{\sigma})(z)$$ $$= (k)_{q}! {k+1 \choose 1}_{q} \partial_{\sigma}^{[k+1]} (z) = (k+1)_{q}! \partial_{\sigma}^{[k+1]}(z) . \quad \Box$$ The next result is important because it describes explicitly the relations between the different rings of twisted differential operators introduced so far. Recall from proposition \[epimon\] that there exists an epi-mono factorization $$\xymatrix@R0cm{ \mathrm D_{A/R,\sigma} \ar@{->>}[r] & \overline{\mathrm D}_{A/R,\sigma} \ar@{^{(}->}[r] & \mathrm D^{(\infty)}_{A/R,\sigma}. }$$ Recall from [@LeStumQuiros15] that the ring $R$ is said to be $q$-flat (resp. $q$-divisible) if $(m)_{q}$ is always regular (resp. invertible) unless $(m)_{q} = 0$ and also that the $q$-characteristic of $R$ is the smallest positive integer $p$ such that $(p)_{q} = 0$, if it exists, and $0$ otherwise. \[thd\] Assume $R$ is $q$-divisible and let $A$ be a $q$-$R$-algebra. Then, 1. \[qrat\] If $q\mathrm{-char}(A) = 0$, we have $$\mathrm D_{A/R,\sigma} = \overline{\mathrm D}_{A/R,\sigma} = \mathrm D^{(\infty)}_{A/R,\sigma}.$$ 2. \[qlitl\] If $q\mathrm{-char}(A) = p >0$, we have $$\mathrm D_{A/R,\sigma}/\partial_{\sigma}^p \simeq \overline{\mathrm D}_{A/R,\sigma} \simeq \mathrm D^{(\infty)}_{A/R,\sigma}/K_{\sigma}^{[p]},$$ where $K_{\sigma}^{[p]}$ is the free $A$-module generated by all $\partial_{\sigma}^{[k]}$ for $k \geq p$. Note that with some extra conventions, assertions \[qrat\]) and \[qlitl\]) could be turned into a single one. We use the formula from corollary \[formd\]. In situation \[qrat\]), then all $q$-integers are invertible in $A$ and the composite map sends the canonical basis $\{\partial_{\sigma}^{k}\}_{k \in \mathbb N}$ of $\mathrm D_{\sigma}$ bijectively onto a basis of $\mathrm D^{(\infty)}_{\sigma}$. The first assertion follows. In situation \[qlitl\]), we have $(p)_{q} = 0$ in $A$ and the element $\partial_{\sigma}^{p} \in \mathrm D_{\sigma}$ is therefore sent to $0$. But since $R$ is $q$-divisible, we have $(m)_{q} \in R^\times$ for $m < p$, and the composite map sends the family $\{\partial_{\sigma}^{k}\}_{k < p}$ of $\mathrm D_{\sigma}$ bijectively onto a basis of $\mathrm D^{(\infty)}_{\sigma}/K_{\sigma}^{[p]}$. $\quad \Box$ 1. The hypothesis in \[qrat\]) is satisfied in the cases of classical differential equations, classical finite difference equations and classical $q$-difference equations. More precisely, it is satisfied for example when 1. $q = 1$ and $R$ is a $\mathbb Q$-algebra, or 2. $q$ is not equal to zero, not a root of $1$ and belongs to a subfield $K$ of $R$. The hypothesis in \[qlitl\]) is satisfied for differential equations and finite difference equations in positive characteristic as well as in the classical quantum case. More precisely, they are satisfied for example when 1. $q = 1$ and $R$ is an $\mathbb F_{p}$-algebra, or 2. $q$ is a non trivial $p$-th root of $1$ ($p$ not necessarily prime) and belongs to a subfield $K$ of $R$, or 3. $q$ is a non trivial $p$-th root of $1$ with $p$ prime (but $q$ does not necessarily belong to a subfield of $R$). 2. Since both $\mathrm D_{A/R,\sigma}$ and $\mathrm D^{(\infty)}_{A/R,\sigma}$ commute with extensions of $R$ (although $\overline{\mathrm D}_{A/R,\sigma}$ does not), and $\mathrm D_{A/R,\sigma}$ always commutes with extensions of $A$, we can sometimes (see the remark following proposition \[locco\]) reduce questions to the generic case $$\label{genc} R = \mathbb Q(t)[s], \quad A = R[x], \quad q = t \quad \mathrm{and} \quad h = s,$$ and work as well over the later. In this case, thanks to the theorem, we can identify $\mathrm D_{A/R,\sigma}$ with $\mathrm D^{(\infty)}_{A/R,\sigma}$. 3. The same proof shows that if $A$ is a twisted $R$-algebra which is only $q$-flat (but not necessarily $q$-divisible), then 1. \[qrat2\] If $q\mathrm{-char}(A) = 0$, then $\mathrm D_{A/R,\sigma} = \overline{\mathrm D}_{A/R,\sigma}$. 2. \[qlitl2\] If $q\mathrm{-char}(A) = p> 0$, then $\mathrm D_{A/R,\sigma}/\partial_{\sigma}^p \simeq \overline{\mathrm D}_{A/R,\sigma}$. The end of this section will be devoted to giving explicit formulas. They are usually quite formal to prove in the ring of twisted differential operators of infinite level and their analog in the twisted Weyl algebra is then easily obtained thanks to theorem \[thd\]. It is useful to have a general formula for the commutation of twisted differential operators with the twisted coordinate $x$: \[commix\] We have $$\forall k \in \mathbb N \setminus \{0\}, \quad \partial_{\sigma}^{[k]} \circ x = \sigma^k(x) \partial_{\sigma}^{[k]} + \partial_{\sigma}^{[k-1]}.$$ We start with the following computation: $$\xi^{(n+1)} = \xi^{(n)}\sigma^n(\xi) = \xi^{(n)}(\tilde x - \sigma^n(x)) = \tilde x \xi^{(n)} - \sigma^n(x)\xi^{(n)},$$ from which we derive $$\tilde x \xi^{(n)} = \xi^{(n+1)} + \sigma^n(x)\xi^{(n)}.$$ We have $$\widetilde{\partial_{\sigma}^{[k]} \circ x} = \tilde \partial_{\sigma}^{[k]} \circ (\mathrm{ Id} \otimes \widetilde{m_x}) \circ \delta.$$ In this formula, $\widetilde{m_x}$ denotes the linearization of multiplication by $x$, which is easily seen to be given by $$\widetilde{m_x}(\xi^{(i)}) = \left\{ \begin{array}{l} x \quad \mathrm{if} \quad i = 0 \\ 0 \quad \mathrm{if} \quad i \neq 0 \end{array} \right. .$$ Now, we know from theorem \[biconf\] (the quantic binomial formula for principal parts) that $$\delta(\xi^{(n)}) = \sum_{i=0}^{n} {n \choose i}_{q} \xi^{(n-i)} \otimes \xi^{(i)}.$$ It follows that $$(\mathrm{ Id} \otimes \widetilde{m_x}) (\delta (\xi^{(n)})) = \tilde x \xi^{(n)}.$$ Putting all these together, we obtain $$\begin{aligned} \widetilde{(\partial_{\sigma}^{[k]} \circ x)} (\xi^{(n)}) &=& \tilde \partial_{\sigma}^{[k]} (\tilde x \xi^{(n)}) \\ &=& \tilde \partial_{\sigma}^{[k]} (\xi^{(n+1)} + \sigma^n(x)\xi^{(n)}) = \left\{ \begin{array}{l} 1 \quad \mathrm{if} \quad n = k-1 \\ \sigma^k(x) \quad \mathrm{if} \quad n = k \end{array} \right. .\end{aligned}$$ And in the end, we get $$\partial_{\sigma}^{[k]} \circ x = \sigma^k(x) \partial_{\sigma}^{[k]} + \partial_{\sigma}^{[k-1]}. \quad \Box$$ \[invgen\] If $x \in A^\times$, then $$\forall k \in \mathbb N \setminus \{0\}, \quad \partial_{\sigma}^{[k]} \circ x^{-1} = \sum_{i=0}^k \frac{(-1)^i}{\prod_{j=0}^i \sigma^{k-j}(x)} \partial_{\sigma}^{[k-i]}.$$ We compute $$\begin{aligned} \sum_{i=0}^k \frac{(-1)^i}{\prod_{j=0}^i \sigma^{k-j}(x)} \partial_{\sigma}^{[k-i]} \circ x &=& \sum_{i=0}^k \frac{(-1)^i}{\prod_{j=0}^i \sigma^{k-j}(x)}(\sigma^{k-i}(x) \partial_{\sigma}^{[k-i]} + \partial_{\sigma}^{[k-i-1]}) \\ &=& \sum_{i=0}^k \frac{(-1)^i}{\prod_{j=0}^{i-1} \sigma^{k-j}(x)} \partial_{\sigma}^{[k-i]} + \sum_{i=0}^{k-1} \frac{(-1)^i}{\prod_{j=0}^i \sigma^{k-j}(x)}\partial_{\sigma}^{[k-i-1]} = \partial_{\sigma}^{[k]}. \quad \Box\end{aligned}$$ \[itform\] We have in $\mathrm D_{A/R,\sigma}^{(\infty)}$, $$\forall k > 0, \quad \partial^{[k]}_{\sigma} \circ x = q^kx \partial^{[k]}_{\sigma} + (k)_{q}h \partial^{[k]}_{\sigma} + \partial_{\sigma}^{[k-1]}.$$ We have in $\mathrm D_{A/R,\sigma}$, $$\forall k > 0, \quad \partial^k_{\sigma} \circ x = q^kx \partial^{k}_{\sigma} + (k)_{q} (h \partial^{k}_{\sigma} + \partial_{\sigma}^{k-1}).$$ The first assertion is simply a reformulation of proposition \[commix\]. For the second one, after a base change, we may reduce to the generic case \[genc\] and thus assume that $R$ is $q$-divisible and $q\mathrm{-char}(R) = 0$. And we may then replace $\mathrm D_{\sigma}$ with $\mathrm D^{(\infty)}_{\sigma}$ in which case we fall back onto the first equality. $\quad \Box$ Note that, in order to prove the second formula, we cannot use directly corollary \[formd\]: our equality takes place into the Weyl algebra and not is not just an assertion about endomorphisms of $A$. We concentrate now on the case $\sigma(x) = qx$. Assume $h = 0$, $q \in R^\times$ and $x \in A^\times$. Then, we have in $\mathrm D^{(\infty)}_{A/R,\sigma}$, $$\forall k \in \mathbb N, \quad \partial_{\sigma}^{[k]} \circ x^{-1} = \sum_{i=0}^k (-1)^i q^{-\frac {(2k-i)(i+1)}2}x^{-i-1} \partial_{\sigma}^{[k-i]}.$$ We have in $\mathrm D_{A/R,\sigma}$, $$\forall k \in \mathbb N, \quad \partial_{\sigma}^{k} \circ x^{-1} = \sum_{i=0}^k (-1)^i q^{-\frac {(2k-i)(i+1)}2} (k)_{q} \cdots (k-i+1)_{q} x^{-i-1} \partial_{\sigma}^{k-i}.$$ The first assertion is a particular case of corollary \[invgen\] and the second one follows by the standard generic argument. $\quad \Box$ Assume that $h = 0$. Then, we have in $\mathrm D^{(\infty)}_{A/R,\sigma}$, $$\forall k, n \in \mathbb N, \quad \partial_{\sigma}^{[k]} \circ x^n = \sum_{i=0}^k q^{(n-i)(k-i)}{n \choose i}_{q} x^{n-i}\partial_{\sigma}^{[k-i]}.$$ We have in $\mathrm D_{A/R,\sigma}$, $$\forall k, n \in \mathbb N, \quad \partial_{\sigma}^{k} \circ x^n = \sum_{i=0}^k q^{(n-i)(k-i)}[i]_{q}!{k \choose i}_{q}{n \choose i}_{q} x^{n-i}\partial_{\sigma}^{k-i}.$$ As usual, the second formula will follow from the first that we prove directly by induction on $n$. We will have $$\begin{aligned} \partial_{\sigma}^{[k]} \circ x^n &=& \sum_{i=0}^k q^{(n-1-i)(k-i)}{n-1 \choose i}_{q} x^{n-1-i}\partial_{\sigma}^{[k-i]} \circ x \\ &=& \sum_{i=0}^k q^{(n-1-i)(k-i)}{n-1 \choose i}_{q} x^{n-1-i}\left(q^{k-i}x \partial_{\sigma}^{[k-i]} + \partial_{\sigma}^{[k-i-1]}\right) \\ &=& \sum_{i=0}^k q^{(n-1-i)(k-i)}{n-1 \choose i}_{q} x^{n-1-i}q^{k-i} x\partial_{\sigma}^{[k-i]} \\ && + \sum_{i=0}^k q^{(n-1-i)(k-i)}{n-1 \choose i}_{q} x^{n-1-i} \partial_{\sigma}^{[k-i-1]} \\ &=& \sum_{i=0}^k q^{(n-i)(k-i)}{n-1 \choose i}_{q} x^{n-i} \partial_{\sigma}^{[k-i]}+ \sum_{i=0}^{k-1} q^{(n-i)(k-i+1)}{n-1 \choose i-1}_{q} x^{n-i} \partial_{\sigma}^{[k-i]} \\ &=& \sum_{i=0}^k q^{(n-i)(k-i)}\left( {n-1 \choose i}_{q} + q^{n-i}{n-1 \choose i-1}_{q}\right) x^{n-i} \partial_{\sigma}^{[k-i]} \\ &=& \sum_{i=0}^k q^{(n-i)(k-i)}{n \choose i}_{q} x^{n-i}\partial_{\sigma}^{[k-i]}. \quad \Box\end{aligned}$$ When $h = 0$, we have $$\forall n \in \mathbb N, \quad \partial^{[k]}_{\sigma}(x^n) = \left\{ \begin{array}{cl} {n \choose k}_{q} x^{n-k} & \mathrm{if} \ n \geq k \\ 0 & \mathrm{otherwise} \end{array} \right.$$ and $$\forall n \in \mathbb N, \quad \partial^{k}_{\sigma}(x^n) = \left\{ \begin{array}{cl} (n)_{q} \cdots (n-k+1)_{q} x^{n-k} & \mathrm{if} \ n \geq k \\ 0 & \mathrm{otherwise} \end{array} \right. .$$ The first assertion follows from the proposition and the second one uses the inclusion $\overline D_{A/R,\sigma} \subset D_{A/R,\sigma}^{(\infty)}$. $\quad \Box$ \[minuso\] Assume that $h = 0$. Then, we have $$\forall n \in \mathbb N \setminus \{0\}, \quad \partial_{\sigma} (x^{n} ) = (n)_{q} x^{n-1}.$$ If $x \in A^\times$ and $q \in R^\times$, then the formula actually holds for all $n \in \mathbb Z$. Only the second part needs a proof: if $x$ is invertible, we have for $n>0$ $$\begin{aligned} 0 &=& \partial_{\sigma}(1) = \partial_{\sigma}(x^nx^{-n}) = \partial_{\sigma}(x^n)x^{-n} + \sigma(x^n) \partial_{\sigma}(x^{-n}) \\ &=& (n)_{q} x^{n-1}x^{-n} + q^nx^n \partial_{\sigma}(x^{-n}) = (n)_{q} x^{-1} + q^nx^n \partial_{\sigma}(x^{-n}),\end{aligned}$$ from which we derive, when $q \in R^\times$, $$\partial_{\sigma}(x^{-n}) = -q^{-n} (n)_{q} x^{-n-1} = (-n)_{q} x^{-n-1}. \quad \Box$$ Formal deformations of twisted differential operators ===================================================== In this section, we study the relation between twisted differential operators relative to various endomorphisms of $A$. We are particularly interested in the comparison of our twisted differential operators with usual differential operators. We assume that there exists a twisted coordinate $x$, that we fix for the rest of the section, and we write $y_{\sigma} := x - \sigma(x)$. Recall that the ring $\mathrm D_{\sigma}^{(\infty)}$ of differential operators (of infinite level) comes with a natural increasing filtration by $A$-submodules $$\mathrm D_{\sigma}^{(\infty)} = \cup_{m=0}^{\infty} \mathrm{Diff}_{m,\sigma}(A),$$ which is called the order filtration. The choice of the twisted coordinate determines a splitting of this filtration. To see this, we let for all $m \in \mathbb N$, $K_{\sigma}^{[m]} \subset \mathrm D_{\sigma}^{(\infty)}$ be the free $A$-module generated by all $\partial_{\sigma}^{[k]}$ for $k \geq m$. Note that this is actually a filtration by left ideals. \[infinf\] The decreasing filtration by the $K_{\sigma}^{[m]}$ is called the ideal filtration on $\mathrm D_{\sigma}^{(\infty)}$. The module of twisted differential operators of infinite level and infinite order on $A$ is $$\widehat {\mathrm D}_{A/R,\sigma}^{(\infty)} = \varprojlim \mathrm D_{\sigma}^{(\infty)}/K_{\sigma}^{[m+1]}.$$ We might again drop the index $A/R$ and write $\widehat {\mathrm D}_{\sigma}^{(\infty)}$. The decreasing filtration of $\widehat {\mathrm D}_{\sigma}^{(\infty)}$ by the closures $\widehat K_{\sigma}^{[m]}$ of the $K_{\sigma}^{[m]}$’s will also be called the ideal filtration . 1. The ideal filtration is separated, which means that we have $\cap_{m} K_{\sigma}^{[m+1]} = \{0\}$ and it follows that $\mathrm D_{\sigma}^{(\infty)} \subset \widehat {\mathrm D}_{\sigma}^{(\infty)}$. Actually, any $\varphi \in \widehat {\mathrm D}_{\sigma}^{(\infty)}$ can be uniquely written as an infinite sum $\sum_{0}^\infty z_{k} \partial_{\sigma}^{[k]}$ with $z_{k} \in A$ (and conversely). In other words, we have the isomorphisms of $A$-modules $$\mathrm D_{\sigma}^{(\infty)}= \bigoplus_{k \in \mathbb N} A \partial_{\sigma}^{[k]} \quad \mathrm{and} \quad \widehat{\mathrm D}_{\sigma}^{(\infty)} = \prod_{k \in \mathbb N} A \partial_{\sigma}^{[k]}.$$ 2. The ideal filtration defines a splitting of the order filtration in the sense that $$\mathrm D_{\sigma}^{(\infty)} = \mathrm{Diff}_{m,\sigma}(A) \oplus K_{\sigma}^{[m+1]} \quad \mathrm{and} \quad \widehat {\mathrm D}_{\sigma}^{(\infty)} = \mathrm{Diff}_{m,\sigma}(A) \oplus \widehat K_{\sigma}^{[m+1]}$$ as $A$-modules. 3. $\widehat {\mathrm D}_{\sigma}^{(\infty)}$ is not a ring in general. More precisely, multiplication on $\mathrm D_{\sigma}^{(\infty)}$ is not continuous for the ideal filtration: we always have $\partial_{\sigma}^{[m]} \to 0$ when $m \to \infty$ but, if $\sigma(x) = qx$ and $x \in A^\times$ for example, we can see that $$\forall m \in \mathbb N, \quad \partial_{\sigma}^{[m]} \circ x^{-1} \equiv \partial_{\sigma}^{[m]}(x^{-1}) = x^{-m-1} \neq 0 \mod K_{\sigma}^{[1]}.$$ If $A$ is any ring and $A[\xi]$ denotes the polynomial ring in one variable $\xi$, then the natural filtration on $\mathrm{Hom}_{A}(A[\xi], A)$ is the decreasing filtration by the kernels of the surjections $$\mathrm{Hom}_{A}(A[\xi], A) \to \mathrm{Hom}_{A}(A[\xi]_{\leq m}, A),$$ where $A[\xi]_{\leq m}$ denotes as before the $A$-module of polynomials of degree at most $m$. The corresponding topology will be called the natural topology of $\mathrm{Hom}_{A}(A[\xi], A)$. Note that $\mathrm{Hom}_{A}(A[\xi], A)$ is separated and complete for the natural topology: $$\varprojlim \mathrm{Hom}_{A}(A[\xi]_{\leq m}, A) = \mathrm{Hom}_{A}(\varinjlim A[\xi]_{\leq m}, A) = \mathrm{Hom}_{A}(A[\xi], A).$$ \[density\] The map $$\xymatrix@R0cm{ A[\xi] \ar[r] & P_{A/R} \\ \xi \ar@{\|->}[r] & \tilde x - x}$$ induces by duality an isomorphism of topological $A$-modules $$\label{dens} \xymatrix@R0cm{ \widehat {\mathrm D}_{A/R,\sigma}^{(\infty)} \ar[r]^-{\simeq} & \mathrm{Hom}_{A}(A[\xi], A).}$$ More precisely, the ideal filtration corresponds to the natural filtration. It follows from corollary \[free\] that the cokernel of the map $$\xymatrix@R0cm{ A[\xi]_{\leq m} \ar@{^{(}->}[r] & A[\xi] \ar[r] & P \ar@{->>}[r] & P_{(n)_{\sigma}}}$$ is generated by $\xi^{(k)}$ for $m < k \leq n$. Moreover, this map is injective when $m \leq n$. Dually, it means that the corresponding map $$\xymatrix@R0cm{\mathrm{Diff}_{n,\sigma}(A) \simeq \mathrm{Hom}_{A}(P_{(n)_{\sigma}}, A) \ar[r] & \mathrm{Hom}_{A}(A[\xi]_{\leq m}, A)}$$ is surjective for $m \leq n$ and that its kernel is exactly $K_{\sigma}^{[m+1]} \cap \mathrm{Diff}_{n,\sigma}(A)$ (it is generated by $\partial_{\sigma}^{[k]}$ for $n \geq k > m$). Taking direct limits on the left, we see that the map $$\xymatrix@R0cm{\mathrm D_{\sigma}^{(\infty)} \ar[r] & \mathrm{Hom}_{A}(A[\xi]_{\leq m}, A)}$$ is surjective with kernel exactly $K_{\sigma}^{[m+1]}$. In other words, we get a canonical isomorphism of $A$-modules $$\xymatrix@R0cm{\mathrm D_{\sigma}^{(\infty)}/K_{\sigma}^{[m+1]} \ar[r]^-{\simeq} & \mathrm{Hom}_{A}(A[\xi]_{\leq m}, A)}.$$ Thus, taking inverse limits on both sides gives the result. $\quad \Box$ 1. By construction, there exists a commutative diagram $$\xymatrix{\mathrm D_{A/R,\sigma}^{(\infty)} \ar@{^{(}->}[r] \ar@{^{(}->}[dd] & \mathrm{End}_{R}(A) \ar[d]^\simeq \\ & \mathrm{Hom}_{A}(P, A) \ar[d] \\ \widehat {\mathrm D}_{A/R,\sigma}^{(\infty)} \ar[r]^-{\simeq} & \mathrm{Hom}_{A}(A[\xi], A) }.$$ 2. We will usually denote by $\tilde \varphi \in \mathrm{Hom}_{A}(A[\xi], A)$ the image of $\varphi \in \widehat {\mathrm D}_{A/R,\sigma}^{(\infty)}$. This is compatible with the previous notation for linearization. In particular, we have $$\forall k, n \in \mathbb N, \quad \widetilde{\partial^{[k]}_{\sigma}}(\xi^{(n)}) = \left\{\begin{array} {cll} 1 & \mathrm{if} \ k = n \\ 0 & \mathrm{otherwise}\end{array} \right. .$$ 3. When $A = R[x]$, we actually get $$\xymatrix@R0cm{\mathrm D_{A/R,\sigma}^{(\infty)} \ar@{^{(}->}[r] & \widehat {\mathrm D}_{A/R,\sigma}^{(\infty)} \ar[r]^-{\simeq} & \mathrm{Hom}_{A}(A[\xi], A) \ar[r]^-{\simeq} & \mathrm{Hom}_{A}(P, A) \ar[r]^-{\simeq} & \mathrm{End}_{R}(A)}$$ and $\widehat {\mathrm D}_{A/R,\sigma}^{(\infty)}$ is a ring in this very special case. 4. There exists a complex analytic analog of the density lemma (for usual differential operators) as explained by Zogman Mebkhout and Luis Narváez in [@MebkhoutNarvaez98]: the ring of algebraic differential operators is dense in the ring of continuous endomorphisms of the structural sheaf. Although it is difficult to give an explicit description of the isomorphism , we can at least show the following: \[compone\] We have $$\forall n \in \mathbb N \setminus \{0\}, \quad \tilde \partial_{\sigma}(\xi^n) = (\sigma(x) - x)^{n-1}.$$ More generally, if $\tau$ is another $R$-algebra endomorphism of $A$, we have $$\forall n \in \mathbb N \setminus \{0\}, \quad \tilde \partial_{\sigma}(\xi^{(n)_{\tau}}) = \prod_{i=1}^{n-1}(\sigma(x) - \tau^i(x)),$$ where $\xi^{(n)_{\tau}} := \xi\tau(\xi) \cdots \tau^{n-1}(\xi)$ denotes the twisted power with respect to the endomorphism $\tau$. By definition, $\tilde \partial_{\sigma}$ is the unique $A$-linear function on $A[\xi]$ such that $\tilde \partial_{\sigma}(\xi^{(n)}) = 1$ for $n = 1$ and $0$ otherwise. We consider now the unique $A$-linear function $u$ on $A[\xi]$ such that $u(\xi^n) = (\sigma(x) - x)^{n-1}$ for $n > 0$ and $u(1) = 0$. We want to show that $u = \tilde \partial_{\sigma}$. The map $u$ may be seen as the composition of division by $\xi$ on $A[\xi]$ (after removing the constant term) and evaluation at $\sigma(x) -x$. But we have $$\xi^{(n)} := \xi\sigma(\xi) \cdots \sigma^{n-1}(\xi),$$ and we know that $\sigma(\xi) = \xi +x - \sigma(x)$. Therefore, it is clear that for $n \geq 2$, we will get $u(\xi^{(n)}) = 0$. Of course, for $n = 0$ we have $\xi^{(n)} = 1$ and we also obtain $0$. Finally, for $n = 1$, we have $\xi^{(n)} = \xi$ and we get $1$. The proof of the second formula follows the same lines. From the first part, we can interpret $\tilde \partial_{\sigma}$ as the composition of division by $\xi$ on $A[\xi]$ and evaluation at $\sigma(x) -x$. We want to apply this to $\xi^{(n)_{\tau}} = \prod_{i=0}^{n-1} \tau^i(\xi)$ and we know that $ \tau^i(\xi) = \xi + x - \tau^i(x)$. Thus, we see that $$\tilde \partial_{\sigma}(\xi^{(n)_{\tau}}) = \prod_{i=1}^{n-1} (\sigma(x) -x + x - \tau^i(x)) = \prod_{i=1}^{n-1}(\sigma(x) - \tau^i(x)). \quad \Box$$ We can actually derive a remarkable consequence of the density lemma: \[dfthm\] If $x$ is a also a $\tau$-coordinate for another $R$-endomorphism $\tau$ of $A$, then there exists an isomorphism of topological $A$-modules that depends only on $x$ $$\xymatrix@R0cm{ \widehat {\mathrm D}_{A/R,\sigma}^{(\infty)} \ar[r]^-{\simeq} & \widehat {\mathrm D}_{A/R,\tau}^{(\infty)}}.$$ More precisely, it is compatible with the ideal filtrations. Recall that the hypothesis is always satisfied when $A$ is a $\tau$-twisted localization of $R[x]$. We may just compose the isomorphism of the density lemma with the inverse of the analogous isomorphism for $\tau$. $\quad \Box$ When the coordinate $x$ is fixed, we may (and will) identify these two topological (or even, filtered) $A$-modules. It is worth mentioning a particular case of this corollary that is of great interest (this is the case $\tau = \mathrm{Id}_{A})$: \[compid\] Assume that $x$ is also an usual coordinate on $A$. Then, there exists a canonical isomorphism $$\xymatrix@R0cm{ \widehat {\mathrm D}_{A/R,\sigma}^{(\infty)} \ar[r]^-{\simeq} & \widehat {\mathrm D}_{A/R}^{(\infty)}.} \quad \Box$$ In theorem 2 of the introduction to [@Pulita14], Pulita shows that, in the $p$-adic world, some differential modules have a natural structure of $\sigma$-module. The main idea is to realize a formal solution of the differential module as formal solution of some $\sigma$-module. This is analog to the way we derive the formal deformation theorem from the density lemma. It seems that the first result in this direction was obtained by André and Di Vizio in [@AndreDiVizio04]. We can give explicit formulas in order to express the twisted derivation from one world as a twisted differential operators in another world: \[changsig\] Assume that $x$ is a also a $\tau$-coordinate for some other $R$-endomorphism $\tau$ of $A$. Then, we have $$\partial_{\sigma} = \sum_{k=1}^{\infty} \left(\prod_{i=1}^{k-1}(\sigma(x) - \tau^i(x))\right)\partial_{\tau}^{[k]}.$$ This follows directly by duality from lemma \[compone\]. $\quad \Box$ \[siginf\] \[corsig\] Assume that $x$ is also a usual coordinate on $A$. Then, $$\label{transform} \partial_{\sigma} = \sum_{k=1}^{\infty} (\sigma(x) -x)^{k-1} \partial^{[k]}$$ and $$\label{transform2} \sigma = \sum_{k=0}^{\infty} (\sigma(x) -x)^{k} \partial^{[k]}.$$ The first assertion follows directly from the proposition and the other one is deduced from the equality $\sigma = 1 - y_{\sigma}\partial_{\sigma}$. $\quad \Box$ 1. In the case $\sigma(x) = qx$, the formulas read $$\partial_{\sigma} = \sum_{k=1}^{\infty} (q-1)^{k-1}x^{k-1} \partial^{[k]} \quad \mathrm{and} \quad \sigma = \sum_{k=0}^{\infty} (q-1)^{k}x^{k} \partial^{[k]}.$$ 2. In the case $\sigma(x) = x + h$, the formulas read $$\partial_{\sigma} = \sum_{k=1}^{\infty} h^{k-1} \partial^{[k]} \quad \mathrm{and} \quad \sigma = \sum_{k=0}^{\infty} h^{k} \partial^{[k]}.$$ Conversely, we may also express the usual derivation in term of twisted derivations. We only do the classical cases: Assume that $x$ is also a usual coordinate on $A$. Then, 1. If $\sigma(x) = qx$, we have $$\partial = \sum_{k=1}^{\infty} (1-q)^{k-1}(k-1)_{q}! x^{k-1}\partial_{\sigma}^{[k]}.$$ 2. If $\sigma(x) = x + h$, we have $$\partial = \sum_{k=1}^{\infty} (-1)^{k-1}h^{k-1}(k-1)!\partial_{\sigma}^{[k]}.$$ In the formula of proposition \[changsig\] we replace $\tau$ by $\sigma$ and $\sigma$ by $\mathrm{Id}$ respectively, in order to obtain $$\begin{aligned} \partial &=& \sum_{k=1}^{\infty} \prod_{i=1}^{k-1}(x -q^ix)\partial_{\sigma}^{[k]} = \sum_{k=1}^{\infty}\prod_{i=1}^{k-1}(1 -q^i)x^{k-1}\partial_{\sigma}^{[k]} \\ &=& \sum_{k=1}^{\infty}(1-q)^{k-1}\prod_{i=1}^{k-1}(i)_{q}x^{k-1}\partial_{\sigma}^{[k]} = \sum_{k=1}^{\infty} (1-q)^{k-1}(k-1)_{q}! x^{k-1}\partial_{\sigma}^{[k]}\end{aligned}$$ in the first case, and $$\partial = \sum_{k=1}^{\infty} \prod_{i=1}^{k-1}(-ih)\partial_{\sigma}^{[k]} = \sum_{k=1}^{\infty} (-h)^{k-1}(k-1)!\partial_{\sigma}^{[k]}$$ in the other one. $\quad \Box$ We may also apply proposition \[changsig\] in order to make more precise the statement of proposition \[pow\]. \[prodf\] Assume that $x$ is also a $\sigma^m$-coordinate for $A$ over $R$, then we have $$\partial_{\sigma^m} = \sum_{k=1}^{m} \left(\prod_{i=1}^{k-1}(\sigma^m(x) - \sigma^i(x))\right)\partial_{\sigma}^{[k]}. \quad \Box$$ One can give a more concrete formula in the quantum situation: \[bigform\] Assume that $x$ is also a $\sigma^m$-coordinate for $A$ over $R$ and that $\sigma(x) = qx$. Then, we have $$\partial_{\sigma^m} = \sum_{k=1}^{m} q^{\frac {k(k-1)}2}(q - 1)^{k-1} (m-1)_{q}\cdots (m-k+1)_{q} x^{k-1} \partial_{\sigma}^{[k]}.$$ We compute the coefficient of $\partial_{\sigma}^{[k]}$ for $1 \leq k \leq m$: $$\begin{aligned} \prod_{i=1}^{k-1}(q^mx - q^ix) &=& \left(\prod_{i=1}^{k-1} q^i\right)\left(\prod_{i=1}^{k-1}(q^{m-i} - 1)\right)x^{k-1} \\ &=& q^{\frac {k(k-1)}2}(q - 1)^{k-1}\left(\prod_{i=1}^{k-1} (m-i)_{q}\right)x^{k-1}. \quad \Box\end{aligned}$$ We will have $ \partial_{\sigma^2} = \partial_{\sigma} + q(q-1)x\partial^{[2]}_{\sigma}. $ Formal confluence ================= We explain here how one can use the quantum Weyl algebra in order to approximate a usual differential operator. We assume that $A$ is a quantum $R$-algebra which means that we are given a twisted coordinate $x$ on $A$ with $\sigma(x) = qx + h$ with $q,h \in R$. Recall from [@LeStumQuiros15a] that the twisted Weyl $R$-algebra $\mathrm D_{\sigma}$ has an increasing filtration by free $A$-modules of finite rank $$\mathrm{Fil}^m\mathrm D_{\sigma} = \bigoplus_{k\leq m} A \partial_{\sigma}^k$$ that is called the order filtration. But it also has a decreasing filtration by free $A$-modules (of infinite rank) $$K_{\sigma}^{m} = \bigoplus_{k\geq m} A \partial_{\sigma}^k$$ that we called the ideal filtration. We will consider the completion $$\widehat {\mathrm D}_{\sigma} = \varprojlim \mathrm D_{\sigma}/K_{\sigma}^{m+1}$$ that also comes with its ideal filtration and we have $\mathrm D_{\sigma} \subset \widehat {\mathrm D}_{\sigma}$. Note that $\widehat {\mathrm D}_{\sigma}$ is the set of all infinite sums $\sum_{0}^\infty z_{k} \partial_{\sigma}^{k}$ with $z_{k} \in A$. In other words, there exists an isomorphism of $A$-modules $$\widehat{\mathrm D}_{\sigma} = \prod_{k \in \mathbb N} A \partial_{\sigma}^k.$$ The $A$-module $\widehat {\mathrm D}_{\sigma}$ is not a ring in general. However, the result holds in the finite quantum characteristic case as we can check right now: If $A$ is $q$-flat and $q\mathrm{-char}(A) = p > 0$, then multiplication is continuous for the ideal topology on $\mathrm D_{A/R,\sigma}$ and turns $\widehat{\mathrm D}_{A/R,\sigma}$ into an $R$-algebra. From the equality (see the remark following theorem \[thd\]) $$\mathrm D_{\sigma}/\partial_{\sigma}^p \simeq \overline{\mathrm D}_{\sigma},$$ and the fact that $\overline{\mathrm D}_{\sigma}$ is a ring, we deduce that the (left) ideal generated by $\partial_{\sigma}^p$ is a two sided ideal. Since multiplication is automatically continuous for the $\partial_{\sigma}^p$-adic filtration, it follows that it is also continuous for the ideal filtration (which is the filtration by left ideals generated by the powers of $\partial_{\sigma}$). $\quad \Box$ \[gdca\] The composite map $$\xymatrix@R0cm{ \mathrm \mathrm D_{A/R,\sigma} \ar@{->>}[r] & \overline{\mathrm D}_{A/R,\sigma} \ar@{^{(}->}[r] & \mathrm D^{(\infty)}_{A/R,\sigma} }$$ is compatible with the ideal filtrations. Moreover, if $R$ is $q$-divisible and $q\mathrm{-char}(R) = 0$, then $\widehat {\mathrm D}_{A/R,\sigma} \simeq \widehat {\mathrm D}_{A/R,\sigma}^{(\infty)}$. This applies in particular when $R$ is a $\mathbb Q$-algebra and $\sigma = \mathrm{Id}$. The first assertion follows from corollary \[formd\] and the second from theorem \[thd\]. $\quad \Box$ We can now state our fist confluence theorem: \[confalg\] Let $R$ be a $\mathbb Q$-algebra, $A$ a $q$-$R$-algebra for some $q \in R$. Assume that the quantum coordinate on $A$ is also a usual coordinate. Then, there exists a canonical map of filtered $A$-modules $$\label{confone} \mathrm D_{A/R,\sigma} \to \widehat {\mathrm D}_{A/R}.$$ Moreover, if $R$ is $q$-divisible, we have 1. If $q\mathrm{-char}(A) = 0$, then the map is injective and $\mathrm D_{A/R,\sigma}$ is dense in $\widehat {\mathrm D}_{A/R}$. Actually, the inclusion $\mathrm D_{A/R,\sigma} \subset \widehat {\mathrm D}_{A/R}$ is strict for the ideal filtrations. 2. If $q\mathrm{-char}(A) = p > 0$, then the map induces an isomorphism $$\left(\mathrm D_{A/R,\sigma}/\partial_{\sigma}^p \simeq \right) \quad \overline {\mathrm D}_{A/R,\sigma} \simeq \mathrm D_{A/R}/\partial^p.$$ The map is simply the composite $$\mathrm D_{\sigma} \to \mathrm D_{\sigma}^{(\infty)} \hookrightarrow \widehat{\mathrm D}_{\sigma}^{(\infty)} \simeq \widehat{\mathrm D}^{(\infty)} \simeq \widehat {\mathrm D}_{A/R},$$ where the next to the last map is the formal deformation isomorphism of corollary \[compid\] and the last one comes from lemma \[gdca\] applied to the case $\sigma = \mathrm{id}$ since $R$ is a $\mathbb Q$-algebra. If we assume that $R$ is $q$-divisible and that $q\mathrm{-char}(A) = 0$, then lemma \[gdca\] tells us that $\mathrm D_{\sigma} = \mathrm D^{(\infty)}_{\sigma}$ as filtered rings (for the ideal filtrations) and we can use corollary \[compid\] again. Finally, when $R$ is $q$-divisible but $q\mathrm{-char}(A) = p >0$, then $(p)_{q} = 0$ in $A$ and all $(m)_{q} \in R^\times$ for $m < p$. We can use the last assertion of theorem \[thd\] and the fact that the isomorphism of corollary \[compid\] is strictly compatible with the filtrations. $\quad \Box$ As in [@LeStumQuiros15a], we denote by $A[T]_{\sigma}$ the twisted polynomial ring associated to $\sigma$: this is the non commutative polynomial ring in $T$ over $A$ with the commutation rule $Tx = \sigma(x)T$. Recall also that the twisted coordinate $x$ is said to be strong is $x - \sigma(x) \in A^\times$. If $R$ is a $q$-divisible $\mathbb Q$-algebra, $q\mathrm{-char}(A) = 0$, $x$ is strong and is also a coordinate for all $\sigma^n$’s, then, the $A$-linear map $$\label{confilm} \xymatrix@R0cm{A[T]_{\sigma} \ar[r] & \widehat{\mathrm D}_{A/R} \\ T^{n} \ar@{\|->}[r] & \sum_{k=0}^{\infty} \frac 1{k!}(\sigma^n(x) -x)^{k} \partial^{k}}$$ has dense image. Compose the isomorphism of theorem 6.13 of [@LeStumQuiros15a] with the map of theorem \[confalg\]. The formula comes from corollary \[siginf\]. $\quad \Box$ The theorem (and its corollary) applies in particular when $q$ is not a root of unity in some field $K$ of characteristic zero, $R$ is an algebra containing $K$, $A = R[x, x^{-1}]$ and $\sigma(x) = qx$. In this situation, $\mathrm D := \mathrm D_{A/R}$ is the non commutative ring $R[x, x^{-1}, \partial]$ with the commutation rule $\partial x = x \partial +1$ and we may see $\widehat{\mathrm D}$ (which is not a ring) as the set of infinite sums $R[x, x^{-1}][[\partial]]$. Also, $\mathrm D_{\sigma} := \mathrm D_{A/R,\sigma}$ is the non commutative ring $R[x, x^{-1}, \partial_{\sigma}]$ with the commutation rule $\partial_{\sigma} x = qx \partial_{\sigma} +1$. The map of the theorem satisfies $$\partial_{\sigma} \mapsto \sum_{k=1}^{\infty} \frac 1 {k!}(q-1)^{k-1}x^{k-1} \partial^{k},$$ and the map of the corollary is given by $$T^{n} \mapsto \sum_{k=0}^{\infty} \frac 1 {k!}(q^{n} -1)^{k} x^{k}\partial^{k}.$$ In the next section, we will have to move between $\sigma$ and powers of $\sigma$ (or more precisely, the other way around, between $\sigma$ and roots of $\sigma$). \[explem\] Let $m \in \mathbb N \setminus \{0\}$. If $(m)_{q}! \in R^\times$, then there exists a unique $A$-linear ring homomorphism $$\xymatrix@R0cm{\mathrm D_{A/R,\sigma^m} \ar[r]^{\iota_{\sigma, m}} & \mathrm D_{A/R,\sigma} \\ \partial_{\sigma^m} \ar@{\|->}[r] & \sum_{k=1}^{m} \frac 1{(k)_{q}!}\left(\prod_{i=1}^{k-1}(\sigma^m(x) - \sigma^i(x))\right)\partial_{\sigma}^{k} }.$$ Moreover, the diagram $$\label{iotdiag} \xymatrix{\mathrm D_{A/R,\sigma^m} \ar[r]\ar[d]^{\iota_{\sigma, m}} & \mathrm D_{A/R,\sigma^m}^{(\infty)} \ar@{^{(}->}[d] \ar@{^{(}->}[r] & \mathrm {End}_{R}(A) \ar@{=}[d] \\ \mathrm D_{A/R,\sigma} \ar[r] & \mathrm D_{A/R,\sigma}^{(\infty)} \ar@{^{(}->}[r] & \mathrm {End}_{R}(A) }$$ is commutative. We may assume (see the second remark following theorem \[thd\]) that we are in the generic situation $$R = \mathbb Q(t)[s], \quad A = R[x], \quad q = t \quad \mathrm{and} \quad h = s.$$ But then, the left horizontal arrows in diagram are bijective. In other words, we can identify the twisted Weyl $R$-algebras with the rings of twisted differential operators of infinite level and use corollary \[prodf\] and corollary \[formd\]. 1. In the case $h = 0$, we have thanks to corollary \[bigform\], the more concrete formula: $$\xymatrix@R0cm{\partial_{\sigma^m} \ar@{\|->}[r] & \sum_{k=1}^{m} \frac { q^{\frac {k(k-1)}{2}}(q - 1)^{k-1}} {(k)_{q}} {m-1 \choose k-1}_{q}x^{k-1} \partial_{\sigma}^{k}. }$$ 2. Assume that $R$ is $q$-divisible. If $q\mathrm{-char}(R) = p > 0$, then the hypothesis is satisfied if and only if $m < p$. Of course if $q\mathrm{-char}(R) = 0$, then the hypothesis is satisfied for any $m \in \mathbb N$. 3. Let $m,n \in \mathbb N \setminus \{0\}$. If $(mn)_{q}!$ is invertible, then $$\iota_{\sigma, mn} = \iota_{\sigma^m,n} \circ \iota_{\sigma,m}.$$ After a base change, we may assume that all $q$-integers are invertible and identify twisted Weyl $R$-algebras with rings of twisted differential operators of infinite level. The assertion then becomes a triviality. $\quad \Box$ \[compuiss\] If $(m)_{q}!$ is invertible in $R$, then the diagram $$\xymatrix{T \ar@{\|->}[d] & A[T]_{\sigma^m} \ar[r] \ar[d] & \mathrm D_{A/R,\sigma^m} \ar[d]^{\iota_{\sigma, m}}, & T \mapsto 1 + (\sigma^m(x) -x)\partial_{\sigma^m} \\ \mathrm T^m & A[T]_{\sigma} \ar[r] & \mathrm D_{A/R,\sigma} , & T \mapsto 1 + (\sigma(x) -x)\partial_{\sigma} }$$ is commutative. Again, we may assume that all $q$-integers are invertible in $R$ and use the second remark following theorem 6.13 of [@LeStumQuiros15a]. $\quad \Box$ Formal confluence in positive quantum characteristic ==================================================== In this section, we extend the formal confluence theorem to the case of positive quantum characteristic. In order to do that, it is necessary to use the $S$-twisted theory where $S$ a filtering (for division) set of positive integers. Thus, we assume here that $A$ is an $S$-twisted $R$-algebra in the sense of [@LeStumQuiros15a] (it is endowed with a compatible system of $n$-th roots $\sigma_{n}$ of $\sigma$ for all $n \in S$). We recall from [@LeStumQuiros15] that a system of roots of $q \in R$ is a compatible family $\underline q := \{q_{n}\}_{n \in S}$ of $n$-th roots of $q$. We call the system $\underline q$ admissible if $$\forall n \in S, \quad (n)_{q^n} \in R^\times.$$ This is a natural condition in order to define the $q$-rational number $(r)_{q}$ for any $r \in \mathbb N\frac 1S$. More precisely, if $r = \frac mn$ with $m \in \mathbb N$ and $n \in S$, then, $$(r)_{q} := \frac {(m)_{q_{n}}}{(n)_{q_{n}}}$$ only depends on $r$ and not on the choice of $m$ and $n$. It is convenient to introduce the following terminology: Let $\{q_{n}\}_{n \in S}$ be a system of roots of $q$ in $R$. Then, $R$ is said to be $\underline q$-divisible if $R$ is $q_{n}$-divisible for all $n \in S$. When the system is admissible, there exists a nice equivalent definition: Let $\{q_{n}\}_{n \in S}$ be an admissible system of roots of $q$. Then $R$ is $\underline q$-divisible if and only if $$\forall r \in \mathbb N\frac 1S, \quad (r)_{q} \in R^\times \cup \{0\}.$$ If $r = \frac mn$ with $m \in \mathbb N$ and $n \in S$, we have $(r)_{q} = \frac {(m)_{q_{n}}}{(n)_{q_{n}}}$. It follows that $(r)_{q} \in R^\times$ (resp. $= 0$) if and only if $(m)_{q_{n}} \in R^\times$ (resp. $= 0$). $\quad \Box$ 1. If $1 - q \in R^\times$, then $\underline q$ is admissible. In particular, if $q \in K$ where $K$ is a subfield of $R$ and $q \neq 0$, then $\underline q$ is admissible. 2. If $\underline q \subset K$ where $K$ is a subfield of $R$, then $R$ is $\underline q$-divisible. For a system $\underline q := \{q_{n}\}_{n \in S}$ of roots $q$, the following are equivalent: 1. $\underline q$ is admissible and for all $r \in \mathbb N\frac 1S \cap [0, 1]$, we have $(r)_{q} \in R^\times$. 2. For all $n \in S$, we have $(n)_{q_{n}}! \in R^\times$. First of all, the second condition also implies that $\underline q$ is admissible and we may therefore make this assumption. If $r = \frac mn$ with $m \in \mathbb N$ and $n \in S$, we know that $(r)_{q} \in R^\times$ if and only if $(m)_{q_{n}} \in R^\times$. Also, we have $r \leq 1$ if and only $m \leq n$. Thus, the second condition, which means that $(m)_{q_{n}} \in R^\times$ whenever $m \leq n$, is equivalent to the requirement that $(r)_{q} \in R^\times$ for $r \leq 1$. $\quad \Box$ Such a system of roots will be called strongly admissible. \[charcri\] If $\underline q$ is an admissible (resp. a strongly admissible) system of roots, and we write for all $n \in S$, $p_{n} := q_{n}\mathrm{-char}(R)$, then we have $$\label{carad} \forall n \in S, \quad p_{n} \nmid n \ (\mathrm{resp.}\ p_{n} > n) \quad \mathrm{or} \quad p_{n} = 0.$$ When $R$ is $\underline q$-divisible, the converse is true. By definition, $q_{n}$ is admissible (resp. strongly admissible) if and only if $(n)_{q_{n}} \in R^\times$ (resp. $(n)_{q_{n}}! \in R^\times$). By definition also, we have $p_{n} \neq 0$ and $p_{n} \mid n$ (resp. $p_{n} \leq n$) if and only if $(n)_{q_{n}} = 0$ (resp. $(n)_{q_{n}}! = 0$). We see that these two conditions are mutually exclusive in general and that they are exhaustive when $R$ is $q_{n}$-divisible. $\quad \Box$ There usually exists strongly admissible systems as the next result shows: Assume that $R$ is $\underline q$-divisible and that $q\mathrm{-char}(R) = p > 0$. Then, if $\underline q$ is a system of $p^n$-th root of $q$, it is strongly admissible. It follows from proposition 1.16 of [@LeStumQuiros15] that for all $n \in S$, we have $q_{p^n}\mathrm{-char}(R) = p^{n+1} > p^n$ and we can apply lemma \[charcri\]. $\quad \Box$ An $x \in A$ is called an $S$-twisted coordinate, or rooted twisted coordinate, (resp. an $S$-quantum coordinate, or rooted quantum coordinate,) if $x$ is a twisted (resp. quantum) coordinate for all $(A, \sigma_{n})$. We call it strong if $x - \sigma_{n}(x) \in A^\times$ for all $n \in S$. Thus, by definition, $x$ is an $S$-quantum coordinate if and only if $$\label{sigen} \forall n \in S, \quad \sigma_{n}(x) = q_{n} x + h_{n} \quad \mathrm{with} \quad q_{n}, h_{n} \in R.$$ When this is the case, we might also say $\underline q$-coordinate and call $A$ a rooted quantum $R$-algebra, an $S$-quantum $R$-algebra or a $\underline q$-$R$-algebra. We call it strong when $x$ is strong. Assume that $x$ is simultaneously a twisted coordinate and a rooted quantum coordinate on $A$ so that holds. Then, $x$ is a quantum coordinate on $A$ and if we write $\sigma(x) = qx + h$ with $q, h \in R$, then $\underline q := \{q_{n}\}_{n \in S}$ is a system of roots of $q$. If this system is admissible, we have $$\forall n \in \mathbb N, \quad h_{n} = \left(\frac 1n\right)_{q}h.$$ If we are given $m \in \mathbb N$ and $n \in S$, we know that $\sigma_{n}^m$ only depends on $r := \frac mn$ and so does $\sigma_{n}^m(x) = q_{n}^mx + (m)_{q_{n}}h_{n}$. It immediately follows that $\underline q$ is a system of roots. Moreover, the case $m = n$ implies that $\sigma(x) = qx + h$ with $q = q_{n}^n$ and $h = (n)_{q^n}h_{n}$ from which we derive the other assertions. $\quad \Box$ 1. If we are given a system $\underline q$ of roots in $R$, we can endow $A := R[x]$ or $A = R[x, x^{-1}]$ with $$\forall n \in S, \quad \sigma_{n}(x) = q_{n} x.$$ In the later case, $x$ is a strong $\underline q$-coordinate if and only if $1 -q \in R^\times$ (and then $\underline q$ is also admissible). 2. More generally, if we are given an admissible system $\underline q$ of roots of $q \in R$ and some $h \in R$, we can endow $R[x]$ with $$\forall n \in S, \quad \sigma_{n}(x) = q_{n} x +\left(\frac 1n\right)_{q}h.$$ Assume that $x$ is a $\underline q$-coordinate on $A$ with $\underline q$ strongly admissible. A rooted twisted differential $A$-module is an $A$-module $M$ endowed with a family of $\partial_{A,\sigma_{n}}$-derivations $\partial_{M,n}$ for all $n \in S$ such that whenever $n,n' \in S$ with $n \| n'$, we have $$\forall s \in M, \quad \partial_{M,n} (s) = \sum_{k=1}^{n'/n} \frac 1{(k)_{q_{n'}}!}\left(\prod_{i=1}^{k-1}(\sigma_{n}(x) - \sigma_{n'}^i(x))\right)\partial_{M,n'}^{k}(s).$$ They form a category that we will denote by $\partial_{A,\underline \sigma}\mathrm{-Mod}^{\mathrm{root}}$. If $A$ is a $\underline q$-$R$-algebra with $\underline q$ strongly admissible, then the rooted twisted Weyl algebra of $A$ is $$\mathrm D_{A/R,\underline \sigma}:= \varinjlim \mathrm D_{A/R,\sigma_{n}}$$ with transition maps (coming from lemma \[explem\]) for $n,n' \in S$ with $n \| n'$ given by $$\xymatrix@R0cm{\mathrm D_{A/R,\sigma_{n}} \ar[r]^{\iota_{\sigma_{n'}, n'/n}} & \mathrm D_{A/R,\sigma_{n'}} \\ \partial_{\sigma_{n}} \ar@{\|->}[r] & \sum_{k=1}^{n'/n} \frac 1{(k)_{q_{n'}}!}\left(\prod_{i=1}^{k-1}(\sigma_{n}(x) - \sigma_{n'}^i(x))\right)\partial_{\sigma_{n'}}^{k} }.$$ \[abag\] Assume that $A$ is a $\underline q$-$R$-algebra with $\underline q$ strongly admissible. Then the rooted twisted differential $A$-modules form an abelian category with sufficiently many injective and projective objects. Actually, if $M$ is a $\mathrm D_{A/R,\underline \sigma}$-module, then the maps $$\xymatrix@R0cm{ M \ar[rr]^{\partial_{M,n}} && M \\ s \ar@{\|->}[rr] && \partial_{M,n} s}$$ turn $M$ into a rooted twisted differential $A$-module and we obtain an equivalence (an isomorphism) of categories $$\mathrm D_{A/R,\underline \sigma}\mathrm{-Mod} \simeq \partial_{A,\underline \sigma}\mathrm{-Mod}^{\mathrm{root}}.$$ This is a consequence of proposition 5.6 of [@LeStumQuiros15a]. $\quad \Box$ Recall from [@LeStumQuiros15a] that $A[T^{\frac 1S}]_{\underline \sigma}$ denotes the non commutative Puiseux polynomial ring with the commutation rule $$\forall m \in \mathbb N, \forall n \in S, \quad T^{\frac mn}x = \sigma_{n}^m(x) T^{\frac mn}.$$ One can also consider the notion of $\underline \sigma_{A}$-module: this is an $A$-module endowed with a compatible family of $\sigma_{A,n}$-linear endomorphisms $\sigma_{M,n}$. There exists an equivalence (an isomorphism) between the category of $A[T^{\frac 1S}]_{\underline \sigma}$-modules and the category of $\underline \sigma_{A}$-modules. Assume that $x$ is a $\underline q$-coordinate on $A$ with $\underline q$ strongly admissible. Then there exists a unique $A$-linear homomorphism of rings $$\label{under} \xymatrix@R0cm{A[T^{\frac 1S}]_{\underline \sigma} \ar[r] & \mathrm D_{A/R, \underline \sigma} \\ T^{\frac 1n} \ar@{\|->}[r] & 1 + (\sigma_{n}(x) -x) \partial_{\sigma_{n}}}$$ inducing a functor $$\label{funcev} \partial_{A,\underline \sigma}\mathrm{-Mod} \to \underline \sigma_{A}\mathrm{-Mod}.$$ If $x$ is a strong $\underline q$-coordinate, then the map is an isomorphism and the functor is an equivalence. This will follow from theorem 6.13 of [@LeStumQuiros15a] once we know that the various maps for different $n$’s are all compatible. More precisely, we have to check that the differential operators $1 + (\sigma_{n}(x) -x) \partial_{\sigma_{n}}$ form a system of roots in $\mathrm D_{\underline \sigma}$. But this follows from proposition \[compuiss\]. $\quad \Box$ \[mainth\] Let $R$ be a $\mathbb Q$-algebra, $\underline q$ a strongly admissible system of roots in $R$ and $A$ a $\underline q$-$R$-algebra. Then, there exists a canonical $A$-linear map $$\label{fqc2} \mathrm D_{A/R, \underline \sigma} \to \widehat {\mathrm D}_{A/R}.$$ If $R$ is $\underline q$-divisible and $\underline q$ is infinite, then the map has dense image. First of all, the map is obtained by taking the direct limit of the maps for all $\sigma_{n}$’s. Now, we set $p_{n} := q_{n}\mathrm{-char}(R)$ for all $n \in S$ and we apply the confluence theorem \[confalg\] to $\sigma_{n}$. If $p_{n} = 0$, we are done. Otherwise, the theorem tells us that the bottom map in the following commutative diagram is surjective: $$\xymatrix{ \mathrm D_{\underline \sigma} \ar[r] & \widehat {\mathrm D} \ar[d] \\ \mathrm D_{\sigma_{n}} \ar@{->>}[r] \ar[u] & {\mathrm D}/\partial^{p_{n}} }.$$ And we proved in lemma \[charcri\] that for all $n \in S$, we have $p_{n} > n$. Since $\underline q$ is infinite, we have $p_{n} \to \infty$ and we see that the image of the upper map is dense. $\quad \Box$ Assume moreover, that $x$ is a strong $\underline q$-coordinate on $A$. Then, the $A$-linear map $$\xymatrix@R0cm{A[T^{\frac 1S}]_{\underline \sigma} \ar[r] & \widehat{\mathrm D}_{A/R} \\ T^{\frac mn} \ar@{\|->}[r] & \sum_{k=0}^{\infty} \frac 1{k!}(\sigma_{n}^m(x) -x)^{k} \partial^{k}}$$ has dense image. The formula comes from corollary \[corsig\] $\quad \Box$ Theorem \[mainth\] (and its corollary) applies in particular when $q$ is a primitive $p$-th root of unity in some algebraically closed field $K$ of characteristic zero, $\underline q$ is a system of $p^n$-th roots of $q$ in $K$, $R$ is an algebra containing $K$, $A = R[x, x^{-1}]$ and $\sigma_{n}(x) = q_{n}x$ (we should actually write $\sigma_{p^n}$ and $q_{p^n}$ but we will try to make the notations easier to use). As we already saw, in this situation, $\mathrm D := \mathrm D_{A/R}$ is the non commutative ring $R[x, x^{-1}, \partial]$ with the commutation rule $\partial x = x \partial +1$ and we may see $\widehat{\mathrm D}$ (which is not a ring) as $R[x, x^{-1}][[\partial]]$. We also want to understand the left hand side and see that $\mathrm D_{n} := \mathrm D_{A/R,\sigma_{n}}$ is the non commutative ring $R[x, x^{-1}, \partial_{n}]$ with the commutation rule $\partial_{n} x = q_{n}x \partial_{n} +1$. The transition maps are given by the rather tricky formulas $$\partial_{n} \mapsto \sum_{k=1}^{p^{n'-n}} \frac { q_{n'}^{\frac {k(k-1)}{2}}(q_{n'} - 1)^{k-1}} {(k)_{q_{n'}}} {p^{n'-n}-1 \choose k-1}_{q_{n'}}x^{k-1} \partial_{n'}^{k},$$ and the map of the theorem satisfies $$\partial_{n} \mapsto \sum_{k=1}^{\infty} \frac 1 {k!}(q_{n}-1)^{k-1}x^{k-1} \partial^{k}.$$ Also, the map of the corollary is given by $$T^{r} \mapsto \sum_{k=1}^{\infty} \frac 1 {k!}(q^{r} -1)^{k} x^{k}\partial^{k}$$ if we set $q^{m/n} := q_{n}^m$. Alexei Belov-Kanel and Maxime Konsevich proved in [@BelovKanelKonsevich07] that the Jacobian conjecture is stably equivalent to the Dixmier conjecture [@Dixmier68]. According to the later, any endomorphism of a Weyl algebra over $\mathbb C$ is an automorphism. And this is still a conjecture even in dimension one. On the other hand, when $q$ is a primitive $p$-th root of unity, then the quantum Weyl algebra $D_{\sigma}$ does not satisfy the Dixmier conjecture but it is an Azumaya algebra. In particular, checking that an endomorphism is an automorphism can be done on the center. As explained by Backelin in [@Backelin11], it is appealing to attack the Dixmier conjecture through quantum Weyl algebras and one can hope that theorem \[mainth\] might provide a tool for this quest. [^1]: Supported by grant MTM2012-35849 from Ministerio de Economía y Competitividad (Spain).
--- abstract: 'We now have a rich and growing set of modeling tools and algorithms for inducing linguistic structure from text that is less than fully annotated. In this paper, we discuss some of the weaknesses of our current methodology. We present a new abstract framework for evaluating natural language processing (NLP) models in general and unsupervised NLP models in particular. The central idea is to make explicit certain adversarial roles among researchers, so that the different roles in an evaluation are more clearly defined and performers of all roles are offered ways to make measurable contributions to the larger goal. Adopting this approach may help to characterize model successes and failures by encouraging earlier consideration of error analysis. The framework can be instantiated in a variety of ways, simulating some familiar intrinsic and extrinsic evaluations as well as some new evaluations.' author: - \| Noah A. Smith[^1]\ School of Computer Science\ Carnegie Mellon University\ Pittsburgh, PA 15213, USA\ `nasmith@cs.cmu.edu` bibliography: - 'lsp.bib' title: Adversarial Evaluation for Models of Natural Language --- Introduction ============ This paper presents a new approach to evaluating computational models of natural language, based on adversarial roles performed by different researchers or their models. We begin in §\[se:current\] with a brief review of current evaluation strategies in NLP. We then turn to coupled adversarial evaluations inspired by perplexity (§\[se:perplexity\]) and the traditional roles of linguists (§\[se:zellig\]). The two-performer setup is formalized in §\[se:formal\]. We then consider the origins of the data and growing awareness of the importance of context on language use (§\[se:context\]) and provide a three-performer setup, in which a third performer manages data selection (§\[se:formal2\]). We close with a few open questions (§\[se:open-questions\]). Current Evaluation Strategies in NLP {#se:current} ==================================== At present, NLP models are primarily evaluated in three ways: intrinsic evaluations, in which model predictions are compared to manually produced “gold-standard” output; extrinsic evaluations, in which output is passed downstream to an application whose performance can in turn be evaluated; algorithm competitions with predefined, formal criteria for success (less typical in NLP but used in some related areas); and (for probabilistic models) perplexity evaluations, in which the model is used to assign a likelihood score to unseen data, and this score is compared to other models’ scores. We assume the reader is familiar with these styles of evaluation and consider their strengths and weaknesses in turn. Intrinsic Evaluations: <span style="font-variant:small-caps;">MatchLinguist</span> {#se:match-linguist} ---------------------------------------------------------------------------------- Models can be evaluated by comparing their predictions on input data to which they have never previously been exposed to the predictions of human experts (known as “gold-standard” linguistic annotations). This is the dominant intrinsic evaluation approach in NLP. In @smith-05c, we introduced the term “<span style="font-variant:small-caps;">MatchLinguist</span>” to refer to the task of automatically reproducing gold-standard linguistic annotations. The strength of intrinsic evaluations is that, once gold-standard annotations are provided, they can be reused forever. Once a scoring algorithm is agreed upon, many researchers can evaluate their models on the same data, making quantitative comparison easy. Unfortunately, this makes it difficult to draw conclusions about how well performance results generalize to other linguistic samples (e.g., in different genres, topics, dialects, languages, etc.). Indeed, some have conjectured that long-term reuse of an annotated test dataset can lead to community-wide “overfitting” to the pecularities of the data and the conventions used in annotating it. ([@wagstaff12], recently expressed concern over this trend in the field of machine learning, emphasizing the gap between such datasets and “real world” problems.) The recent trend of developing new, small testing datasets, often in a range of languages or genres, helps to alleviate this problem (e.g., the CoNLL depedency parsing shared tasks; [@buchholz-06]). A major problem with intrinsic evaluations is that they assume the phenomenon of interest is already well-enough understood that linguistic experts have identified the best representation and trained annotators to produce it accurately. Anyone who has worked on an annotation project, however, knows that interaction with the data always leads to evolution within the annotation scheme. Intrinsic evaluation further commits the fallacy that human evaluations are worthy of replication. Annotators are only human, and we have very restricted ways of evaluating them (e.g., inter-annotator agreement). These annotator evaluations are often incomplete, ignoring major factors like the kind and amount of training annotators have been subjected to and the learning curve of the annotators. A model that succeeds at the <span style="font-variant:small-caps;">MatchLinguist</span> task can be said to have reproduced what a particular set of annotators, with a particular kind of training, on a particular kind of data, within a particular amount of time, would generate on the test set. Our view is that drawing stronger conclusions about the quality of such a model may be too bold. Even if we accept human annotations as correct, the intrinsic evaluation strategy only permits comparison between models that produce similar output. We cannot use it to test two divergent theories of linguistic structure without introducing more automation or manual effort.[^2] We also cannot use it to test models of the interaction of different levels of structure (e.g., morphology and syntax), unless all levels of interest are part of the human annotation effort or all models make use of the same preprocessing mechanisms, whether manual or automatic. The more we are forced to incorporate pre- and post-processing in order to evaluate our models, the more narrow our claims must be. Finally, the cost of employing linguistic experts to annotate data is often cited as one of the main motivating factors for unsupervised NLP. When we consider that any annotation project used to evaluate an unsupervised NLP model has been necessarily limited in how much data annotators could annotate and how many iterations they could make over the data, it seems that using these cost-constrained annotations as a gold-standard to match is misguided. Unsupervised automatic learners should be able to reason about far more data far more consistently; should matching what resource-limited humans can do really be our aim? As noted by Alex Clark (personal communication), linguists really perform two tasks in creating an annotated dataset, and the labor is likely divided. One task is defining the formalism: what is the set of analyses that are possible for each input? The other is selecting the correct one for each input. Supervised NLP models focus only on the latter task, while unsupervised NLP models may perform both tasks, depending on the underlying assumptions. Though it has not been consistently articulated this way, and not all NLP researchers are likely to agree, perhaps we should consider the goal of doing linguistics—describing and explaining the phenomena of human language, or defining formalisms, as above—in ways that unaided humans cannot. Indeed, as long as the mark of success for an unsupervised linguistic learner is to closely match what annotation scheme designers and annotators believe they already know about language, we cannot claim that <span style="font-variant:small-caps;">MatchLinguist</span>-evaluations of unsupervised NLP models have anything to do with advancing the scientific study of language. Extrinsic Evaluations: Passing the Buck --------------------------------------- Extrinsic evaluations are attractive because they allow NLP modelers to make claims of “usefulness” about their models. Real-world applications that use NLP models of various kinds include machine translation systems, search engines, information extraction systems, and question answering systems. Insofar as evaluation of these systems’ quality is uncontroversial, there is little to be said against an argument for the usefulness of a model whose output improves the downstream state of the art. Unfortunately, system evaluation remains fraught with debate for most of these downstream applications. There is also a practical concern: evaluating an NLP model in a downstream system requires the ability to incorporate that model’s functionality within such a system. The open-source versions of such applications do not generally provide a “plug-and-play” architecture for linguistically annotated input, and if they do, there are strong assumptions about what kind of annotation is to be provided. How to use a particular linguistic annotation within any given application is itself a research question. Further, as in intrinsic evaluations, models that make use of different kinds of representations will not generally be comparable in downstream applications, since much will depend on the process of incorporating the annotations into the application. Finally, applications change fast. There is value in having stable mechanisms to compare models; yet the community tends to show little interest in performance gains in a downstream application that is no longer the state of the art, since the results may not generalize to newer, better systems. All is not hopeless, and extrinsic evaluations should continue to provide evidence for model quality. However, we are not optimistic that downstream applications can serve as the primary evaluation mechanism for NLP models, due to these challenges of access and stability. Algorithm Competitions ---------------------- Some research agendas lead naturally to the design of competitions in which a well-defined problem is stated formally and benchmarks to test an algorithm’s success are constructed by experts. A notable example is the Omphalos competition [@starkie04], in which competitors constructed context-free grammar learning algorithms. Theoretical and practical matters were taken quite seriously; datasets were designed to be provably sufficient for identifying the language but beyond the capabilities of the state of the art at the time. Some of the benchmarks were constructed around natural language phenomena. Though our proposed adversarial evaluations here seek to drive natural language modeling research, not formal language acquisition, much of what was done by the Omphalos designers and competitors focused on the construction of negative examples, which also play a key role here.[^3] Perplexity ---------- The idea of using model likelihood on test data to compare probabilistic models arose in the speech recognition community, where it was applied to the evaluation of language models. It provides a simple way to compare any models that properly assign probability mass to linguistic data. Perplexity evaluations were mostly abandoned in the 1990s when it became clear that perplexity reductions did not correlate with word error rate reductions on speech recognition tasks. In general, it is widely known that having a good probablistic model of data need not have anything to do with having a model that performs well in intrinsic or extrinsic evaluations. Perplexity’s use as a scientific tool is less controversial, though it is not widespread or widely accepted in computational linguistics today, with the possible exception of the Bayesian topic modeling subcommunity [see, e.g., @blei-03].[^4] There are also some key difficulties. First, only probabilistic models define perplexity scores. While probabilistic modeling has many attractions, requiring that researchers adopt that framework in order to compare with other work is unnecessarily exclusionary. Second, two models’ perplexity scores are only comparable if they define exactly the same event space. In practice, this means prior agreement on the vocabulary and on handling of out-of-vocabulary terms. This must be done with great care, because events that are assigned very low probability by a model can have a large effect on the model’s perplexity score. (Assigning zero, in particular, leads to infinite perplexity. Perplexity offers no way to rank two models that have infinite perplexity, no matter how sharp their differences on the non-zero-probablity instances.) Focusing on perplexity can lead to over-attention to smoothing algorithms, the details of which may be less important in large-data settings [@brants-07]. And finally, many models in use today involve latent structures, so that perplexity—which is calculated by marginal inference—can only be calculated approximately. Conclusions based on approximate perplexity comparisons, with each researcher deciding on his or her own approximations, are suspect at best. Improving Perplexity: Claude, the Chooser {#se:perplexity} ========================================= Perplexity, in its most general form, requires the performer to define a probability distribution $p$ over some event space $\mathcal{X}$. During the evaluation period, a series of events $\langle x_1, \ldots, x_N\rangle$ are assumed to be drawn i.i.d. from the “true” distribution, which we denote by $p^\ast$. The estimated perplexity is given by: $$\exp_2 {\displaystyle \left( - \frac{1}{N} \sum_{n=1}^N \log_2 p(x_n) \right)} \hphantom{\sum} \approx \exp_2 {\displaystyle \left( - \sum_{x\in \mathcal{X}} p^\ast(x) \log_2 p(x) \right)}$$ (On the right is the true perplexity, approximated on the left using a test sample. We use $\exp_2 (a)$ to more clearly denote $2^a$.) As we have noted, the event space $\mathcal{X}$, is often an infinitely large discrete space (e.g., the space of strings for a given alphabet). Performers must assign nonzero probability to all $x \in \mathcal{X}$, and be able to compute that probability, to be able to compete. Suppose we replace the calculation with a choice between two elements of $\mathcal{X}$, $x$ and $y$, the former being true data from $p^\ast$, and the latter being contrived or synthesized data. These two elements would be presented in random order, hiding the provenance of each. Performers might use probabilistic models to make this decision (e.g., choosing $x$ iff $p(x) > p(y)$, and choosing $y$ otherwise), but they need not do so. Any approach could be applied to make the choice. Insofar as the ability to distinguish real data from contrived data is of scientific or practical interest, we would prefer a model with greater average accuracy on this binary task. We will conflate the engineer of such a model and the model itself, calling both “Claude.”[^5] Claude takes as input two instances from $\mathcal{X}$—sentences in the language modeling case—denoted by $x$ and $y$. $x$ is assumed to be drawn from a true linguistic sample, and the other, $y$, to be contrived by an adversary who is given access to $x$, and whom we call “Zellig.”[^6] We will return to Zellig in §\[se:zellig\], for now taking it for granted that Zellig’s role can be meaningfully performed. We remark on a few observations about this task: - The accuracy score is easy to calculate, objective, and does not hinge on any human input beyond the choice of the test data $\langle x_1, \ldots, x_N\rangle$ and the machinations of Zellig to construct confusion instances $\langle y_1, \ldots, y_N\rangle$. - Comparing Claude to an alternative, competing model “Chloe” is straightforward, regardless of their internal operations and representations of the data. In particular, they need not use the same theory (or any theory), and they need not use probabilistic models. They only need to distinguish true data from contrived data. - Zellig’s role is crucial. If Zellig creates $y$ through some “safe,” trivial operation on $x$ (e.g., replacing common words with common words of the same syntactic category, or, worse, just copying $x$; or selecting instances from a corpus that closely approximates the same $p^\ast$ whence $x$ is drawn), then for reasonably large $N$ no Claude will be able to achieve better than 50% accuracy. On the other hand, if Zellig is built to be completely ignorant (e.g., sampling from a character $n$-gram distribution), it should be easy for any Claude to achieve very high performance. Zellig, Transformer of Data {#se:zellig} =========================== It quickly becomes clear that the quality of a Zellig must be defined in terms of contemporary Claudes (and vice versa). A good Zellig, in short, is one that stumps the Claudes of the day, but not all to the same degree. In some sense, Zellig is like an extrinsic task evaluation, except that rather than taking a model of language’s predictions as output, we imagine that it challenges that model to be aware of phenomena that hold in real linguistic data but not in corrupted versions of those data. We consider next three kinds of Zelligs, each suggesting a different research goal that is interesting regardless of its role in stumping Claudes. Human Zellig ------------ A human linguistic expert who has a theory of language might manually corrupt a real linguistic utterance $x$ to create an ill-formed similar utterance $y$. The minimal pair then represents a prediction of Zellig’s theory. Supporting evidence for the theory might come from a fluent (or native-speaker) human Claude, who selects which of the two utterances is ill-formed. In this setup, if Claude performs well (across many instances), Zellig’s theory is given some credence. This is simply an experiment for testing claims about well-formedness in natural languages. A more useful variation might compare Zellig to another expert, Zelda, to see whose theory better predicts native speaker judgments. This alters the role of a linguist from annotator (§\[se:match-linguist\]) to creative illustrator. Rather than constructing theories that seek to account “horizontally” for all phenomena at a particular level of description in a natural language (e.g., syntax), the linguistic expert is free to consider “vertical” interactions among any levels at all that are appropriate to identifying selected phenomena in a language. Model of Language Zellig ------------------------ It is, of course, a small step to imagine that human Zellig would write a program to perform the $x \mapsto y$ transformation. Indeed, many existing models of language can be used to construct a Zellig. For example, a probabilistic language model might be queried to find a string that has high probability and low (but nonzero) “distance” from $x$: $$y = \operatorname{arg\,max}_{x' \in \mathcal{X} : 1 \le \Delta(x', x) \le \delta} p(x')$$ where $\Delta(x', x)$ might be the Hamming distance or some other metric on strings.[^7] A model of linguistic constraints might identify the constraints holding in $x$, then make a change that violates one of them. A little-discussed property that any computational model of language might be expected to have is the ability to produce instances in violation of the underlying theory. We propose that explicit construction of algorithms for this use-case is motivated as a new way to validate models in computational linguistics, and further may lead to new insights about computational models of linguistic phenomena. Note further that the same model, if it provides algorithms for both kinds of queries, might serve as Zellig or Claude in different evaluations. Text-Generating System Zellig {#se:generation} ----------------------------- Another kind of Zellig can be constructed as an NLP system that generates text as output. For example, suppose that each $x_n$ is a sentence in English that was translated (by a human) from French. Assume $x_n$ comes packaged with metadata, which we will denote $m_n$, which is comprised of the original French sentence. If Zellig is a machine translation system, then $y_n$ will be an automatic translation of French sentence $m_n$. Another scenario might consider question answering: $m_n$ is a question, $x_n$ its human answer, and $y_n$ the answer from a system. Two versions might be considered here, one where Claude observes $m_n$ (encouraging evaluation of adequacy of translation or correctness of question answering), and one where it is kept hidden (judging only fluency). The setup therefore provides a new way to perform system evaluations, exploiting models of language that seek to pass distinguishability tests. Defining the Two-Performer Evaluation {#se:formal} ===================================== ![The basic setup for the $Z$-task and the $C$-task (see §\[se:formal\]). Colors and shapes were chosen to match the graphics used in the talk on June 7, 2012. Here $M$ is not made available to Claude, but in some settings it might be. \[fig:basic-setup\]](basic-adversarial-eval) A formal definition of the basic adversarial evaluation follows; Figure \[fig:basic-setup\] provides an illustration. We assume a random source of linguistic instances, the probability distribution $p^\ast$. Each instance $x_n$ is drawn from $p^\ast$. Optionally, $p^\ast$ defines a joint distribution over the instance random variable $X$ and a metadata random variable $M$. We assume that instances are generated at fixed periodic intervals of length $t$. The evaluation involves two performers, Zellig and Claude, who—though in an adversarial relationship—are not in direct competition with each other. Zellig can be compared to other performers of the $Z$-task, defined below, using the same sample and Claude. Claude can be compared to other performers of the $C$-task, defined below, using the same sample and Zellig. On each iteration (indexed by $n$), Zellig takes $x_n$ (and optionally $m_n$) and constructs an object $y_n$, purportedly from the support of $p^\ast$. If there is metadata $m_n$, then Zellig should seek $y_n$ that is well-paired with $m_n$. It is to Zellig’s advantage to choose $y_n \not = x_n$. Zellig must perform all necessary computation within time $t$, before the next iteration.[^8] This is the $Z$-task. On each iteration, the pair $(x_n, y_n)$ is permuted uniform-randomly and presented to Claude. Claude must guess which element is the false instance $y_n$; all necessary computation must be within time $t$, before the next iteration.[^9] We denote by $z_n$ Claude’s choice. (It is helpful to think of Claude as one step behind Zellig, so that at an arbitrary point in time, Zellig is generating $y_n$ and Claude is guessing between $x_{n-1}$ and $y_{n-1}$.) This is the $C$-task. Over $N$ instances, the score is defined by: $$S = \frac{1}{N} \sum_{n=1}^N \boldsymbol{1}\{ z_n = y_n \} \hphantom{\sum} \approx \mathbb{E}_{p^\ast(X)}[ \boldsymbol{1}\{Z = Y\} ] \label{eq:score}$$ For two competing performers of the $Z$-task, the one achieving the lower* $S$ is the winner; this is a Zellig who more successfully deceived Claude. For two competing performers of the $C$-task, the one achieving the higher $S$ is the winner; this is a Claude who has more successfully distinguished real data from corrupted data. #### A connection to cryptography. {#crypto-paragraph} Our $C$-task bears some similarity to a particular notion of security of an encryption scheme called “indistinguishability of encryptions” [@bellare98].[^10] To highlight the similarity, we will employ Claude as an analyst and Zellig as an encryption oracle within a particular kind of attack known as “chosen ciphertext.” In this evaluation, Claude chooses two plaintext messages and sends both to Zellig. Zellig chooses one at his discretion, encrypts it, and sends the ciphertext to Claude. Zellig succeeds—and the scheme declared secure—if Claude can do no better than chance at guessing which message Zellig chose. Returning to our linguistic setup, for Zellig to have this ability would be evidence for “strong linguistic knowledge.” Much like the notion of “provable security,” “strong linguistic knowledge” should perhaps be regarded with skepticism; a scheme can be considered secure only until it is broken, and a Zellig remains respectable only as long as the state-of-the-art Claudes cannot consistently perform well on his output. (Note, however, that the evaluation we have proposed does not allow Claude any control over the inputs to Zellig.) #### Spam detection. Several members of the audience at the June 7 talk noticed the similarity between Zellig and Claude’s activities and the adversarial relationship between spammers and spam detection software. Of course, spammers are constrained by the speech act they seek to execute through the message $y$, and they have no analogue to “$x$,” unlike Zellig. Successful spam detection systems presumably exploit this (and metadata $m$) heavily. (For an interesting recent discussion of spammer strategy, specifically considering the linguistic choices involved, see [@herley12].) #### Game theory. We have deliberately avoided discussing the proposed evaluation in game-theoretic terms. On reading a draft of this proposal, economist Bryan Routledge exclaimed, “I want this data,” seeing long-term transcripts of the choices by Claudes and Zelligs as inherently interesting in studying the dynamics of “co-evolution” in evolutionary game theory [@weibull95]. We leave this possible point of exploration for future work. On Context {#se:context} ========== The reader may have noticed the introduction of a largely underspecified element, metadata $m_n$ on each iteration, in §\[se:generation\]. The importance of context—encoded in our setup as metadata—to interpretation and generation of language has been noted with increasing intensity in recent discourse about NLP. Context can include well-studied variables like the dialect or genre in which language is produced, or simply “co-text” (a term used in a recent presentation by Graeme Hirst to refer to nearby text), or farther-removed representations of the situation in which the text arose.[^11] In recent research efforts, we and others have made the prediction of contextual information from text a task of its own, often predicting future contextual variables from text in a forecasting setup (e.g., [@kogan-09]). A strong statement of the importance of context is to claim that $p^\ast$ is such an over-simplification of reality as to be useless. Indeed, all corpora currently used to construct and evaluate NLP systems come with some description of the provenance of the text. The community already views the construction of contextualized linguistic resources as a valuable research effort; what we lack are frameworks for objectively evaluating the quality of such datasets or their relevance to scientific or engineering efforts, and the early incorporation of this information into our models. The evaluation framework proposed here offers a first step toward imposing the same kind of rigorous evaluation on data selection methods as on data modeling methods. We noted that one role of linguistic experts in this framework is as a human Zellig performer who contrives corrupted instances $y$ from observed linguistic instances $x$. By introducing a third performer, called “John,”[^12] we propose another role for linguistic experts—the curation of linguistic datasets with contextual descriptions, and the construction of systems to perform this task. We have reached a time when raw text data is available in massive amounts, often with metadata as objets trouvés; the collection and further description of such data (adding to the metadata) naturally feeds the adversarial evaluations we have proposed so far. A useful by-product of John’s performance is the generation of metadata that enables error analysis. Many researchers desire understanding of the kinds of systematic “mistakes” that NLP systems make, but we have very few methodological tools for this kind of characterization. Many researchers resort to fine-grained statistics on errors or selection of illustrative examples, but these do little to show the way forward for future research. Correlating errors to well-defined phenomena marked in metadata may be a more useful tool in gaining an understanding of what a given model (performing either as Zellig or as Claude) solves or does not solve. Defining the Three-Performer Evaluation {#se:formal2} ======================================= ![The full setup for the $J$-task, $Z$-task, the $C$-task (see §\[se:formal2\]). Colors and shapes were chosen to match the graphics used in the talk on June 7, 2012. Here $M$ is not made available to Claude, but in some settings it might be. \[fig:full-setup\]](full-adversarial-eval) The full three-performer adversarial evaluation is illustrated in Figure \[fig:full-setup\]. Claude and Zellig are exactly as before (§\[se:formal\]). We introduce performer John, who replaces “$p^\ast$” as a source of pairs $(x_n, m_n)$. John’s contribution is to collect data that reveal phenomena on contextualized natural language, or to implement algorithms that collect such data. Researchers performing this task—the $J$-task—are expected to justify the relevance of the selection method for scientific exploration and/or engineering NLP systems. Further, John can be compared to another $J$-task performer “Jennifer” by selecting a Zellig-Claude pair and measuring the scores $S$ as achieved before. John is said to outperform Jennifer, given Zellig and Claude, if Zellig’s task has been made harder with John’s data than with Jennifer’s (i.e., a higher score results). ------------------- ------------------------------------------------ ---------- $C$-task (Claude) distinguish data from non-data high $S$ $Z$-task (Zellig) generate corruptions of data low $S$ $J$-task (John) select data to exemplify phenomena of interest high $S$ ------------------- ------------------------------------------------ ---------- : Summary of the three performers. Any evaluation on the three tasks ($C$-task, $Z$-task, and $J$-task) requires the presence of the other two performers. Just as evaluations on multiple datasets with different properties are often used to make stronger arguments in NLP, an evaluation can be strengthened by considering a range of other performers. For example, in evaluating Claude, we might compare his performance against a baseline Chloe on a range of evaluations $\mathcal{Z} \times \mathcal{J}$, where $\mathcal{Z}$ is a set of existing Zelligs and $\mathcal{J}$ a set of existing Johns. Higher-level analysis can be performed by relating $S$ to properties of the Zelligs or Johns. The wider the range of other performers, the more confident we can be that an evaluation result is not due to idiosyncrasies. The discussion has been fairly abstract; we have deliberately avoided making assumptions about what kinds of resources a performer might have access to constructing the algorithm or model to perform a task. The original idea was conceived out of skepticism toward evaluations for unsupervised NLP models, but we believe the framework is appropriate regardless of the level of supervision. Indeed, the framework forces us to differentiate two different kinds of supervision: 1. Supervision from the <span style="font-variant:small-caps;">MatchLinguist</span> perspective, in which expert annotations are provided in support of the task. 2. Supervision within the task: - In the $C$-task, observations of tuples $(m_n, x_n, y_n)$, with $X$ and $Y$ labeled as such (rather than randomly permuted), for a given John-Zellig pair. - In the $Z$-task, observations of a given Claude’s choices in response to the generated $y_n$, given each $(m_n, x_n)$ pair from a given John. - In the $J$-task, observations of the generated $y_n$ from a given Zellig and $z_n$ from a given Claude in response to each $(m_n, x_n)$ produced by the performer. We call a round of evaluation $n$ “transparent” from the perspective of a given performer if that performer can see the other performers’ actions clearly in the round.[^13] From each performer’s perspective, <span style="font-variant:small-caps;">MatchLinguist</span> supervision, though perhaps useful, is indirect. From a learning perspective, it is the second kind of supervision that is expected to give the most information. Given the framework, it is easy to explain supervised, semi-supervised, and unsupervised versions of the evaluation. We must simply specify the schedule of observations—transparent and non-transparent—that occur before evaluation takes place. A few interesting cases include: - zero observation rounds prior to evaluation; - a fixed number of transparent rounds (“supervised”) prior to evaluation; - a fixed number of transparent rounds followed by a fixed number of non-transparent rounds (“semi-supervised”) prior to evaluation; or - a fixed number of non-transparent rounds (“unsupervised”) prior to evaluation. - Orthogonal to all of the above, performers might adapt their performance during the evaluation’s non-transparent rounds. Regardless of which framework is selected, the explanation of any performer should clarify what resources were used to construct it, and how, as in current NLP research. For frameworks that involve adaptation, reporting how the score changes over time (e.g., as a time series) would be useful for comparing convergence rates. Finally, we suggest again that any of the roles might be played by humans. Such an exercise might be useful in establishing human “upper bounds” (risking the problems underlying <span style="font-variant:small-caps;">MatchLinguist</span>), or in training any of the performers. An example suggested by Amber Wilcox-O’Hearn is a human Claude who provides supervision for a supervised learner Zellig. Open Questions {#se:open-questions} ============== #### Collusion. We have assumed that, for clarity’s sake, collusion among performers should not be allowed. However, collusion between any pair might lead to more challenging evaluations for the third performer and might be worth considering. #### Cheating. Is it possible to cheat? Validity of evaluations in this framework may rest on limiting the resources available to some performers. We believe, for example, that it is possible to cheat if John does not have access to external data sources unavailable to Zellig and Claude. #### Should we do it? A reasonable concern, raised by Dan Bikel, is that the collective actions of a community of Johns, Zelligs, and Claudes might not lead to improved models of natural language. The ideas laid out here are intended to refocus our evaluations so as to build better models of the phenomena inherent in language. Yet it is not hard to imagine that researchers would collectively over-attend to $S$ (Equation \[eq:score\]) and lose sight of those phenomena. We therefore propose a modest start. A few straightforward John, Zellig, and Claude performers should be publicly released, perhaps through an API allowing inspection of all data, algorithms, and scores. If the API is extended to allow new performers to join in and test performance, we conjecture that the adversarial framework will begin to be used to provide evidence for the quality of newly developed models. Whether this evidence is judged meaningful by the research community will depend, of course, on the particulars. We believe, though, that a critical assessment of our evaluation practices, and the introduction of some new ones, can only benefit future research. \[lastpage\] [^1]: The ideas in this paper were first presented in a talk at the NAACL-HLT Workshop on Inducing Linguistic Structure, June 7, 2012, invited by organizers Phil Blunsom, Trevor Cohn, and Joao Graça; many thanks to members of that audience for helpful comments and thoughtful questions. Thanks also to those with whom conversations about these matters have sharpened my thinking: Fred Jelinek, Dan Bikel, Jordan Boyd-Graber, Alex Clark, Jaime Carbonell, Chris Dyer, Kevin Gimpel, Geoff Gordon, Lori Levin, Adam Lopez, André Martins, Behrang Mohit, Brendan O’Connor, Kemal Oflazer, Philip Resnik, Bryan Routledge, Nathan Schneider, Yanchuan Sim, Amber Wilcox-O’Hearn, and Dani Yogatama. Errors are, of course, the responsibility of the author. [^2]: For an example of this problem, consider the literature on unsupervised part-of-speech tagging. A whole range of evaluation scores for this problem exist, each proposing a different way of dealing with the incommensurability of categories that come from linguistic annotators or various unsupervised learning models. [^3]: We gratefully acknowledge Alex Clark for bringing this to our attention. [^4]: Even in that community, the usefulness of perplexity has been brought into question; @chang09 found that topic models with better perplexity may infer relatively less semantically meaningful topics. [^5]: After Claude Shannon (1916–2001), the father of information theory. [^6]: After Zellig Harris (1909–1992), linguist and methodologist of science. [^7]: In past work evaluated within the <span style="font-variant:small-caps;">MatchLinguist</span> paradigm, we considered functions that generated large “neighborhood” sets of strings similar to $x$ but perturbed in ways expected to corrupt linguistic quality. The corrupting function was heuristic and served the definition of a “contrastive” objective function for learning. See, for example, @smith-05c. [^8]: If Zellig does not provide $y_n$ in time $t$, a reasonable default is to set $y_n = x_n$, thereby giving a free point to Claude. [^9]: If Claude does not make a choice in time $t$, then a uniform-random choice should be made. [^10]: We gratefully acknowledge Amber Wilcox-O’Hearn for introducing this connection in personal communication. [^11]: Social media platforms offer a rich set of possibilities for the last of these, since messages are broadcast from an identifiable individual with a history, to a set of identifiable individuals connected to her, at a known timestamp, etc. [^12]: After John Sinclair (1933–2007), a corpus linguist. [^13]: An extremely adversarial variant of the evaluation might allow all performers some transparent rounds. While entertaining, we believe such a scenario begins to lose attraction as a way to objectively compare systems in a highly controlled, understandable setting.
--- abstract: 'For each $r$, $0\leq r \leq m$, it is presented the class of quaternary linear codes $\mathcal{LRM}(r,m)$ whose images under the Gray map are binary codes with parameters of Reed-Muller $RM(r,m)$ code of order $r$.' author: - \| Faina I. Solov’eva[^1]\ Sobolev Institute of Mathematics\ pr. ac. Koptyuga 4, Novosibirsk 630090\ Russia\ e-mail: sol@math.nsc.ru title: 'On $\mathbb{Z}_4$-linear Reed-Muller like codes' --- Preliminaries ============= First consider some necessary definitions and notions. Consider the ring $\mathbb{Z}_4$ of integers modulo $4$. The set $\mathbb{Z}_4^n$ is a module with addition operation over the ring $\mathbb{Z}_4$. The [Lee weight]{}, $w_L(\cdot)$, of a quaternary vector is the sum of weights of its coordinate positions: $$w_L(0)=0, w_L(1)=w_L(3)=1, w_L(2)=2.$$ The [Lee distance]{}, $d_L(\cdot\,,\cdot)$, between any quaternary vectors $x, y\in \mathbb{Z}_4^n$ is defined as $d_L(x, y)=w_L(x-y)$. The set $\mathbb{Z}_4^n$ is a metric space with respect to the Lee metric. A [quaternary code]{} of length $n$ is a subset of the metric space $\mathbb{Z}_4^n$. A quaternary code of length $n$ is [linear]{} if it is a subgroup of the additive group of the ring $\mathbb{Z}_4^n$. We use further capital letters for binary codes and calligraphic for quaternary. Let us remind the standard maps $\alpha$, $\beta$ and $\gamma$ from $\mathbb{Z}_4$ to $\mathbb{Z}_2$: $$\begin{array}[b]{c\|ccc} \mathbb{Z}_4 & \alpha & \beta & \gamma\\[1pt] \hline ~0~\rule{0pt}{12pt} & ~0~ & ~0~ & 0\\ 1 & 1 & 0 & 1\\ 2 & 0 & 1 & 1\\ 3 & 1 & 1 & 0 \end{array}.$$ These maps can be extended in the usual way to maps from $\mathbb{Z}_4^n$ to $\mathbb{Z}_2^n.$ The [Gray map]{} $\phi\colon \mathbb{Z}_4^n \to \mathbb{Z}_2^{2n}$ is defined by $$\phi(x)=(\beta(x),\gamma(x)),\quad \mbox{for any}\ x\in \mathbb{Z}_4^n.$$ It is well known that $\phi$ is an isometry of the metric spaces $\mathbb{Z}_4^n$ and $\mathbb{Z}_2^{2n}$. Two quaternary codes, $\mathcal{C}$ and $\mathcal{D}$, of length $n$ are [equivalent]{} if there exist a vector $x\in\mathbb{Z}_4^n$, a permutation $\pi$ of $n$ coordinate positions and an inversion $\sigma$ on $n$ coordinates such that $C=\pi(\sigma (D))+x$. A binary code is called [$\mathbb{Z}_4$-linear]{} with respect to the Gray map if there exists an equivalent code $C$ such that its preimage $\phi^{-1}(C)$ is linear. There are several known classes of nonlinear binary codes with good properties which can be represented as linear quaternary codes. Among them there are such prominent codes as Preparata, Kerdock, Delsarte-Goethals, Goethals-Delsarte, some perfect codes, some Hadamard codes, see the list of references [@N1989; @KN1992; @HKCSS1994; @NK1996; @K2000; @K2001; @W1997]. In [@K2000; @K2001] the classifications of $\mathbb{Z}_4$-linear perfect and $\mathbb{Z}_4$-linear Hadamard codes are presented. It is established that for any $n=2^k, n\geq 16$ the number of nonequivalent $\mathbb{Z}_4$-linear perfect (Hadamard) codes is $\lfloor(k+1)/2\rfloor$. All quaternary linear codes whose images under the Gray map are perfect codes can be described using Mollard construction, see [@K2000]. The representation of $\mathbb{Z}_4$-linear Preparata codes is done in [@T2005]. Using switching approach it is established that the set of all quaternary linear Preparata codes of length $n=2^m$, $m$ odd, $m\ge 3$, is nothing more than the set of codes of the form $\mathcal{H_{\lambda,\psi}}+\mathcal{M}$ with $$\mathcal{H_{\lambda, \psi}}=\{y+T_{\lambda}(y)+S_{\psi}(y)\mid y\in H^n\},\quad \mathcal{M}=2H^n,$$ where $T_{\lambda}(\cdot)$ and $S_{\psi}(\cdot)$ are vector fields of a special form defined over the binary extended linear Hamming code $H^n$ of length $n$. An upper bound on the number of nonequivalent quaternary linear Preparata codes of length $n$ is obtained, namely, $2^{n\log_2 n}$. There are several papers devoted to quaternary Reed-Muller $RM(r,m)$ codes of order $r$, $0\leq r \leq m$. In [@HKCSS1994] it is established that binary Reed-Muller $RM(r,m)$ codes of order $r$, $r \in \{0,1,2,m-1, m\}$ are $\mathbb{Z}_4$-linear and conjectured that all other Reed-Muller codes are not $\mathbb{Z}_4$-linear. The conjecture is proved in [@HLK1998]. In [@HKCSS1994] the class of quaternary codes $\mathcal{QRM}(r,m)$ for each $r$, $0\leq r \leq m$ is introduced. The image of the code $\mathcal{QRM}(r,m)$ under the map $\alpha$ (see the definition of the map above) is linear Reed-Muller $RM(r,m)$ code for all $r$, $0\leq r \leq m$. The class of the codes includes the quaternary linear Kerdock codes and its dual the quaternary linear Preparata code from [@HKCSS1994]. The generalization of the result is given in [@BFP2005]. The class of the codes obtained in [@BFP2005] includes all the quaternary linear Kerdock codes and the quaternary linear Preparata codes. Thus, another representation of the quaternary linear Preparata codes is given. The images of all these codes under the map $\alpha$ are also linear Reed-Muller codes. In [@PR1997] the additive Reed-Muller code $\mathcal{ARM}(r,m)$ of order $r$, $0\leq r \leq m$ is defined. The code is an additive subgroup of $\mathbb{Z}_2^{k_1}\times \mathbb{Z}_4^{k_2},$ $k_1=2^{m-1},\, k_2=2^{m-2}, m\geq 2$ and it is announced that its binary image is linear for $r\in \{0,1,m-1,m\}$ and nonlinear for $r=m-2, m > 3$. Reed-Muller-like codes ====================== In this section for every integer $r$, $r \in \{0,1,\ldots ,m\}$, we construct the class of quaternary linear codes of length $2^{m-1}$, code distance $d=2^{m-r}$ and size $2^k$, where $$k = 1+ \left( \begin{array}{c} m \\ 1 \end{array} \right) + \cdots + \left( \begin{array}{c} m \\ r \end{array} \right).$$ The image of any such code under the Gray map is a binary (not necessary linear) code with parameters of Reed-Muller $RM(r,m)$ code of order $r$. Let $v=(v_1,\ldots,v_m)$ range over $\mathbb{Z}_2^m$. The binary [Reed-Muller code $RM(r,m)$ of order $r$]{} is generated by all binary vectors of length $2^m$ corresponding to the Boolean functions $f(v)$ equaled to monomials of degree not more than $r$. The code has the following parameters: $\bullet$ length of the code is $n=2^m$; $\bullet$ the size of the code is $2^k$, where $k = 1+ \left( \begin{array}{c} m \\ 1 \end{array} \right) + \cdots + \left( \begin{array}{c} m \\ r \end{array} \right);$ $\bullet$ the code distance $d=2^{m-r}.$ It is known, see [@MWSl], that the binary Reed-Muller code $RM(r,m)$ of order $r$ can be described by Plotkin (doubling) construction: $$RM(r,m) = \{(x,x+y) \mid x \in RM(r,m-1), y \in RM(r-1,m-1)\}.$$ Binary not necessary linear code whose parameters coincide with parameters of the binary linear Reed-Muller code $RM(r,m)$ of order $r$ we will call [Reed-Muller-like code]{} of order $r$. We are going to prove that among them there are $\mathbb{Z}_4$-linear codes. Such codes have some regular properties. For example, all such binary codes are transitive. Preimages of them under the Gray map are quaternary codes with parameters: $\bullet$ length of the code is $n=2^{m-1}$; $\bullet$ the size of the code is $2^k$, where $k = 1+ \left( \begin{array}{c} m \\ 1 \end{array} \right) + \ldots +\left( \begin{array}{c} m \\ r \end{array} \right);$ $\bullet$ the code distance $d=2^{m-r}.$ We will denote any such quaternary linear code by $\mathcal{LRM}(r,m)$, its binary image – by $LRM(r,m)$. We construct a sequence of quaternary linear $\mathcal{LRM}(r,m)$ codes that includes the following classes of quaternary linear codes: the quaternary repetition code $\mathcal{LRM}(0,m)$, some quaternary linear Hadamard codes from [@K2001], some quaternary linear extended perfect codes from [@K2000; @K2001], full-even weight code $\mathcal{LRM}(m-1,m)$ and $\mathcal{LRM}(m,m)=\mathbb{Z}_4^{2^m}$. We construct the $\mathcal{LRM}(r,m)$ codes by induction on $m$, where $m=\log n, m>1$. For $m=1$ there exist the following trivial quaternary linear codes: a\) the quaternary linear code $\mathcal{LRM}(0,1)=\mathcal{RM}(0,1)=\{(0),(2)\},$ its binary image is $LRM(0,1)=\phi(\mathcal{LRM}(0,1))=RM(0,1)=\{(0,0),(1,1)\};$ b\) the quaternary linear code $\mathcal{LRM}(1,1)=\mathcal{RM}(1,1)=\{(0),(1),(2),(3)\},$ its binary image is $LRM(1,1)=\phi(\mathcal{LRM}(1,1))=RM(1,1)=\{(0,0),(0,1),(1,1),((1,0)\}.$ Let us consider also more interesting case $m=2$. Here we have the following quaternary linear codes and their binary images under the Gray map: a\) the quaternary linear code $\mathcal{LRM}(0,2)=\mathcal{RM}(0,2)=\{(0,0),(2,2)\}$ and its binary image $LRM(0,2)=\phi(\mathcal{LRM}(0,2))=RM(0,2)=\{(0,0,0,0),(1,1,1,1)\};$ b\) the quaternary linear code $$\mathcal{LRM}(1,2)=\mathcal{RM}(0,2)=\{(0,0),(1,1),(2,2),(3,3),(1,3),(3,1),(0,2),(2,0)\}$$ and its binary image full-even weight code of length 4: $LRM(1,2)=\phi(\mathcal{LRM}(1,2))=RM(1,2);$ c\) the quaternary linear code $\mathcal{LRM}(2,2)=\mathbb{Z}_4^2$ and its binary image $$LRM(2,2)=\phi(\mathcal{LRM}(2,2))=RM(2,2).$$ Let $\mathcal{LRM}(r,m-1)$ and $\mathcal{LRM}(r-1,m-1)$ be any two quaternary linear codes with parameters $$(n=2^{m-2}, \, 2^k, \, d=2^{m-r-1})\,\,\, \mbox{and} \,\,\, (n=2^{m-2}, \, 2^{k^{\prime}}, \, d=2^{m-r}),$$ where $$k=\sum_{i=0}^{r} \left( \begin{array}{c} m-1 \\ i \end{array} \right), k^{\prime}=\sum_{i=0}^{r-1} \left( \begin{array}{c} m-1 \\ i \end{array} \right).$$ It is not difficult to show that Plotkin construction applied to these codes gives us a quaternary linear $\mathcal{LRM}(r,m)$ code of order $r$, i.e. $$\mathcal{LRM}(r,m)= \{(x,x+y) \mid x \in \mathcal{LRM}(r,m-1), y \in \mathcal{LRM}(r-1,m-1)\}.$$ So we get the following result. For any $r$, $0\leq r \leq m$, $m\geq 1$, the set $\mathcal{LRM}(r,m)$ is a quaternary linear code with parameters $$(n=2^{m-1}, \, 2^k, \, d=2^{m-r}),\, \mbox{where} \,\, k=\sum_{i=0}^{r} \left( \begin{array}{c} m \\ i \end{array} \right),$$ whose image under the Gray map is a binary code with parameters of Reed-Muller $RM(r,m)$ code of order $r$. Taking into account that there exist quaternary linear Hadamard codes, all quaternary linear extended perfect codes from [@K2000; @K2001] whose binary images under the Gray map are nonlinear codes and all $RM(r,m)$ codes of order $r$, $r \in \{3,\ldots, m-2\}$ are not $\mathbb{Z}_4$-linear we get a sequence of quaternary linear codes $\mathcal{LRM}(r,m)$ of order $r$ such that all their binary images are nonlinear codes with parameters of Reed-Muller codes $RM(r,m)$ for any order $r\in \{3,\ldots, m-2\}$. Let us show that the codes from Theorem 1 do not equivalent to codes from [@BFP2005] for $r\in \{3,\ldots, m-2\}$. According to [@BFP2005] a quaternary linear code $\mathcal{QRM}(r,m-1)$ of length $2^{m-1}$ has size $2^{2k}$, where $k=\sum_{i=0}^{r} \left( \begin{array}{c} m - 1\\ i \end{array} \right)$. From this fact and (1) we conclude that any quaternary linear codes $\mathcal{LRM}(r,m)$ and $\mathcal{QRM}(r,m)$ of the same length have different sizes, so they do not equivalent to each other. All binary images of the codes $\mathcal{LRM}(r,m)$ under the Gray map are nonlinear codes with parameters of Reed-Muller codes $RM(r,m)$ of order $r\in \{3,\ldots, m-2\}$, but all binary images of codes $\mathcal{QRM}(r,m)$ are linear $RM(r,m)$ codes of the same order. Therefore their binary images having the same parameters are also nonequivalent to each other. [99]{} Borges J., Fernandes C., Phelps K. T., Quaternary Reed-Muller codes, IEEE Trans. Inform. Theory, 2005, V. 51, N. 7, P. 2686–2691. Borges J., Phelps K. T., Rifà J., and Zinoviev V. A., On $Z_4$-Linear Preparata-like and Kerdock-like Codes, IEEE Trans. Inform. Theory, 2003, V. 49, N. 11, P. 2834–2843. Calderbank A. R., Cameron P. J., Kantor W. M., and Seidel J. J., $\mathbb{Z}_4$-Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets, Proc. London Math. Soc., 1997, V. 75, P. 436–480. Hammons A. R., Jr., Kumar P. V., Calderbank A. R., Sloane N. J. A., and Solé P., The $\mathbb{Z}_4$-Linearity of Kerdock, Preparata, Goethals, and Related Codes, IEEE Trans.Inform. Theory, 1994, V. 40, N. 2, P. 301–319. Hou X.-D., Lahtonen J. T., Koponen S., The Reed-Muller code $R(r,m)$ is not $Z_4$-linear for $3\leq r \leq m-2$. IEEE Trans. Inform. Theory, 1998, V. 44, N. 2, P. 798–799. Krotov D. S., $\mathbb{Z}_4$-Linear Perfect Codes, Diskr.Analiz Issled. Operatsii, Ser. 1, 2000, V. 7, N. 4, P. 78–90. Krotov D. S., $\mathbb{Z}_4$-Linear Hadamard and Extended Perfect Codes, in Proc. Int. Workshop on Coding and Cryptography, 2001, Paris, France, P. 329–334. Kuzmin A. S., Nechaev A. A., Construction on error-correcting codes using linear recurrences, J. of Math. Sciebces, 1992, V. 47, N. 5, P. 183–184. MacWilliams F. J., Sloane N. J. A., The theory of error-correcting codes, Amsterdam: North-Holland, 1977. Nechaev A. A., Kerdock codes in a cyclic form, Discrete Math. 1989, V. 1, N. 4, P. 123–139. Nechaev A. A., Kuzmin A. S., $\mathbb{Z}_4^n$-linearity, two approaches, Proc. of Fourth Int. Workshop on Algebraic and Comb. Coding Theory, Sozopol, Bulgaria. September 1996, P. 112–115. Pujol J., Rifa J., Additive Reed-Muller codes, Proc. of Int. Symp. on Inform. Theory, Ulm, Germany, 1997, P. 508. Tokareva N. N., Representation of $\mathbb{Z}_4$-Linear Preparata Codes Using Vector Fields, Problems of Inform. Transm., 2005, V. 41, N. 2, P. 113–124. Wan Z.-X., Quaternary Codes, Singapore: World Scientific, 1997. [^1]: This research was partially supported by the project N512 of the Russian education program “Development of scientific level of Universities”.
--- abstract: 'With the recent possible signal of dark matter from the CDMS II experiment, the $Z''$ mass of a new version of the dark left-right gauge model (DLRM II) is predicted to be at around a TeV. As such, it has an excellent discovery prognosis at the operating Large Hadron Collider.' --- UCRHEP-T486\ February 2010 [Bound on $Z'$ Mass from CDMS II\ in the Dark Left-Right Gauge Model II\ ]{} : One year ago, we proposed [@klm09] that dark-matter fermions (scotinos) are naturally present in an unconventional left-right gauge extension of the standard model (SM) of particle interactions, which we call the dark left-right model (DLRM). It is a nonsupersymmeric variation of the alternative left-right model (ALRM) discussed already 23 years ago [@m87; @bhm87]. One important difference of both the DLRM and the ALRM with the conventional left-right model (LRM) [@dgko91] is the fact that tree-level flavor-changing neutral currents [@gw77] are naturally absent so that the $SU(2)_R$ breaking scale may easily be at around a TeV, allowing both the charged $W_R^\pm$ and the extra neutral $Z'$ gauge bosons to be observable at the large hadron collider (LHC). Interesting phenomenology of $Z'$ decay into scalar bosons in the DLRM has just recently been discussed [@adhm10]. In this paper, we propose a new variant of this extension which we call DLRM II. \[Other more exotic variants are also possible [@m09].\] Instead of having Majorana scotinos as dark matter, we now have Dirac scotinos. Their interactions with nuclei through the $Z'$ are thus relevant for understanding the recent result of the dark-matter direct-search experiment CDMS II [@cdms09]. It will be shown that the $Z'$ mass may indeed be around a TeV, and its discovery prognosis at the LHC is excellent. : Consider the gauge group $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)$. The conventional leptonic assignments are $\psi_L = (\nu,e)_L \sim (1,2,1,-1/2)$ and $\psi_R = (\nu,e)_R \sim (1,1,2,-1/2)$. Hence $\nu$ and $e$ obtain Dirac masses through the Yukawa terms $\overline{\psi}_L \Phi \psi_R$ and $\overline{\psi}_L \tilde{\Phi} \psi_R$, where $\Phi = (\phi_1^0, \phi_1^-; \phi_2^+, \phi_2^0) \sim (1,2,2,0)$ is a Higgs bidoublet and $\tilde{\Phi} = \sigma_2 \Phi^* \sigma_2 = (\overline{\phi_2^0},-\phi_2^-;-\phi_1^+, \overline{\phi_1^0})$ transforms in the same way. Both $\langle \phi_1^0 \rangle$ and $\langle \phi_2^0 \rangle$ contribute to $m_\nu$ and $m_e$, and similarly $m_u$ and $m_d$ in the quark sector, resulting thus in the appearance of tree-level flavor-changing neutral currents. Suppose the term $\overline{\psi}_L \tilde{\Phi} \psi_R$ is forbidden by a symmetry, then the same symmetry may be used to maintain $\langle \phi_1^0 \rangle = 0$ and only $e$ gets a mass through $\langle \phi_2^0 \rangle \neq 0$. At the same time, $\nu_L$ and $\nu_R$ are not Dirac mass partners, so they could in fact be completely different particles with independent masses of their own. Whereas $\nu_L$ is clearly the neutrino we observe in the usual weak interactions, $\nu_R$ can now be something else entirely. Here we rename $\nu_R$ as $n_R$ and show that it may in fact be a scotino, i.e. a fermionic dark-matter candidate. In our previous proposal [@klm09], we imposed a new global $U(1)$ symmetry $S$ in such a way that the spontaneous breaking of $SU(2)_R \times S$ will leave the combination $L = S - T_{3R}$ unbroken. We then showed that $L$ is a generalized lepton number, with $L=1$ for the known leptons, and $L=0$ for all known particles which are not leptons. Here we consider instead the case $L = S + T_{3R}$. Our model is nonsupersymmetric, but it may be rendered supersymmetric by the usual procedure which takes the SM to the MSSM (minimal supersymmetric standard model). Under $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1) \times S$, the fermions transform as shown in Table 1. Note the necessary appearance of the exotic quark $h$, which will turn out to carry lepton number as well. Fermion $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)$ $S$ ---------------------- ----------------------------------------------------- -------- $\psi_L = (\nu,e)_L$ $(1,2,1,-1/2)$ $1$ $\psi_R = (n,e)_R$ $(1,1,2,-1/2)$ $3/2$ $\nu_R$ $(1,1,1,0)$ $1$ $n_L$ $(1,1,1,0)$ $2$ $Q_L = (u,d)_L$ $(3,2,1,1/6)$ $0$ $Q_R = (u,h)_R$ $(3,1,2,1/6)$ $-1/2$ $d_R$ $(3,1,1,-1/3)$ $0$ $h_L$ $(3,1,1,-1/3)$ $-1$ : Fermion content of proposed model. The scalar sector consists of one bidoublet and two doublets: $$\Phi = \pmatrix{\phi_1^0 & \phi_2^+ \cr \phi_1^- & \phi_2^0}, ~~~ \Phi_L = \pmatrix{\phi_L^+ \cr \phi_L^0}, ~~~ \Phi_R = \pmatrix{\phi_R^+ \cr \phi_R^0}.$$ Their assignments under $S$ are listed in Table 2. Scalar $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)$ $S$ ------------------------------------------- ----------------------------------------------------- -------- $\Phi$ $(1,2,2,0)$ $-1/2$ $\tilde{\Phi} = \sigma_2 \Phi^* \sigma_2$ $(1,2,2,0)$ $1/2$ $\Phi_L$ $(1,2,1,1/2)$ $0$ $\Phi_R$ $(1,1,2,1/2)$ $1/2$ : Scalar content of proposed model. The Yukawa terms allowed by $S$ are then $\overline{\psi}_L \Phi \psi_R$, $\overline{\psi}_L \tilde{\Phi}_L \nu_R$, $\overline{\psi}_R \tilde{\Phi}_R n_L$, $\overline{Q}_L \tilde{\Phi} Q_R$, $\overline{Q}_L \Phi_L d_R$, and $\overline{Q}_R \Phi_R h_L$, whereas $\overline{\psi}_L \tilde{\Phi} \psi_R$, $\overline{n}_L \nu_R$, $\overline{Q}_L \Phi Q_R$, and $\overline{h}_L d_R$ are forbidden. Hence $m_e$, $m_u$ come from $v_2 = \langle \phi_2^0 \rangle$; $m_\nu$, $m_d$ come from $v_3 = \langle \phi_L^0 \rangle$; and $m_n$, $m_h$ come from $v_4 = \langle \phi_R^0 \rangle$. This structure shows clearly that flavor-changing neutral currents are guaranteed to be absent at tree level. As it stands, both the neutrino $\nu$ and the scotino $n$ are Dirac fermions, and lepton number $L$ is conserved. If we now introduce a soft term $\nu_R \nu_R$ which breaks $L$ by two units, then $\nu_L$ gets a Majorana mass through the canonical seesaw mechanism, as is usually assumed. As for $n$, it remains a Dirac fermion, being protected by a residual global $U(1)$ symmetry, under which $n$, $W_R^+$ transform as 1, and $h$, $\phi_1^{0,-}$ transform as $-1$. : Since $e$ has $L=1$ and $n$ has $L=2$, the $W_R^+$ of this model must have $L = S + T_{3R} = 0 + 1 = 1$. This also means that unlike the conventional LRM, $W_R^\pm$ does not mix with the $W_L^\pm$ of the SM at all. This important property allows the $SU(2)_R$ breaking scale to be much lower than it would be otherwise, as explained already 23 years ago [@m87; @bhm87]. Let $e/g_L = s_L = \sin \theta_W$ and $s_R = e/g_R$, with $c_{L,R} = \sqrt{1-s_{L,R}^2}$, then $g_B = e/\sqrt{c_L^2-s_R^2}$ and the neutral gauge bosons of the DLRM (as well as the ALRM) are given by $$\pmatrix{A \cr Z \cr Z'} = \pmatrix{s_L & s_R & \sqrt{c_L^2-s_R^2} \cr c_L & -s_L s_R/c_L & -s_L \sqrt{c_L^2 - s_R^2}/c_L \cr 0 & \sqrt{c_L^2 - s_R^2}/c_L & -s_R/c_L} \pmatrix{W_L^0 \cr W_R^0 \cr B}.$$ Whereas $Z$ couples to the current $J_{3L} - s_L^2 J_{em}$ with coupling $e/s_L c_L$ as in the SM, $Z'$ couples to the current $$J_{Z'} = s_R^2 J_{3L} + c_L^2 J_{3R} - s_R^2 J_{em}$$ with coupling $g_{Z'} = e/s_R c_L \sqrt{c_L^2-s_R^2}$. The masses of the gauge bosons are given by $$\begin{aligned} && M_{W_L}^2 = {e^2 \over 2 s_L^2} (v_2^2 + v_3^2), ~~~ M_Z^2 = {M_{W_L}^2 \over c_L^2}, ~~~ M_{W_R}^2 = {e^2 \over 2 s_R^2} (v_4^2 + v_2^2), \\ && M_{Z'}^2 = {e^2 c_L^2 \over 2 s_R^2 (c_L^2 - s_R^2)} (v_4^2 + v_2^2) - {s_L^2 s_R^2 M_{W_L}^2 \over c_L^2(c_L^2 - s_R^2)},\end{aligned}$$ where zero $Z-Z'$ mixing has been assumed, using the condition [@bhm87] $v_2^2/(v_2^2+v_3^2) = s_R^2/c_L^2$. : The $Z'$ couplings to $u$, $d$, $n$ (in units of $g_{Z'}$) are given by $$\begin{aligned} && u_L = -{1 \over 6} s_R^2, ~~~ u_R = {1 \over 2} c_L^2 - {2 \over 3} s_R^2, ~~~ u_V = {1 \over 4} c_L^2 - {5 \over 12} s_R^2, \\ && d_L = -{1 \over 6} s_R^2, ~~~ d_R = {1 \over 3} s_R^2, ~~~ d_V = {1 \over 12} s_R^2, \\ && n_L = 0, ~~~ n_R = {1 \over 2} c_L^2, ~~~ n_V = {1 \over 4} c_L^2.\end{aligned}$$ The effective Lagrangian for elastic scattering of the scotino $n$ off quarks is then given by $${\cal L} = {g^2_{Z'} n_V \over M^2_{Z'}} (\bar{n} \gamma_\mu n)(u_V \bar{u} \gamma^\mu u + d_V \bar{d} \gamma^\mu d).$$ In the original DLRM [@klm09], $n$ is a Majorana scotino, so it does not contribute to the s-wave elastic spin-independent scattering cross section in the nonrelativistic limit. Here $n$ is a Dirac scotino, so it will contribute. Let $$f_P = g^2_{Z'} n_V (2 u_V + d_V)/M^2_{Z'}, ~~~ f_N = g^2_{Z'} n_V (u_V + 2 d_V)/M^2_{Z'},$$ then its elastic cross section per nucleon is given by [@cls09] $$\sigma_0 = {4 m_r^2 \over \pi} {[Z f_P + (A - Z) f_N]^2 \over A^2},$$ where $Z$ and $A$ are the atomic and mass numbers of the target nucleus, and $m_r = m_n m_P/(m_n + m_P) \simeq m_P$. The CDMS II collaboration [@cdms09] observed two possible signal events with an expected background of $0.6 \pm 0.1$. Using $^{73}$Ge, i.e. $Z = 32$ and $A - Z = 41$, as a representative estimate of $\sigma_0$, this result could also be considered as an upper bound, i.e. $$\sigma_0 = {\pi \alpha^2 m_P^2 (105 c_L^2 - 137 s_R^2)^2 \over (146)^2 s_R^4 (c_L^2 - s_R^2)^2 M^4_{Z'}} < 3.8 \times 10^{-8}~{\rm pb},$$ which occurs at $m_n = 70$ GeV. : We consider the range $e^2 < s_R^2 < c_L^2 - e^2$, where the lower bound corresponds to $g_R=1$ and the upper bound to $g_B=1$. The values of $g_{Z'}$ and $\Gamma_{Z'}/M_{Z'}$ are plotted in Fig. 1(a) and (b), where $Z'$ is assumed to decay only into SM fermions. ![(a) $g_{Z'}$ vs $s_R^2$. (b) $\Gamma_{Z'} / M_{Z'}$ vs $s_R^2$ for SM fermions decay products only in the cases $M_{Z'} = 500$ GeV (blue solid) and $M_{Z'} \to \infty$ (red dashed).[]{data-label="fig:plot1"}](gZ2plot.ps "fig:"){width="45.00000%"} ![(a) $g_{Z'}$ vs $s_R^2$. (b) $\Gamma_{Z'} / M_{Z'}$ vs $s_R^2$ for SM fermions decay products only in the cases $M_{Z'} = 500$ GeV (blue solid) and $M_{Z'} \to \infty$ (red dashed).[]{data-label="fig:plot1"}](Z2width.ps "fig:"){width="45.00000%"}\ (a) (b) We compute the production and decay of $Z'$ to $e^+e^-$ at the Tevatron as a function of $M_{Z'}$ for various values of $s_R^2$ and compare it to data [@Aaltonen:2008vx] at $E_{\rm cm} = 1.96$ TeV and an integrated luminosity of 2.5 fb$^{-1}$ in Fig. 2(a). We then plot the exclusion limits on $M_{Z'}$ from both the new CDMS II data and the Tevatron as a function of $s_R^2$ in Fig. 2(b). Note that the CDMS II bound is stronger than the Tevatron bound for $s_R^2 < 0.5$. Note also that due to the accidental cancellation in the numerator of $\sigma_0$ in Eq. (12), the observed events at CDMS II cannot be interpreted as signals of dark matter in this model if $s_R^2 > 0.5$, because they would be excluded by the Tevatron data. ![(a) Lower bound on the $Z'$ mass in this model from Tevatron dielectron search. (b) $M_{Z'}$ vs $s_R^2$ from the CDMS II (blue dashed) and Tevatron (red solid) bounds. The dotted segments assume a simple extrapolation of the Tevatron data.[]{data-label="fig:collider"}](CDFconstraint.ps "fig:"){width="45.00000%"} ![(a) Lower bound on the $Z'$ mass in this model from Tevatron dielectron search. (b) $M_{Z'}$ vs $s_R^2$ from the CDMS II (blue dashed) and Tevatron (red solid) bounds. The dotted segments assume a simple extrapolation of the Tevatron data.[]{data-label="fig:collider"}](MZ2limit.ps "fig:"){width="45.00000%"}\ (a) (b) Given that $M_{Z'}$ is allowed to be in the TeV range, its discovery prognosis is excellent at the LHC. We show in Fig. 3 its discovery reach (assuming $E_{\rm cm} = 14$ TeV) by 10 dilepton events (either dielectron or dimuon) which satisfy the following basic cuts on their transverse momenta, rapidities, and invariant mass: $p_T > 20$ GeV (each lepton), $\|\eta\| < 2.4$ (each lepton), $\|M_{\ell {\bar \ell}} - M_{Z'}\| < 3 \Gamma_{Z'}$. Using these cuts, the dominant SM background from $\gamma/Z$ (Drell-Yan) is negligible. With an integrated luminosity of 1 fb$^{-1}$, the $Z'$ of DLRM II with $M_{Z'} \sim 2$ TeV may then be discovered at the LHC. ![Luminosity for $Z'$ discovery by 10 dielectron events at LHC. Small circles are Tevatron limits.[]{data-label="fig:LHC"}](Lfor10events.ps){width="55.00000%"} : In this model, the dark-matter relic abundance is presumably determined by the annihilation $n \bar{n} \to Z' \to$ SM fermions. The thermally averaged cross section multiplied by relative velocity is approximately given by $$\langle \sigma v_{rel} \rangle_{Z'} = {g^4_{Z'} c_L^4 m_n^2 \sum_f (f_L^2 + f_R^2) \over 32 \pi (4 m_n^2 - M_{Z'}^2)^2},$$ where the sum over fermions should include a factor of 3 for quarks and an overall factor of 3 for families. Fixing the above at 1 pb as a typical value to satisfy the requirement of dark-matter relic abundance, it can easily be shown that for $m_n=70$ GeV, the required $M_{Z'}$ is very much below the CDMS II bound. \[For example, for $s_R^2=0.4$, $M_{Z'} = 267$ GeV would be required.\] In other words, the $n \bar{n} \to Z'$ annihilation cross section would be too small to account for the observed dark-matter relic abundance. To remedy this situation, the mechanism proposed in the original DLRM may be invoked, i.e. $n \bar{n} \to l^- l^+$ through $\Delta_R^+$ exchange. However, this requires adding the $SU(2)_R$ scalar triplet $(\Delta_R^{++},\Delta_R^+, \Delta_R^0)$, which is not necessary in our present version and thus not very much motivated. The alternative is to consider a larger value of $m_n$. The CDMS II bound on $\sigma_0$ is very well approximated in the range $0.3 < m_n < 1.0$ TeV by the expression $$\sigma_0 < 2.2 \times 10^{-7}~{\rm pb}~(m_n/1~{\rm TeV})^{0.86}.$$ Using this on the right-hand side of Eq. (12), we plot in Fig. 4(a) and (b) the $M_{Z'}$ bounds for $m_n=400$ and 600 GeV, as well as the solutions of $M_{Z'}$ (with $M_{Z'} > 2m_n$) to Eq. (13) for 1 pb. We see that there are indeed consistent solutions (where the solid curve is higher than the dash curve) for a range of $s_R^2$ in each case. If $m_n$ falls below 300 GeV, then there is no solution because $M_{Z'}$ would then be excluded by the Tevatron bound. We note also that only a modest resonance enhancement is needed from the denominator of Eq. (13). The $n \bar{n}$ annihilation to $l^+l^-$ through $W_R$ exchange also contributes to the dark-matter relic abundance, but its value is an order of magnitude less, i.e. $$\langle \sigma v_{rel} \rangle_{W_R} = {3 g^4_R m_n^2 \over 64 \pi (m_n^2 + M_{W_R}^2)^2}.$$ ![(a) For $m_n = 400$ GeV, the CDMS II bound on $M_{Z'}$ (blue dashed) and the value of $M_{Z'}$ (red) from $\langle \sigma v_{rel} \rangle_{Z'} = 1$ pb vs $s_R^2$; (b) same as in (a) for $m_n = 600$ GeV.[]{data-label="fig4"}](mn400.eps "fig:"){width="45.00000%"} ![(a) For $m_n = 400$ GeV, the CDMS II bound on $M_{Z'}$ (blue dashed) and the value of $M_{Z'}$ (red) from $\langle \sigma v_{rel} \rangle_{Z'} = 1$ pb vs $s_R^2$; (b) same as in (a) for $m_n = 600$ GeV.[]{data-label="fig4"}](mn600.eps "fig:"){width="45.00000%"}\ (a) (b) : Unlike the original DLRM, where a scalar triplet $(\Delta_R^{++},\Delta_R^+,\Delta_R^0)$ may mediate lepton flavor violating processes such as $\mu \to eee$ at tree level, and must be forbidden by hand, the DLRM II is safe because it has no such interactions. Nevertheless, lepton (as well quark) flavor violation occurs in one loop in the $SU(2)_R$ sector, in complete analogy to that of the SM in the $SU(2)_L$ sector. The branching fraction of $\mu \to e \gamma$ is then $$B(\mu \to e \gamma) = {3 \alpha \|\delta_R\|^2 \over 64 \pi} \left( {s_L^2 M_{W_L}^2 \over s_R^2 M_{W_R}^2} \right)^2 < 1.2 \times 10^{-11},$$ where the experimental upper bound has also been displayed, and $\delta_R$ is the analog of the well-known suppression factor $\delta_L = \sum_i U^_{ei} U_{\mu i} (m_{\nu_i}^2/M_{W_L}^2)$ in the SM. For $s_R^2 = s_L^2$, we have $M_{W_R} = 1.5$ TeV, then $\|\delta_R\| < 0.116$. Since the flavor structure of scotino mixing and their mass-squared differences are unknown, this upper bound could be saturated, and the observation of $\mu \to e \gamma$ may be imminent. The same holds for other lepton flavor violating processes such as $\mu - e$ conversion in nuclei. Note that the contribution to the muon anomalous magnetic moment here is about $10^{-10}$, well below the experimental sensitivity. A more comprehensive study, including contributions to $D^0 - \bar{D}^0$ mixing [@m88], will be given elsewhere. : As the LHC begins its operation, one of its first possible discoveries could be a $Z'$ through the process $q \bar q \to Z' \to l^+ l^-$. There are many $Z'$ models, and some of them could also be invoked [@Lee:2008em] to explain the CDMS II results. However, the coupling of the dark matter to the $Z'$ in these models is in general not related to the $Z'$ leptonic couplings. Here they are intimately connected and predicted as a function of only $s_R^2$. In fact, if we assume $s_R^2 = s_L^2$ (i.e. left-right symmetry), then there is no free parameter. Our numerical analysis in this paper is only a rough estimate for illustration, but it points to the important assertion that the $Z'$ interactions in this model are fixed with respect to direct dark-matter search and the detection of $Z'$ itself at an accelerator. In these exciting times of having both the functioning LHC and ongoing dark-matter search experiments, the dark-matter mystery in astroparticle physics may be near a solution. : The work of S.K. was partially supported by the Science and Technology Development Fund (STDF) Project ID 437 and the ICTP Project ID 30. This work was also supported by the U. S. Department of Energy Grants No. DE-AC02-98CH10886 (HL) and No. DE-FG03-94ER40837 (EM). [99]{} S. Khalil, H. S. Lee, and E. Ma, Phys. Rev. [D79]{}, 041701(R) (2009). E. Ma, Phys. Rev. [D36]{}, 274 (1987). K. S. Babu, X.-G. He, and E. Ma, Phys. Rev. [D36]{}, 878 (1987). For a comprehensive treatment, see for example N. G. Deshpande [et al.]{}, Phys. Rev. [D44]{}, 837 (1991). S. L. Glashow and S. Weinberg, Phys. Rev. [D15]{}, 1958 (1977). A. Aranda, J. L. Diaz-Cruz, J. Hernandez-Sanchez, and E. Ma, arXiv:1001.4057 \[hep-ph\]. E. Ma, Phys. Rev. [D79]{}, 117701 (2009). Z. Ahmed [et al.]{} \[CDMS Collaboration\], arXiv:0912.3592 \[astro-ph\]. See, for example, Q.-H. Cao, I. Low, and G. Shaunghnessy, arXiv:0912.4510 \[hep-ph\]. T. Aaltonen [et al.]{} \[CDF Collaboration\], Phys. Rev. Lett. [102]{}, 031801 (2009). E. Ma, Mod. Phys. Lett. [A3]{}, 319 (1988). See, for example, H. S. Lee, AIP Conf. Proc. [1078*]{}, 569 (2009) \[arXiv:0808.3600 \[hep-ph\]\].
--- abstract: 'We have studied the electronic states in 1–5 layers thick Ag films on V(100), by means of ab initio density functional calculations. Due to the mismatch of the electronic structure of Ag and V, quantum well states of both sp and d character localized on Ag films are formed. We find that the hybridization of the Ag quantum well states with the V orbitals is nevertheless important, and must be taken into account in order to fully understand the observed properties, in particular the energies and the dispersion of the photoemission peaks in ARPES experiments.' author: - 'P. Lazi'' c' - 'Ž. Crljen' - 'R. Brako' title: 'Localization and hybridization of electronic states in thin films of Ag on V(100) ' --- \[introduction\]Introduction ============================ Ultrathin metallic films grown on a different metal substrate show interesting structural and electronic properties. Some grow in registry with the substrate, while others form a variety of commensurate or incommensurate structures, develop grain boundaries, etc. Of particular interest are the systems which exhibit quantum size effects (QSE), where the electronic states in the films, with thickness typically from one up to several tens of layers, show quantized motion in the direction perpendicular to the surface plane. Quantum well (QW) states or quantum well resonances can be formed, depending on the degree of mismatch between the electronic structure of the two metals, and the properties can vary a lot over the extent of the two-dimensional surface Brillouin zone (BZ). Several experimental techniques can be used to probe the electronic properties of thin metallic films, and in particular the quantum well states. For an overview see, e.g., a recent review paper.[@Mil-02] Most commonly used are angle-resolved photoemission spectroscopy (ARPES) and the scanning tunneling microscopy (STM). ARPES has been used mostly in the normal emission mode, i.e. in the direction perpendicular to the surface, but angularly resolved spectra in off-normal directions make it possible to study the band structure along any direction of the surface Brillouin zone.[@Joh-94; @Mat-02] The photoemission spectroscopy probes a macroscopic surface region, determined by the size of the incoming photon beam, which means it is best suited for systems of uniform thickness. A well known example of this kind are silver films on a Fe(100) whisker, which can be grown up to 120 macroscopically uniform atomic layers.[@Pag-99] STM is a nanoscopically local method, and offers the possibility to study local structures, such as islands, chains, dots, etc. With the scanning tunneling spectroscopy (STS) technique, in which the differential conductance ($dI/dV$) is measured, one can reliably determine the energy of quantized electronic states in the range of approximately 1 eV below and above the Fermi level. The constant current STM technique can also give important information about the surface electronic structure, in particular by direct imaging of standing waves formed around surface defects in the low bias voltage (and usually low temperature) regime, and taking the Fourier transform. This method is known as the Fourier transform scanning tunneling microscopy (FT-STM).[@Spr-97] In this paper we study the properties of thin films of silver on a V(100) surface. The films can be grown pseudomorphically, i.e. in registry with the substrate, up to a thickness of at least ten layers. It has been recently shown that, by using an appropriate deposition and annealing procedure, films of very high quality can be obtained even on V(100) surface with oxygen-induced ($5\times1$) reconstruction, because silver atoms displace oxygen and lift the reconstruction.[@Kra-03a] The silver films grow as (111) layers of fcc structure, slightly tetragonally distorted due to a small mismatch of interatomic distances of the two lattices. The electronic properties of this system are very interesting, due to the profound difference of the band structure of the two metals. In bulk vanadium, the s bands extend approximately from 2.5 to 7 eV below the Fermi level, while the d bands exist between the Fermi level down to 3 eV. The order is reverse in bulk silver which has d bands between 3.5 and 6 eV below the Fermi level, and the broad sp bands extend from the Fermi level down to 7 eV. This symmetry mismatch is particularly important in the center of the two-dimensional Brillouin zone of the V–Ag (100) interface, where it amounts to a [symmetry gap]{} which prevents hybridization of certain electron bands across the interface. In ultrathin Ag films on V(100), it has a consequence that both the sp orbitals of Ag close to the Fermi level and d orbitals at larger binding energies form QW states which are mostly localized within the Ag layers. In normal ARPES, which selects electronic states at the center of the two-dimensional Brillouin zone, the QW states appear as strong narrow peaks.[@Val-96; @Kra-03b] A number of approaches have been used in the past in order to describe the electronic properties, in particular the QW states, in ultrathin metallic films. Quasi-one-dimensional models, such as a square well potential [@Kra-01] or the phase accumulation model [@Lin-87] have been successfully used to to interpret the energies of QW states. More sophisticated methods have also been used, such as the tight-binding approach [@Smi-94] and layer-Korringa-Kohn-Rostoker approach.[@Ern-02] In a few systems the QW states have been investigated by self consistent density functional calculations.[@Car-00; @Wei-03] There are strong reasons to use the ab initio methods. First, there are no adjustable parameters, and a wide range of calculated structural and electronic properties offer the possibility of a detailed comparison with experiments. Also, quantities such as the expected STM profiles and the amplitudes of the wavefunctions of the QW states, which cannot be obtained in simple approaches, can be calculated. In a recent paper we have reported our ab initio calculations of the structure of the Ag/V(100) system.[@Kra-03a] We used the density functional theory (DFT) approach and calculated the structure and the total energy of silver films on the V(100) surface from one to five layers thick. The calculated interlayer distances, step heights, work functions and other quantities were in excellent agreement with the experimental findings. In this work we use the same method to calculate the [electronic]{} properties of the Ag/V(100) system and compare the results with photoemission experiments. \[studies\] Experimental studies of A/V(100) films ================================================== STM experiments give important information on structural properties of Ag/V(100) films, e.g. the disappearance of the oxygen-assisted ($1\times5$) reconstruction of the bare V surface upon silver adsorption and annealing, the step heights between Ag terraces of different number of layers, etc. In STM images the Ag terraces appear smooth, with small corrugation of square symmetry corresponding to the atomic structure. In a recent experiment standing waves around impurities [@Kra-03c; @Kra-05; @Kra-04] have been observed, which indicates that the motion of electrons with energies around the Fermi level is largely two-dimensional, confined to the Ag films. This is one of the rare examples where the oscillatory wave patterns at metal surfaces have been observed at room temperature. In photoemission spectra along the surface normal the quantum well states of predominantly sp character appear as strong well defined peaks in the energy region between the Fermi level and 2.5 eV below it. They are narrow and the position of the maximum does not depend on the energy of the incident light, which means that they are localized on the silver films. The peak position is a characteristic of the number of Ag layers, and the appearance of one or more peaks in the photoemission spectra can be used to determine the thickness and the degree of perfection of the films. Off-normal photoemission measurements make it possible to determine the dispersion relation of the QW states.[@Huf-95] Away from the center of the surface Brillouin zone, i.e. at finite $k_{\|\|}$, the photoemission peaks of the QW states broaden and the dependence upon $k_{\|\|}$ often deviates from the parabolic free-electron behavior. Several of the theoretical approaches mentioned in the Introduction have been used to treat the sp QW states of Ag/V(100) films, including a simple square potential,[@Mil-99] the phase accumulation model,[@Mil-99] the layer-Korringa-Kohn-Rostoker approach,[@Ern-02] and the tight-binding calculations.[@Kra-03b] These calculations have made it possible to assign the experimentally observed QW states to different families according to the number of nodes of the wavefunction, for up to eight layers or so.[@Ern-02] The energy region from 2.5 to 7 eV below the Fermi level in photoemission is dominated by features derived from d bands of Ag. A number of sharp peaks are observed in normal photoemission, implying that some d orbitals form QW states localized within silver films. \[calculation\]Calculations =========================== The density functional calculations were performed using the DACAPO program with ultrasoft pseudopotentials for the Perdew-Wang exchange-correlation functional, in the generalized gradient approximation (GGA). The computational procedure has been described in considerable details in Ref. , and here we only give a brief overview. Seven layers of vanadium were used to describe the bcc V(100) substrate, on which one to five silver layers were added. Since periodic boundary conditions were used in all three spatial directions, a layer of vacuum of around 15 [Å]{} was added on top of silver. The structure was allowed to fully relax in the direction perpendicular to the surface, except for the three bottom V layers, which were kept fixed at the bulk interatomic distance. We used a mesh of $12\times12$ $k$ points in the directions along the surface plane. Once the full selfconsistency of the relaxed structure was achieved, we made additional calculations with many more $k$ points (typically 36 from the center to the edge of the BZ) along the two high-symmetry directions of the surface Brillouin zone, in order to obtain the dispersion relations and the wavefunctions of the Kohn-Sham eigenstates in more detail. Since in these calculations the $k$ points do not cover uniformly the whole two-dimensional Brillouin zone, the potential determined beforehand in selfconsistent calculations is used. We use the calculated electronic eigenstate energies and wavefunctions for comparison with experiments, in particular photoemission. However, the Kohn-Sham density functional approach is designed primarily for an accurate calculation of the total energy of the system. By varying the positions of the atoms one can determine the equilibrium atomic configuration, the interatomic potential around the equilibrium, etc. The electronic eigenstates are obtained as an intermediate result and their quality and suitability for the description of various electronic processes is not guaranteed. The method is particularly unreliable for empty electronic states above the Fermi level, which do not contribute towards the total energy of the system and are therefore not included in the energy functional. In fact, it is possible to perform the DFT calculations using just enough orbitals to accommodate all the electrons, at the expense of the rate of convergence, in which case the empty states will be obviously inadequate. Even if sufficiently many orbitals are included, the vacuum layer separating the films in the direction perpendicular to the surface may not be large enough to prevent the coupling of states above the Fermi level due to a finite probability of tunneling through the vacuum. Care should be exerted if empty states are used e.g. in the analysis of STM images. Nevertheless, it has repeatedly been found that, if adequate precaution is used, the calculated electronic eigenstates yield a good description of the electronic structure of the solid. From comparison with other calculations we have concluded that our calculation indeed gives good electronic structure, apart from some d bands of silver being too close to the Fermi level, which will be discussed in detail later on. Apparently, the d bands have been a problem in a number of DFT calculations of the electronic structure of noble metals. \[results\]Results and comparison with experiment ================================================= The Kohn-Sham eigenstates ------------------------- We first made density functional calculations of an unsupported Ag monolayer (i.e. a layer of silver atoms surrounded on both sides by vacuum) of the same geometry as Ag atoms in a 1ML film adsorbed on V(100). The energies of the Kohn-Sham eigenstates are shown in Fig. \[monolayer\], along the high symmetry directions of the surface Brillouin zone $\bar{\Gamma}$–$\bar{X}$ and $\bar{\Gamma}$–$\bar{M}$ The states between 2.5 and 4.5 eV binding energy with relatively flat dispersion curves are d bands, while the s band shows a free-electron like dispersion and crosses the Fermi level at around $k_{\|\|}=0.8$ Å$^{-1}$ in both calculated directions. Around the center of the Brillouin zone the s orbitals also contribute some weight to the $\textrm{d}_{zz}$ band, second from below, which is possible as both belong to the same completely symmetric representation of the symmetry group at $\bar{\Gamma}$ point. Next we calculated the Kohn-Sham energy bands of bulk vanadium and projected them onto the directions relevant for the surface calculations. In Fig.\[Vbulk\] (a) are shown the bands along the $\Gamma$–$\Delta$–H direction, i.e. the $k_z$ direction in our surface calculations, which projects onto the $\bar{\Gamma}$ point of the (100) vanadium surface Brillouin zone. In Fig.\[Vbulk\] (b) we show the bands in the high symmetry directions of the surface Brillouin zone, for 20 values of $k_z$. There are no true energy gaps at or near the $\bar{\Gamma}$ point. However, according to Fig.\[Vbulk\] (a) in the energy region between 2.3 eV and the Fermi level only states of $\Delta_2$ symmetry exist. Consequently, the electronic states of silver adlayers at $\bar{\Gamma}$ which are totally rotationally symmetric with respect to the $z$ axis (i.e. s and $\textrm{d}_{zz}$) which fall in this energy range cannot hybridize with the substrate bands. This is the so called [symmetry gap]{} which allows the existence of quantum well states in adsorbate layers even when the substrate has no true energy gaps. Of course, the symmetry incompatibility ceases to be exactly true as soon as we move away from the $\bar{\Gamma}$ point. Thus we expect that the photoemission peaks from such adsorbate QW become broader when measured away from the normal direction, i.e. for a finite $k_{\|\|}$. This is unlike the QW states on substrates with a true energy gap, where narrow photoemission peaks can be observed for large $k_{\|\|}$, until they hit either the Fermi level or the edge of a projected band. At binding energies larger than approximately 3 eV the converse is true. According to Fig.\[Vbulk\], only electronic states of the $\Delta_1$ symmetry exist at $\bar{\Gamma}$ point, which implies that there is a symmetry gap for d states other than $\textrm{d}_{zz}$ and the respective QW states may occur. Finally, we come to the results for the combined vanadium–silver systems. The dispersion relations of the eigenstates for 1 ML and 2 ML Ag films on seven layers of V(100), calculated as explained in section \[calculation\], are shown in Figs. \[1layer\] and \[2layers\]. The thin dotted lines in Fig. \[1layer\] are the results for bare vanadium substrate consisting of seven layers, which are similar to the bulk dispersion relations projected onto the surface, Fig.\[Vbulk\] (b), but more coarsely spaced because of the small number of layers. The full lines are the results of the calculation with the silver adlayer. It is easy to recognize the electronic states which are localized mostly in vanadium, as they run very close to the dotted lines of the clean V, while the states associated with silver do not have a corresponding dotted line. The states which at $k_{\|\|}=0$ have a large weight on silver atoms are shown by thick lines around the center of the Brillouin zone. The different line styles denote states of different symmetry, which has been deduced from the density of states projected onto atomic orbitals. In the energy range between the Fermi level and 2 eV we have gone a step further, and calculated the integral over the silver atoms of the density of various eigenstates, for the full range of $k_{\|\|}$ vectors. The results are shown as lines of various thickness, the thick segments corresponding to states localized at silver atoms by more than around 35%. The reasoning behind this approach is that the calculated integrals are roughly proportional to the intensity of the photoemission peaks in ARPES experiments, since photoelectrons originating from deeper layers have a small probability of leaving the solid without being scattered. The most prominent among the thick lines are the quantum well states with energy of 1.65 eV for 1 ML and 0.66 eV for 2 ML films at $\bar{\Gamma}$ point. They appear as strong narrow peaks in ARPES experiments, as discussed in the following subsection. Comparing the states of 1 ML Ag films on vanadium, Fig. \[1layer\], with the states of the unsupported Ag monolayer, Fig. \[monolayer\], it is evident that the Ag d bands (except $\textrm{d}_{zz}$), drawn by dashed lines, are pulled down to lower energies, while the shape of the dispersion curves remains quite similar. This can be attributed to the fact that the effective potential well confining the states within the Ag film is wider in the Ag/V(100) case than in the (rather unrealistic) self-standing Ag film, but otherwise the d states do not hybridize significantly with the vanadium substrate. The situation is very different in the case of sp and $\textrm{d}_{zz}$ derived bands, which fall largely within the energy range of the vanadium s band and hybridize strongly with it. Around the center of the surface Brillouin zone the weight of the Ag sp orbitals is transferred mostly to the quantum well state at binding energy 1.65 eV mentioned in the preceding paragraph, and to the state at 6.56 eV, just below the lower edge of the vanadium s band. The $\textrm{d}_{zz}$ state (with some contribution of Ag sp orbitals) hybridizes strongly with the vanadium s band and contributes to several states, shown as thick grey lines. Their number would increase if we used more than 7 layers of V, while their weight on Ag atoms would decrease. Finally, in the limit of a thick vanadium substrate a broad Ag $\textrm{d}_{zz}$–V s resonance would appear. A qualitative interpretation of the band structure of 2 ML Ag films, Fig. \[2layers\], compared to the 1 ML results is that each d state localized on silver splits into two. The sp-derived quantum well state with highest energy, well localized on Ag, is now at 0.66 eV below the Fermi level. Quantum well states and resonances ---------------------------------- In order to investigate into detail the character of quantum well states and resonances, we have made intensity plots of the amplitudes of the wavefunctions of some Kohn-Sham eigenstates. The wavefunctions shown in Fig. \[eigenfunctions\] are for 2 ML Ag films on V, but the results for 1 ML Ag films are similar, although with fewer nodes along the $z$ direction. We plotted the wavefunctions at the center of the surface Brillouin zone $\bar{\Gamma}$ where they can be expressed as a real function by factoring out a constant complex phase. The wavefunction of the sp QW state at $k_{\|\|}=0$, $E=0.66$ eV, is particularly interesting. It is almost translationally invariant in the directions of the surface plane, with one node, as predicted by the classification of states in the simple potential well models, e. g. as shown in Fig. 10 (d) in Ref. . However, it has an unexpectedly large amplitude on the first one or two vanadium layers, which is contributed by orbitals of $\textrm{d}_{zz}$ symmetry. In fact, the (approximate) projection of the density of states onto atomic orbitals of various symmetry shows that the wavefunction consists of approximately half Ag sp and half V $\textrm{d}_{zz}$ orbitals. Although large on the first V layer, the contribution of V orbitals decreases quickly and vanishes within a couple of V layers, since these orbitals are pulled out from their ‘natural’ energies (i.e. the respective vanadium d bands) and do not propagate into the vanadium bulk. In Fig. \[ag2density\] we show the square of the wave function of the 2 ML Ag QW state, for three values of the wavevector $k_{\|\|}$ along the $\bar\Gamma$–$\bar{\mathrm M}$ direction. The functions are shown along the $z$ coordinate at a particular $x_0$ and $y_0$, chosen in such a way that the line does not intersect any pseudopotential sphere used in our calculation. This was done to avoid the strong variation of the wavefunctions close to the atomic cores, but apart from that the choice of $x_0$ and $y_0$ is not significant, since the amplitudes of the wavefunctions are almost translationally invariant, as can be seen in the density plot of the QW state at $k_{\|\|}=0$ in Fig. \[eigenfunctions\] (d). Fig. \[ag2density\] confirms the quantum well state character of the wave function at $\bar\Gamma$, i.e. $k_{\|\|}=0$, where the [symmetry gap]{} is fully effective. At $k_{\|\|}=0.37$ Å$^{-1}$, which is just inside the grey box in Fig. \[2layers\], the wavefunction has a much decreased density outside the first Ag atom, and extends up to the third and fourth V layer. Finally, at $k_{\|\|}=0.57$ Å$^{-1}$ the wavefunction is located mostly on the first two V layers, but extends through the rest of the slab and has a tail into the vacuum on the other side of our Ag/V structure. Thus the state has lost its QW character, and can be more appropriately described as an interface resonance. These results show that although simple well potential or phase accumulation models may successfully reproduce the energies of the QW states at the $\bar{\Gamma}$ point with an appropriate choice of parameters, only ab initio calculations can give accurate information about the degree of hybridization with V orbitals, in particular away from the center of the surface Brillouin zone, which can be essential for photoemission cross sections and other properties. Comparison with experiments --------------------------- In Fig. \[PE-exp\] we have plotted the experimental spectra of normal photoemission from a monolayer of Ag on V(100) [@Mil-04] and calculated energies at the $\bar{\Gamma}$ point of electronic states localized on the silver adlayer. The experimental spectrum was measured at the ELETTRA synchrotron in Trieste. The sample was at room temperature, and the energy resolution of the measurement was around 30–50 meV. [@Kra-03b] The prominent peak at 1.65 eV binding energy has been assigned to the Ag sp quantum well state, and the one around 5.2 eV to the Ag d QW state. The FWHM of both peaks is close to 400 meV. In Ref. a careful analysis of spectra taken in a similar experiment, in which the energy resolution was around 10 meV and the temperature could be varied, showed that at 60 K the 2 ML sp QW peak width is around 360 meV, which was decomposed into various contributions. Based on those results, we can attribute around 200 meV of the line width in Fig. \[PE-exp\] to the finite lifetime of the hole (this contribution depends roughly quadratically upon the binding energy of the QW state), 150 meV to impurity and phonon scattering, and the remaining 30–50 meV to the experimental resolution. In Ref. it was found that a large part of the apparent width of the d QW peak around 5.2 eV comes from the fact that it is a doublet, with the two components split by around 120 meV. The splitting is due to the spin-orbit coupling, which is not included in our calculations. Another peak which can be attributed to a d QW state occurs at 4.7 eV binding energy. It is quite weak in the spectrum shown here but has a larger intensity at other photon energies. It is a singlet, with a width around 95 meV.[@Kra-03b] The results of our calculations are shown as lines of various style and height. The height indicates the degree of localization of the electronic state on the silver adlayer, which gives a rough estimate of the expected photoemission strength. Thus the state of s symmetry at 1.65 eV is shown with a full height line, since it lies in the symmetry gap of vanadium and cannot propagate into the bulk, although it has a strong component on the first V layer, as already discussed for the 2 monolayer QW state. Similarly, the Ag d states of $\textrm{d}_{xy}$ and $\textrm{d}_{x^2-y^2}$ symmetry at larger binding energies also form QW states due to symmetry mismatch with substrate bands, as do the $\textrm{d}_{xz}$, $\textrm{d}_{yz}$ states which are shown with a line of double height since they are degenerate at $k=0$. The states derived from Ag $\textrm{d}_{zz}$ orbitals, however, are shown as half-height grey lines, since they readily hybridize with the s band of the vanadium substrate. These states have also a small admixture of Ag s orbitals. Finally, we have denoted the state at 6.56 eV also by a black bar, i.e. as a QW state of s symmetry, because in our calculation the wavefunction decreases approximately exponentially on the deeper vanadium layers. However, our calculation with only seven layers of vanadium underestimates the width of the s band of V, and a more appropriate description of this state may be that of a s resonance at the bottom of the vanadium s band. The calculated energy of the s QW state at $E=1.65$ eV agrees well with the peak in the experimental spectrum. The energies of d quantum well states, however, are around 1 eV too high compared to the experimental peaks. There is no reason for any shift of the experimental photoemission peaks of the d states compared to sp, and this discrepancy appears to be a deficiency of the density functional calculation. It is indeed known that local density functional calculations tend to place the d bands of silver (and other noble metals as well) too close to the Fermi energy, as discussed in Ref. and references therein. We obtain good agreement for the energies of the sp QW states at $\bar{\Gamma}$ point for thicker Ag films as well. In Table \[table\] we show the calculated energies of the highest state of sp symmetry, for films from 1 to 5 monolayers of Ag. In all cases, except possibly for 3 ML, the energies fall in the symmetry gap discussed earlier. The results are compared with the values measured in normal photoemission.[@Kra-01] The agreement is very good, considering the finite width and the temperature dependence of the experimental peaks. There is no direct measurement of the QW state in 3 ML Ag films, which are probably structurally unstable.[@Kra-03a] However, in Ref. photoemission spectra have been reported from Ag films on V(100) of nominal coverage of 2.5 ML, which presumably consist of regions of 2 ML and 3 ML coverage. In Fig. 4 of that reference the broad peak around $2.3$ eV, which is the upper edge of the vanadium s band, is assigned to photoemission from a QW state or resonance associated with the 3 ML regions, in good agreement with our calculated value. The dispersion of Ag QW states can be compared with off-normal ARPES experiments. In Fig. \[2layers\] we have marked by a grey box the $E-k_{\|\|}$ region in the $\bar{\Gamma}$–$\bar{M}$ direction in which the angle resolved PES spectra of 2 ML Ag films on vanadium have been reported.[@Kra-03c; @Kra-05] Comparing the thick lines in our calculation to the density plot of the off-normal photoemission intensity in Fig. [6.14]{} of Ref. and Fig. 2 of Ref. we see that the agreement is excellent. Both the quadratic dispersion of the QW state around the $\bar{\Gamma}$ point and the subsequent bending down, as well as the transfer of photoemission intensity to the vanadium-induced states which cross the Fermi level at finite $k_{\|\|}$ are seen in the experimental spectra. It is interesting that our calculations find a quite different behavior in the $\bar{\Gamma}$–$\bar{X}$ direction, where a state with binding energy slightly larger than the QW state acquires a significant weight on the surface atoms soon after leaving the $\bar{\Gamma}$ point of the Brillouin zone. This direction, however, has not been investigated by ARPES experiments. In Refs. standing wave patterns around surface defects have been observed in STM images taken on 1–5 ML thick Ag films on V(100) at room temperature. The defects were mostly point scatterers, and in some cases also surface steps. Such patterns are known to originate from the interference of electrons with predominantly two-dimensional character, usually surface states,[@Has-93] on localized potentials. This is a rare example of standing waves in a room temperature experiment, as most other observations have been done at low temperatures. It indicates a strongly enhanced two-dimensionality of electronic states in silver overlayers with energies around the Fermi level at large values of the wavevector $k_{\|\|}$. In the unsupported silver monolayer, Fig. \[monolayer\], the Fermi level is crossed by the s band at around $k_{\|\|}=0.8$ Å$^{-1}$. For 1 ML and 2 ML Ag films on V(100), Figs. \[1layer\] and \[2layers\], states largely localized at the surface (thick lines) also appear in this range of $k_{\|\|}$. Strong hybridization with vanadium states makes the whole picture rather complicated, and a quantitative interpretation of the STM experiments is not possible at present. \[conclusions\] Conclusions =========================== Our calculations of the electronic states in few monolayers thick silver films on V(100) substrate have shown that there is a strong localization of silver-derived states within the Ag layer, due to the mismatch of the electronic structure of the two metals. The localization is particularly noticeable at the $\bar{\Gamma}$ point of the surface Brillouin zone, where quantum well states of both s character (close to the Fermi level) and d character (at more than 3 eV binding energy) appear on Ag films. States of $\textrm{d}_{zz}$ symmetry, on the other hand, readily hybridize with vanadium s electrons, which are symmetry compatible. Hybridization effects also become more important away from the center of the Brillouin zone. We have obtained good agreement with photoemission experimental spectra, both in the direction normal to the surface and with the available measurements in off-normal directions. This work was supported by the Ministry of Science and Technology of the Republic of Croatia under contract No. 0098001. We acknowledge useful discussions with M. Milun, P. Pervan and M. Kralj, and their kind permission to reproduce unpublished experimental spectra. [00]{} M. Milun, P. Pervan, and D.P. Woodruff, Rep. Prog. Phys. 65, 99 (2002). P.D. Johnson, K. Garrison, Q. Dong, N.V. Smith, Dongqi Li, J. Mattson, J. Pearson, and S.D. Bader, Phys. Rev. B 50, 8954 (1994). I. Matsuda, T. Ohta, and H.W. Yeom, Phys. Rev. B 65, 085327 (2002). J.J. Paggel, T. Miller, and T.-C. 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--- abstract: 'We propose the semiclassical quantization for complicated electron systems governed by a many-band Hamiltonian. An explicit analytical expression of the corresponding Berry phase is derived. This impact allows us to evaluate the Landau magnetic levels when the rigorous quantization fails, for instance, for bilayer graphene and graphite with the trigonal warping. We find that the magnetic breakdown can be observed for the certain type of classical electron orbits.' author: - 'A.Yu. Ozerin' - 'L.A. Falkovsky' title: ' Berry phase, semiclassical quantization and Landau levels' --- The most accurate investigation of the band structure of metals and semiconductors is studying the Landau levels in magneto-transport and magneto-optical experiments. However, the theoretical solution of the band problem in magnetic fields cannot often be exactly found. A typical example is presented by graphene layers. For bilayer graphene and graphite, the effective Hamiltonian is a $4\times4$ matrix giving four energy bands. Fig. \[cr\_se\] shows nearest two bands of the level structure together with semiclassical orbits. The trigonal warping described by the effective Hamiltonian with a relatively small parameter $\gamma_3$ provides an evident effect (see right panel). Another important parameter is the gate-tunable bandgap $U$ in bilayer graphene. In this situation, the quantization problem cannot be solved within a rigorous method. To overcome this difficulty one can use a perturbation theory, however this theory becomes quite complicated for the many-band Hamiltonian. Alternatively, the semiclassical quantization can be applied. Thus, we can use the Bohr-Zommerfeld condition as $$\frac{c}{e\hbar B}S(\varepsilon)= 2\pi\left[n+\frac{\mathcal T}{4}+\delta(\varepsilon)\right]\,. \label{on}$$ Here $S(\varepsilon)$ is the cross-section area of the electron orbit in the ${\bf k}$ space for the energy $\varepsilon$ in absence of the magnetic field [B]{} and for the constant momentum projection $k_z$ on the magnetic field, $n$ is an integer supposed to be large. $\mathcal T$ is the number of the smooth turning points on the electron orbit. There are two smooth turning points for the Landau levels and only one for skipping electrons reflected by the hard edge. The goal of this letter is an explicit analytical expression for the $\delta(\varepsilon)-$phase within the band scheme of the matrix Hamiltonian. The semiclassical approach is used for the magnetic field normal to the layered system when the quantization of in-layer momentum components is only essential and the size of the Fermi surface is small compared with the Brillouin zone size. We illustrate our results for bilayer graphene. Notice, that the $\delta(\varepsilon)-$phase depends on the energy and can be taken in the interval $0\leq\|\delta\|\leq1/2$. If the spin is neglected, $\delta=0$ and $\mathcal T=2$ for the Landau levels, and $\delta=1/2$ and $\mathcal T=2$ for monolayer graphene. In these two cases, the semiclassical result coincides with the rigorous quantization and it is closely connected with the topological Berry phase [@Be]. This $\delta-$phase was evaluated for bismuth in Ref. [@Fal], preceding Berry’s work by almost two decades, and it was considered again for bismuth in Ref. [@MS]. For graphite, the semiclassical quantization was applied in Ref. [@Dr]. However, in the general case, the evaluation of the $\delta-$phase is still attracted a widespread interest [@TA; @CU; @KEM; @PM; @PS; @LBM; @ZFA]. The problem under consideration is described by the Hamiltonian in the band representation $$\label{hami} {(\bf V\cdot\tilde{k}} +\Gamma-\varepsilon)\Psi=0\,,$$ where the column $\Psi$ consists of functions corresponding with a number of bands included and is labelled by the band subscript which we omit together with the matrix subscripts on $\Gamma$ and ${\bf V}$; a summation over them is implied in Eq. (\[hami\]). Matrices $\Gamma$ and ${\bf V}$ are the first two terms in a series expansion of the Hamiltonian in the power of quasi-momentum $\bf{k}$. In the magnetic field, the momentum operator ${\bf\tilde{k}}$ depends on the vector-potential ${\bf A}$ by means the Peierls substitution, $${\bf\tilde{k}}=-i\hbar\nabla-e{\bf A}/c,$$ providing the gauge invariance of the theory. The magnetic field can also enter explicitly describing the magnetic interaction with the spin of a particle. However, for the graphene family, the magnetic interaction is weak and omitted here. A simple example of Eq. (\[hami\]) is given by the graphene monolayer. There are two sublattices in it, and Eq. (\[hami\]) is represented by a $2\times2$ matrix if the spin of carriers is neglected. Another example considered below is bilayer graphene with the $4\times4$ matrix Hamiltonian. For the monolayer and bilayer graphene, both ${\bf V}$ and ${\bf {\tilde k}}$ are two-dimensional vectors, e.g., with $x$ and $y$ components. We seek for $\Psi$ in the form $$\nonumber \Psi=\Phi \exp{(is/\hbar)}\,, \label{be}$$ where the function $s$ is assumed to be common for all components of the column $\Psi$. The equation for $\Phi$ is reduced to $$\begin{aligned} \label{ham1}[{\bf V\cdot (k}-i\hbar\nabla) +\Gamma-\varepsilon]\Phi=0\,,\\ \label{k} \text {with} \quad {\bf k}=\nabla s-e{\bf A}/c\,.\end{aligned}$$ The function $\Phi$ is expanded in series of $\hbar/i$: $$\Phi=\sum_{m=0}^{\infty}\left(\frac{\hbar}{i}\right)^m \varphi_m\,.$$ Comparing the terms involving the same powers of $\hbar$ in Eq. (\[ham1\]) we have $$\label{ham3}({\bf V \cdot k} +\Gamma-\varepsilon)\varphi_m=-{\bf V \nabla}\varphi_{m-1}\,.$$ For $m=0$, we get a homogeneous system of algebraic equations $$\label{ham4}({\bf V\cdot k} +\Gamma-\varepsilon)\varphi_0=0\,$$ which has a solution under the condition $$\text{Det}({\bf V\cdot k} +\Gamma-\varepsilon)=0\,. \label{edm}$$ This equation determines the classical electron orbit, $\varepsilon(k_x,k_y)=\varepsilon$, in presence of the magnetic field while the electron energy $\varepsilon$ is constant. At the same time, the equation yields the electron dispersion equation with ${\bf k}$ as the momentum without any magnetic field. In 3d case, the electron dispersion depends as well on the momentum projection $k_z$ on the magnetic field and our scheme can be implied in this case without the expansion in $k_z$. It is convenient to choose the vector-potential in the Landau gauge $A_x=-By , A_y=A_z=0$ in such a way that the Hamiltonian does not depend on the $x-$coordinate. Then, the $x-$momentum component $K_x$ becomes a conserved quantum number and the function $s$ in Eq. (\[k\]) can be written as $$s=xK_x+\sigma(y)\,.\label{s}$$ The equations (\[k\]) are reduced to $$k_x=K_x+\frac{e}{c}By\,,\quad k_y=\frac{d\sigma}{dy}.$$ These equations enable us to use the variable $k_x$ instead of $y$ and to obtain $$\begin{aligned} \sigma(k_x)=\frac{c}{eB}\int\limits^{ k_x} k_y(k_x') dk_x'\,, \label{si}\end{aligned}$$ where $k_y$ as a function of $k_x$ is determined by the dispersion equation (\[edm\]). The eigenfunction column obeying Eq. (\[ham4\]) can be multiplied by the scalar function $C$ common for all elements of the column $$\varphi_0\rightarrow C\varphi_0\,$$ where $\varphi_0$ is any eigen-column of Eq. (\[ham4\]). The function $C$ is determined by Eq. (\[ham3\]) with $m=1$. Left-to-right multiplying both sides of this equation by $\varphi^_0$ and using the Hamiltonian hermiticity, i.e. the complex conjugations of Eq. (\[ham4\]), we get the consistency condition $$\label{ham5} \varphi^_0{\bf V\cdot\nabla}(C\varphi_0)=0\,,$$ where the derivative with respect to $y$ (i.e. to $k_x$) is only to be taken. The left hand-side of this equation can be written as $$\frac{1}{C}\frac{d C}{dk_x}+\frac{1}{2\varphi^_0V_y\varphi_0} \frac{d\varphi^V_y\varphi}{dk_x}+\frac{i}{\varphi^_0V_y\varphi_0}\text{Im}\, \varphi^_0V_y\frac{d\varphi_0}{dk_x}$$ Using the identity $\varphi^_0{\bf V}\varphi_0=\varphi^_0\varphi_0{\bf v}$ with the electron velocity ${\bf v}=\partial \varepsilon/\partial {\bf k}$, one can write the solution of Eq. (\[ham5\]) as $$\label{c} C=c_0(\varphi^_0\varphi_0 v_y)^{-1/2}\exp(-i\theta)\,,$$ where $$\theta=\text {Im}\int\frac{dk_x}{\varphi^_0\varphi_0 v_y} \varphi^_0V_y\frac{d\varphi_0}{dk_x} \label{thet}$$ and $c_0$ is the normalization factor. The quantization condition can be written as usual from the requirement that the wave function has to be single-valued. Continuing Eqs. (\[si\]), (\[c\]), and (\[thet\]) along the orbit and making the bypass in the complex plane around the turning points where $v_y=0$ to obtain the decreasing solutions in the classically unaccessible region, one obtains $\mathcal{T}=2$ and $\delta-$phase as a contour integral along the classical orbit $$\label{on1} \delta(\varepsilon)=\frac{1}{2\pi}\text{Im} \oint\frac{dk_x}{\varphi^_0\varphi_0 v_y} \varphi^_0V_y\frac{d\varphi_0}{dk_x}\,.$$ Using the Hamiltonian hermiticity, after the simple algebra (see Ref. [@Fal]), Eq. (\[on1\]) can be rewrite as $$\delta(\varepsilon)=\frac{1}{4\pi}\text{Im} \oint\frac{dk}{\varphi_0^\varphi_0 v} \varphi^_0\left[{\bf V}\times\frac{d}{d{\bf k}}\right]_z\varphi_0$$ called usually the Berry phase. Now let us calculate the $\delta-$phase for bilayer graphene. In simplest case, the effective Hamiltonian can be written (see, for instance Refs. [@PP; @GAW]) as $$H(\mathbf{k})=\left( \begin{array}{cccc} U \, & q_{+} \,& \gamma_1 \, & 0\\ q_{-} \,& U \, & 0\,& 0\\ \gamma_1 \, &0 \,&-U \, &q_{-}\\ 0 \,& 0 \,&q_{+} \,&-U \end{array}\right) , \label{ham}$$where the parameter $U$ describes the tunable gap, $\gamma_1$ is the nearest-neighbor hopping integral energy, the matrix elements are expanded in the momentum $k_{\pm}=\mp ik_x-k_y$ near the $K$ points of the Brillouin zone, and the constant velocity parameter $v$ is incorporated in the notation $q_{\pm}=vk_{\pm}$. Here, the orbit is the circle defined by Eq. (\[edm\]), written in the following form $$[(U+\varepsilon)^2-q^2][(U-\varepsilon)^2-q^2]-\gamma_1^2(\varepsilon^2-U^2)=0\,. \label{disp}$$ The eigenfunction ${\mathbf \varphi_0}$ of the Hamiltonian (\[ham\]) can be taken as $${\mathbf \varphi_0}=\left( \begin{array}{c} (U-\varepsilon)[(\varepsilon+U)^2-q^2]\\ q_{-}[q^2-(\varepsilon+U)^2]\\ \gamma_1(U^2-\varepsilon^2)\\ \gamma_1q_{+}(U-\varepsilon)\end{array}\right) ,\label{fun}$$ with the norm squared $$\begin{aligned} \varphi_0^\varphi_0=[(\varepsilon+U)^2-q^2]^2[(\varepsilon-U)^2+q^2]\nonumber\\ +\gamma_1^2(\varepsilon-U)^2 [(\varepsilon+U)^2+q^2]\,.\label{norm}\end{aligned}$$ The derivatives for Eq. (\[on1\]) are calculated along the trajectory where the energy $\varepsilon$ and consequently the trajectory radius $q$ are constant. The equation (\[disp\] ) has only one solution for $q^2$ if $\|U\|<\|\varepsilon\|<\sqrt{U^2+\gamma_1^2}.$ First, let us consider this case. \(i) [there is only one orbit at given energy $\varepsilon$ with the radius squared]{} $$q^2=U^2+\varepsilon^2+\sqrt{4U^2\varepsilon^2+(\varepsilon^2-U^2)\gamma_1^2}\,.$$ The matrix $V_y=\partial H/\partial k_y$ in Eq. (\[on1\]) has only four nonzero elements $V_{y12}=V_{y21}=V_{y34}=V_{y43}=-1$. Using Eqs. (\[disp\]) and (\[fun\]), we find $$\text{Im}\,\varphi_0^V_y\frac{d\varphi_0}{dk_x}= 4U\varepsilon(U-\varepsilon)[(\varepsilon+U)^2-q^2]\,.\label{sver}$$ This expression is constant on the trajectory as well as $\varphi_0^\varphi_0$, Eq. (\[norm\]). Therefore, in order to find $\delta$, Eq. (\[on1\]), we have to integrate along the trajectory $$\oint \frac{dk_x}{v_y}\,.$$ This integral equals $-dS(\varepsilon)/d\varepsilon$, where $S(\varepsilon)=\pi q^2$ is the cross-section area, Eq. (\[on\]), with $$\nonumber \frac{dS(\varepsilon)}{d\varepsilon}=\pi\varepsilon\frac{2(q^2+U^2-\varepsilon^2)+\gamma_1^2}{q^2-U^2-\varepsilon^2}\,. \nonumber\label{dsec}$$ Now we have to substitute Eqs. (\[norm\]), (\[sver\]), and (\[dsec\]) into Eq. (\[on1\]). Thus, we find the Berry phase $$\delta(\varepsilon)=\frac{-\varepsilon U}{q^2-\varepsilon^2-U^2}= \frac{-\varepsilon U}{\sqrt{4U^2\varepsilon^2+(\varepsilon^2-U^2)\gamma_1^2}} \label{del1}$$ shown in Fig. \[disg\], where $\delta-$phase of bilayer graphene with trigonal warping is also shown, the detailed calculations will be elsewhere published. For the ungaped bilayer, $U=0$, the Berry phase $\delta(\varepsilon)=0$. The Berry phase depends on the energy and $\delta=\mp 1/2$ at $\varepsilon =\pm U$. At the larger energy, $\varepsilon\gg U$, the Berry phase $\delta\rightarrow \mp U/\gamma_1$. Substituting Eq. (\[del1\]) in the semiclassical quantization condition, Eq. (\[on\]), and solving the equation obtained for $\varepsilon$, we get energy levels as a function of the magnetic field. We have to notice that the Landau numbers $n_L$ listed in Fig. \[comp\] do not coincide with the numbers $n$ in the semiclassical condition (\[on\]). The rigorous quantization shows that there are only one Landau level with $n_L=0$ and three Landau levels with $n_L=1$ [@Fa]. These levels are not correctly described within the semiclassical approach. However, for $n_L\geq 2$, there are levels in all four bands $s$ (two nearest bands with $s=2,3$ are shown in Fig. \[comp\]). They correspond with the quantum number $n = n_L - 1$, and the semiclassical levels become in excellent agreement with the rigorous solution for the larger $n$. \(ii) [for]{} $\|U\|/\sqrt{1+(2U/\gamma_1)^2}<\|\varepsilon\|<\|U\|\,,$ [at the given energy, there are two orbits with the radius squared]{} $$q_{1,2}^2=U^2+\varepsilon^2\pm r\,, \text{where}\quad r=\sqrt{4U^2\varepsilon^2+(\varepsilon^2-U^2)\gamma_1^2}\,.$$ This is an effect of “the mexican hat”. Then we seek for the general solution as a sum of two solutions $\varphi_0^j(1)$ and $\varphi_0^j(2)$ corresponding to these two contours, $$\varphi_0^j=C_1\varphi_0^j(1)+C_2\varphi_0^j(2)$$ with two scalars $C_1$ and $C_2.$ Instead of Eq. (\[ham5\]) we have a system of two equations written in the $2\times2$ matrix form as follows $$a\frac{dC}{dq_x}+bC=0 \label{c12}$$ where the notations of the matrix elements are introduced $$a_{ik}=\varphi_0^(i)V_y\varphi_0(k)\,, \quad{\displaystyle b_{ik}=\varphi_0^(i)V_y\frac{d\varphi_0(k)}{dq_x}}\,.$$ The off-diagonal matrix elements $a_{ik}$ vanish, $a_{12}=a_{21}=0$. Thus, the first equation of the system (\[c12\]) becomes $$2q_{1y}r\frac{dC_1}{dq_x}+(2i\varepsilon U-rq_x/q_{1y})C_1 +i(2\varepsilon U+r)C_2=0\,,$$ and the second equation can be obtained with the index replacement $1\leftrightarrow2$ and $r\rightarrow -r$. These equations can be simplified with the substitution $$C_i= \tilde{C}_i(q_i^2-q_x^2)^{-1/4}\,.\label{pm}$$ For the new functions $\tilde{C}_i$, we get the equation system $$\begin{array}{c}{\displaystyle q_{1y}\frac{d\tilde{C}_1}{dq_x}+iE\tilde{C}_1 +i\sqrt{\frac{q_{1y}}{q_{2y}}}(E+\frac{1}{2})\tilde{C}_2=0\,,}\\{\displaystyle q_{2y}\frac{d\tilde{C}_2}{dq_x}-iE\tilde{C}_2 -i\sqrt{\frac{q_{2y}}{q_{1y}}}(E-\frac{1}{2})\tilde{C}_1=0\,}, \end{array}$$ where the parameter $q_{iy}=\sqrt{q_{i}^2-q_x^2}, \quad i=1,2$ and $E=\varepsilon U/r$. For the minimum of conduction band (maximum of valence band), where $r\rightarrow 0$, there is a simple limit, $$q_{1y}\frac{d\tilde{C}_1}{dq_x}-\frac {i}{2}\tilde{C}_1=0\quad \text {with}\quad \tilde{C}_2=-\tilde{C}_1\,.$$ Solving this equation, one gets $$\tilde{C}_1=c_0\exp\left(\frac{i}{2}\arcsin{\frac{q_x}{q_1}}\right)\,.\label{tc}$$ Going with $q_x$ along the trajectories and making the bypass in the complex plane around the turning points $q_x=\pm q_1$ and $q_x=\pm q_2$, we see that both $C_1$ and $C_2$ acquire from two turning points in Eq. (\[pm\]) the additional phase $-\pi$ with $\mathcal {T}=2$. At the same time, we have $-1/2$ from Eq. (\[tc\]) for $\delta-$phase. Thus, at the boundaries of the narrow interval considered, the $\delta-$phase obtains the same value, $\delta=-1/2.$ Taking into account the phases of the functions $\varphi_0^j(i)$, we see, that the area $S(\varepsilon)$ in Eq. (\[on\]) can play the different role. In weak magnetic fields, slower oscillations with the smaller $S(\varepsilon)$ corresponding to $q_2$ should be observed in oscillating phenomena. However, when the magnetic field becomes larger and the semiclassical condition is fulfilled only for the larger cross-section $S(\varepsilon)$, calculated with $q_1$, the larger frequency oscillations should be observed. This is nothing but the magnetic breakdown [@CF] which should be utilized if the chemical potential belongs to the interval where the effect of “the mexican hat” appears. In conclusion, our study shows that the semiclassical approach gives a powerful tool for probing the electron magnetic properties in metals. The Berry phase depending on the energy can be calculated and observed even for complicated band scheme. The method presented here should be useful for many electron systems. We thank I. Luk’yanchuk for helpful discussions. 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Zhang, M.M. Fogel, D.P. Arovas, Phys. Rev. B 84, 075451 (2011). B. Partoens, F.M. Peeters, Phys. Rev. B 74, 075404 (2006). A. Grüneis, C. Attaccalite, L. Wirtz, H. Shiozawa, R. Saito, T. Pichler, A. Rubio, Phys. Rev. B 78, 205425 (2008). L.A. Falkovsky Phys. Rev. B 84, 115414 (2011). M.N. Cohen, L.M. Falikov, Phys. Rev. Lett. 7, 231 (1961).
--- abstract: \| A code, , has been developed which computes, in the two-dimensional case, the initially circular orbits of guiding centra in an arbitrary axisymmetric potential with an arbitrary, weak perturbing potential in solid body rotation. This perturbing potential is given by its Fourier expansion. The analytic solution solves the linear epicyclic approximation of the equations of motion. To simulate the motion of interstellar matter and to damp the Lindblad resonances, we have in these equations introduced a friction which is proportional to the deviation from circular velocity. The corotation resonance is also damped by a special parameter. The program produces, in just a few seconds, orbital and density maps, as well as line of sight velocity maps for a chosen orientation of the galaxy. We test by comparing its results with previous simulations and observations from the literature, which gives satisfactory agreement. The aim is that this program should be a useful complement to elaborate numerical simulations. Particularly so are its abilities to quickly explore the parameter space, to construct artificial galaxies, and to isolate various single agents important for developing structure of interstellar matter in disc galaxies. author: - \| N. Piñol-Ferrer$^{1}$[^1], P. O. Lindblad$^{1}$, K. Fathi$^{1}$\ $^1$Stockholm Observatory, Department of Astronomy, Stockholm University, AlbaNova Center, 106 91 Stockholm, Sweden\ bibliography: - 'mybib.bib' title: Analytic gas orbits in an arbitrary rotating galactic potential using the linear epicyclic approximation --- \[firstpage\] Galaxies: kinematics, Galaxies: barred, Galaxies: spiral. Introduction ============ Since the emergence of high spatial resolution two-dimensional velocity field for galaxies, the interpretation of observed velocity fields has become one of the key concerns of the extragalactic scientific community (e.g., @Boulesteix1987; @Adam1989; @Bacon1992; @Bacon2001; @Allington1997; @Hernandez2003). Of particular importance is the class of galaxies which host bars and/or spiral arms, as they are estimated to be the most abundant types of galaxies in the local Universe [@Loveday1996]. The theoretical foundation for understanding the kinematics and dynamics of spiral galaxies has been worked out over the last century. Bertil @Lindblad1963 [@Lindblad1964] conceived a picture of circulation of stars between the spiral arms of a quasi-steady rotating spiral potential. Simultaneously, @LinShu1964 presented their density wave theory approaching the problem from a different theoretical point of view. Subsequently, @Shu1973 and @Roberts1979 derived solutions for the circulation of the interstellar medium through such a spiral potential and predicted large scale galactic shocks along the spiral arms. These processes make bars and/or spiral arms potential actors to redistribute angular momentum which will lead to dramatic effects such as intense star formation, build up of bulges, or onset of nuclear activity. In turn, the host galaxies may undergo dramatic secular evolution on time-scales compared to the galaxy dynamical time (see @Kormendy2004 for a comprehensive discussion). Beginning with @Holmberg1941 in the pre-computer era and in the 1950s with electronic computers (e.g. P.O. @Lindblad1960), a large number of simulations of development and evolution of structure in galaxies have been made with a variety of computer codes. These studies have often aimed at establishing the details of the flow of gas in and around bars (eg. @Athanassoula1992b), as well as the mechanisms which trigger starbursts and nuclear activity in the centers of galaxies (e.g., @Shlosman1989). However, they often involve extensive computational power leading to relatively long computing times. Several successful attempts have been carried out in order to deliver analytic solutions for observable parameters for some of the effects involved in the above questions (Sakhibov & Smirnov 1989; @Canzian1993; @Wada1994; @Lindblad1994; @Schoenmakers1997; @Wong2004; @Fathi2005; @Byrd2006; @vandeVenfathi2010), though they have all exclusively treated one specific morphological feature in each analytic model. While a linear analysis can be a guide for a physical understanding of complicated numerical results from simulations, in these analytic studies, bars and spiral arms have been accounted for separately, and interpretations have been based on marginalizing the effect of the other features. A notable difference between these results and numerical simulations is that the stepwise integration in a detailed simulation requires appreciable computing times, while for analytical calculations the computing time is negligible. In this paper, we present an analytic solution within the epicyclic approximation, in which we introduce an arbitrary gravitational potential and a damping coefficient for adequate appearance of the corotation resonance. Our solution has been coded as a development of the program <span style="font-variant:small-caps;">Epic</span> [@Lindblad1994] which was based on the first order epicyclic approximation. The current code, , derives the response of interstellar matter, originally in circular orbits, to the gravitational potential in a matter of few seconds, which makes it an efficient code for surveying a large parameter space. In section \[sec:method\], we present the analytical solution for the epicyclic theory with a generic potential, and we describe our code . We compare results generated by with @Wada1994 in section \[sec:models\], and in section \[sec:new\], we demonstrate a standard case of a barred galaxy. In section \[sec:1365\] we compare results obtained by with observations and simulations made for the galaxy NGC 1365. Finally, we conclude in section \[sec:conclusions\]. Method {#sec:method} ====== The epicyclic description of nearly circular stellar orbits in a circularly symmetric galaxy was developed by B. @Lindblad1927 in order to theoretically explain the observed velocity ellipsoid in the Milky Way galaxy. Later on, B. @Lindblad1958 introduced a rotating perturbing potential in the theory and pointed out the resonances that carry his name. P.O. Lindblad & P.A.B. Lindblad (1994), as well as @Wada1994, introduced gas dynamical friction in the epicyclic approximation. This form of damping the motions would make the first order approximation valid also over the Lindblad resonances, but still not over the corotation resonance. The potential we are considering in our problem is: $$\Phi (r,\theta) = \Phi_0 (r) + \Phi_1 (r,\theta) \label{eq:pot}$$ where ($r,\theta)$ are the polar coordinates in a frame co-rotating with the potential at the pattern speed frequency . $\Phi_0(r)$ is the axisymmetric potential, and $\Phi_1 (r,\theta)$ is an arbitrary perturbing potential developed as a Fourier series $$\centering \Phi_1 (r,\theta) = - \sum_{m=1}^{\rm n} \left[ \rm P_{ \emph \rm c,\emph m} (\emph r) \cos\left( \emph m\,\theta\right) + P_{\rm s,\emph m}(\emph r) \sin\left(\emph m\,\rm \theta\right)\right] \label{eq:fourierpot}$$ or $$\centering \Phi(r,\theta)=\Phi_0(r)-\sum_{\emph m=1}^{\rm n} \Psi_{\emph m}(r) \cos\emph m\left(\theta-\vartheta_{\emph m}(r)\right)$$ where $$\centering \Psi_{\emph m}^2(r)=\rm P_{ \emph \rm c,\emph m}^2 + P_{\rm s,\emph m}^2 \,\,\,\,\,\,\,\,\text{ and }\,\,\,\,\,\,\,\, \tan(\emph m \vartheta_{\emph m})=\rm P_{ \emph \rm c,\emph m} / P_{\rm s,\emph m}$$ We consider small deviations, $\xi$ and $\eta$, from circular motion following the usual notation, where $$\begin{aligned} r&=r_0+\xi \\ \theta&=\theta_0 + (\Omega-\Omegap) \, t + \frac{1}{r_0} \eta \label{eq:coor}\end{aligned}$$ and $\Omega(r_0)$ is the angular frequency of circular motion. In addition, we introduce a frictional force proportional to the velocity deviation from circular motion with the coefficient $-2\lambda$. As @Wada1994 points out, this is analogous to the Stokes’ formula [eg. @Yih1977 p. 365] where the drag on a sphere moving slowly in a viscous fluid is proportional to the first power of the velocity. The equations of motion, as derived in appendix \[ap:theory\], can then be written as: $$\begin{aligned} \centering \ddot{\xi} + 2 \lambda \dot{\xi} - 2\Omega \dot{\eta} - 4 \Omega A \xi &= - \frac{\partial \Phi_1}{\partial r} =\nonumber\\ =&\sum_{m=1}^{n} \big[ C_m \cos m(\theta - \vartheta_m) + E_m \sin m(\theta - \vartheta_m)\big] \label{eq:eqofmotionfric1}\\ \ddot{\eta} + 2 \Omega \dot{\xi} + 2 \lambda \dot{\eta} + 4 \lambda A \xi &= - \frac{1}{r}\frac{\partial \Phi_1}{\partial \theta} = - \sum_{m=1}^{n} D_m \sin m(\theta - \vartheta_m) \label{eq:eqofmotionfric2}\end{aligned}$$ where $$C_m=\frac{d\Psi_m}{dr}; \,\,\,\,\,\,D_m=m\frac{\Psi_m}{r}; \,\,\,\,\,\, E_m=m\Psi_m\frac{d\vartheta_m}{dr} \nonumber$$ The solution for the motion of the guiding center will then be: $$\begin{aligned} \xi = \sum_{m=1}^{n} \big[ d_m \cos m(\theta-\vartheta_m) + e_m \sin m(\theta -\vartheta_m) \big] \label{eq:xisol}\\ \eta= \sum_{m=1}^{n} \big[ g_m \sin m(\theta-\vartheta_m) + f_m \cos m(\theta -\vartheta_m) \big]\label{eq:etasol} \end{aligned}$$ where the amplitudes $d_m$, $e_m$, $g_m$ and $f_m$ are given in appendix \[ap:theory\]. The code: {#subsec:epic} ---------- The code is an extension of the code developed by @Lindblad1994 and computes the analytic solution (\[eq:xisol\]) and (\[eq:etasol\]), as well as its corresponding density and velocity maps, generated by the arbitrary potential. derives the axisymmetric potential, $\Phi_0$, from a given rotational velocity curve and the perturbing potential, $\Phi_1$, is introduced by its Fourier decomposition, and (eq. \[eq:fourierpot\]). In addition to the velocity curve and the parameters of the potential, another free input parameter is a constant pattern speed of the potential, . The damping coefficient of the frictional force, $\lambda$, which was introduced in the analysis to simulate orbits of interstellar matter, is assumed to have a linear tendency with radius. This linearity is determined by the given values of $\lambda$ at the outer inner Lindblad resonance (oILR) and at the outer Lindblad resonance (OLR) of the system (it is taken at 0 kpc if no ILR is present and if no OLR is present, it is taken at $r_{max}$ in the rotation curve). These two values are also given as input parameters in . This definition of the damping coefficient, $\lambda$, that allows it to vary along the radius, is desirable since it should depend on the gas density. As can be seen from eqs. \[eq:d\]-\[eq:f\], the introduction of the friction coefficient $\lambda$ has damped the amplitudes and eliminated the singularities at the Lindblad resonances where $\kappa^2 - \omega_m^2 = 0$. However, the singularity at corotation, $\omega_m=0$, remains. As shown by @BinneyTremaine2008 [Ch 3.3.3(b)], the stellar motions close to corotation turn into pendulum like oscillations around the Lagrange equilibrium points $L_4$ and $L_5$. With increasing distance from corotation, the angular amplitude increases until the orbit reaches the equilibrium points $L_1$ and $L_2$ and flips into a circumcentral orbit (see Figs. 16 and 26 in P.A.B. @Lindblad1996 [hereafter LLA96]). In the simulations of LLA96 the points $L_4$ and $L_5$ represent density minima and $L_1$ and $L_2$ density maxima along the corotation radius. Actually, our equations of motion (\[eq:xisol\]) and (\[eq:etasol\]) are no longer valid around corotation because when inserting $\theta$ from eq. (\[eq:coor\]) into the differential equations (\[eq:eqofmotionfric1\]) and (\[eq:eqofmotionfric2\]) we have assumed $\eta/r_0 << (\Omega-\Omega_p)$ which is not true close to corotation where $\Omega-\Omega_p$ approaches zero. We then see from eq. (\[eq:eqofmotionfric2\]) that for small $\xi$, $\dot{\xi}$ and $\dot{\eta}$ we get a pendulum like oscillation in $\eta$, and we should not get a linearized equation. P.O. @Lindblad1960 has pointed out that a time dependent potential can damp all resonances, including the corotation resonance. For a potential that varies as $e^{\gamma t}$, besides the twisting of the orbits around corotation, $\omega_m$ in the denominator of eqs. (\[eq:d\]) to (\[eq:f\]) is replaced by $\sqrt(\omega_m^2+\gamma^2)$. In similarity to this, in the corotation singularity is avoided by replacing the factor $\omega_m$ in the denominator of eqs. (\[eq:d\])-(\[eq:f\]) by $\omega_{\epsilon}$, where $$\begin{aligned} \frac{1}{\omega_{\varepsilon}}=\frac{\omega_m}{\omega_m^2+\varepsilon_m^2}\end{aligned}$$ where $\varepsilon_m=m\varepsilon$ and $\varepsilon$ is an additional input parameter in the code. As illustrated in Fig. \[fig:epsplot\], the amplitudes of the motion of the guiding center, growing large and of opposite sign on both sides of corotation, are smoothed and brought to zero at the exact resonance distance. $1/\omega_{\varepsilon}$ reaches a maximum of $1/2\varepsilon_m$ at the distance $\omega_m=\varepsilon_m$ from corotation. This means that the stellar elliptical orbits, trapped around corotation, in our case collapses to circular rotation at the corotation radius. This may not be an entirely inappropriate approximation to the solution in the corotation case. ![Damping of the amplitudes $\xi$ and $\eta$ at corotation. Solid line: Variation of the amplitudes with $\omega$ according to the first order epicyclic theory with $\varepsilon=0$. Dashed line: Variation of the amplitudes introduced in .[]{data-label="fig:epsplot"}](epsplot.jpg){width="48.00000%"} The derivation of the density is made applying the continuity equation. Using the equation (F-5) in @BinneyTremaine2008, in our notation the ratio between the perturbed and unperturbed surface densities will be $$\frac{\rho}{\rho_0} = 1 - \xi- \frac{1}{r}\frac{\partial \eta}{\partial \theta} - \frac{\partial \xi}{ \partial r} \label{eq:density}$$ where we have assumed the unperturbed surface density $\rho_0$ to have a flat distribution. generates a number of supporting plots and diagrams like the rotation curve, the circular frequencies showing the resonances, $\omega$ vs $\kappa$, the perturbing potential, the total potential, radial and tangential perturbing forces, the phase variation of the perturbing potential, orbits, densities, as well as line of sight velocity fields for various position angles of the line of nodes. The resolution in $r$ required by in the tabulated values of the Fourier components of the perturbing potential is related to the increase of the phase shifts $\vartheta_m$ with $r$. It is convenient to express this relation in terms of the wavelength of the mode $m=2$ in the spiral structure. If $\lambda_m$ is the wavelength of mode $m$, we have approximately for a tightly wound spiral $$\begin{aligned} \frac{\rm d\vartheta_{\emph m}}{\rm d \emph r} = \frac{2\pi}{\emph m \lambda_m} = \frac{\pi}{\lambda_2} \label{eq:k}\end{aligned}$$ Our requirement for to interpret the potential tables $\rm P_{ \emph \rm c,\emph m}(r)$, $\rm P_{\rm s,\emph m}(r)$ is $\Delta(m\vartheta_m)<\pi/2$ or $$\begin{aligned} \Delta r<\frac{\lambda_2}{ 2\,n\, } \label{eq:condition}\end{aligned}$$ where $\Delta r$ is the radial increment in the potential table and $n$ is the maximum value of $m$. Finally, we can choose between counter-clock and clock wise rotation for an easier comparison with observed cases. vs Wada’s (1994) model {#sec:models} ======================= @Wada1994 [hereafter W94] independently provided an analytical model, similar to <span style="font-variant:small-caps;">Epic</span>, representing the behavior of a non-self-gravitating gas in a rotating potential with a weak bar-like distortion. Like the original version of <span style="font-variant:small-caps;">Epic</span> [@Lindblad1994] there was no solution for the corotation region. In the model Wada used the Toomre potential for the axisymmetric potential, and for the perturbation the potential given by @Sanders1977 $$\Phi(R,\psi)=\Phi_0(R) ( 1 + \Phi_{\rm b}(R ) \cos{2\psi})$$ $$\Phi_0 = - \sqrt{\frac{27}{4}} \frac{a \,v^2_{\rm max}}{\sqrt{R^2+a^2}} \;\;\;\;\; \& \;\;\;\;\; \Phi_{\rm b} = \varepsilon_0 \, a \frac{R^2}{(R^2+a^2)^{1.5}} \label{eq:wadapot}$$ In these equations, $v_{\rm max}$ is the maximum rotational velocity, $a$ is the core radius and $\varepsilon_0$ is a parameter which represents the strength of the bar potential. We test our code by using the same potential, circular frequency and parameters as W94. W94, however, chooses the friction to be proportional to the velocity in the radial direction only. considers a core radius, $a$, equal to 1 and a maximum rotational velocity, $v_{\rm max}$, equal to $(4/27)^{1/4}$, simplifying the expressions of $\Phi_0$ and $\Phi_{\rm b}$. W94 assumed a weak bar strength, $\varepsilon_0$, equal to 0.05 and a pattern speed equal to 0.1. \ \ We use in this section the same potentials and derive the input needed for from them (the rotational velocity curve and the Fourier components of the perturbed potential). We have also used the same pattern speed of 0.1, a friction coefficient of 0.02 at the oILR and 0.01 at the OLR and a corotation softening coefficient of 0.02. We have located the bar horizontally, equally to its position in W94, and consider a counter-clockwise rotation. Using these input parameters for our code, we have derived the resonance radia at the same radia as W94: inner/outer Lindblad resonance (iILR and oILR) at 0.90 and 2.22 respectively, corotation (CR) at 4.53 and outer Lindblad resonance (OLR) at 6.09 (see Fig. 1 in W94). We present the orbits generated by in the upper panel of Fig. \[fig:wada94mod1\], corresponding to the ones presented in Fig. 4 in W94. The leading arms around the iILR and trailing arms around oILR are well presented in both cases. damps as well the orbits around corotation, allowing us to present orbits until radia further than CR (see middle panel in Fig. \[fig:wada94mod1\]). estimates the density map generated by the input potential in the lower panel in Fig. \[fig:wada94mod1\], showing the continuation of the spiral arms out to the OLR. : a demonstration {#sec:new} ================= \[fig:modpo\_pota\] \[fig:modpo\_potb\] We demonstrate in this section a case to illustrate what is able to produce. The rotation curve in this case, shown in Fig. \[fig:rotdemos\], is that given by two superimposed exponential disks with masses of $11.5 \times 10^9$ and $2.87 \times 10^9$ and with scale lengths of 0.194 kpc and 1.15 kpc respectively (reasonable values for the inner part of barred galaxies, @Lindblad2010). To this, is added an ad hoc perturbing bar potential, described by its Fourier components of cosine of 2, 4, 6, 8 and 10 times $\theta$ for each value of the radius, as seen in Fig. \[fig:modpo\_pota\] (left graph). This results in the total potential shown in Fig. \[fig:modpo\_potb\] (right graph). The relative bar strength $\|\Phi_1\|_{\rm min}/\|\Phi_0\|_{\rm min}$ is equal to 0.026. Fig. \[fig:modpo\_ome\] shows the variations of $\Omega$, $\Omega\pm \kappa/2$ and $\Omega\pm \kappa/4$ with radius. Our assumed pattern velocity of 70 , which places the oILR close to the minimum of the perturbing potential, is shown as a straight line in the figure, and the resonances are shown by the vertical lines. The iILR lies very close to the center, oILR at 1.33 kpc, CR at 2.26 kpc, and OLR at 3.03 kpc. Computing the orbits, we have assumed the damping coefficients constant at $\lambda=60$ and $\varepsilon=60$ . The resulting orbits are seen in Fig. \[fig:modpo\_orb\]. The damping coefficients have been adjusted such that the orbits, presenting a laminar flow pattern, do not cross. This is also a condition for to be able to produce meaningful density and velocity maps. This choice means that these coefficients and the strength of the bar are correlated. The densities, as given by are shown in Fig. \[fig:modpo\_denvel\] (left graph), and the line of sight velocities in the plane of the galaxy as seen with a position angle of the line of nodes of 0 is given in Fig. \[fig:modpo\_denvel\] (right graph). We see how the orbits well inside the oILR are orientated perpendicular to the bar but twisting against the direction of rotation. At the oILR they are tilted about 45 against the position of the bar, and close to corotation they are elongated along the bar. The orbits show sharp kinks on the leading side of the bar. This is connected to large velocity jumps at the corresponding positions and strong enhancements of the intensities parallel to the bar on the leading side from the ILR to CR. Such dust lanes along the leading side of the bar are well-known common features in barred galaxies (e.g. @Athanassoula1992b). From corotation the response splits and a trailing spiral arm continues out to OLR. The very faintest density contours close an ellipse between CR and OLR elongated perpendicular to the bar. Using the same case, we have varied $\Omega_p$ to observe changes in the response. The density and velocity maps generated by for $\Omega_p$ equal to 60 and to 80 are shown in Fig. \[fig:case6080\]. We have used a constant damping coefficient $\lambda$ and $\varepsilon$ equal to 60 for the slow and fast potential. The trailing arms extending from CR and going towards OLR are shorter and fainter as the potential gets slower. Moreover, a faster bar generates more prominent, and also more tightly wound, dust lanes inside CR, with larger velocity jumps, as shown in Fig. \[fig:case6080\], right graphs. When the pattern speed gets larger, the resonance distances decrease. As seen from Fig. \[fig:modpo\_potb\] (left graph), the CR and OLR move to stronger perturbing amplitudes, which is partly the reason for the change of response. With an observed bar potential and observed structures similar experiments should help to estimate the pattern velocity and the true positions of the resonances. \[fig:den60\] \[fig:vel60\]\ \[fig:den80\] \[fig:vel80\] applied to NGC 1365 {#sec:1365} ==================== To evaluate the limitations and abilities of we want to compare it to hydrodynamic simulations of a nearby barred galaxy. For this, the strongly barred Seyfert galaxy NGC 1365 (Fig. \[fig:1365\]) would be suitable. This galaxy has been extensively observed (see the review by P.O. @Lindblad1999) and a hydrodynamic simulation of the gas flow pattern was derived by LLA96. Also @Halphangc1365 observed and modeled the gas flow in the bar. ![image](ngc1365op.jpg){width="0.9\linewidth"} A total hybrid velocity field has been derived by @Lindbladpo1996 based on the HI velocity field of @HIngc1365 and complemented with optical long slit spectra. We have here accepted the rotation curve based on this as derived by LLA96 (Fig. \[fig:input\]). It agrees closely with the rotation curve given by @Halphangc1365. We have derived the perturbed potential from the total surface density following the analysis using Bessel functions described in @BinneyTremaine2008 [Ch. 2.6.2] and LLA96 (their Appendix A). For this analysis, we used the Fourier series decomposition of the total surface density, obtained by LLA96 from a J-band image as given in their Figs. 7 and 9, and a normalized triangle density distribution along the vertical direction, with a $z_0$ scale parameter equal to 1 kpc. We smoothed the rotational velocity curve and the Fourier components along the radius and interpolated the curves in order to achieve enough resolution for the $\lambda_2$ criterion of the spiral potential (eq. \[eq:condition\]). We use a clock-wise rotation in order to reproduce the observed galaxy. When comparing with observations, we have taken the results given by , located in the galactic plane, and transformed them into the sky plane. For this purpose, we have used a position angle, PA, equal to 40 and an inclination angle, $i$, of 42 (@HIngc1365, also used in LLA96). In addition, we have assumed the systemic velocity to be 1632 and a scale of 0.1 kpc/arcsec. Comparing with observations and simulations ------------------------------------------- The hydrodynamic simulations presented in LLA96 are performed using a flux-splitting 2nd-order code FS2, solving the flow of an ideal isothermal non-viscous gas [@vanAlbada1985]. LLA96 applies two different potentials derived from NGC 1365: bar only and bar + spiral. We have followed the same procedure in order to compare the results. ### Bar potential When deriving the bar only potential, only the cosine contributions to the Fourier series has been considered, and only up to a radius of around 100 arcsec, where the phase shift of the different modes start a steep increase (see Figs. 8, 9 and 11 in LLA96). The bar is positioned horizontally along the x-axis (Fig. \[fig:inputbar\], upper left). As in LLA96, we have used a pattern speed of 20 . As illustrated in Fig. \[fig:inputbar\] (lower left), this pattern speed fixed the resonance radia to: the inner inner Lindblad resonance (iILR) at 0.4 kpc, the outer inner Lindblad resonance (oILR) at 2.7 kpc, the corotation radius (CR) at 14.4 kpc and the outer Lindblad resonance (OLR) at 21 kpc. We have used a constant friction coefficient $\lambda$ of 7 , and a corotation damping coefficient $\varepsilon$ of 5 . We are using an amplitude for the perturbing potential $\rm A_{\rm bar}=0.5$, half the amplitude used in LLA96. Thus, we have a potential with a bar strength, $\|\Phi_1\|_{\rm min}/\|\Phi_0\|_{\rm min}$, of 0.01. Increasing the bar amplitude, means that we have to increase the damping coefficients correspondingly to avoid self-crossing orbits, that are not realistic in a gaseous flow, and the results will be nearly the same. This may imply that the bar in NGC 1365 is actually too strong to be dealt with . \ Using these input parameters, we obtained with the orbits, densities and velocity field shown in Fig. \[fig:inputbar\] and Fig. \[fig:outputbar\]. The orbits are seen face-on. The density and velocity maps are projected on the sky plane, for comparison with the results of the simulations derived in LLA96 as well as with the observations. We see from Fig. \[fig:inputbar\] (lower right) that the orbits in the region around oILR take elliptical shape, twisting in the counter rotation direction from being orientated at large angles to the bar, over 45 degrees at the oILR, to more elongated with the bar outside this resonance. Inside corotation matter circulates clockwise with respect to the bar, and the twisting and increasing deviation from elliptical shape gives rise to density increase forming a small trailing spiral across the oILR. This straightens out to a lane along the leading edge of the bar with sharp shock-like velocity jumps over the lane. The influence of the inner 1:4 resonance, that occurs around R = 9 kpc, can also be seen in the orbits. Even the 1:6 resonance further out is discernible in the slight crowding of orbits. Strong crowding of orbits are seen at CR at the Lagrangian points L1 and L2 at the end of the bar and, with slightly spiral form, at the OLR in a direction perpendicular to the bar. Compared to the orbits of the BM model of LLA96 (their Fig. 16), we see that between oILR and CR the BM orbits are more elongated and more orientated along the bar than ours. The 1:4 resonance is clearly seen also here. Around corotation the BM model displays the pendulum like motion referred to before and which is not reproduced by the linear approximation of . The interstellar matter density map (Fig. \[fig:outputbar\] left graph), now projected on the sky plane, should be compared with the optical image of the galaxy (Fig. \[fig:1365\]) and LLA96 (their Fig. 15). As expected from the orbits, the map shows a lane on the leading side of the bar that bends into spiral shape over the oILR. At the position of the 1:4 resonance a faint spur is seen, a strong arm appears at the end of the bar along the corotation radius, and a faint trailing arm extends out to the OLR. All these features have their correspondence in the BM density map of LLA96 (their Fig. 15). Fig. \[fig:outputbar\], right graph, shows the innermost region of the EPIC5 density map, where, as expected, a small nuclear leading spiral is seen across the iILR. The velocity map, to be compared with LLA96 (their Fig. 20) is shown in Fig. \[fig:outputbar\] (middle graph). The map is projected on the sky plane with a line of nodes in PA 40 and an inclination between the galaxy symmetry plane and the sky plane of $i$ = 42. The main difference, comparing the result against the observations and simulations, is that the twisting of the zero velocity contour towards the bar is much weaker in the case, which is connected with the different shapes of the orbits in the bar discussed above. This may partly be due to the difficulty of to accept a very strong bar, as in NGC 1365, without getting crossing orbits. The behavior of the velocity contours inside the oILR are similar to that in LLA96, and the sharp velocity jumps, of the order of 100 , in the lanes on the leading side of the bar are seen in all cases. The twisting of the contours along the spiral arms with velocity jumps on the inner side of the arms inside corotation are seen both in the simulations and in the results. ![image](den1365bar.jpg){width="0.33\linewidth"} ![image](vel1365bar.jpg){width="0.33\linewidth"} ![image](innerspiral.jpg){width="0.33\linewidth"} ### Bar + Spiral potential {#subsec:bar+spiral} In the case of the bar + spiral potential we have taken the sine and cosine contribution of the Fourier series of the first three even modes until a radius of around 220 arcsec, just as described in LLA96. The resulting potential is seen in Fig. \[fig:inputspiral\]. We have used a pattern speed of 18 (found for this potential by LLA96) and placed the bar horizontally along the x-axis. This pattern speed makes the resonance radia to be: the inner inner Lindblad resonance (iILR) at 0.38 kpc, the outer inner Lindblad resonance (oILR) at 3.1 kpc, the corotation radius (CR) at 15.8 kpc and the outer Lindblad resonance (OLR) at 23.5 kpc. We have chosen the coefficients $\lambda$ and $\varepsilon$ and parameter $A_{bar}$ the same as in the bar only case. The main difference between this case and the bar only case is that the spiral features are led much easier through corotation, which can be seen both in the orbits, Fig. \[fig:inputspiral\] lower right graph, and the densities, Fig. \[fig:outputspiral\] upper left graph. This was noticed also by LLA96. \ ![image](den1365spiral.jpg){width="0.48\linewidth"} ![image](vel1365spiral.jpg){width="0.48\linewidth"}\ ![image](inner100.jpg){width="0.48\linewidth"} Fabry-Perot interferometry of NGC 1365, covering the H$\alpha$ regime has been obtained by @Halphangc1365. By kind permission we have got access to their total H$\alpha$ intensity data, which are reproduced in Fig. \[fig:outputobs\] (left map). In Fig. \[fig:outputobs\] (right map) we have for comparison color coded the intensities from Fig. \[fig:outputspiral\]. Clearly the model agrees well with the H$\alpha$ structure. The hot spots in the nuclear region and the active Seyfert nucleus is of course not reproduced by . The velocity field obtained with is shown in Fig. \[fig:outputspiral\], together with the optical radial velocity field of NGC 1365 (inner 100 arcsec) taken from LLA96. In the innermost part, well inside the oILR, the contours indicate rapid rotation in nearly circular orbits. Between oILR and CR we again see how the isovelocity contours are concentrated to the leading edge of the bar. In the case the contours are squeezed to a shock-like jump of about 100 km/s. The optical map cannot make such a sharp jump because it is smoothed due to the limited spatial resolution given by the limited set of long slits on which it is based. Over corotation we see similar shock behavior on the inner side of the spiral arms. The residual map obtained by subtracting the rotational velocity, Fig. \[fig:input\], from the model velocity field is presented in Fig. \[fig:m-r\]. In this Fig. \[fig:m-r\] is also presented the residual velocities taken from @HIngc1365. They subtracted from the observed HI velocity map, the rotational velocity derived from their observations considering a warp disc model after a radius of 250 arcsec. The general behavior is similar in the two cases with shocks along the bar and spiral arms. It is informative to compare the map with the orbital map in Fig. \[fig:inputspiral\] (lower right) to envision how the orbital circulation results in the observed line of sight velocity map. ![image](denfp1365.jpg){width="0.48\linewidth"} ![image](denmodel1365.jpg){width="0.48\linewidth"} ![image](m-r.jpg){width="0.48\linewidth"} ![image](m-rHI_kambiz.jpg){width="0.4\linewidth"} Conclusions {#sec:conclusions} =========== cannot fully replace elaborate hydrodynamical simulations. solves the equations of motion in a solid body rotating, time independent galactic potential. This is done in the first order epicyclic approximation. computes the deformation of the initially circular orbits of the guiding center, the velocity fields, and the structure created. pictures a steady state created by a weak perturbing potential, it does not give the time evolution of galactic structure. Due to the degeneracy of the strength of the gravitational potential and damping coefficients, we can only derive relative values of the density. Our comparison with a more elaborate simulation, as well as observations, show that it still can reproduce structural and dynamic properties. One of the advantages with , involving an analytic solution, is the speed by which it can survey the parameter space. Further, the possibilities, by varying parameters, to isolate artificially various agents important for forming structure of galaxies will be of importance for understanding the formation of galactic structure. will be a useful complement to time consuming, elaborate simulations. We would like to thank to J. A. Sellwood and R. Zánmar Sánchez for letting us use their H$\alpha$ data of NGC 1365. We also want to thank to the reviewer, Gene Byrd, for his useful comments. NP-F acknowledges financial support from NOTSA, and KF is supported by the Swedish Research Council (Vetenskapsrådet). Perturbations in the epicyclic approximation {#ap:theory} ============================================ We present in this paper a solution of the motion of a mass particle in an axisymmetric potential, $\Phi_0(r)$, with a weak non-axisymmetric perturbation, $\Phi_1(r,\theta)$. We assume that the potential is rotating with a pattern speed , and we study the system in a corotating frame. We express the potential in polar coordinates: $$\Phi(r,\theta)=\Phi_0(r)+\Phi_1(r,\theta)=\Phi_0(r) - \sum_{m=1}^{n} \Psi_m(r) \, \cos m(\theta - \vartheta_m(r)) \label{eq:potentialap}$$ where $\vartheta_m$ is the spiral phase. The rotation of the axisymmetric potential is about the center of gravity, which we have assumed to be at rest. Thus one has to be cautious in the presence of strong $m=1$ perturbation, when the center of mass could be significantly displaced. In a corotating coordinate system we can write the equations of motions (see also @BinneyTremaine2008, p.189): $$\begin{aligned} \ddot{r}-r\dot{\theta}^2&=-\frac{\partial \Phi}{\partial r} + 2 r \dot{\theta}\Omega_p + r \Omega_p^2 \\ r\ddot{\theta}+2\dot{r}\dot{\theta}&=-\frac{1}{r}\frac{\partial \Phi}{\partial \theta}-2\dot{r}\Omega_p\end{aligned}$$ We introduce $\xi(t)$ and $\eta(t)$ as deviations from circular motion and write: $$\begin{aligned} \centering r&=r_0+\xi \\ \theta&=\theta_0 + (\Omega - \Omegap) \,t + \frac{1}{r_0} \eta\end{aligned}$$ where $\Omega$ is the circular angular velocity at a radius $r_0$. Assuming $\xi$ and $\eta$ to be small, and linearizing the equations of motion by neglecting higher order terms of $\xi$ and $\eta$, we get: $$\begin{aligned} \centering \ddot{\xi} - 2 \Omega\dot{\eta} - 4 \Omega A \xi &= - \frac{\partial \Phi_1}{\partial r} \label{eq:eqofmotion1}\\ \ddot{\eta} + 2 \Omega \dot{\xi} & = - \frac{1}{r} \frac{\partial \Phi_1}{\partial \theta} \label{eq:eqofmotion}\end{aligned}$$ where $A$ is the Oort constant, $A=-r/2 \,\,\, d\Omega/dr$. To describe the motion of a gaseous medium, we introduce a frictional force proportional to the deviation from circular motion with a damping coefficient, $2\lambda$ (see @Wada1994, @Lindblad1994). With the introduction of this frictional force the equations of motion, eq. (\[eq:eqofmotion1\]) and (\[eq:eqofmotion\]), are transformed into the equations (\[eq:eqofmotionfric\]). $$\begin{aligned} \centering \ddot{\xi} + 2 \lambda \dot{\xi} - 2\Omega \dot{\eta} - 4 \Omega A \xi &= - \frac{\partial \Phi_1}{\partial r} = \sum_{m=1}^{n} \big[ C_m \cos m(\theta - \vartheta_m) + E_m \sin m(\theta - \vartheta_m)\big] \nonumber\\ \ddot{\eta} + 2 \Omega \dot{\xi} + 2 \lambda \dot{\eta} + 4 \lambda A \xi &= - \frac{1}{r}\frac{\partial \Phi_1}{\partial \theta} = - \sum_{m=1}^{n} D_m \sin m(\theta - \vartheta_m) \label{eq:eqofmotionfric}\end{aligned}$$ where $$\begin{aligned} \centering C_m = \frac{d\Psi_m}{dr}; \;\;\;\;\; &D_m=m\frac{\Psi_m}{r}; \;\;\;\;\; E_m=m\Psi_m\frac{d\vartheta_m}{dr} \nonumber\end{aligned}$$ Provided that we are not close to corotation, where $\Omega=\Omega_p$, we can assume that $\eta/r_0<<(\Omega-\Omega_p)$, what makes $$\theta\sim\theta_0 + (\Omega - \Omegap) \,t \label{eq:aprox}$$ with $\kappa^2=4\Omega^2-4\Omega A$, we can write the full solution of the now linearized eqs. (\[eq:eqofmotionfric\]) as: $$\begin{aligned} \xi &= c e^{-\alpha\lambda t} \cos \kappa(t-t_0) + \sum_{m=1}^{n} \big[ d_m \cos m(\theta-\vartheta_m) + e_m \sin m(\theta -\vartheta_m) \big] \label{eq:xisolap}\\ \eta&= - \frac{2\Omega}{\kappa}c e^{-\beta\lambda t} \sin \kappa(t-t_0) + \sum_{m=1}^{n} \big[ g_m \sin m(\theta-\vartheta_m) + f_m \cos m(\theta -\vartheta_m) \big]\label{eq:etasolap}\end{aligned}$$ $\alpha$ and $\beta$ are functions of $\Omega$ and $\kappa$, and $c$ is an arbitrary constant. The second terms on the left side of eqs. (\[eq:xisolap\]) and (\[eq:etasolap\]) give the forced oscillation due to the perturbing force. The first terms give the damped oscillation with the epicyclic frequency $\kappa$ around these guiding centra. We will leave out these latter terms in what follows and just consider the motions of the guiding center. We introduce these eq. (\[eq:xisolap\]) and eq. (\[eq:etasolap\]) into (\[eq:eqofmotionfric\]), getting a system of equations with the 4$n$ unknown amplitudes, $d_m$, $e_m$, $g_m$ and $f_m$: $$\begin{aligned} -(\omega_m^2 + 4 \Omega A ) d_m - 2 \omega_m \Omega g_m &= C_m -2\lambda \omega_m e_m \nonumber\\ -(\omega_m^2 + 4 \Omega A ) e_m + 2 \omega_m \Omega f_m &= E_m +2\lambda \omega_m d_m \nonumber\\ 2\omega_m\Omega d_m + \omega_m^2 g_m &= D_m - 2\lambda \omega_m f_m + 4 \lambda A e_m \nonumber\\ 2\omega_m\Omega e_m - \omega_m^2 f_m &= - 2\lambda \omega_m g_m - 4 \lambda A d_m \label{eq:system}\end{aligned}$$ where $\omega_m = m\,(\Omega-\Omegap)$. The final solution is shown below. $$\begin{aligned} d_m&=\frac{(\kappa^2-\omega_m^2)(\omega_m C_m + 2 \Omega D_m) - 2 \lambda (\kappa^2+\omega_m^2)E_m - 4\lambda^2 (\omega_m C_m -2\Omega D_m) - 8\lambda^3 E_m}{\omega_\varepsilon \left[ \left(\kappa^2-\omega_m^2\right)^2 + 8\lambda^2\left(\kappa^2+\omega_m^2\right) + 16 \lambda^4 \right]} \label{eq:d} \\ \nonumber\\ g_m&=\frac{-(\kappa^2-\omega_m^2)\left[2\Omega\left(\omega_m C_m +2\Omega D_m \right) - \left(\kappa^2 - \omega_m^2\right) D_m\right] + 4 \lambda \left[\left(\kappa^2-\omega_m^2\right)A+2\omega_m^2\Omega\right]E_m }{\omega_\varepsilon^2 \left[ \left(\kappa^2-\omega_m^2\right)^2 + 8\lambda^2\left(\kappa^2+\omega_m^2\right) + 16 \lambda^4 \right]} \nonumber \\ &\;+\frac{- 4 \lambda^2\left[\left(\omega_m C_m + 2 \Omega D_m \right) \left(2\Omega - 4 A\right) - \left(\omega_m^2+3\kappa^2-8\Omega^2\right)D_m\right] + 16 \lambda^3 A E_m}{\omega_\varepsilon^2 \left[ \left(\kappa^2-\omega_m^2\right)^2 + 8\lambda^2\left(\kappa^2+\omega_m^2\right) + 16 \lambda^4 \right]} \label{eq:g}\\ \nonumber\\ e_m &=\frac{\omega_m^2\left(\kappa^2-\omega_m^2\right)E_m + 2 \lambda \left[ \left( \kappa^2 + \omega_m^2\right) \left( \omega_m C_m + 2 \Omega D_m \right) - 2 \Omega \left(\kappa^2-\omega_m^2 \right) D_m \right] - 4 \lambda^2 \omega_m^2 E_m + 8 \lambda^3 \omega_m C_m}{\omega_\varepsilon^2 \left[ \left(\kappa^2-\omega_m^2\right)^2 + 8\lambda^2\left(\kappa^2+\omega_m^2\right) + 16 \lambda^4 \right]} \label{eq:e} \\ \nonumber\\ f_m&=\frac{2\omega_m\left(\kappa^2-\omega_m^2\right)\Omega E_m + 2 \lambda \left[4 \omega_m \Omega \left( \omega_m C_m + 2 \Omega D_m\right) + \left(\kappa^2-\omega_m^2\right)\left(2AC_m - \omega_m D_m\right)\right]+8\lambda^2 \omega_m E_m \left(\Omega-2A\right) + 8 \lambda^3 \left(2AC_m+\omega_m D_m\right) }{\omega_\varepsilon^2 \left[ \left(\kappa^2-\omega_m^2\right)^2 + 8\lambda^2\left(\kappa^2+\omega_m^2\right) + 16 \lambda^4 \right]} \label{eq:f}\end{aligned}$$ At the Lindblad resonances $\kappa^2-\omega_m^2=0$, and we can see how the friction terms are damping the resonances when $\lambda\neq 0$. At corotation $\omega_m=0$. To damp this resonance, as explained in Section \[sec:method\] in the paper, we have replaced $\omega_m$ in the denominator of eqs. (\[eq:d\]) to (\[eq:f\]) by $\omega_\varepsilon$, where $$\begin{aligned} \frac{1}{\omega_{\varepsilon}}=\frac{\omega_m}{\omega_m^2+\varepsilon_m^2}\end{aligned}$$ where $\varepsilon_m = m\varepsilon$ and $\varepsilon$ is an additional parameter. [^1]: E-mail: npi@astro.su.se
--- abstract: 'We study the quantum phase transition (QPT) in a non-Hermitian Tavis-Cummings (TC) model of experimentally accessible parameters, which is engineered with two drive fields applied to an ensemble of two-level systems (TLSs) and a cavity, respectively. When the two drive fields satisfy a given parameter-matching condition, the coupled cavity-TLS ensemble system can be described by an effective standard TC Hamiltonian in the rotating frame. In this ideal Hermitian case, the engineered TC model can exhibit the super-radiant QPT with spin conservation at an experimentally-accessible critical coupling strength, but the QPT is however spolied by the decoherence. We find that in this non-Hermitian case, the QPT can be recovered by introducing a gain in the cavity to balance the loss of the TLS ensemble. Also, the spin-conservation law is found to be violated due to the decoherence of the system. Our study offers an experimentally realizable approach to implementing QPT in the non-Hermitian TC model.' author: - 'Guo-Qiang Zhang' - Zhen Chen - 'J. Q. You' title: 'Experimentally Accessible Quantum Phase Transition in a non-Hermitian Tavis-Cummings Model Engineered with Two Drive Fields' --- Introduction.—Quantum phase transition (QPT) has been widely studied in various quantum systems (see, e.g., [@Mebrahtu12; @Hwang15; @Hwang16; @Zhou16; @Wang14; @Aharony96; @Sachdev93; @Harris08; @Zhang13; @Zhang17; @Lv18; @He15]), because of its fundamental importance in many-body physics and potential applications in quantum technologies [@Sachdev11]. Among them, the super-radiant QPT was predicted [@Hepp73; @Wang73; @Hioe73; @Hepp73-1] in, e.g., the Dicke model [@Dicke54], which involves the collective interaction between an ensemble of two-level systems (TLSs) and the quantized field in a cavity. In the thermodynamic limit of a large number of TLSs, the ground-state properties of the Dicke model, such as the excitations in the TLS ensemble, can display an abrupt change related to the QPT in the system [@Emary03; @Emary03-1; @Ye11], when continuously varying the collective coupling strength around a critical value. As required to be comparable to the frequencies of the TLS ensemble and the cavity mode, this critical coupling strength is difficult to achieve experimentally. Moreover, the no-go theorem due to the squared electromagnetic vector potential also hinders the presence of the super-radiant QPT in the Dicke model [@Rzazewski75; @Birula79; @Rabl16]. Due to these difficulties, the nonequilibrium QPT (i.e., the simulated QPT) has been proposed [@Dimer07; @Nagy10; @Zou14] and observed [@Baumann10; @Baumann11; @Baden14] in cavity quantum electrodynamics (QED) systems by engineering an effective Dicke Hamiltonian. In the experiment, most quantum systems can only reach the strong-coupling regime [@Xiang13; @Kurizki15], where the Dicke model can be reasonably reduced to the Tavis-Cummings (TC) model [@Tavis68] via the rotating-wave approximation (RWA) [@Walls94], i.e., ignoring the counter-rotating coupling terms between the TLS ensemble and the cavity mode. Similar to the Dicke model, the TC model is another archetypal model known to exhibit the super-radiant QPT [@Castanos09]. Nevertheless, QPT in the TC model can occur only in the regime of extremely strong coupling, which is unaccessible for a realistic system and in which the RWA actually breaks down [@Walls94]. In addition, the decoherence inevitably occurring in the realistic system can spoil the QPT in the TC model [@Keeling10; @Larson17; @Soriente18; @Feng15]. Therefore, it is of great importance to explore the nonequilibrium QPT by engineering an effective experimentally-accessible TC Hamiltonian [@Feng15; @Zou13; @Zhu20]. In this Letter, we study the QPT in a driven quantum system consisting of a TLS ensemble coupled to a cavity mode. To reduce the critical coupling strength of the TC model, we use two external fields with the same frequency to drive the TLS ensemble and the cavity, respectively. By choosing proper drive-field parameters, an effective TC model can be rebuilt in the rotating frame, with the effective frequencies of the TLS ensemble and the cavity mode being their respective frequency detunings from the drive fields. In the ideal [Hermitian]{} case (i.e., without decoherence in the system), we can demonstrate the QPT in the driven system, where the corresponding critical coupling strength is tunable (via varying the drive-field frequency) and experimentally accessible. However, decoherence inevitably occurs in the system and the QPT of the TC model is spoiled in this [non-Hermitian]{} case [@Keeling10; @Larson17; @Soriente18; @Feng15]. We find that the QPT in the non-Hermitian TC model can be recovered by using a cavity with gain to [balance]{} the loss of the TLS ensemble. In sharp contrast to the spin conservation in the ideal Hermitian case, the spin-conservation law is found to be violated in the non-Hermitian case [@Torre16; @Kirton17]. Moreover, we propose to implement this non-Hermitian TC model with a hybrid circuit-QED system. To the best of our knowledge, this is the first proposal to engineer an experimentally accessible [non-Hermitian]{} TC model that can have QPT in the presence of decoherence in the system. ![(a) Schematic representation of the engineered TC model with a TLS ensemble strongly coupled to a cavity mode, where two drive fields with the same frequency are applied to the ensemble and the cavity mode, respectively. (b) Implementation of the proposed TC model using a hybrid circuit-QED system composed of an ensemble of NV centers in diamond interacting with an active coplanar-waveguide resonator, where the gain of the resonator results from an auxiliary flux qubit controlled by two microwave fields [@Quijandria18]. Also, two fields with the same frequency drive the NV ensemble and the resonator, respectively.[]{data-label="figure1"}](figure1.pdf){width="45.00000%"} QPT in the ideal Hermitian case.—The proposed system consists of $N$ TLSs (e.g., spins) in a cavity, each with the same transition frequency $\omega_{s}$ and coupled to a cavity mode with coupling strength $\lambda_{s}$ \[see Fig. \[figure1\](a)\]. This system can be described in a RWA by the TC model (we set $\hbar=1$) $$\label{TC} H_{\rm TC}=\omega_{c}a^{\dag}a+\omega_{s}J_{z}+\frac{\lambda}{\sqrt{N}}(a^{\dag}J_{-}+aJ_{+}),$$ where $a$ ($a^{\dag})$ is the annihilation (creation) operator of the cavity mode with resonant frequency $\omega_c$, $J_{x}, J_{y}$ and $J_{z}$ are the collective spin operators of the TLS ensemble with raising and lowering operators $J_{\pm}=J_{x} \pm iJ_{y}$, and $\lambda=\lambda_{s}\sqrt{N}$ is the collective coupling strength between the TLS ensemble and the cavity mode. This Hamiltonian has a conserved parity [@Castanos09], $[H_{\rm TC},\Pi]=0$, where $\Pi=\exp{[i\pi(a^{\dag}a+J_z+N/2)]}$. In the ideal case without decoherence in the system, the TC model exhibits a QPT at the critical coupling strength $\lambda = \lambda_{c} \equiv \sqrt{\omega_{s}\omega_{c}}$ in the thermodynamic limit $N \rightarrow +\infty$ [@Castanos09]. However, this QPT is spoiled by the decoherence of the system [@Keeling10; @Soriente18; @Larson17; @Feng15]. Also, it is extremely difficult to demonstrate the QPT due to the inaccessibility of the very large critical coupling strength $\lambda_{c}$ in a realistic system. To solve these problems, we first manage to reduce the critical coupling strength by applying two drive fields of the same frequency $\omega_{d}$ to the cavity and the TLS ensemble, respectively. This corresponds to adding the drive Hamiltonian $H_d=\Omega_{a}(a^{\dag}e^{-i\omega_d t}+ae^{i\omega_d t}) +(\Omega_{J}/\sqrt{N})(J_{+}e^{-i\omega_d t}+J_{-}e^{i\omega_d t})$ to the Hamiltonian in Eq. (\[TC\]). Here $\Omega_{a}$ is the Rabi frequency between the drive field and the cavity mode, and $\Omega_{J}\equiv\Omega_{s}\sqrt{N}$ is the collective Rabi frequency between the drive field and the TLS ensemble, where $\Omega_{s}$ is the Rabi frequency between the drive field and each TLS. In a rotating reference frame with respect to the frequency $\omega_{d}$ of the two drive fields, the total Hamiltonian of the system can be converted to $$\begin{aligned} \label{TC-d} H^{(d)}_{\rm TC}&=&\Delta_{c} a^{\dag}a + \Delta_{s}J_{z}+\frac{\lambda}{\sqrt{N}}(aJ_{+}+a^{\dag}J_{-}) +(\Omega_{a}a^{\dag}+\Omega_{a}^{}a) \nonumber\\ & &+\frac{1}{\sqrt{N}}(\Omega_{J}J_{+}+\Omega_{J}^{}J_{-}),\end{aligned}$$ where $\Delta_{c(s)} \equiv \omega_{c(s)}-\omega_{d}$ ($>0$) is the frequency detuning of the cavity mode (TLS ensemble) relative to the corresponding drive field. By introducing a displacement $a= A+\alpha$ and $a^{\dag}= A^{\dag}+\alpha^{}$ with $\alpha=-\Omega_{a}/\Delta_{c}$, i.e., a translation transform, the above Hamiltonian becomes $$\begin{aligned} \label{TC-u} H_{\rm TC}^{(d)}&=&\Delta_{c} A^{\dag}A + \Delta_{s}J_{z}+\frac{\lambda}{\sqrt{N}}(AJ_{+}+A^{\dag}J_{-}) \nonumber\\ & & +\frac{1}{\sqrt{N}}\left[(\Omega_{J}+\lambda\alpha)J_{+}+(\Omega_{J}^{}+\lambda\alpha^{})J_{-}\right],\end{aligned}$$ where $A^{\dag}$ and $A$ are also bosonic creation and annihilation operators obeying $[A,A^{\dag}]=1$. When the two drive fields satisfy the parameter-matching condition $\Omega_{a}/\Omega_{J}=\Delta_{c}/\lambda$, the effects of these two drive fields cancel each other and the Hamiltonian is reduced to $$\label{eff} H_{\rm TC}^{(d)}=\Delta_{c} A^{\dag}A + \Delta_{s}J_{z}+\frac{\lambda}{\sqrt{N}}(AJ_{+}+A^{\dag}J_{-}),$$ which has the same form as the standard TC model in Eq. (\[TC\]), but the critical coupling strength $\lambda = \lambda_{c} \equiv \sqrt{\Delta_{s}\Delta_{c}}$ becomes experimentally accessible by reducing $\Delta_c$ ($\Delta_s$) via tuning the drive-field frequency $\omega_d$. Like the standard TC model in Eq. (\[TC\]), the effective TC model in Eq. (\[eff\]) also exhibits a normal (super-radiant) phase with unbroken (broken) parity symmetry when $\lambda<\lambda_{c}$ ($\lambda>\lambda_{c}$), and the TLS ensemble in the thermodynamic limit has the critical behaviors [@Zou13] $$\label{order-Jz} \begin{split} \frac{\langle J_{z}\rangle}{(N/2)} =\Bigg\{ \begin{array}{cc} -1, &~~~ \lambda < \lambda_{c};\\ -\lambda_{c}^{2}/\lambda^{2}, &~~~ \lambda \geq \lambda_{c}, \end{array} \end{split}$$ and $$\label{order-J-} \begin{split} \frac{\langle J_{-}\rangle}{(N/2)} =\Bigg\{ \begin{array}{cc} 0, &~~~ \lambda < \lambda_{c};\\ (1-\lambda_{c}^{4}/\lambda^{4})^{1/2}, &~~~ \lambda \geq \lambda_{c}. \end{array} \end{split}$$ This gives $\langle J_{z}\rangle \sim \|\lambda-\lambda_{c}\|^{\nu_{z}}$ and $\langle J_{-}\rangle \sim \|\lambda-\lambda_{c}\|^{\nu_{-}}$ around $\lambda=\lambda_{c}$, with critical exponents $\nu_{z}=1$ and $\nu_{-}=1/2$. The displaced cavity field can also display a QPT because $\langle A \rangle=-\lambda\langle J_{-}\rangle/(\sqrt{N}\Delta_{c})$, i.e., $\langle A^{\dag}A \rangle$ behaves as $\langle A^{\dag}A \rangle\sim \|\lambda-\lambda_c\|^{\nu_{A}}$, with $\nu_{A}=1$. Therefore, $\langle a^{\dag}a \rangle\sim\|\lambda-\lambda_c\|^{\nu_a}$, with $\nu_{a}=1/2$ [@SM], owing to $\langle a \rangle \equiv \langle A \rangle+\alpha=-\lambda\langle J_{-}\rangle/(\sqrt{N}\Delta_{c})-\Omega_{a}/\Delta_{c}$. Note that two suitable drive fields are required to demonstrate the QPT. If only one finite drive field is applied (i.e., either $\Omega_{a}\neq 0$ but $\Omega_{J}=0$ or $\Omega_{J} \neq 0$ but $\Omega_{a} = 0$), the effective Hamiltonian of the driven system does not preserve the parity symmetry [@Emary04]. In this case, the system can only tend to exhibit the critical behavior when the sole drive field becomes extremely weak [@Zou13]. QPT in the non-Hermitian case with gain.—When the decoherence of the system is considered, the system becomes non-Hermitian. We use a Langevin equation approach [@Walls94] to study the critical behavior of the non-Hermitian system. With the Hamiltonian in Eq. (\[TC-d\]), the dynamics of the driven system is governed by the following Langevin equations: $$\begin{aligned} \label{Langevin} \dot{a}&=&-i(\Delta_{c}-i\kappa_{c})a-i\frac{\lambda}{\sqrt{N}}J_{-}-i\Omega_{a}, \nonumber\\ \dot{J_{-}}&=&-i(\Delta_{s}-i\gamma_{\perp})J_{-}+i\frac{2\lambda}{\sqrt{N}}J_{z}a +i2J_{z}\frac{\Omega_{J}}{\sqrt{N}},\\ \dot{J_{z}}&=&-i\frac{\lambda}{\sqrt{N}}(aJ_{+}-a^{\dag}J_{-})-i\frac{1}{\sqrt{N}}(\Omega_{J}J_{+}-\Omega_{J}^{}J_{-}) -\gamma_{\parallel}\bigg(\frac{N}{2}+J_{z}\bigg),\nonumber\end{aligned}$$ where $\kappa_{c}$ is the decay rate of the cavity, and $\gamma_{\perp}$ ($\gamma_{\parallel}$) is the transversal (longitudinal) relaxation rate of the TLS ensemble. The operator $\mathcal{O}$ ($=\{a,J_{-},J_{z}\}$) can be written as a sum of its expectation value $\langle \mathcal{O} \rangle$ and fluctuation $\delta \mathcal{O}$, i.e., $\mathcal{O}=\langle \mathcal{O} \rangle+\delta \mathcal{O}$. At the steady state $\langle \mathcal{\dot{O}} \rangle=0$, it follows from Eq. (\[Langevin\]) that $$\label{steady} \begin{split} &(\Delta_{c}-i\kappa_{c})\langle a\rangle+\frac{\lambda}{\sqrt{N}}\langle J_{-}\rangle+\Omega_{a}=0,\\ &(\Delta_{s}-i\gamma_{\perp})\langle J_{-}\rangle-\frac{2\lambda}{\sqrt{N}}\langle J_{z}\rangle \langle a \rangle -2\langle J_{z}\rangle\frac{\Omega_{J}}{\sqrt{N}}=0,\\ &\frac{\lambda}{\sqrt{N}}(\langle a\rangle \langle J_{+}\rangle-\langle a^{\dag}\rangle \langle J_{-}\rangle)+\frac{1}{\sqrt{N}}(\Omega_{J}\langle J_{+}\rangle-\Omega_{J}^{}\langle J_{-}\rangle)\\ & -i\gamma_{\parallel}\bigg(\frac{N}{2}+\langle J_{z}\rangle\bigg)=0, \end{split}$$ where a mean-field approximation is applied to the two-operator terms [@Krimer19]. Substituting the first equation in Eq. (\[steady\]) into the second and third equations in Eq. (\[steady\]) to eliminate $\langle a\rangle$, we have $$\label{spin-1} \begin{split} &\bigg[1+\frac{\lambda^{2}}{(\Delta_{c}\Delta_{s}-\kappa_{c}\gamma_{\perp})-i(\Delta_{c}\gamma_{\perp}+\Delta_{s}\kappa_{c})} \frac{\langle J_{z}\rangle}{(N/2)}\bigg]\langle J_{-}\rangle=0,\\ &\frac{\kappa_{c}\lambda^{2}}{\Delta_{c}^{2}+\kappa_{c}^{2}}\frac{\langle J_{+}\rangle \langle J_{-}\rangle}{(N/2)^{2}} +\gamma_{\parallel}\bigg(1+\frac{\langle J_{z}\rangle}{(N/2)}\bigg)=0, \end{split}$$ when the parameter-matching condition in the non-Hermitian case, $\Omega_{a}/\Omega_{J}=(\Delta_{c}-i\kappa_{c})/\lambda$, is used. Obviously, Eq. (\[spin-1\]) has only a set of [trivial]{} solutions $\langle J_{z}\rangle/(N/2)=-1$ and $\langle J_{-}\rangle=0$. This verifies that the decoherence of the system ruins the QPT occurring in the standard Hermitian TC model [@Keeling10; @Soriente18; @Larson17; @Feng15]. ![(a) $\langle J_{z}\rangle/(N/2)$ and (b) $\langle J_{-}\rangle/(N/2)$ versus the reduced coupling strength $\lambda/\Delta_{c}$ under the parameter-matching condition of the two drive fields. For the (black) solid curve, $\kappa_{c}=\gamma_{\perp}=\gamma_{\parallel}=\kappa_{g}=0$, while $\kappa_{c}=\gamma_{\perp}=1$, $\gamma_{\parallel}=0.1$ and $\kappa_{g}=\kappa_{g}^{(0)} = \kappa_{c}+\gamma_{\perp}\Delta_{c}/\Delta_{s}$ for the (red) dashed curve. Other parameters are $\lambda=8$, $\Omega_{a}/\sqrt{N}=1$, and $\Delta_{s}=\Delta_{c}$.[]{data-label="figure2"}](figure2.pdf){width="48.00000%"} To recover the QPT, we manage to introduce a gain medium in the dissipative cavity. With the gain included, we obtain the same equations as Eqs. (\[Langevin\])-(\[spin-1\]), but $\kappa_{c}$ is replaced by $\kappa\equiv\kappa_{c}-\kappa_{g}$, where $\kappa_{g}$ is the gain rate of the cavity owing to the gain medium. The parameter-matching condition becomes $\Omega_{a}/\Omega_{J}=(\Delta_{c}-i\kappa)/\lambda$, and Eq. (\[spin-1\]) is converted to $$\label{spin-2} \begin{split} &\bigg[1+\frac{\lambda^{2}}{(\Delta_{c}\Delta_{s}-\kappa\gamma_{\perp})-i(\Delta_{c}\gamma_{\perp}+\Delta_{s}\kappa)} \frac{\langle J_{z}\rangle}{(N/2)}\bigg]\langle J_{-}\rangle=0,\\ &\frac{\kappa\lambda^{2}}{\Delta_{c}^{2}+\kappa^{2}}\frac{\langle J_{+}\rangle \langle J_{-}\rangle}{(N/2)^{2}} +\gamma_{\parallel}\bigg(1+\frac{\langle J_{z}\rangle}{(N/2)}\bigg)=0. \end{split}$$ To have the QPT, we need $\Delta_{c}\gamma_{\perp}+\Delta_{s}\kappa \equiv \Delta_{c}\gamma_{\perp}-\Delta_{s}(\kappa_{g}-\kappa_{c})=0$, which gives the required gain rate $\kappa_{g}^{(0)} = \kappa_{c}+\gamma_{\perp}\Delta_{c}/\Delta_{s}$. Now, the parameter-matching condition $\Omega_{a}/\Omega_{J}=(\Delta_{c}-i\kappa)/\lambda$ is reduced to $\Omega_{a}/\Omega_{J}=\Delta_{c}(1+i\gamma_{\perp}/\Delta_{s})/\lambda$. When $\kappa_{g}=\kappa_{g}^{(0)}$, besides the set of trivial solutions $\langle J_{z}\rangle/(N/2)=-1$ and $\langle J_{-}\rangle=0$, Eq. (\[spin-2\]) has also another set of [nontrival]{} solutions $$\label{Jm} \frac{\langle J_{z}\rangle}{(N/2)}=-\frac{\lambda_{c}^{2}}{\lambda^{2}},~~~~~~ \frac{\langle J_{-}\rangle}{(N/2)}=\frac{\lambda_{c}}{\lambda} \bigg[\bigg(1-\frac{\lambda_{c}^{2}}{\lambda^{2}}\bigg)\frac{\gamma_{\parallel}}{\gamma_{\perp}}\bigg]^{1/2},$$ with $\lambda_{c}$ modified as $$\lambda_{c} \equiv \sqrt{\Delta_{c}\Delta_{s}(1+\gamma_{\perp}^{2}/\Delta_{s}^2)}.$$ Here $\langle J_{-}\rangle$ is assumed to be real for simplicity. As shown in Fig. \[figure2\], when varying the critical coupling strength $\lambda_{c}$ from $\lambda < \lambda_{c}$ to $\lambda > \lambda_{c}$, the proposed driven system exhibits a QPT from the normal phase with $\langle J_{z}\rangle/(N/2)=-1$ and $\langle J_{-}\rangle=0$ to the super-radiant phase with $\langle J_{z}\rangle$ and $\langle J_{-}\rangle$ given in Eq. (\[Jm\]). In the non-Hermitian case with gain, $\langle J_{z}\rangle/(N/2)$ has the same behavior as in the Hermitian case \[see Fig. \[figure2\](a)\]. In fact, around $\lambda=\lambda_c$, $\langle J_{z}\rangle \sim \|\lambda-\lambda_{c}\|^{\nu_{z}}$, with $\nu_{z}=1$. Figure \[figure2\](b) shows that in the non-Hermitian case with gain, $\langle J_{-}\rangle/(N/2)$ is much reduced in the regime of super-radiant phase, but it can be analytically derived from Eq. (\[Jm\]) that around $\lambda=\lambda_c$, $\langle J_{-}\rangle/(N/2)$ still exhibits the same critical behavior as in the Hermitian case: $\langle J_{-}\rangle \sim \|\lambda-\lambda_{c}\|^{\nu_{-}}$, with $\nu_{-}=1/2$. ![(a) Mean photon number $\langle a^{\dag}a\rangle/N$ and (b) characteristic quantity $F$ of the TLS ensemble versus the reduced coupling strength $\lambda/\Delta_{c}$, where $\kappa_{c}=\gamma_{\perp}=\gamma_{\parallel}=\kappa_{g}=0$ for the (black) solid curve, and $\kappa_{c}=\gamma_{\perp}=1$, $\gamma_{\parallel}=0.1$ and $\kappa_{g}=\kappa_{g}^{(0)} = \kappa_{c}+\gamma_{\perp}\Delta_{c}/\Delta_{s}$ for the (red) dashed curve. Other parameters are the same as in Fig. \[figure2\].[]{data-label="figure3"}](figure3.pdf){width="48.00000%"} From the first equation in Eq. (\[steady\]), but with $\kappa_{c}$ replaced by $\kappa = \kappa_{c}-\kappa_{g}^{(0)} = -\gamma_{\perp}\Delta_{c}/\Delta_{s}$, we have $$\label{a} \langle a\rangle=-\frac{\lambda}{\sqrt{N}\Delta_{c}(1+i\gamma_{\perp}/\Delta_{s})}\langle J_{-}\rangle -\frac{\Omega_{a}}{\Delta_{c}(1+i\gamma_{\perp}/\Delta_{s})}.$$ With $\gamma_{\perp}=0$, it reduces to the result in the ideal Hermitian case. In Fig. \[figure3\](a), we plot $\langle a^{\dag}a\rangle/N$ versus the reduced coupling strength $\lambda/\Delta_{c}$ in both the ideal Hermitian case and the non-Hermitian case with gain. In these two cases, the results for $\langle a^{\dag}a\rangle/N$ globally look different, but we can analytically derive that $\langle a^{\dag}a\rangle/N$ in the non-Hermitian case with gain also exhibits the same critical behavior as in the Hermitian case, i.e., around $\lambda=\lambda_c$, $\langle a^{\dag}a\rangle/N \sim \|\lambda-\lambda_{c}\|^{\nu_{a}}$, with $\nu_{a}=1/2$ [@SM]. Without decoherence in the system ($\kappa_c=\gamma_{\perp}=\gamma_{\parallel}=0$), $\Omega_{a}/\Omega_{J}=(\Delta_{c}-i\kappa_{c})/\lambda$ is reduced to the parameter-matching condition in the ideal Hermitian case, $\Omega_{a}/\Omega_{J}=\Delta_{c}/\lambda$, and the second equation in Eq. (\[spin-1\]) becomes trivial. In this Hermitian case, the results in Eqs. (\[order-Jz\]) and (\[order-J-\]) are obtained by solving the first equation in Eq. (\[spin-1\]) and using the spin-conservation law $\langle J_{x}\rangle^2+\langle J_{y}\rangle^2+\langle J_{z}\rangle^2=(N/2)^{2}$ [@Soriente18; @Zhu20]. In contrast, when the decoherence of the system is considered, the spin-conservation law is usually [violated]{}, with $\langle J_{x}\rangle^2+\langle J_{y}\rangle^2+\langle J_{z}\rangle^2 < (N/2)^{2}$ [@Torre16; @Kirton17]. To characterize this violation, we define a characteristic quantity $F=(\langle J_{x}\rangle^{2}+\langle J_{y}\rangle^{2}+\langle J_{z}\rangle^{2})/(N/2)^{2}$. Obviously, $F=1$ in the ideal Hermitian case owing to the spin-conservation law. For the non-Hermitian case with gain, it is easy to verify that $F=1$ in the regime of normal phase, but $$\label{} F= \frac{\lambda_{c}^{2}}{\lambda^{2}}\bigg(1-\frac{\lambda_{c}^{2}}{\lambda^{2}}\bigg)\frac{\gamma_{\parallel}}{\gamma_{\perp}} +\frac{\lambda_{c}^{4}}{\lambda^{4}}$$ in the regime of super-radiant phase, where the ratio $\gamma_{\parallel}/\gamma_{\perp}$ between the longitudinal- and transversal-relaxation rates plays a significant role. In the non-Hermitian case with gain, $F$ monotonically decreases when $\lambda/\Delta_{c}>1.008$ (corresponding to $\lambda > \lambda_{c}$) \[see the red dashed curve in Fig. \[figure3\](b)\]. Possible implementation.—In the experiment, the proposed model with gain can be implemented using a hybrid circuit-QED system composed of a dissipative TLS ensemble, such as the nitrogen-vacancy (NV) centers in diamond, coupled to an active coplanar-waveguide resonator \[see Fig. \[figure1\](b)\]. The transition frequency of NV centers can be tuned by an external magnetic field and the number of NV centers in the sample can be $N \sim 10^{12}$ [@Kubo10; @Amsuss11], approaching the thermodynamic limit of the proposed system. The hybrid system is usually in the strong-coupling regime and can be described by a standard TC Hamiltonian in Eq. (\[TC\]). To engineer an active resonator, one can harness an auxiliary flux qubit transversely coupled to the resonator with a coupling strength $g_{a}$ and longitudinally driven by two microwave fields [@Quijandria18]. After eliminating the degree of freedom of the auxiliary qubit, the effective gain rate $\kappa_{g}$ of the resonator can be tuned from 0 to, e.g., $0.2g_{a}$ (i.e., $2\pi\times 6 $ MHz for $g_{a}/2\pi=30$ MHz) by varying the amplitudes and frequencies of the two microwave fields [@Quijandria18]. Moreover, as shown in Fig. \[figure1\](b), two additional drive fields with Rabi frequencies $\Omega_{a}$ and $\Omega_{J}$ pump the NV-center ensemble and the resonator, respectively. When the parameter-matching condition $\Omega_{a}/\Omega_{J}=\Delta_{c}(1+i\gamma_{\perp}/\Delta_{s})/\lambda$ is achieved for these two drive fields, an experimentally accessible TC model is then engineered and the relevant quantities for demonstrating the QPT are governed by Eqs. (\[spin-2\]) and (\[a\]). Discussions and Conclusions.—The special case with only one drive field was studied in Refs. [@Zou13; @Emary04], which is related to the cavity or the TLS ensemble pumped by a drive field, i.e., Eq. (\[TC-d\]) with $\Omega_{a}\neq 0$ but $\Omega_{J}=0$ [@Zou13] or $\Omega_{J} \neq 0$ but $\Omega_{a} = 0$ [@Emary04]. At a finite strength of this drive field, the biased TC model does not preserve the parity symmetry due to the bias term and the QPT disappears in the system [@Emary04]. Only in the weak drive-feld limit \[i.e., either $\Omega_{a} \rightarrow 0$ when $\Omega_{J}=0$ or $\Omega_{J} \rightarrow 0$ when $\Omega_{a} = 0$ in Eq. (\[TC-d\])\] can the model tend to have the QPT [@Zou13]. For a realistic system, the cusp-like QPT behavior is however much smoothened by the inevitable cavity loss and the dissipation of the TLS ensemble even in the weak drive-feld limit [@Feng15]. In our proposal, when the two drive fields satisfy the parameter-matching condition, we can obtain an effective TC Hamiltonian with parity symmetry \[i.e., Eq. (\[eff\])\]. This TC Hamiltonian can exhibit QPT at an experimentally-accessible critical coupling strength and the weak drive-field limit is not necessary. Even with the decoherence of the system, the QPT behavior is still achievable by harnessing an active cavity (instead of a passive cavity). Very recently, the QPT in a TC model induced by a single [squeezed]{} drive field was investigate [@Zhu20], where the effective Hamiltonian is a biased TC model with a two-photon drive term, corresponding to Eq. (\[TC-d\]) with $\Omega_{a}a^{\dag}+\Omega_{a}^{}a$ replaced by $\Omega_{a}a^{{\dag}2}+\Omega_{a}^{}a^{2}$ and $\Omega_{J}=0$. Using a squeezing transformation, this biased TC Hamiltonian can be transformed to an anisotropic Dicke Hamiltonian [@Qin18]. Experimentally, if a [pure]{} squeezed drive field, $a^{{\dag}2}+a^{2}$, cannot be achieved, as discussed above, any finite [unsqueezed]{} part of the single drive field can ruin the QPT. In our study, the obtained effective Hamiltonian in Eq. (\[eff\]) is a standard TC model without the bias term and no squeezed drive field is required. Also, the spin-conservation law is used in Ref. [@Zhu20]. It is different from our non-Hemitian case in which the spin-conservation law is found to be violated [@Torre16; @Kirton17]. In conclusion, we have studied the QPT in a TC model engineered with two drive fields applied to the TLS ensemble and the cavity, respectively. In the ideal Hermitian case without decoherence, the QPT can occur at an experimentally-accessible critical coupling strength, but it is spoiled by the decoherence of the system. In this non-Hermitian case, we find that the QPT can be recovered by harnessing a gain in the cavity to balance the loss of the TLS ensemble. In sharp contrast to the spin conservation in the ideal Hermitian case, the spin-conservation law is however found to be violated in our non-Hermitian case. Moreover, we propose to implement this non-Hermitian TC model using a hybrid circuit-QED system. 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--- abstract: 'In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.' author: - Benedikt Kolbe - 'Myfanwy E. Evans' title: 'Isotopic tiling theory for hyperbolic surfaces[^1] ' --- Introduction ============ The enumerative approaches of Delaney-Dress tiling theory [@DRESS1987] in the two-dimensional hyperbolic plane have facilitated a novel investigation of three-dimensional Euclidean networks, where hyperbolic tilings of triply-periodic minimal surfaces (TPMS) are used for an enumeration of crystallographic nets in $\mathbb{R}^3$ [@Sadoc1989; @Nesper2001; @Robins2005; @Robins2006; @Ramsden2009; @Castle2012]. By relating in-surface symmetries of the TPMS to ambient Euclidean symmetries [@Perez2002; @Hyde2014], the problem of graph enumeration and characterisation in $\mathbb{R}^3$ is transformed to a two-dimensional problem in equivariant tiling theory. The idea is that tilings of the hyperbolic plane can be reticulated over the surface to give a Euclidean geometry to the tile boundaries. This idea has been explored in several contexts over the past $30$ years, including standard hyperbolic tilings by disk-like tiles with kaleidoscopic symmetry [@Ramsden2003; @Ramsden2009], infinite tiles with network-like boundaries [@Hyde2000; @hyderams; @evansper1; @evansper3; @Kirkensgaard2014], and infinite tiles with geodesic boundaries [@evansper2]. The approach is motivated by the confluence of minimal surface geometry and the structural chemistry of zeolites and metal-organic frameworks [@Hyde1991; @Hyde1993; @Andersson1984; @Chen2001]. The enumeration of such hyperbolic tilings reduces down to a problem of enumerating all embeddings of graphs on the orbifold associated to the symmetry group of a tiling, as well as a suitable notion of equivalence among different tilings. Delaney-Dress tiling theory provides a systematic approach to the complete enumeration of combinatorial equivalence classes of tilings in simply connected spaces. Computer implementations of algorithms based on Delaney-Dress tiling theory can exhaustively enumerate the combinatorial types of equivariant tilings in space forms [@Huson1993]. This gives us a description of all combinatorially distinct tilings of an orbifold. For our purposes, we require an understanding of the distinct ways in which this combinatorial structure can be embedded on the orbifold, which in turn represent isotopically distinct tilings of the hyperbolic plane. For example, the Stellate orbifolds 2223 and 2224 can be decorated by a simple combinatorial structure, however this simple structure can manifest as an infinite set of isotopically distinct embedded hyperbolic tilings [@evansper1; @evansper2; @Pedersen2017; @Pedersen2018]. The classification of embedded combinatorial structures is precisely what this paper will address. We will generalize the Delaney-Dress combinatorial tiling theory to classify all isotopically distinct equivariant tilings of any hyperbolic surface of finite genus, possibly nonorientable, with boundary, and punctured. We consider here the $2$-dimensional case, however, the related classifications for higher dimensional hyperbolic orbifolds is also briefly discussed. Our approach is constructive and therefore allows, in theory, a complete enumeration of such classes of tilings. These results will facilitate the future development of the EPINET database (Euclidean patterns in non-Euclidean tilings) [@epinet], the central resource of the enumeration of Euclidean structure through hyperbolic tilings. Throughout this paper, we make heavy use of the notion of orbifolds [@thurston] and mapping class groups [@primermcgs]. The connection of equivariant isotopic tiling theory and mapping class groups is novel, however, there is a well-known connection between the Teichmuller space of Riemann surfaces of genus $g$ and certain tilings of the hyperbolic plane with $4g$ sided geodesical polygons that we will use as inspiration [@primermcgs]. We will also derive some algorithms to enumerate all equivariant tilings on a hyperbolic Riemann surface in its uniformized metric. Note that this also produces tilings for other Riemannian surfaces by uniformizing the metric within its conformal equivalence class. This paper is structured into five main sections which cumulatively build the connection between equivariant isotopic tiling theory and mapping class groups. We clarify several smaller questions along the way, building intuition of previous results in a new context. We begin with section \[sec:orbs\] (Symmetry Groups of Tilings and Orbifolds), where we recapitulate the notion of two-dimensional developable orbifolds and expand the framework to incorporate more general classes of orbifolds with punctures and boundary. This is followed by section \[sec:ddtheory\] (Isotopic Tiling Theory) where we generalize combinatorial Delaney-Dress tiling theory to encode isotopically distinct tilings of surfaces in terms of generators of the symmetry group. In the brief section \[sec:outs\] (Outer Automorphisms) we will elucidate the connection between outer automorphisms and the generators that encode isotopically distinct tilings. Then, having laid all the groundwork, we will introduce the mapping class group (MCG) of orbifolds in section \[sec:MCGorbifold\] and prove fundamental results facilitating its applications to tiling theory. In section \[sec:bhtheory\], we establish relations between the spaces of tilings of covering spaces and lastly, we highlight how some of the algebraic properties of MCGs relate to the isotopy classes of tilings they correspond to. This paper represents the theoretical foundation for an enumeration of isotopy classes of tilings on surfaces. The implementation of these results will appear in [@BenMyf2] and are of inherent interest in the natural sciences. As a result, the first part of this paper does not follow the tradition of a theorem with subsequent proof style of exposition that most math papers stick to. Instead, before the proofs get too technical, we did our best to explain the ideas using a more descriptory exposition. Symmetry Groups of Tilings and Orbifolds {#sec:orbs} ======================================== We begin with orbifolds [@Conway2002; @thurston; @adem2002orbifolds]. Let $\mathcal{X}$ be a $2$D space form, i.e. a simply connected Riemannian manifold $\mathcal{X}$ with constant sectional curvature. We only work with developable orbifolds, which means that the orbifold $\mathcal{O}$ is the quotient space $\mathcal{X}/\Gamma$, where $\Gamma\subset \operatorname{Iso}(\mathcal{X})$ is a discrete subgroup. The difference between $\mathcal{X}/\Gamma$ as a topological space and as an orbifold is that for the orbifold structure, one retains the information concerning $\Gamma$ and can reconstruct the topological space $\mathcal{X}$ from $\mathcal{X}/\Gamma$ and vice versa. The group $\Gamma$ is called the fundamental group of the orbifold $\mathcal{O}.$ In the classical orbifold setting, $\Gamma$ is required to act cocompactly. We will only require the codomain to have finite area in its uniformized metric, i.e. the metric induced by $\mathcal{X}.$ In particular, we are interested in the case $\mathcal{X} = \mathbb{H}^2$, where $\Gamma$ is a NEC group (non-Euclidean crystallographic group), or a hyperbolic orbifold group. The hyperbolic case is also the only case that admits infinitely many isomorphism types of hyperbolic groups $\Gamma$, i.e. there are infinitely many non-diffeomorphic hyperbolic orbifolds. Here, diffeomorphisms are understood as diffeomorphisms of orbifolds as defined in [@thurston] but note that two hyperbolic $2$-orbifolds are diffeomorphic if and only if their symmetry groups are isomorphic. Let $\mathcal{O}$ be a $2$D orbifold. We can identify the symmetry groups using Conway’s orbifold symbol, as described below, but extended by generators for the non-classical features our orbifolds might have, i.e. hyperbolic transformations $H_i$ of $\mathbb{H}^2$, corresponding to non-mirror boundary components of $\mathcal{O}$ and parabolic transformations $P_j$ corresponding to punctures. The diffeomorphic structure of $\mathcal{O}$ is determined by the Conway symbol for its fundamental group $\Gamma:= A\cdots H_i\cdots P_j\cdots \star abc\cdots\times\cdots\circ\cdots$. There are generators for the translations associated to each handle, given by $X$ and $Y$, and going around a handle in an oriented way corresponds to the curve associated to the commutator $\alpha:=[X,Y]=XYX^{-1}Y^{-1}.$ There are also generators for each gyration point of order $A$, and for a curve $\gamma$ going around the gyration point once we have $\gamma^A=1,$ where we interpret the curve as a deck transformation [@ratcliffe2013foundations]. For each mirror we have the usual Coxeter group relations, which depend on the angles of the intersecting mirrors. However, somewhat more subtle, in the case where the interior of the orbifold contains nontrivial features, we actually need to choose one mirror per mirror boundary component that we give two generators $P$ and $Q$, ordered in positive orientation corresponding to its two mirror halves and one generator $\lambda$ for the curve that goes around this boundary component once in positive orientation. In the literature, $\lambda$ is known as the connecting generator for this mirror boundary component. We then add the relation $P=\lambda^{-1} Q \lambda$. Next, going around a cross-cap corresponds to a generator $\omega$ with $Z^2=\omega$, where $Z$ corresponds to the curve entering the cross-cap once. There is one global relation for an orbifold, namely, the product of all Greek letters (plus the nonclassical elements) in the above has to be trivial, i.e. $$\begin{aligned} \label{eq:globalrel} \gamma...\Pi_iH_i\Pi_jP_j\lambda...\omega...\alpha...=1.\end{aligned}$$ We shall refer to this presentation as the standard presentation of the fundamental group of $\mathcal{O}$. To standardize notation, we can also assume that in the presence of a crosscap, all handles are replaced by two crosscaps each [@conwayzip]. We will only consider the geometric generators of the above form for orbifold groups in this paper. Note that there is a description of the deck transformations in $\Gamma$ as homotopy classes of curves on the orbifold $\mathcal{O}$ [@ratcliffe2013foundations; @Conway2002], which we already used above in the description of $\omega$, and which will be important to us. The elements of an orbifold fundamental group $\Gamma$ can be assigned types according to their algebraic properties and their action on the hyperbolic plane. Similar to [@Maclachlan1975], we define the type of an element in $\Gamma$ as follows. Torsion elements that preserve the orientation of $\mathbb{H}^2$ are the elliptic transformations of a given order. Mirrors represent torsion elements of order $2$ that reverse the orientation. Orbifold groups like $\Gamma$, when viewed as conformal transformations of the upper half plane $U\subset \mathbb{C}$, have an associated limit set $\Lambda\subset \mathbb{R}$. The complement of $\Lambda$ in $\mathbb{R}$, $C$, has more than one connected component if $\mathcal{O}$ has a boundary. The conjugates of powers of the hyperbolic transformations associated to the boundary components of $\mathcal{O}$ map some component of $C$ to itself and are called boundary hyperbolic. Also, there are the parabolic transformations corresponding to the punctures and the orientation preserving as well as the reversing hyperbolic transformations associated to the genus of a surface. All of these designations give the type of an element in $\Gamma.$ We call an automorphism of $\Gamma$ type preserving if it preserves the types of all elements in $\Gamma.$ Isotopic Tiling Theory {#sec:ddtheory} ====================== Tesselations of $\mathbb{H}^2$ can be described using combinatorial tiling theory [@DRESS1987; @Huson1993]. Combinatorial tiling theory classifies all possible equivariant combinatorial types of tilings on space forms $\mathcal{X}$. It deals with the case that each tile is a closed and bounded disc and the symmetry group of the tiling acts cocompactly. A set $\mathcal{T}$ of such topological discs in $\mathcal{X}$ is called a tiling if every point $x\in\mathcal{X}$ belongs to some disc (tile) $T\in \mathcal{T}$ and if for every two tiles $T_1$ and $T_2$ of $\mathcal{T}$, $T_1^0\cap T_2^0=\emptyset,$ where $S^0$ denotes the interior of a set $S.$ All tilings in this paper will be assumed to be locally finite, meaning that any compact disc in $\mathcal{X}$ meets only a finite number of tiles. More specifically, combinatorial tiling theory considers a barycentric subdivision of the tiles and subsequent tracking of the combinatorics of the resulting chamber system to give the coordination sequence of the tiling [@DRESS1987]. We call a point that is contained in at least $3$ tiles a vertex, and the closures of connected components of the boundary of a tile with the vertices removed edges. The only exception to this are two-fold rotational centers of symmetry, which we will also consider to be vertices, although they don’t contribute to the chamber system. Let $\mathcal{T}$ be a tiling of $\mathcal{X}$ and $\Gamma$ be a discrete subgroup of $\operatorname{Iso}(\mathcal{X})$. If $\mathcal{T}=\gamma\mathcal{T}:=\{\gamma T\|t\in\mathcal{T}\}$ for all $\gamma\in\Gamma$, then we call the pair $(\mathcal{T},\Gamma)$ an equivariant tiling and $\Gamma$ its symmetry group. We call two tiles $T_1,T_2\in \mathcal{T}$ equivalent or symmetry-related if there exists $\gamma\in\Gamma$ s.t. $\gamma T_1=T_2.$ We call the subgroup of $\Gamma$ that leaves invariant a particular tile $T\in\mathcal{T}$ the stabilizer subgroup $\Gamma_T.$ A tile is called fundamental if $\Gamma_T$ is trivial and we call the whole tiling fundamental if this is true for all tiles. An equivariant tiling is called tile-, edge-, or vertex-$k$-transitive, if the number of equivalence classes under the action of the symmetry group is $k.$ Note that the above definitions do not require $\Gamma$ to be the maximal symmetry group for the tiling $\mathcal{T}$. A fundamental tile-1-transitive equivariant tiling (fundamental tiling for short, where this doesn’t lead to confusion) $(\mathcal{T}, \Gamma)$ has a single type of tile that is a fundamental domain for $\Gamma$ and any fundamental domain for $\Gamma$ also gives rise to such a tiling. The above represents the framework for combinatorial tiling theory as outlined in [@Huson1993]. We will generalize the theory to work with more general symmetry groups and also non-simply connected spaces. We will now detail a slightly different point of view using orbifolds and a concrete realization of a symmetry group $\Gamma$ in $\operatorname{Iso}(\mathcal{X})$, where we particularly emphasize the case $\mathcal{X}=\mathbb{H}^2$. One can view tilings as combinatorial structures or classes of decorations on orbifolds. This is based on the simple observation that any tesselation has the symmetry group of a developable orbifold. The underlying topological space of $\mathcal{O}$ can be extracted from any fundamental domain for $\Gamma$ in $\mathbb{H}^2,$ with appropriate edge identifications corresponding to the action of the generators of $\Gamma$ on the fundamental domains boundary. The action of $\Gamma$ also gives rise to a fundamental transitive tiling. Each fundamental transitive tile can also be interpreted as a (bordered) fundamental domain and can thus each be seen as a possible canvas, on which we can draw any orbifold decoration (after getting rid of the boundary edges, if necessary). Note that the tiling that results from the drawing can exhibit a higher degree of symmetry than what the symmetry group of the equivariant tiling suggests. Of course, a drawing on a fundamental domain corresponds to a piecewise linear embedding of a graph into the orbifold. Geometrically, we first identify a set $C$ of curves such that cutting $\mathcal{O}$ along $C$ produces a fundamental domain, with the edge identifications (from Poincar' e’s theorem) yielding the given presentation of $\Gamma$ [@Lucic1991; @Lucic1990]. For the decorations, one now draws arbitrary curves on $\mathcal{O}$. Then, cutting open $\mathcal{O}$ along $C$ and imposing $\Gamma$ produces any tiling with symmetry group $\Gamma$. However, not all decorations produce an equivariant tiling by closed discs. In the language of Delaney-Dress tiling theory, each chamber system encoding an equivariant tiling with symmetry group $\Gamma$ essentially corresponds to a triangulation of $\mathcal{O}=\mathbb{H}^2/\Gamma.$ [^2] When viewing tilings as combinatorial decorations on orbifolds, it becomes natural to consider the more general situation of finite volume orbifolds and thus more general symmetry groups for the tilings than for classical Delaney-Dress tiling theory. There are a number of ways of approaching this problem. In [@DRESS1987], the statements made for the case of bounded tiles work just as well for an equivariant tiling theory for symmetry groups with cusps[^3]. For the theory to incorporate such punctures, one still looks at tesselations in terms of chamber systems and geometric cell complexes and treats the cusps as marked points belonging to the surface, but when embedding these into a manifold to obtain a tesselation, one needs to remove the cusps before embedding. Geometrically, the idea corresponds to pushing the punctures to the boundary of the unit circle in the Poincaré model for $\mathbb{H}^2.$ Alternatively, a puncture in the orbifold, corresponding to a parabolic transformation in $\mathbb{H}^2$, can be seen as the limit of a sequence of gyration points of increasing order, with order $\infty$. This is in line with the Conway notation for orbifolds. From this point of view, the tilings for finite volume orbifolds with punctures are attained as limits of tilings for orbifolds where the puncture is a gyration point of increasing order. We will therefore treat the cusped case in the same way as the classical case, except where we highlight differences, but with order $\infty$ singular points. The theory for orbifolds with boundary can be dealt in the following way. We can replace all boundaries with mirrors, with each mirror/boundary component contributing a $-1$ to the Euler characteristic. The resulting orbifold induces a tesselation of $\mathbb{H}^2$. We now apply combinatorial tiling theory as before, and simply remove the mirrors, replacing the mirror symmetries with the hyperbolic transformations that represent the boundary deck transformations in $\operatorname{Iso}(\mathbb{H}^2).$ The boundary of the tesselated submanifold of $\mathbb{H}^2$ is then a result of the boundary and symmetry operations of the original smaller orbifold it covers. For simply connected spaces and combinatorial classes of tilings, starting from the fundamental tilings, all other equivariant tilings with the same symmetry group are obtained by using GLUE and SPLIT operations [@delone78; @Huson1993]. The different combinatorial types of a fundamental domain for a given classical orbifold were classified in [@Lucic1991]. Using the Delaney-Dress ($D$)-symbol, one can give unique names to the combinatorial structures on $2$-orbifolds that represent tilings on space forms, which can be used for enumeration purposes [@Delgado-Friedrichstilings]. We will subsequently focus on fundamental tilings. Given the generators, described in section \[sec:orbs\], of a symmetry group $G\subset \operatorname{Iso}(\mathbb{H}^2),$ the $D$ symbol describes how the group acts on the associated chamber system of a tiling [@DRESS1987], where the chambers are triangles in a triangulation of the orbifold. A fundamental tiling is obtained from a fundamental domain for $G$, with the given generators acting on its boundary edges. By the Poincar' e theorem, a set of (geometric) generators of $G$ all map part of the fundamental tile’s boundary to itself to yield a presentation of the symmetry group, and the $D$ symbol tells us in which way. Even if we restrict to geodesically bordered tiles, the $D$ symbol only defines a tiling up to shearing the fundamental domain. We now explain a fundamental observation that is one of the starting points of our investigation. The Teichmueller space $T(G)$ is the space of type preserving, discrete faithful representations in $\operatorname{PGL}(2,\mathbb{R})$ (for orientable orbifold groups, one usually restricts to $\operatorname{PSL}(2,\mathbb{R})$) of the abstract hyperbolic group $G$ with standard presentation, modulo conjugation by elements in $\operatorname{PGL}(2,\mathbb{R})$. This space carries a natural topology, namely the subspace and subsequent quotient topology of $\operatorname{Hom}(G,\operatorname{PGL}(2,\mathbb{R})),$ which itself is endowed with the compact-open topology. The topology of $G$ is the discrete one and $\operatorname{PGL}(2,\mathbb{R})$ carries the topology it inherits from its usual structure as a Lie group. It is well-known [@thurston] that $T(G)$ is (component-wise) homeomorphic to $\mathbb{R}^{k}$ for some $k$. The generalization to our more general orbifolds just means that we expand, by ideal hyperbolic triangles, the collection of primitive orbifolds with unique hyperbolic structures that assemble to produce more complicated orbifolds and continue with the same arguments presented in [@thurston]. This implies that two different sets of generators as in section \[sec:orbs\] for $G$ in $\operatorname{Iso}(\mathbb{H}^2)$ can be continuously deformed into one another in $\mathbb{H}^2$. The small caveat here is that for orientable $G$, there are two representations with opposite orientation (see section \[sec:outs\]) in $\operatorname{PGL}(2,\mathbb{R})$, so the connectedness of $T(G)$ is only true for representations of the same orientation. The importance of the above is the observation that during the process of continuously deforming one representation of $G$ into another, the combinatorial structure of the chamber system associated to the tiling remains invariant. Therefore, perhaps somewhat surprisingly, there is a set of combinatorial instructions for how to decorate the fundamental domain to produce a particular tiling from the generators is independent of the particular representation of $G$ in $\operatorname{PGL}(2,\mathbb{R})$. This set of instructions can be read off the $D$ symbol, see figure \[fig:2224\] for an example. Within a combinatorial class of fundamental tiles, we can interpret the other fundamental tiles with different positions for the generators as obtained by shearing the original one. This deformation can in fact be realized by a quasi-conformal mapping and induces a metric on the orbifold. In particular, one can calculate the metric properties (diameter,...) of a given orbifold realization, properties which are potentially of interest to the natural sciences. We will now turn to an important example that we will use to highlight the general situation. Consider the fundamental $4g$ polygon of a closed hyperbolic Riemann surface $S$ of genus $g$ with given hyperbolic metric. The construction of a tiling starts from a given point $x\in S$ which is the base point of the generating curves $\{\gamma_i\}_{i=1}^{2g}$ for the fundamental group of $S$. Within each homotopy classes for the closed curves $\gamma_i$, there is a unique geodesical representative. Cutting the surface along these geodesics produces a hyperbolic tile and tesselation. The combinatorial structure of the associated hyperbolic tiling is uniquely determined, as there are $4g$ copies of the fundamental tile around every vertex, which does not depend on how the edge identifications are realized, i.e. on the presentation of $\pi_1(S)$. In particular, there is only one associated combinatorial class of fundamental tiling with $4g$ edges. On the other hand, the choice of base point for the construction of a fundamental domain for the generators produces a plethora of metrically distinct fundamental tilings, which are all derived from the same point in $T(G).$ What different types of fundamental tilings can we create in this way? Any other tiling starts from a different point $p\in S$, and there is a path $c$ connecting $x$ and $p$. The path can be extended to an isotopy of $S$. In this way, $c$ uniquely determines a homeomorphism up to isotopies from $(S,x)$ to $(S,p)$, following results relating to the point-push map in [@primermcgs]. In particular, if one fixes a reference set of generators of $\pi_1(S)$, the resulting isotopy extension only leaves the set of generators invariant on $S$ if the path $c$ induces a trivial homeomorphism up to isotopy. This means that if we fix what the generators $\{\gamma_i\}$ map to in $\operatorname{Iso}(\mathbb{H}^2),$ there is only one isotopy class of tilings associated to $S$ and $\{\gamma_i\}$. The combinatorial information needed to produce the corresponding tiling is then simply given by any base point needed to construct the associated fundamental domain. The case $p=x$ is of particular interest. For nontrivial curves $c$, the generators in $\operatorname{Iso}(\mathbb{H}^2)$ change, but because the induced automorphism on $\pi_1(S)$ by any such curve $c$ is always inner, by the Dehn-Nielsen-Baer theorem [@primermcgs], it corresponds to the isotopically trivial homeomorphism of the surface and does not change the tiling in $\mathbb{H}^2$. The same line of reasoning works for more general orbifolds $\mathcal{O}$ and their fundamental groups $G$, possibly with repeated use of the above arguments for more than one randomly chosen point, and explains why within a fixed set of generators for $\pi_1(\mathcal{O})\subset \operatorname{Iso}(\mathbb{H}^2)$ and combinatorial type of fundamental tiling with the generators acting on its boundary, the isotopy type of decoration does not depend on the choice of random points required in the construction, up to inner automorphisms of the chosen generators. Note that this does not preclude different sets of generators with the same combinatorial decorations to yield the same isotopy class of fundamental tilings, because the decoration of an orbifold that gives rise to a fundamental tiling is in general not sufficiently complicated to keep track of such changes. A class of examples is discussed in section \[sec:results\] below. Sufficiently complicated decorations where this cannot happen always exist, as we shall see. A related idea to produce all combinatorial types of fundamental domains is to randomly choose points of $\mathbb{H}^2$ w.r.t. which the Dirichlet fundamental domains eventually produce all combinatorial types of fundamental tilings for any given group $G$ (by theorem $3.3$ of [@Lucic1991]). However, the construction of the Dirichlet fundamental domain is insensitive to the generators chosen for $G$. Generally, the edges of the fundamental tile can be given purely in terms of the generators, as edges connecting symmetry points, or randomly chosen points. The random points show up in the triangulation that is the chamber system of the orbifold when a vertex is not located at an increased symmetry site. This situation can be read off the $D$ symbol. Using this approach to fundamental tilings from the chamber system related to the $D$-symbols gives a completely algebraic/combinatorial way of producing the fundamental tilings from the generators of $G.$ In practice, this invariant description in terms of generators comes from simply producing a combinatorial version of a tiling from the $D$-symbol and then placing the vertices in the associated decoration accordingly, see figure \[fig:2224\]. In doing so, the vertices have to be given in terms of their positions relative to the generators.[^4] [0.31]{} ![Fundamental tilings with symmetry group $2224$.[]{data-label="fig:2224"}](2224_114_fd.png "fig:"){width="\textwidth" height="\textwidth"} [0.31]{} ![Fundamental tilings with symmetry group $2224$.[]{data-label="fig:2224"}](2224_114_primgens.png "fig:"){width="\textwidth" height="\textwidth"} [0.31]{} ![Fundamental tilings with symmetry group $2224$.[]{data-label="fig:2224"}](2224_114_othergens.png "fig:"){width="\textwidth" height="\textwidth"} As an illustration (figure \[fig:2224\]), consider the hyperbolic orbifold group $G=2_12_22_34_4$, where the subscripts track the positions of the generators in figure \[fig:2224ft1\]. The placements of the generators in $\mathbb{H}^2$ in figure \[fig:2224ft1\] allows a fundamental tiling for the supergroup $\star 2224$ simply by considering the convex hull in $\mathbb{H}^2$ of the indicated points $1$ to $4$. Now, there are two ways that $2224$ sits inside $\star 2224$. One is obtained by reducing the symmetry of the $4$-fold rotation point, corresponding to figure \[fig:2224ft1\]. The other is obtained by doing the same to a $2$-fold rotation center, as has been done in figure \[fig:2224ft2\]. These are the only two combinatorial classes of fundamental tilings in $\mathbb{H}^2$ for $2224$ [@Lucic1991]. By the above discussion, we can combinatorially give a description of the edges belonging to the fundamental tiling. In figure \[fig:2224ft1\], consider the rotations corresponding to the generators $r_1,...,r_4,$ with centers $c_1,...,c_4$. Because the tiling is obtained by doubling the fundamental tiling of $\star 2224$, it is straightforward to see that the corners/increased symmetry points on the polygon’s boundary correspond clockwise, starting at $c_1$, to the points $c_1,c_2, c_3, c_4, r_4(c_3), r_4r_3r_4^{-1}(c_4)$. This procedure readily generalizes to arbitrary stellate orbifolds, i.e. those with only rotations for generators. Given any generators $r_1,...,r_4$ in $\operatorname{Iso}(\mathbb{H}^2)$, this description of edges defines a fundamental tiling, in this case with totally geodesic edges, regardless of the generators placement in $\mathbb{H}^2$. Similarly, for the fundamental tiling of figure \[fig:2224ft2\], the edges are given by hyperbolic lines connecting the points $c_1,c_2, r_1(c_2),r_3^{-1}(c_4),c_3,c_4.$ Figure \[fig:2224othergens\] illustrates that this relation for the edges still holds and we obtain a fundamental tiling with symmetry group $G$, this time without the additional symmetries that we used to find the end points of the edges of the fundamental tile on whose boundary the generators are positioned. This example illustrates that it can be very helpful to look at versions of the equivariant tiling in question that exhibit more symmetries than the given symmetry group. As mentioned in the introduction, we want to classify equivariant tilings of a hyperbolic Riemann surface $S$ in its uniformized metric, i.e. given a fundamental hyperbolic polygon of $S$ in $\mathbb{H}^2$, we want to find all ways of equivariantly tiling it, with fixed symmetry group $G\subset \operatorname{Iso}(S)\subset \operatorname{Iso}(\mathbb{H}^2).$ We consider equivariant tilings with the same symmetry group that are isotopic in $S$ equivalent. As far as isotopic tiling theory is concerned, it is not enough to consider just the abstract group $G$ and the associated $D$ symbols in our more general setting. Instead, it is important to use the method of producing fundamental tilings from $D$ symbols along with specific generators for $G$ as outlined above. There is a way to carefully choose only those sets of ‘locations’ for generators for $G$ that yield a priori different fundamental tilings of $S$ (see sections \[sec:outs\] and \[sec:bhtheory\] below). We will see in section \[sec:bhtheory\] that these different sets of locations for generators for $G$ give rise to non-isotopic tilings of $S$, for any decoration that is sufficiently complicated. It is a well-known fact that any closed curve on $S$ has a unique geodesic representative in its isotopy class, so we will only work with piece-wise geodesic decorations, with breaks occuring only at vertices. Consider tiling the genus $3$ fundamental polygon of the Riemann surface $S$ in $\mathbb{H}^2$ with symmetry group $\star 246$. There are three different versions of the $22222$ subgroup that are supergroups of $\pi_1(S)$. By Hurwitz’ theorem, there is a smallest (area-wise) possible hyperbolic group $G_0$ that is a supergroup of $\pi_1(S)$ and all three versions of $22222$ will be a subgroup of $G_0$ and we see that $\star 246=G_0.$ Each version of $22222$ now has to be treated independently of the others when classifying all isotopy classes of equivariant tilings on $S$. Indeed, the fundamental tilings for every possible set of generators for each of these groups are non-isotopic as tilings on $S$ (see section \[sec:bhtheory\] below). Before we go on to introduce new tools for tackling the new challenge of finding appropriate sets of generators for the symmetry groups of tilings, we would like to point out that the GLUE and SPLIT operations’ validity remains unchanged in this new setting. One could ask if two different sets of generators $S_1, S_2$ for the same group that produce different fundamental tilings lead to the same tiling of $S$ after a sequence of such operations. If this were the case, then firstly the sequence of operations would be different since they are invertible. However, this would mean that these two different sequences of operations, each applied only to tilings derived from $S_1$ would yield combinatorially equivalent tilings. These are equivalent in the classical Delaney-Dress tilings theory, so no additional ambiguity emerges by us distinguishing between tilings associated to different sets of generators for the symmetry groups. In particular, it is, in a way, very natural to consider the isotopy classes of tilings w.r.t. a set of generators for the symmetry group. Furthermore, this result is very important for enumerative isotopic tiling theory. The Group of Outer Automorphisms {#sec:outs} ================================ Let $G$ be a group. The group of all automorphisms of $G$ is denoted by $\operatorname{Aut}(G)$. Conjugation by any element $g\in G$ induces an automorphism $c_g(\tilde{g}):=g\tilde{g}g^{-1}$ for $\tilde{g}\in G$ . Such automorphisms are traditionally called inner automorphisms, and the normal subgroup of all of them is denoted by $\operatorname{Inn}(G):=\{c_g\|g\in G\}.$ The picture is easiest to understand and yet very general in the case where $G$ is a group of automorphisms of a space in some category. For concreteness and notational convenience we think of isometries of a Riemannian manifold $M$. Roughly speaking, after having chosen a fundamental domain $D$ for $G$ on which the generators of $G$ act, $M$ breaks up into copies of $D$. An inner morphism corresponds instead to building up $M$ starting from another copy of $D$. For this reason, one is often actually interested in the group of outer automorphisms $\operatorname{Out}(G):=\operatorname{Aut}(G)/\operatorname{Inn}(G).$ Consider now the case where $G$ is a hyperbolic orbifold group. In its universal covering space $\mathbb{H}^2$, one finds special points of increased symmetry, i.e. invariant subsets, corresponding to a geometric realisation of the elements of $G$. In some cases, these points correspond to a singleton that is fixed under some element of $G$, while in other cases, they are submanifolds that are left invariant by an element of $G$. By the location of generators of $G$, we really mean the locations in $\mathbb{H}^2$ of the invariant subsets. We fix a set of generators $G_1\subset\operatorname{Iso}(\mathcal{X})$ for $G$ and consider subsets $S$ of the elements of $G\subset\operatorname{Iso}(\mathcal{X})$ with $\|S\|=\operatorname{rank}(G)$. We are interested in the following question: When do we have $<S>=G$? Interpreted within the context of group automorphisms starting from $G_1$, which will correspond to the identity morphism, we see that they are exactly those subsets $S$ that correspond to an element of $\operatorname{Aut}(G)$. Note, however, that we are not interested in the full group of automorphisms. Instead, we will restrict our attention to the subgroup of type preserving automorphisms. This restriction is exactly what is needed to ensure that the combinatorics of general tilings are invariant when given as decorations of the associated orbifold (see theorem \[thm:mcgout\] below). Any tesselation with symmetry group $G$ is clearly invariant under an inner automorphism of $G$. The converse is also true - the inner automorphisms of $G$ are the only orientable automorphisms that leave invariant any decoration of fundamental domains for compact orbifold groups $G$. We will prove a version of this statement in proposition \[prop:BHgen\] below. One way to think about this is to look at the relation between orbifold group elements and curves on the orbifold, which in turn can be interpreted as decorations lifted to the universal cover. Thus, when a sufficiently complicated decoration of the fundamental domain is invariant w.r.t. an (orientation preserving) homeomorphism of the underlying orbifold, the underlying homeomorphism must be isotopically trivial in the orbifold because it fixes all curves and therefore orbifold elements. This means that it corresponds to an inner automorphism of $G$ by theorem \[thm:mcgout\] below. In case of noncompact orbifolds, this statement is only true for geometric automorphisms. A geometric automorphism is one that is realized by a homeomorphism of $\mathbb{H}^2$, see definition \[def:geomauto\] below. A nongeometric automorphism can change the combinatorial structure of the decoration of the orbifold. We therefore are not interested in the full group of outer automorphisms, because in the general case of orbifolds with boundaries or punctures, the designation of the type of the generator as a hyperbolic translation or a boundary parabolic transformation is important to us. When decorating an orbifold, thereby producing tesselations, we want the number and different types of generators to be fixed so as to preserve the original combinatorial structure of the tesselation. Another important point is that changing the orientation of an orientable decorated orbifold does not impact the locations of the decorations on the orbifold and therefore the associated tesselation remains invariant. Therefore, we are also only interested in a representative automorphism out of the class of orientation preserving or reversing geometric automorphisms. While orientation is a geometric notion, there is an algebraic analogue [@Zieschang1966] that captures the intuition of the geometric notion, so it makes sense to talk about the orientation of automorphisms of abstractly defined groups. The original surface decorating problem hereby reduces to the study of the group of outer automorphisms of a hyperbolic orbifold symmetry group. We are now prepared to formulate a result that highlights the importance of the $2$D setting. The Mostow rigidity theorem implies that the deformation space of finite volume hyperbolic structures on an orbifold of dimension $\ge 3 $ is a singleton. In particular, $\operatorname{Out}(\mathcal{O})$ is trivial and once we have chosen generators for the symmetry group, there is no way to obtain other generating sets via a geometric automorphism. In effect, this means that the combinatorial tiling theory for such non-simply connected hyperbolic manifolds is the same as classical combinatorial tiling theory, which does not take into account different sets of generators and all possible isotopy classes of tilings can be attained by randomly choosing points w.r.t. which one produces the Dirichlet fundamental domain. The Mapping Class Group of an Orbifold {#sec:MCGorbifold} ====================================== Our goal is to classify all of the relevant locations for generators for hyperbolic orbifold groups in $\operatorname{Iso}(\mathbb{H}^2)$ that lead to different tilings when decorated in an invariant way, according to $D$-symbols as outlined in section \[sec:ddtheory\]. Having laid all the groundwork, we now introduce the mapping class group (MCG) of orbifolds and prove fundamental results facilitating its applications to tiling theory. Let $\mathcal{O}$ be a not necessarily orientable compact hyperbolic $2$-orbifold, possibly with finitely many punctures and some boundary components. Denote by $O$ its underlying topological surface with weighted marked points at conical singularities of order equal to the assigned weight. Punctures can be treated as part of $O$ by assigning the weight label $\infty$, see section \[sec:ddtheory\]. Every mirror in $\mathcal{O}$ represents a part of a boundary component in $O$, corresponding to the set of points in its singular locus. Finally, we give all points in the singular locus of some mirror the same label $M,$ and call $O$ together with its labels the labelled underlying surface of $\mathcal{O}$. Let $\Sigma$ be the singular locus of $\mathcal{O}$, i.e. the (not necessarily isolated) branch point set of the covering $\mathbb{H}^2\to \mathbb{H}^2/\pi_1(\mathcal{O})$ and denote $O_0:=O-\Sigma.$ We consider $\operatorname{Hom}(\mathcal{O}),$ the group of homeomorphisms of $O$ that leave invariant the features of $O$, i.e. leave invariant the weight assigned to marked points, map punctures to punctures, mirrors to mirrors and other boundary components to other boundary components. Endowing $\operatorname{Hom}(\mathcal{O})$ with the compact-open topology turns it into a topological group. Denoting by $\operatorname{Hom}_0(\mathcal{O})$ the connected component of the identity in $\operatorname{Hom}(\mathcal{O}),$ we define the mapping class group (MCG) of the orbifold $\mathcal{O}$ as $\operatorname{Mod}(\mathcal{O}):=\operatorname{Hom}(\mathcal{O})/\operatorname{Hom}_0(\mathcal{O}).$ In other words, $\operatorname{Mod}(\mathcal{O})$ is the group of isotopy classes of the homeomorphisms in $\operatorname{Hom}(\mathcal{O})$, where at every step of an allowed isotopy the corresponding map is in $\operatorname{Hom}(\mathcal{O})$. In contrast to standard ways of defining the MCG, this means that all isotopies are free in the sense that they can twist around the boundary. In particular, mirrors do not individually need to be invariant under an isotopy within a connected component of mirrors. For any $[g]\in\operatorname{Mod}(\mathcal{O})$ that does not permute boundary components, there is thus always a representative $f\in \operatorname{Hom}(\mathcal{O})$ that fixes the boundary pointwise if $g$ is orientation preserving (if $\mathcal{O}$ is orientable). See also [[@Epstein1966 Theorem 5.6]]{}. There is thus no difference in how boundaries and punctures are treated. The MCG is usually defined using homotopies instead of isotopies. For orientable $\mathcal{O}$, [[@Maclachlan1975 Lemma 2]]{} shows that $f\in \operatorname{Hom}(\mathcal{O})$ is homotopic in $O_0$ to the identity if and only if they are isotopic in $\operatorname{Hom}(\mathcal{O})$ as defined above. The arguments in the proof are based almost exclusively on results by Epstein in [@Epstein1966], whose results are also proved for nonorientable surfaces and not necessarily hyperbolic ones. Now, all ordinary boundary components of $O$ are disjoint from labelled ones representing mirrors. Therefore, mirrors are treated in the same way as boundary components but disjointly and Epstein’s proof remains correct word for word. Thus, we have \[lem:homiso\] Let $[f],[g]\in \operatorname{Mod}(\mathcal{O})$. Then $[f]=[g]$ if and only if $f$ and $g$ are homotopic in $O_0$. Let $\Gamma=\pi_1(\mathcal{O})$, with natural projection map $p:\mathbb{H}^2\to\mathcal{O}=\mathbb{H}^2/\Gamma$. Consider a connected component $Z$ of $\mathbb{H}^2- p^{-1}(\Sigma)$, where $\Sigma,$ as above, denotes the singular locus of $\mathcal{O}$. Then $p:Z\to O_0$ is a non-branched and regular cover of connected topological spaces. Furthermore, $\pi_1(O_0)$ has generators $X_i$ corresponding to curves around the isolated points of the singular locus. In particular, $\pi_1(O_0)$ has the same number of generators as $\pi_1(\mathcal{O})$ in its standard presentation if one excludes the mirror symmetries from the latter. We now choose appropriate base points $z_0$ and $x_0$ for $Z$ and $O_0$ such that $p(z_0)=x_0$, so we can talk about concrete subgroups of the fundamental groups involved. The groups of deck transformations $\tilde{\Gamma}$ of the cover $p:Z\to O_0$ and $\pi_1(O_0)$ are related by $\tilde{\Gamma}=\pi_1(O_0)/\pi_1(Z)$. Here, we interpret $\pi_1(Z)$ as a subgroup of $\pi_1(O_0)$ in the usual way, i.e. under the push forward of $p_\sharp$ as $p_\sharp(\pi_1(Z))$. Clearly, $\pi_1(Z)$ equals the normal closure of the elements $X_i^{o_i}$, where the $o_i$ are the orders of the $X_i$ in $\mathcal{O}$, since these are exactly the relations imposed on the generators of $\pi_1(Z)$ when passing over to $\pi_1(O_0)$ with $p$. Let $f\in\operatorname{Hom}(\mathcal{O}),$ then, by construction, $f:O_0\to O_0$ preserves the order of branching of $p$. We therefore have that $f_\sharp(X_i^{o_i})\in \pi_1(Z).$ This is exactly the criterion (see [@Hatcher prop. $1.33$]) for the map $f\circ p$ to lift to a map $f_1:Z\to Z. $ We will check that $f_1$ can be uniquely extended to the closure of $Z$ in $\mathbb{H}^2$ and then, if necessary by reflections, to a map $f^$ on all of $\mathbb{H}^2$, following the arguments in [@Maclachlan1975 p. 500]. Take a small neighborhood $U$ of one of the punctures in $Z$. Then $f_1(U)$ has infinite cyclic fundamental group, meaning it is either a punctured disc or an annulus. The case of the annulus cannot be true, because $Z\backslash f_1(U)=f_1(Z\backslash U)$ is connected and the annulus also has to be bounded because an unbounded annulus would contain infinitely many punctures. Thus, $f_1$ permutes the punctures and can be extended to the closure of $Z$. The extension is unique and the only ambiguity here stems from lifting $f$ to $f_1$, but two such lifts are related by a deck transformation in $\Gamma.$ We obtain an automorphism $\alpha$ of $\Gamma$ by $\gamma\mapsto f^\gamma (f^)^{-1},$ where the group relations are easily checked. Below we will need the following theorem, which is proved in [@Marden2015] for orientable orbifolds. \[thm:idisot\] Suppose $f\in\operatorname{Hom}(\mathcal{O})$ and let $f^$ be its lift like above. Then the induced automorphism $\alpha$ is the identity automorphism of $\pi_1(\mathcal{O})$ if and only if $f$ is isotopic in $O_0$ to the identity mapping. We will see below in the proof of theorem \[thm:mcgout\] that homotopic mappings in $O_0$ yield the same automorphism of $\Gamma$, which deals with one direction. The proof of the other direction requires careful study of the proof in [@Marden2015]. Indeed, the proof works in a similar way as presented there, but we need to exchange one of the key ingredients. The following lemma replaces lemma $1$ in [@Marden2015]. Suppose $S$ is a surface like above (possibly obtained, like $O_0$, from a surface with features $O$) and $g$ is a homeomorphism of $S$. If there exists an arc $c$ from a point $0 \in S$ to $g(0)$ such that $\alpha$ is homotopic to $cg(\alpha)c^{-1}$ for all simple closed curves $\alpha$ based at $0$ that are disjoint from the boundary, then $g$ is homotopic in $S$ to the identity. In [@Bers1960], the corresponding statement for orientable, closed surfaces is proved using presentations of the surface in $\mathbb{H}^2.$ All of the arguments used there also work for the surfaces with features that we study, as long as we keep in mind the following. First, the orbifold fundamental group has a natural interpretation in terms of orbifold loops. Second, the construction of the lift of a map $f$ in [@Bers1960 p. 20] has to be replaced by the construction of $f^$ given above. Lemma \[lem:homiso\] then yields the isotopy of the theorem. Lastly, recall that the fundamental group $\pi_1(S)$ of a surface is generated by simple closed curves, also in the orbifold case. The MCG of a space is often studied by looking at the action of the homeomorphism classes on isotopy classes of curves. For example, let $\mathcal{O}=2222a$, with $a\ge 2$, then $\operatorname{Mod}(\mathcal{O})$ is one of two different types of groups. If $a=2$, $\operatorname{Mod}(\mathcal{O})=\operatorname{Mod}(S_5)$, the usual MCG of the $5$-punctured sphere with punctures $p_1,...,p_5$ corresponding to the hyperbolic rotations $r_1,...,r_5$. If $a>2$, $\operatorname{Mod}(\mathcal{O})$ is the subgroup of $\operatorname{Mod}(S_5)$ corresponding to those homeomorphism classes that fix the conical singularity $a$. It is well-known that the elements of finite order in $\pi_1(2222a)$ are characterisic, i.e. are preserved as a set under automorphisms. If, moreover, an automorphism $\alpha$ is type preserving and orientation preserving, $\alpha(r_i)=tr_jt^{-1}$ [@Zieschang1966]. It is impossible that this kind of transformation sends an elliptic transformation to a nontrivial power of itself. Indeed, assume that $r^d=trt^{-1}$ for some $d>1$. Then $r^{d-1}=trt^{-1}r^{-1}=[t,r]$ is elliptic. However, by [@Friedrich2008 pp. 191-193], the commutator of an elliptic transformation with any other transformation cannot be elliptic. Together with the general statements in theorem \[thm:mcgout\] below, the above reasoning implies that these types of transformations never yield an automorphism of the whole orbifold group, even if they do yield ones of the local group. This generalizes an observation made in [@evansper1] and [@evansper2], and discussed in more detail in [@myfphd], whereby the placement of generators in certain domains of $\mathbb{H}^2$ is prohibited. Any automorphism of $\pi_1(\mathcal{O})$ with conical singularities can be assigned an orientation with the expected property that all orientation-preserving automorphisms form a subgroup of index $2$ in all automorphisms [@Zieschang1966]. In addition, type and orientation-preserving automorphisms map the elliptic generators to conjugates of elliptic generators of the same order [@Zieschang1966]. We denote with $\operatorname{Aut}^+(\pi_1(\mathcal{O}))$ the subgroup of orientation and type preserving automorphisms, which contains all inner automorphisms, and with $\operatorname{Out}^+(\pi_1(\mathcal{O}))$ the corresponding subgroup of outer automorphisms. The well-known Dehn-Nielsen-Baer theorem can be generalized to $\operatorname{Mod}(\mathcal{O})\cong \operatorname{Out}^+(\pi_1(\mathcal{O}))$ [@Maclachlan1975]. We will prove the following, in much the same way, by providing an explicit isomorphism. \[thm:mcgout\] Let $\mathcal{O}$ be a nonorientable hyperbolic orbifold. Then the MCG $\operatorname{Mod}(\mathcal{O})$ defined above is isomorphic to $\operatorname{Out}_t(\pi_1(\mathcal{O}))$, the group of type preserving outer automorphisms. If $\mathcal{O}$ is orientable, then the MCG $\operatorname{Mod}(\mathcal{O})$ is isomorphic to $\operatorname{Out}^+(\pi_1(\mathcal{O}))$, the group of orientation and type preserving automorphisms. We only need to worry about the nonorientable case. Define a morphism $\varphi: \operatorname{Mod}(\mathcal{O})\to \operatorname{Out}_t(G)$ by $\varphi(f)(\gamma):=f^\gamma(f^)^{-1}$ for $\gamma\in G$, where $f^$ is the lift of $f$ defined above. Notice that the ambiguity of $\tilde{f}$ and therefore $f^$ means that $\varphi$ is only defined up to inner automorphisms. Moreover, two isotopic maps in $\operatorname{Hom}(\mathcal{O})$ yield the same image in $\operatorname{Out}_t(G)$, so $\varphi$ is well-defined on isotopy classes. Indeed, we can assume w.l.o.g. that $f$ is isotopic to the identity and fixes some base point $x_0\in O_0$, and repeat the arguments found in the proof of theorem $1$ in [@Maclachlan1975], with the addition of the arguments found in [@Zieschang1966 p. 152] to conclude that we also obtain the orientation-reversing automorphisms of $G$ by the orientation-reversing lift of an orientation-reversing homeomorphism $f$. In [@Macbeath1967 Theorem 3], it is proved that any automorphism of a hyperbolic orbifold group with compact codomain is realized geometrically, i.e. induced by a homeomorphism of $\mathbb{H}^2.$ The proof there can be extended to finite area orbifolds using the uniqueness and existence of an extremal quasi-conformal mapping within an isotopy class of homeomorphisms of the hyperbolic plane as given in [@Abikoff1980 p. 59, Theorem 2]. The only difference in the proof then is that instead of reducing to the case of a compact surface by passing over to a finite index subgroup, by the positive resolution of the Fenchel conjecture in [@fox52; @fconjecture; @Chau], we pass over to the fundamental group of a possibly punctured and bordered orientable surface. This means that instead of every automorphism being realized geometrically as in the compact case, we obtain the statement that only the type preserving ones are realized, as this is the case for surfaces with boundaries and punctures. This last statement, instead of using the original Dehn-Nielsen-Baer theorem for compact surfaces, employs theorem $8.8$ from [@primermcgs] instead, which on account of us allowing homeomorphisms that are not the identity on the boundary holds for surfaces with boundary as well, as long as the automorphisms considered are type preserving. We thus conclude that all type preserving automorphisms of $G$ are realised geometrically and therefore $\varphi$ is surjective. For injectivity, assuming that $f$ lifts to a homeomorphism $f^$ that induces an inner automorphism, there is a lift of $f$ that is the identity automorphism on $G.$ Theorem \[thm:idisot\] concludes the proof.[^5] While theorem \[thm:mcgout\] is an important result, it is as of yet unclear how to use this isomorphism in general for practical purposes. The same proof holds in the Euclidean case, where the surjectivity of the homomorphism is true for the same basic reasons that it is true for hyperbolic orbifolds.[^6] From the proof of theorem \[thm:mcgout\] and the fact that geometric automorphisms are type preserving, we also obtain the following. \[prop:comporbs\] For a compact orbifold $\mathcal{O}$, missing the boundary hyperbolic elements and punctures, every automorphism of $\pi_1(\mathcal{O})$ is realized geometrically, so the MCG $\operatorname{Mod}(\mathcal{O})$ is isomorphic to either the group of all outer automorphisms $\pi_1(\mathcal{O})$ or just the orientation preserving ones, depending on whether or not $\mathcal{O}$ is orientable. Lifts of Mapping Class Groups {#sec:bhtheory} ============================= In an effort to relate the MCGs of some surfaces to the MCGs of covers of the surface, Birman-Hilden theory was introduced [@Birman1973]. The idea is the following. Given a covering map $p:S \to X$ of surfaces, one may look at fiber-preserving homeomorphisms $f:S \to S$ that for all $x\in X$ map the fibers $p^{-1}(x)$ to $p^{-1}(y)$ for some $y\in X$. If this is the case, then $f$ induces a homeomorphism on $X$. Conversely, if a homeomorphism $f$ on $X$ lifts to a map and therefore a homeomorphism $\tilde{f}$ on $S$, $\tilde{f}$ must be fiber-preserving. If for any two fiber-preserving homeomorphisms on $S$ that are homotopic as maps on $S$, there is a homotopy passing only through fiber-preserving homeomorphisms, then we say that $p$ has the Birman-Hilden property. The importance of this notion is that the MCGs for surfaces are defined through homotopies and in order to relate the MCGs of both spaces, one needs to ensure that only isotopic homeomorphisms of $X$ lift to isotopic homeomorphisms of $S$. It is known [@Zieschang1973 Theorem $11.1$] that if $p$ is a finite-sheeted branched regular covering map of orbifolds, then $p$ has the Birman-Hilden property. As such, Birman-Hilden theory concerns itself with the well-definedness of a lift of an isotopy class of a map to an isotopy class in the covering space. However, this leaves open the question of the existence of a lift of a representative of an isotopy class of maps. We will also investigate the question of existence of lifts of homeomorphisms of orbifolds to their covering spaces.[^7] Let $p:\mathcal{O}_1\to \mathcal{O}$ be a covering map of orbifolds. Any hyperbolic orbifold (with the exception of non-developable orbifolds), $\mathcal{O}$, possibly with punctures and non-empty boundary, can be presented as $\mathbb{H}^2/\Gamma$ where $\Gamma=\pi_1(\mathcal{O})$ is a discrete subgroup of $\operatorname{Iso}(\mathbb{H}^2)$. We have that $\mathbb{H}^2\to \mathbb{H}^2/\Gamma$ is a regular branched cover, where the branch locus is a (possibly non-discrete) nowhere dense set in $\mathcal{O}$. Similarly, we have $\mathcal{O}_1=\mathbb{H}^2/\Gamma_1$ and we naturally have $\Gamma_1\subset\Gamma,$ with each of these groups acting as a group of deck transformations on the universal covering by $\mathbb{H}^2$. We are only interested in finite covers, which translates to $\Gamma_1$ having finite index in $\Gamma$, equal to the degree of $p$. For compact orientable surfaces, it is well-known that any finite index subgroup of the fundamental group is isomorphic to the fundamental group of a covering surface, whereas any infinite index subgroup is free. We will start the subsequent discussion with results whose proofs do not, as far as we know, appear in the literature but can be carried out with well-known methods in the field. First we need the following definition, the notion of which was touched upon in section \[sec:outs\]. \[def:geomauto\] We call an automorphism $\alpha$ of an orbifold group $\Gamma$ (with standard presentation) geometrical, if there exists a homeomorphism $f$ of $\mathbb{H}^2$ that is $\Gamma$ fiber-preserving w.r.t. the universal covering of the orbifold by $\mathbb{H}^2$ and induces $\alpha$ via $\alpha(\gamma)=f\gamma f^{-1}$, where $\gamma\in \Gamma\subset\operatorname{Iso}(\mathbb{H}^2)$ if interpreted as a group of deck transformations on $\mathbb{H}^2$. Homeomorphisms satisfying $f\Gamma f^{-1}=\Gamma$ always yield geometric automorphisms. This is equivalent to $f$ inducing a homeomorphism of the orbifold $\mathbb{H}^2/\Gamma.$ On the other hand, any homeomorphism $f$ of a surface $\mathbb{H}^2/\Gamma$ lifts to a homeomorphism $\tilde{f}$ of the universal covering space $p:\mathbb{H}^2\to\mathbb{H}^2/\Gamma$ that is $\Gamma$ fiber preserving by applying the lifting criteria in [@Hatcher prop. $1.33$] and uniqueness of lifts to the map $f\circ p$ and $f^{-1}\circ p$ (possibly with continuous extensions for the branch points). The same lifting condition also holds for coverings by orbifolds, as the singular locus can be ignored for the existence of a lift away from branch points, only to be reinstated later by Riemann’s removable singularity theorem. Recall, also, that theorem \[thm:mcgout\] implies that geometric automorphisms are exactly those that are type preserving. We now state a further corollary of theorem \[thm:mcgout\]. \[cor:geoautsubgr\] A geometric automorphism of an orbifold group $\Gamma$ that induces an automorphism on an orbifold subgroup $S\subset \Gamma$ induces a geometrical automorphism on $S$. For the following theorem we will mostly follow the proof of theorem $8.2$ in [@Zieschang1973], but produce a slightly stronger result. \[thm:submcgs\] Let $G$ be the symmetry group of a hyperbolic orbifold $\mathcal{O}$ and $G_1$ a subgroup of finite index that is not cyclic. Then a geometrical automorphism $\alpha$ of $G_1$ is induced by a $G$ fiber-preserving homeomorphism in $\operatorname{Hom}(\mathcal{O})$ iff $\alpha$ is induced by an automorphism $\hat{\alpha}$ of $G$. If $\alpha$ is induced by a $G$ fiber-preserving homeomorphism $f$, then $f$ induces a homeomorphism on the orbifold $\mathbb{H}^2/G$ as well as, by assumption, on $\mathbb{H}^2/G_1,$ and thus induces an automorphism $\hat{\alpha}$ of $G$ that stabilizes $G_1$ in $G$, which proves one direction. For the other direction, first consider the situation for the at most index $2$ subgroup $\tilde{N}\subset G$ that contains only orientation preserving elements. We further pass to a finite index normal subgroup $N\subset \tilde{N}$ of $G$, which we can take to be the fundamental group of a possibly punctured and bordered orientable surface. Now let $\alpha$ be induced by a homeomorphism $h$ of $\mathbb{H}^2$ such that $\alpha (n)=h\circ n\circ h^{-1}\; \forall n\in N,$ which w.l.o.g. can be chosen to be the uniquely determined extremal quasi-conformal mapping of $\mathbb{H}^2$ satisfying this relation. This statement remains true even for orientation reversing homeomorphisms of $\mathbb{H}^2$ by the uniqueness of the extremal maps, since they can be given by a quasi-conformal map composed with an orientation reversing isometry of $\mathbb{H}^2,$ which leaves invariant the dilatation. Now define for arbitrary $g\in G$ $$\begin{aligned} \varphi=\hat{\alpha}(g)hg^{-1}.\end{aligned}$$ For $n\in N$ we obtain $$\begin{aligned} \varphi(n)&=\hat{\alpha}(g)hg^{-1}(n)=\hat{\alpha}(g)h(g^{-1}ng)g^{-1}=\hat{\alpha}(g)h(g^{-1}ng)g^{-1}\\ &=\hat{\alpha}(g)\alpha(g^{-1}ng)hg^{-1}=\alpha(n)\hat{\alpha}(g)hg^{-1}=\alpha(n)\varphi.\end{aligned}$$ The fourth equality uses $g^{-1}ng\in N$, since $N$ is normal. Now, $\alpha(n)$ and $n$ act as isometries on $\mathbb{H}^2$, hence leave the dilatation of $\varphi$ invariant, so by the uniqueness of extremal maps we obtain $\varphi=h$ and thus $\hat{\alpha}(g)=h(g)h^{-1}.$ Since $g$ was arbitrary, $h$ preserves $G$-fibers. However, by $\hat{\alpha}(g)=h\circ g \circ h^{-1}$, this also shows that $h$ preserves the fibers of any intermediate subgroup between $N$ and $G$ and, in particular, $G_1$, which proves the theorem. The next is a slight generalization of the result of [@Maclachlan1975 Lemma 11]. \[lem:uniqueext\] If some automorphism $\alpha$ of the hyperbolic orbifold group $G$ induces an automorphism $\alpha\|_N$ of a noncyclic subgroup $N\subset G$, then there is only one extension of $\alpha\|_N$ to $G$, i.e. if $\alpha\|_{N}=\operatorname{id}_{N}$ then $\alpha=\operatorname{id}_{G}.$ Recall that a characteristic subgroup $C$ of $G$ is a subgroup that is invariant under all automorphisms of $G$. This means that $C\subset \varphi(C)$ $\forall \varphi\in\operatorname{Aut}(G)$ and thus also $C\subset\varphi^{-1}(C),$ i.e. $\varphi(C)\subset C$ so that any $\varphi\in\operatorname{Aut}(G)$ induces an element of $\operatorname{Aut}(C).$ It is well-known that every finitely presented group contains a finite index subgroup that is characteristic. Let $g\in G$. By passing to a finite index subgroup, we first assume that $N$ is normal in $G.$ Then for any $n\in N$ we have $gng^{-1}=\alpha(gng^{-1})=\alpha(g)n\alpha(g)^{-1}, $ i.e. $g^{-1`}\alpha(g)$ commutes with every element of $N$. The condition that $N$ is not a cyclic subgroup of $G$ now implies that $g=\alpha(g),$ because a nontrivial element in $G$ commutes only with elements of a cyclic subgroup it is a part of [@Friedrich2008]. Now let $S\subset G$ be an arbitrary subgroup, which contains a characteristic subgroup $C$ of $G$. Any automorphism of $G$ induces one on $C$ and, since $C$ is normal in $S$ and $G$, the extension to both of them is unique. So, $\alpha\|_C$ extends uniquely to $\alpha\|_S$ and to $\alpha$. If there was some other extension of $\alpha\|_S$ to an automorphism of $G$, then this would contradict the uniqueness of the extension of $\alpha\|_C$ to $\alpha$. Note that lemma \[lem:uniqueext\] remains valid within the class of geometrical automorphisms. We now give a short proof of the Birman-Holden property for general orbifold groups, which is somewhat different than that in [@Zieschang1973]. \[prop:BHgen\] Let $S\subset G$ be a finite index subgroup of the hyperbolic orbifold group $G$ that is not cyclic. If a $G$-fiber preserving homeomorphism $\varphi$ of $\mathbb{H}^2$ is $S$-fiber isotopic to the identity, then $\varphi$ is $G$-fiber isotopic to the identity. By assumption, $\varphi$ induces an automorphism $\alpha$ of $G$, which induces $\operatorname{id}_{S}$ on the subgroup $S$, so by lemma \[lem:uniqueext\] $\alpha=\operatorname{id}_{G},$ which by theorem \[thm:mcgout\] implies that $\varphi$ is $G$ fiber isotopic to the identity. An important technical consequence of this Birman-Hilden property in terms of isotopic tiling theory is that two sufficiently complicated tilings that arise from decorations w.r.t. a non-conjugate pair of sets of generators are never isotopic in $S,$ even if they are topologically not distinct, i.e. they are related by a homeomorphism of $S$. Indeed, two sufficiently complicated decorations from a non-conjugate pair of sets of generators for $\Gamma$ are isotopically distinct in $\mathbb{H}^2/\Gamma$ by the arguments outlined in section \[sec:outs\], so by theorem $2$ the map exchanging these decorations must be nontrivial. Now, by proposition \[prop:BHgen\] above, as tilings of a Riemann surface that covers $\mathbb{H}^2/\Gamma$, the two tilings cannot be related by an isometry. As another result of the Birman-Hilden property, given a covering of orbifolds $\mathcal{O}_1\to\mathcal{O}_2$, with groups $G_1$ and $G_2$ respectively, a homeomorphism class in $\mathcal{O}_2$ lifts to one in $\mathcal{O}_1$, iff there is a lift of a representing homeomorphism. By theorem \[thm:submcgs\], this is equivalent to the corresponding automorphism of $G_2$ restricting to an automorphism of $G_1.$ In other words, a geometric automorphism of $G_1$, induced by a map $f$ such that $fG_1f^{-1}=G_1$, is extendable to $G_2$, iff $f$ satisfies $fG_2f^{-1}=G_2.$ Consider the subgroup $\mathcal{L}\subset \operatorname{Hom}(\mathcal{O}_2)$ of homeomorphisms of $\mathcal{O}_2$ that lift to homeomorphisms on $\mathcal{O}_1$ and set $A:=\operatorname{Hom}(\mathcal{O}_2)/\mathcal{L}.$ Two elements $f\mathcal{L},g\mathcal{L}\in A$ are equal iff $fg^{-1}\in\mathcal{L}.$ This implies that the induced automorphisms $A_f,$ $A_g$ of $G_2$ satisfy $A_f(G_1)=A_g(G_1).$ Said in another way, there are as many equivalence classes in $A$ as there are isomorphic versions $G_1$ in $G_2$ that get exchanged by automorphisms of $G_2$. In this situation, $A_f(G_1)$ has the same index in $G_2$ for all $f$. Since $G_2$ is finitely generated, there are only a finite number of subgroups in $G_2$ of a given index, so we obtain the following. \[prop:finitelifts\] Given the situation of the last paragraph, there are at most finitely many homeomorphism classes with representatives that do not lift to a finite covering space. The contents of this section open up possible investigations into more refined questions relating to isotopic tiling theory on a Riemannian surface $S$. For example, lemma \[lem:uniqueext\] tells us that elements of the MCG that are supported in a particular subsurface give rise to automorphisms that leave invariant a subset of the generators. It is well-known that the MCG of any surface has generators that are supported in subsurfaces. Furthermore, one is often in a situation where one is only interested in a subclass of all isotopy classes of tilings of a given surface, in which case theorem \[thm:submcgs\] facilitates the investigation. Proposition \[prop:finitelifts\] tells us that from a perspective of tilings as graph embeddings, there is only a finite number of topologically distinct graph embeddings into $S$ that can be produced by changing the isotopy class of the embedding of the graph into the orbifold. The results furthermore add to the duality of the description of the MCG as a group of geometric transformations and as a group of algebraic transformations. In particular, the following important related questions can be examined from an algebraic or a geometric point of view. - Which isotopically distinct tilings with the same symmetry group $G$ are related by a homeomorphism of $S$? - How does an element of the MCG of an orbifold relate to the MCG of a covering orbifold? Note that in most cases these questions do not have a generic answer and depend on the set up, i.e. the conformal structure on $S$ and the tiling. Finite order elements of the MCG ================================ Theorem \[thm:mcgout\] implies that the MCG does not depend on the orders of the torsion elements of the orbifold group and as a result, abstract results on MCGs are sometimes important for applications. In our definition of the MCG, where homeomorphisms are allowed to change the boundary, surfaces with boundary do not necessarily have torsion free MCGs, in contrast to the classical situation. An important technical aspect of the EPINET enumerative project is that many tilings of the hyperbolic surface $S$ in question are related by isometries of the surface that lift to symmetries of $\mathbb{R}^3$. When producing nets in $\mathbb{R}^3$, one often only wants one representative of these. A realization of $\mathcal{O}$ as a quotient space with symmetry group $\pi_1(\mathcal{O})$ induces a metric and conformal structure on $\mathcal{O}$. A finite order element $f$ of $\operatorname{Mod}(\mathcal{O})$ lifts to a transformation on $\mathbb{H}^2$ that induces an element of $\operatorname{Hom}(\mathcal{O})$ for every power of $f$. Now, suppose $f$ acts as an isometry of $\mathbb{H}^2$ after lifting. Now, $f$ clearly satisfies $fgf^{-1}\in \pi_1(\mathcal{O}),$ so, since $f$ is an isometry, $f\in \mathcal{N}(\pi_1(\mathcal{O})),$ the normalizer in $\operatorname{Iso}(\mathbb{H}^2)$. Now, suppose that $h\in \operatorname{Iso}(\mathbb{H}^2)$ acts trivially by conjugation on $G=\pi_1(\mathcal{O})$. Then, it would have to fix all of the fixed points on the unit circle at infinity of the hyperbolic translations in $G$. Every hyperbolic orbifold sits inside a surface with two independent translations and therefore, $h$ fixes $4$ points on the unit circle and must be the identity, so we see that $\mathcal{N}(G)$ injects into $\operatorname{Aut}(G),$ where $G$ itself acts as inner automorphisms of $G$. Now, $fGf^{-1}\subset G$ implies that $f$ preserves $G$ orbits as $x \sim y$ implies $fy=fgx=\tilde{g}fx$ for some elements $g,\tilde{g}\in G$ and therefore $f$ induces a conformal automorphism of the orbifold $\mathbb{H}^2/G$. Conformality of the map can be checked locally with the help of Riemann’s removable singularity theorem. Clearly, $G$ itself acts trivially on the space of its orbits and lifting arguments imply that we actually get an isomorphism of groups $\operatorname{CAut}(\mathbb{H}^2/G)\equiv\mathcal{N}(G)/G$, where $\operatorname{CAut}(\mathbb{H}^2/G)$ is the group conformal automorphisms. In particular, we see that the normalizer is discrete and that $f$ can be interpreted as an element of a hyperbolic supergroup of $G.$ Note, however, that different conformal structures of $\mathbb{H}^2/G$ can give rise to different towers of supergroups of $G$, which is particularly important because in some cases, the MCG is generated by torsion elements. Now, the condition that $f$ acts as an isometry of $\mathbb{H}^2$ depends solely on the conformal structure induced by the realization of $\mathcal{O}$. The proof in [@primermcgs Theorem $7.1$] of the Nielsen Realization Problem for cyclic subgroups of the MCG of classical surfaces works in the same way, almost word for word for our more general surfaces. The essential ingredient is the contractibility of Teichm" uller space, which is also true in our setting. The idea is to find a point in Teichm" uller space that is fixed by the induced action of the periodic MCG element to show the existence of a metric on $\mathcal{O}$ such that $f$ acts as an isometry, in which case its lift to $\mathbb{H}^2$ does too. This observation partially settles the Nielsen realization problem for orbifold MCGs, see [@primermcgs] for further details in the classical setting. The above discussion is of considerable importance for $3$D net enumeration, since the triply periodic minimal surfaces that account for physically interesting tilings have finite groups of symmetries that are induced by ambient isometries and therefore yield the same $3$D structure. In such applications, the finite order elements of the MCG are the candidates for transformations one might be interested in disregarding. For example, the finite orders of elements of the MCG $M_n$ of an $n$-times punctured sphere were studied in [@Gillette1967], [@Murasugi1982]. The result is that $m$ is the order of an element in the MCG if and only if $m$ divides $n$, $n-1$, or $n-2$, and there is even an explicit form of representatives of conjugacy classes of all finite order elements using the standard representation for these groups from braid groups. Using these, one can enumerate the cosets giving rise to symmetry classes of isotopy classes of tilings with rotational symmetry on a given Riemmanian surface $S$. For example, the group $22222$ has the MCG of a $5$ punctured sphere and the finite order element with order $3$ corresponds to the $3$-fold rotation in $\star 246$ around a $6$-fold gyration point. Summary and Implications for Applications {#sec:results} ========================================= We have developed a classification of all isotopically distinct equivariant tilings of a hyperbolic surface $S$ of finite genus, possibly nonorientable, with boundary, and punctured. First, we find the smallest (in terms of area) possible symmetry group of $S$, which corresponds to a symmetry group $G_0$ of the hyperbolic fundamental polygon belonging to $S$. This smallest symmetry group $G_0$ exists as a consequence of generalizations of the classical Hurwitz theorem [@oikawa]. There are finitely many possible symmetry groups $G$ for tilings such that $G_0\subset G\subset \pi_1(S)$ because orbifold groups are finitely generated. Given such a $G$, we choose a set of geometrical generators of cardinality $\operatorname{rank}(G).$ From these generators, we obtain a set of fundamental tilings with symmetry group $G$ as a decoration of the associated orbifold $\mathcal{O}.$ The decoration is specified up to isotopy by a combinatorial description from the Delaney-Dress symbol of the tiling. The mapping class group $\operatorname{Mod}(\mathcal{O})$ of $\mathcal{O}$ naturally acts on the set of sets of generators of cardinality $\operatorname{rank}(G).$ Thus, starting from the classical Delaney-Dress symbol for the fundamental tiling with the starting set of generators, one obtains all other isotopically distinct fundamental equivariant tilings with symmetry group $G$ by repeated applications of $\operatorname{Mod}(\mathcal{O})$. For each of the resulting fundamental tilings, we independently apply the GLUE and SPLIT operations exactly in the same way as in the classical setting to eventually produce all equivariant tilings with symmetry group $G$. One caveat here is that in some examples, for example when a fundamental tile for a group generated entirely by rotations of the same order is obtained by doubling a fundamental tile for the index $2$ supergroup that is generated by reflections, it is possible to find two different sets of generators that are nonconjugate but act on the boundary of the same fundamental domain. In such situations, it is possible that the tiling associated to a decoration of the orbifold is unchanged by an element of the MCG. In case of a fundamental tiling, one only needs to retain one of the two versions of the tiling to produce all tilings associated to the symmetry group. Such a situation can only occur if the decoration corresponding to the fundamental tiling fails to detect the changes the generators undergo and the new set of generators is a permutation of the old set. Thus, by identifying exactly how they are permuted, one can eliminate this ambiguity by getting rid of the MCG element that realizes this permutation. While $G_0$ is the smallest symmetry group commensurate with $S$, this group depends entirely on the hyperbolic finite area metric on $S$. Without reference to any specific hyperbolic structure, there are many possible chains of subgroups that yield potential symmetry groups of $S$. For example, the group $\star 2226$ appears as the smallest fundamental domain of the $H$ surface in [@RobinsHsurface]. However, this group does not appear at all as a symmetry group of the $P$ surface in [@Robins2004]. Both surfaces are of genus $3$. Also, $\star 246$ has no hyperbolic supergroups, even though $\star 237$ is smaller. An essential ingredient in any assignment of MCG elements to the isotopy classes of tilings comes from the fact that our MCGs have solvable word problem, which one can prove for our more general surfaces in almost the same way as has been done in [@primermcgs] for closed surfaces, using the Alexander method. This allows an unambiguous and complete enumeration of all isotopy classes of tilings on hyperbolic Riemann surfaces by an enumeraton of MCG elements, a project which we pursue in [@BenMyf2]. We thank Vanessa Robins, Stephen Hyde, and Stuart Ramsden of the Australian National University for detailed discussion and guidance. This research was funded by the Emmy Noether Programme of the Deutsche Forschungsgemeinschaft. B.K was supported by the Deutscher Akademischer Austauschdienst for a research stay at the Australian National University. [10]{} \[1\][[\#1]{}]{} urlstyle \[1\][DOI \#1]{} Epinet. <http://epinet.anu.edu.au/>. 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[^2]: While the ideas also work for higher dimensional orbifolds, decorating fundamental domains with codimension $1$ drawings quickly becomes unwieldly and its description lies outside the scope of this paper for general orbifolds, so the pictures we describe works best in the $2$D setting. [^3]: See [@DRESS1987 Section $4$d)] in particular [^4]: Most of this reasoning also works for higher dimensional hyperbolic orbifolds. As a result of Mostow’s rigidity theorem for developable hyperbolic orbifolds (see [@Prasad1973], [@Mostow1968], [@Boileau2004]) there is only one set of generators for the corresponding group and any way to produce all combinatorially distinct fundamental tilings works, without the need to reference specific generators. [^5]: A similar proof in a very different context is outlined in [@fujiwara]. [^6]: The only difference is that the theorems on quasi-conformal maps turn into statements about affine maps. These were known much earlier than their respective hyperbolic counterparts. [^7]: The following discussion is cast for hyperbolic orbifolds. For a description of the Euclidean case, refer to [@Zieschang1973 $\S 9$].
--- author: - 'rédigé par Weizhe <span style="font-variant:small-caps;">Zheng</span>' date: \| Mémoire de Master 2année\ réalisé sous la direction du Professeur Luc Illusie title: 'Théorème de Gabber d’indépendance de $l$' --- On expose ici le de Gabber d’indépendance de $l$ pour la cohomologie d’intersection d’un schéma propre équidimensionnel sur le spectre d’un corps fini (\[th.IH\]). On suit [@ind] à très peu près. Je remercie vivement mon directeur, Luc Illusie, qui m’a beaucoup aidé. Suites récurrentes linéaires ============================ On aura besoin de certaines propriétés élémentaires des suites. Soient $P\subset\Z$ une partie non vide stable par addition par $\N$ et $E$ un corps. On désigne par $E^P$ l’ensemble des suites $(a_n)_\ndP$ à valeurs dans $E$ (des applications de $P$ vers $E$). On pose $$\begin{array}{rrcl} T:& E^P &\to& E^P\\ &(a_n)_\ndP \mt (a_{n+1})_\ndP. \end{array}$$ \[pr.Rl\] Soit $a=(a_n)_{n\in P} \in E^P$. Les conditions suivantes sont équivalentes : Il existe un polynôme $Q(X) \in E[X]$ à constant non-nul tel que $Q(T)a=0$. il existe $r\ge 0$, $c_0,\dots,c_r\in E$, $c_0c_r \neq 0$ tels que $$\sum_{i=0}^r c_i a_{n+i}=0 \text{ pour tout } n\in P.$$ le sous-$E$- $E[T]a\subset E^P$ est et $T$ en est un automorphisme. \(a) $\Leftrightarrow$ (b) Si on pose $Q(X)=\sum_{i=0}^r c_i X^i$, alors $(Q(T)a)_n=\sum_{i=0}^r c_i a_{n+i}$. \(a) et (b) $\Rightarrow$ (c) $E[T]a$ est un sous-$E$-, (a). L’action de $T$ en est injective, (b). \(c) $\Rightarrow$ (a) On peut prendre comme $Q(X)$ le polynôme mininal de $T$ sur $E[T]a$. Une suite $a\in E^P$ est dite récurrente linéaire si elle satisfait aux conditions équivalentes de \[pr.Rl\]. L’ensemble des suites récurrentes linéaires se note $\Rl(P,E)$. Pour tout $\alpha\in E^$, $(\alpha^n)_\ndP\in \Rl(P,E)$. \[cor.Rl\] $\Rl(P,E)$ est un sous-$E$-de $E^P$. L’de restriction $\Rl(\Z,E)\to \Rl(P,E)$ est bijective. Soit $E'$ une extension de $E$. Si $a\in \Rl(P,E')$ prend toutes ses valeurs dans $E$, alors $a\in \Rl(P,E)$. \(i) ((ii)) est clair d’après la condition (a) ((b)). Pour (iii), on utilise la condition (c). On a $E[T]a \otimes_E E' \simeq E'[T]a$, donc $E[T]a$ est sur $E$. On a une injection $E[T]a \har E'[T]a$, ce qui donne l’injectivité de l’action de $T$ sur $E[T]a$. Notations et rappels ==================== Pour tout nombre premier, on choisit une clôture algébrique $\Qlb$. Soit $F$ un corps de $\chi_F \neq l$. Pour un schéma $X$ séparé sur $\Spec F$, on définit les $\Qlb$-faisceaux constructibles sur $X$ et la catégorie dérivée $\Dbc(X,\Qlb)$ ([@WeilII 1.1] et [@Ekedahl]). Les dérivées $\Dbc(-,\Qlb)$ sont stables par les six opérations $Rf_!, Rf_, f^, Rf^!, \Lt, \RcHom$, et par le foncteur dualisant $D$. On normalise le dernier par $$D_X K = \RcHom(K,Ra^!\Qlb)$$ où $a: X\to\Spec F$. On désigne par $K(X,\Qlb)$ le groupe de Grothendieck des $\Qlb$-. Pour un $\Qlb$-faisceau $\cF$ sur $X$, on note $[\cF]$ sa classe dans $K(X,\Qlb)$. L’application $\cF\mapsto [\cF]$ se prolonge en une surjective ((X,)) && K(X,)\ K=\_[q]{} (-1)\^q \[\^q(K)\]. On désigne parfois l’image de $K$ encore par $K$. Les six opérations sur $\Dbc(-,\Qlb)$ induisent les homomorphismes entre les groupes $K(-,\Qlb)$. Perversité ---------- On prend la perversité autoduale $p$ et la $(\pDle0,\pDge0)$ sur $\Dbc(X,\Qlb)$ comme dans [@BBD 4.0]. On rappelle que pour $K\in \Dbc(X,\Qlb)$, $$K\in \pDle0 \iff \fH^q(i_x^K)=0, \forall x\in X, \forall q>-\delta(x),$$ $$K\in \pDge0 \iff \fH^q(i_x^!K)=0, \forall x\in X, \forall q<-\delta(x),$$ où $i_x$ est l’inclusion de $\{x\}$ dans $X$, $i_x^!=j^Ri^!$ pour $\{x\} \sto[j] \adhx \stackrel{i}{\hookrightarrow} X$, $\delta(x)=\dim\adhx$ . Le foncteur dualisant $D_X$ échange $^p D^{\le 0}$ et $\pDge0$. Si on note $d=\dim X$, alors $\Dle{-d} \subset \pDle0 \subset \Dle0$, donc $\Dge0 \subset\pDge0 \subset \Dge{-d}$ par l’orthogonalité . On note $\Per(X,\Qlb)$ la des pervers sur $X$. Tout objet de cette est de longueur finie , donc \[eq.K=K(P)\] K(X,)=K((X,)) est le groupe abélien libre engendré par les classes d’isomorphie des pervers simples. Soit $j: U \har X$ une immersion ouverte. Alors l’extension intermédiaire d’un faisceau pervers $K\in \Per(U,\Qlb)$ par $j$ se calcule par la formule $j_{!}K = {}^p \tau^Y_{\le-1} Rj_K$ , où $Y\stackrel{i}{\hookrightarrow} X$ est le fermée . Si $f: K_1 \to K_2$ est un morphisme dans $\Per(U,\Qlb)$, alors $j_{!}f = \ptle[Y]{-1} Rj_* f$.[^1] Notons que $$\ensdr{K\in\Dbc(X,\Qlb)}{i^K\in \Dle{-d-1}_Y} \subset \ensdr{K\in\Dbc(X,\Qlb) }{ i^K\in \pDle{-1}_Y } ,$$ où $d=\dim Y$. On a donc un morphism de foncteurs $$\tYle{-d-1} \to \ptYle{-1} : \Dbc(X,\Qlb) \to \Dbc(X,\Qlb).$$ \[pr.j!\\] Si le morphisme $\tYle{-d-1} Rj_ K_\beta \to \ptYle{-1} Rj_* K_\beta$ est un isomorphisme pour chaque $\beta=1,2$, alors on peut faire des identifications $j_{!} K_\beta = \tYle{-d-1} Rj_ K_\beta$, $\beta=1,2$ et $j_{!} f = \tYle{-d-1} Rj_ f$. Si $Y$ est lisse purement de dimension $d$ et les $\fH^e i^Rj_K_\beta$, $e \ge -d$, sont lisses pour $\beta=1,2$, alors l’hypothèse de est vraie. \(i) On a un diagramme commutatif $$\xymatrix{\tYle{-d-1} Rj_* K_1 \ar[d]_{\tYle{-d-1} Rj_* f}\ar[r] & \ptYle{-1} Rj_* K_1\ar[d]^{\ptYle{-1} Rj_* f}\\ \tYle{-d-1} Rj_* K_2 \ar[r] & \ptYle{-1} Rj_* K_2}$$ (ii) Dans le cas général, on peut calculer $j_{!}$ en itérant \[pr.j!\\]. \[cor.j!\\] Supposons $F$ parfait. Soient $A$ un ensemble fini, $(l_\alpha)_{\alpha\in A}$ une famille de nombres premiers $\neq \chi_F$, $(K_\al)_\aldA \in \prod_{\aldA} \Per(U,\Qlalb)$. Alors il existe $n\ge 0$ et des immersions ouvertes $$U=U_0 \har[j_1] U_1 \har[j_2] \cdots \har[j_n] U_n=X$$ vérifiant les conditions suivantes Pour $1\le m \le n$, $Y_m = U_m \bsl U_{m-1}$ (équipé de la structure de sous-schéma réduit induite) est lisse sur $F$ et purement de dimension $d_m$, $d_1 > d_2 > \cdots > d_n \ge 0$, $K_\al\m = \tle[Y_m]{-d_m-1} Rj_{m} K_\al\mm$, et $\fH^e(i_m^! K_\al\m)$ est lisse sur $Y_m$, $\forall e\in \Z, \aldA$, où $i_m: Y_m \har U_m$, $K_\al\m = j_{m!} K_\al\mm$, $K_\al^{(0)} = K_\al$. Donc $$j_{!}K_\al = K_\al^{(n)} = \tle[Y_n]{-d_n -1}Rj_{n} \cdots \tYle[1]{-d_1 -1} Rj_{1} K_\al, \quad \forall \aldA.$$ Soient $\al_1, \al_2 \in A$ tels que $l_{\al_1} = l_{\al_2}$ et $f:K_{\al_1} \to K_{\al_2}$ un morphisme. Alors $$j_{!}f = \tle[Y_n]{-d_n -1}Rj_{n} \cdots \tYle[1]{-d_1 -1} Rj_{1} f.$$ Si, en outre, $U$ est lisse sur $F$ purement de dimension $d>\dim(X-U)$ et $\fH^e(K_{\al_\beta})$ est lisse sur $U$, $\forall e\in \Z$, $\beta=1,2$, alors $\forall \beta=1,2$, $K_\alb\m$ est concentré en degrés $[-d,-d_m-1]$ pour $1\le m \le n$, et j\_[!\]{}K\_&=& Rj\_[n\]{} Rj\_[1\]{} K\_,\ j\_[!\]{}f &=& Rj\_[n\]{} Rj\_[1\]{} f. La condition (iii) ne sera pas utilisée dans la suite. ([@Tian 1.5 ?]) On montre (i) et (ii) en même temps. (iii) en suit. $U\subsetneqq X$. Par sur $\dim(X-U)$, il suffit de montrer qu’il existe des immersions ouvertes $$U\har[j_1] U_1 \har[j'] X$$ telles que $Y_1 = U_1\bsl U$ (réduit) soit lisse sur $F$ et purement de dimension $d_1 > \dim (X-U_1)$, $j_{1!} K_\al = \tYle[1]{-d_1-1} Rj_{1} K_\al$, $j_{1!} f = \tYle[1]{-d_1-1} Rj_{1} f$, et que $\fH^e(i_1^! j_{1!} K_\al)$ soit lisse sur $Y_1$, $\forall e\in \Z$, $\aldA$, où $i_1: Y_1 \har U_1$. On pose $Y=X\backslash U$ et lui donne la structure de schéma réduit induite. On prend un ouvert $W$ de $Y$ tel qu’il soit lisse sur $F$, $\dim(Y-W) <\dim Y$ et que $\fH^e(j_W^* i^* Rj_K_\al)$, $\fH^e(j_W^ Ri^! j_{!}K_\al)$ soient lisses sur $W$ pour tout $e\in \Z$ et tout $\aldA$, où $i: Y\har X$, $j_W: W\har Y$. On pose $U_1= U\cup W$. Alors $Y_1 = W$ est lisse sur $F$ et purement de dimension $d_1=\dim Y >\dim (X-U_1)$, car $X-U_1 = Y-W$. On a le d’immersions $$\xymatrix{U \ar@(ur,ul)[rr]^j \ar[r]^{j_1} & U_1 \ar[r]^{j'} &X\\ & Y_1 \ar[u]^{i_1} \ar[r]^{j_W} & Y\ar[u]^{i}}$$ Notons que $$j_W^i^Rj_ = i_1^* j'^* Rj'_{} Rj_{1}= i_1^* Rj_{1},$$ $$j_W^Ri^!j_{!} = Ri_1^! j'^ j'_{!} j_{1!} = Ri_1^! j_{1!}.$$ Donc $\fH^e(i_1^Rj_{1}K_\al)$, $\fH^e(i_1^! j_{1!} K_\al)$ sont lisses sur $Y_1$, $\forall e\in \Z$, $\aldA$. D’après \[pr.j!\\], $j_{1!} K_\al = \tYle[1]{-d_1-1} Rj_{1} K_\al$, $\aldA$ et $j_{1!} f = \tYle[1]{-d_1-1} Rj_{1} f$. On fixe un corps fini $k=\F_q$, $q=p^\nu$, et une clôture $\kb$. Désormais, sauf mention expresse du contraire, on travaille sur $k$ et on ne considère que des schémas séparés sur $\Spec k$. Pour un tel schéma $X$, on désigne par $\| X \|$ l’ensemble des points fermés de $X$. Soit $l$ un nombre premier $l\nmid q$. Pour un $\Qlb$-$\cF$ sur $X$ et $x\in \|X\|$ (avec un point $\xb$ de $X$ localisé en $x$), le Frobenius $\Fr_x\in\Gal(k(\xb)/k(x))$ agit sur la fibre $\cF_{\xb}$ par transport de structure. On pose $$L_x(\cF,t) = \det(1-t^{\deg x}\Fr_x, \cF_{\xb})^{-1} \in \Qlb(t).$$ On peut le voir comme un de $1+t\Qlb [[t]]$, et on a alors \[eq.Lx\] L\_x(,t) = (\_[n1]{} (\_x\^n,\_) t\^[nx]{}/n). La de $L_x(\cF,t)$ et la formule (\[eq.Lx\]) s’étendent à $K(X,\Qlb)$ par additivité, et l’homomorphisme K(X,) && \_[x\|X\|]{} (1+t)\ K (L\_x(K,t))\_[x\|X\|]{} est injectif [@Laumon 1.1.2]. Pureté ------ On renvoie à [@WeilII 6.2] pour la notion de complexe pur (mixte). Soit $w$ un entier. On rappelle que les complexes mixtes de poids $\le w$ sont stables par $f^$ et $Rf_!$, et que les complexes mixtes de poids $\ge w$ sont stables par $Rf^!$ et $Rf_$. On note $\Dbm$ la des complexes mixtes. On note $\Perm$ ($\Per_w$) la des pervers mixtes (purs de poids $w$), et $\Km$ ($K_w$) son groupe de Grothendieck. Alors $\Km$ est aussi le groupe de Grothendieck de $\Dbm$. D’après [@BBD 5.1.7(ii) (5.3.1)], $\Perm$ ($\Per_w$) est une sous-épaisse de $\Per$, donc $\Km$ ($K_w$) est le sous-groupe abélien libre de $K$ (\[eq.K=K(P)\]) engendré par les classes d’isomorphie des pervers simples mixtes (purs de poids $w$). Par conséquent $\Km= \oplus_{w\in\Z}K_w$. Plus généralement, pour tout intervalle (éventuellement non borné) $I\subset \Z$, on peut définir $\Per_I$ et $K_I$. On a $K_I= \oplus_{w\in I}K_w$. \[th.pHi\] Soit $K\in \Dbm(X,\Qlb)$. Pour que $K$ soit de poids $\le w$ ($\ge w$), il faut et il suffit que chaque $\pH^iK$ soit de poids $\le w+i$ ($\ge w+i$). \[th.Gabber\] Soit $f$ un morphisme quasi-fini de schémas séparés sur $\Spec k$. Alors $f_{!}$ préserve $\Per_I(-,\Qlb)$. \[cor.pds\] On garde les notations et les hypothèses de (pour $F=k$). Soit $\aldA$. Si $K_\al \in \Per_{\le w}(U,\Qlalb)$, alors pour $1\le m \le n$, $\fH^e(Rj_{m}K_\al\mm)$ est mixte de poids ponctuels $\le w-d_m-1$, $\forall e\le -d_m-1$. Si $K_\al \in \Per_{\ge w}(U,\Qlalb)$, alors pour $1\le m \le n$, $\fH^e(i_m^ Rj_{m}K_\al\mm)$ est mixte de poids ponctuels $\ge w-d_m+1$, $\forall e\ge -d_m$. $K_\al\m = \tYle[m]{-d_m-1} Rj_{m}K_\al\mm$. \(i) D’après \[th.Gabber\], $K_\al\m \in \Per_{\le w}(U_m, \Qlalb)$. Pour $e \le -d_m-1$, $$\fH^e(Rj_{m}K_\al\mm) = \fH^e(K_\al\m)$$ est mixte de poids $\le w+e \le w-d_m-1$. \(ii) D’après \[th.Gabber\], $K_\al\m \in \Per_{\ge w}(U_m, \Qlalb)$. On a un triangle $$Ri_m^! K_\al\m \to i_m^ K_\al\m \to i_m^Rj_{m} K_\al\mm \to,$$ déduit du triangle $Ri_m^!M\to i_m^M \to i_m^Rj_{m}j_m^M\to$ pour $M\in \Dbc(U_m,\Qlalb)$. Pour $e\ge -d_m$, $\fH^e(i_m^Rj_{m}K_\al\mm) =\fH^{e+1}(Ri_m^! K_\al\m)$ est donc mixte de poids ponctuels $\ge w+e+1\ge w-d_m+1$, car $Ri_m^! K_\al\m$ est mixte de poids $\ge w$ et à de cohomologie lisses. \[lm.t\] Soient $D$ une triangulée munie d’une $(\PDle0, \PDge0)$, $K\sto[f] L \sto[g] M \sto[h]$ un triangle dans $D$, $d\in \Z$ le cobord $\PH^d(h) : \PH^d(M) \to \PH^{d+1}(K)$ soit nul. Alors on a un diagramme des 9 \[9dia.1\] dont les colonnes sont des triangles distingués canoniques. On complète le $$\xymatrix{\Ptled K \ar[d]\ar[r]^{\Ptled f} &\Ptled L\ar[d]\\ K\ar[r]^f &L}$$ en un diagramme des 9 \[9dia.2\] dont les deux premières colonnes sont des trianlges distingués canoniques. La première (troisième) ligne implique que $M_1\in \PDle d$ ($M_2\in \PDge d$). Le diagramme des suites exactes longues donne alors un $$\xymatrix{&\PH^d M\ar[d]\ar[r]^{0}& \PH^{d+1}K\ar[d]\\ 0\ar[r]& \PH^d M_2 \ar[d]\ar[r] &\PH^{d+1}K \\ &0}$$ qui implique $\PH^d M_2 =0$, donc $M_2\in {}^PD^{>d}$. En utilisant [@BBD 1.1.9], on voit alors que (\[9dia.2\]) s’identifie à un diagramme de la forme (\[9dia.1\]). Soit $f:Z\to X$ un morphisme quasi-fini. Alors le foncteur $$f_{!} : \Per_{[w,w+1]}(Z,\Qlb) \to \Per_{[w,w+1]}(X,\Qlb)$$ est exact. Le cas $f$ fini étant trivial, on peut supposer que $f$ soit une immersion ouverte $j:U\har X$. On se donne une suite exacte courte $0\to K_1 \sto[g_1] K_2 \sto[g_2] K_3 \to 0$ dans $\Per_{[w,w+1]}(U,\Qlb)$, et on cherche à montrer que la suite déduite $0\to j_{!}K_1 \to j_{!}K_2 \to j_{!}K_3 \to 0$ est exacte. On prend des $U_m$, $0\le m \le n$, comme dans \[cor.j!\\] et on montre l’exactitude de \[eq.se\] 0K\_1K\_2K\_30 par récurrence sur $m$. Le cas $m=0$ est tautologique. On suppose l’exactitude de (\[eq.se\]) établie pour $m-1$, $1\le m\le n$. Alors on a un triangle $$Rj_{m} K_1\mm \sto[Rj_{m} g_1\mm] Rj_{m} K_2\mm \sto[Rj_{m} g_2\mm] Rj_{m} K_3\mm \sto[h].$$ \[cor.j!\\], K\_&=& Rj\_[m\]{} K\_, =1,2,3,\ g\_&=& Rj\_[m\]{} g\_, =1,2. Par , $\tYle[m]{-d_m-1}= \Ptle{-d_m-1}$, où $P$ est la sur $U_m$ obtenue par recollement des $t$-structures $(\Dbc(U_m,\Qlb),0)$ et $$(\Dle0(Y_m,\Qlb),\Dge0(Y_m,\Qlb)).$$ Notons \^[-d\_m-1]{}(Rj\_[m\]{}K\_3) &=& i\_[m\]{}\^[-d\_m-1]{}(i\_m\^\Rj\_[m\]{}K\_3),\ \^[-d\_m]{}(Rj\_[m\]{}K\_1) &=& i\_[m\]{}\^[-d\_m]{}(i\_m\^\Rj\_[m\]{}K\_1). \[cor.pds\], $\fH^{-d_m-1}(i_m^ Rj_{m} K_3\mm)$ est mixte de poids ponctuels $\le w-d_m$, $\fH^{-d_m}(i_m^ Rj_{m} K_1\mm)$ est mixte de poids ponctuels $\ge w-d_m+1$. Donc $\PH^{-d_m-1}(h)=0$. \[lm.t\], on a alors un triangle $$K_1\m \sto[g_1\m] K_2\m \sto[g_2\m] K_3\m \to,$$ qui donne l’exactitude de (\[eq.se\]). \[rm.\[f!\\]\] L’opération $f_{!}$ sur $\Per_w(-,\Qlb)$ induit donc un homomorphisme des groupes de Grothendieck $K_w(-,\Qlb)$, et on définit un homomorphisme $f_{!}$ sur $\Km(X,\Qlb)$ en prenant la somme directe. On a alors un $$\xymatrix{\Ob(\Per_{[w,w+1]}(Z,\Qlb))\ar[d]\ar[r]^{f_{!}} & \Ob(\Per_{[w,w+1]}(X,\Qlb))\ar[d]\\ \Km(Z,\Qlb)\ar[r]^{f_{!}} & \Km(X,\Qlb)}$$ Quand $f$ est fini, cette coïncide avec le $f_$ virtuel restreint à $\Km$. \[pr.dec1\] Soient $K\in \Per_w(X,\Qlb)$, $i: Y\hookrightarrow X$ une immersion fermée et $j: U \har X$ l’ouvert . Alors $K$ admet une unique décomposition $K=j_{!} K'\oplus i_* K''$, où $K'\in \Per_w(U,\Qlb)$, $K''\in\Per_w(Y,\Qlb)$. \[pr.Extnul\] Soient $K,L\in \Dbm(X,\Qlb)$ purs de poids $w$. Alors le morphisme $\Hom(K,L[1]) \to \Hom(K_\kb,L_\kb[1])$ est nul. \[cor.dec\] Soit $K\in \Dbm(X,\Qlb)$ pur de poids $w$. Alors $$\det(1-t\Fr, H^i(X_\kb,K_\kb)) = \prod_{e\in \Z} \det\!\left(1-t\Fr, H^i(X_\kb, (\pH^eK_\kb)[-e])\right).$$ Pour tout $d\in \Z$, on a un morphisme de $$\xymatrix{\fr^* \ptle{d} K_\kb\ar[d]_{\Fr}\ar[r]& \fr^* K_\kb \ar[d]_{\Fr}\ar[r] & \fr^* \ptpg d K_\kb\ar[d]_{\Fr} \ar[r]^(.7)0 &\\ \ptle d K_\kb \ar[r]& K_\kb\ar[r] &\ptpg d K_\kb\ar[r]^(.7)0 &}$$ où la nullité des flèches de degrés $1$ découle de \[th.pHi\] et de \[pr.Extnul\]. Alors on a un morphisme de $$\xymatrix{\RG(\Xkb, \ptle{d} K_\kb)\ar[d]_{\Fr}\ar[r]& \RG(\Xkb, K_\kb) \ar[d]_{\Fr}\ar[r] & \RG(\Xkb, \ptpg d K_\kb)\ar[d]_{\Fr} \ar[r]^(.7)0 &\\ \RG(\Xkb,\ptle d K_\kb) \ar[r]& \RG(\Xkb,K_\kb)\ar[r] &\RG(\Xkb,\ptpg d K_\kb)\ar[r]^(.7)0 &}$$ Donc $$\begin{gathered} \det(1-t\Fr, H^i(X_\kb,K_\kb)) \\ = \det(1-t\Fr, H^i(X_\kb, \ptle{d} K_\kb)) \det(1-t\Fr, H^i(X_\kb, \ptpg{d} K_\kb)).\end{gathered}$$ $(E,I)$-compatibilité ===================== Soit $E$ un corps, $I$ une partie de $$\ensdr{\lio}{ l\nmid q \text{ un nombre premier, } \iota: E\hookrightarrow\Qlb \text{ un plongement de corps}}.$$ On dit qu’un système $(t_\lio)_\lidI \in \prod_\lidI \Qlb$ est $(E,I)$-compatible (ou $E$-compatible s’il n’y a pas de confusion à craindre) s’il existe $c\in E$ $t_\lio = \iota(c)$ pour tout $\lio\in I$. \[rm.Rl\] Si $P$ est comme dans §1, $(a_\lio)_\lidI \in \prod_\lidI \Rl(\Z,\Qlb)$ $(a_{\lio,n})_\lidI$ $E$-compatible $\forall n\in P$, alors $(a_{\lio,n})_\lidI$ est $E$-compatible $\forall n\in \Z$, \[cor.Rl\](ii) et (iii). Soit $X$ un schéma séparé sur $k=\F_q$. On dit qu’un système $(K_\lio)_\lidI \in \prod_\lidI K(X,\Qlb)$ est $(E,I)$-compatible (ou $E$-compatible) si pour tout $x\in\|X\|$ et tout $n\ge 1$, $(\Tr(\Fr_x^n, (K_\lio)_\xb))_\lidI \in \prod_\lidI \Qlb$ est $(E,I)$-compatible. D’après (\[eq.Lx\]), $(K_\lio)$ est $E$-compatible pour tout $x\in \|X\|$, il existe $s_x\in E[[t]]$ $L_x(K_\lio,t) = \iota(s_x)$ pour tout $\lidI$. Ici on a étendu $\iota$ en un plongement d’anneaux $E[[t]]\to \Qlb [[t]]$. Les systèmes $E$-compatibles forment un sous-anneau de $\prod_\lidI K(X,\Qlb)$. $(\Qlb)_\lidI$ est . Plus , pour $b\in E^$ $\iota(b)$ soit une unité $l$-adique $\forall \lidI$, le système $(\Qlb^{(\iota(b))})_\lidI$ [@WeilII 1.2.7] est , car les traces locales sont $\iota(b)$. Stabilités ---------- La $E$-compatibilité est stable par les six opérations et donc par le foncteur dualisant. \[th.6op\] Soient $f:X\to Y$ un morphisme de schémas séparés sur $k$ et $(K_\lio)_\lidI\in \prod_\lidI K(X,\Qlb)$ un système $(E,I)$-compatible sur $X$. Alors $(Rf_K_\lio)_\lidI, (Rf_!K_\lio)_\lidI \in \prod_\lidI K(Y,\Qlb)$ sont des systèmes $(E,I)$-compatibles sur $Y$. On a des résultats similaires pour $f^, Rf^!$ et pour $\Lt, \RcHom, D$. Les résultats pour $\Lt$ et $f^$ sont triviaux. Pour $Rf_!$ on utilise la formule des traces $$\Tr(\Fr_y,(Rf_! K_\lio)_\yb) = \sum_{x\in X_y(\F_{q^n})} \Tr(\Fr_x, (K_\lio)_\xb), \forall y\in Y(\F_{q^n}), n\ge 1.$$ Il reste à montrer le résultat pour $D$. On prend un système $(K_\lio)_\lidI$ . Il suffit de voir la de $(\Tr(\Fr_x,(DK_\lio)_\xb))_\lidI$ pour chaque $x\in X(\F_{q^n})$, $n\ge 1$. Le problème est local. Par dévissage, que $X=A$ soit une variété abélienne et $x=0_A \in A(k)$ soit l’origine. On définit $f_\lion, g_\lion : A(k)\to \Qlb$ pour $n\ge 1$ par f\_(a) &=& \_[bA(), T\_n(b)=a]{} (\_b, (K\_)\_),\ g\_(a) &=& \_[bA(), T\_n(b)=-a]{} (\_b, (DK\_)\_), où $T_n : A(\Fqn) \to A(\F_q)$ est la trace. Il suffit de voir la de $(g_{\lio,1}(0))$. On sait la de $(f_\lion(0))_\lidI$ par hypothèse et on veut démontrer la de $(g_\lion(0))_\lidI$. \[rm.Rl\], il suffit donc de trouver pour chaque $\lidI$ une $s_\lio\in \Rl(\Z,\Qlb)$ telle que $s_\lion = f_\lion(0)$, $s_{\lio,-n} = g_\lion(0)$ pour tout $n\ge 1$. $\lio$ étant fixé, on ne l’indique plus dans les indices. Pour une fonction $f:A(k)\to \Qlb$, on définit la transformée de Fourier par $$\cF(f)(\rho) = \sum_{a\in A(k)} f(a)\rho (a),$$ où $\rho: A(k) \to \Qlb^$ est un caractère. On va montrer que pour tout $\rho$, il existe $(S_n(\rho))_{n\in \Z} \in \Rl(\Z,\Qlb)$ telle que $S_n(\rho)=\cF(f_n)(\rho)$, $S_{-n}(\rho)= \cF(g_n)(\rho)$ pour $n\ge 1$. On prend $s_n = \cF^{-1}(S_n)(0)$ pour tout $n\in\Z$. Alors $s\in \Rl(\Z,\Qlb)$ \[cor.Rl\](i). En effet, par la formule des traces, $$\cF(f_n)(\rho) = \Tr(\Fr^n, R\Gamma(A_\kb, K\Lt \fL_\rho)),$$ $$\cF(g_n)(\rho) = \Tr(\Fr^n, R\Gamma(A_\kb, DK\Lt \fL_{\rho^{-1}})),$$ où $\fL_\rho$ est le $\Qlb$-faisceau lisse de rang 1 correspondant à $\rho$ [@SGA4d Sommes trig.]. Le résultat découle donc de la dualité entre $R\Gamma(A_\kb, K\Lt \fL_\rho)$ et $$R\Gamma(A_\kb, DK\Lt \fL_{\rho^{-1}}).$$ Pour les détails, on renvoie à [@ind §3]. Pour tout schéma $X$ séparé sur $k$, $$\Tr(\Fr,R\Gamma(X_\kb,\Q_l))$$ est dans $\Q$ (et même dans $\Z$, \[th.int\]) et indépendant de $l$. On prend $E=\Q(\zeta_p)$ et on choisit un $I$. On fixe un caractère additif non-trivial $\psi_0: \F_p \to E^$ et on pose $\psi_\lio: \iota \circ \psi_0 \circ \Tr_{\F_q/\F_p} : k \to \Qlb^$, $\lidI$. On considère $A=\mathbb{A}^1_k$ et $\fL_{\psi_\lio}$ sur $A$. Alors $(\fL_{\psi_\lio})_\lidI$ est un système . Soient $V=\mathbb{A}^d_k$, $V'$ son dual, $f: V\times V'\to A$ l’accouplement canonique, et $(K_\lio)_{\lidI} \in \prod_\lidI \Dbc(V,\Qlb)$ un système . \[th.6op\], les transformées de Fourier-Deligne $$\cF_{\psi_\lio}K_\lio = Rp_{2!} (p_1^ K_\lio \Lt f^\fL_{\psi_\lio})[d]$$ forment un système sur $V'$, où $p_1:V\times V'\to V$, $p_2: V\times V'\to V'$ sont des projections. La $E$-compatibilité est également stable par l’extension intermédiaire virtuelle (\[rm.\[f!\\]\]) par un morphisme quasi-fini pour les $\Km$. \[th.f!\\] Soient $f:Z\to X$ un morphisme quasi-fini de schémas séparés sur $k$, $(K_\lio)_\lidI\in \prod_\lidI \Km(Z,\Qlb)$ un système $E$-compatible sur $Z$. Alors $(f_{!}K_\lio)_\lidI\in \prod_\lidI \Km(X,\Qlb)$ est un système $E$-compatible sur $X$. D’après \[th.6op\], que $f$ soit une immersion ouverte $j:U\har X$. $\sharp I=1$ ou $2$. (a) Cas où il existe $w\in \Z$ vérifiant $K_\lio\in K_w(U,\Qlb)$ pour tout $\lidI$. On écrit $K_\lio= [L_{\lio,1}]-[L_{\lio,2}]$, où $L_{\lio,\al}\in \Per_w(U,\Qlb)$, $\al=1,2$, $\lidI$. On applique \[cor.j!\\] (pour $A=I\times\{1,2\}$) et on pose $K_\lio\m=[L_{\lio,1}\m]-[L_{\lio,2}\m]$, $0\le m\le n$. Alors $j_{!}K_\lio = K_\lio^{(n)}$. On montre la de $(K_\lio\m)_\lidI$ par récurrence sur $m$. Le cas $m=0$ est vide. On suppose la de $(K_\lio\mm)_\lidI$ établie, $1\le m \le n$. $L_{\lio,\al}\m = \tYle[m]{-d_m-1} Rj_{m} L_{\lio,\al}\mm$, $\al=1,2$. Pour tout $x\in \|U_{m-1}\|$, $L_x(K_\lio\m,t) = L_x(K_\lio\mm,t)$. \[cor.pds\], pour tout $y\in \|Y_m\|$, $L_y(K_\lio\m,t)$ peut être extrait de $L_y(Rj_{m}K_\lio\mm,t)$ comme la partie de poids $\le w-d_m-1$, $\lidI$. D’après \[th.6op\], $(Rj_{m}K_\lio\mm)_\lidI$ est $E$-compatible. Donc $(K_\lio\m)_\lidI$ l’est aussi. \(b) Cas général. Le résultat découle de (a) et de \[lm.p\], car par , $$f_{!}= \bigoplus_{w\in \Z} i_{w,X} f_{!} p_{w,U},$$ avec des notations de \[lm.p\]. \[lm.p\] Soit $w$ un entier. La projection $p_{w,X}: \Km(X,-)\to K_w(X,-)$ et l’inclusion $i_{w,X}: K_w(X,-)\to\Km(X,-)$ préservent la . Le résultat pour $i_w$ est trivial. On montre le résultat pour $p_w$ simultanément pour tout $w\in\Z$. $\sharp I=1,2$. \(a) Un cas spécial. On suppose $X$ lisse sur $k$ purement de dimemsion $d$. Soit $(K_\lio)_\lidI\in \Km(X,\Qlb)$ avec $K_\lio = \sum_{w\in\Z}K_{\liow}$, $K_\liow= [L_{\lio,w,1}]-[L_{\lio,w,2}]$, où $L_{\lio,w,\al}\in\Per_w(X,\Qlb)$ et $\fH^e(L_{\lio,w,\al})$ lisse sur $X$, $\forall e\in\Z$, $\al=1,2$, $\wiZ$, $\lidI$. Alors $L_x(K_\liow,t)$ peut être extrait de $L_x(K_\lio,t)$ comme la partie de poids $w-d$, $\forall x\in \|X\|$. Donc $\forall \wiZ$, les $p_w(K_\lio) = K_\liow$, $\lidI$, forment un système . \(b) Cas général. $X$ réduit. On fait une récurrence noethérienne. Le résultat pour $p_{w,\emptyset}$ étant trivial, on suppose le résultat pour $p_{w,Y}$ établi pour tout fermé $Y\subsetneqq X$ (réduit). On prend $(K_\lio)_\lidI \in \prod_\lidI \Km(X,\Qlb)$ un système . On pose $K_\lio = \sum_{w\in\Z}K_{\liow}$, $K_\liow= [L_{\lio,w,1}]-[L_{\lio,w,2}]$ où $L_{\lio,w,\al}\in \Per_w(X,\Qlb)$, $\al=1,2$. On prend un ouvert non vide $j:U\har X$ lisse sur $k$ purement de dimemsion $d$ $\fH^e(j^L_{\lio,w,\al})$ soit lisse sur $U$, $\forall e\in\Z$, $\al=1,2$, $\wiZ$, $\lidI$, et $i:Y\har X$ le fermé . Alors $\forall \wiZ$, $(j^K_\liow)_\lidI$ est un système sur $U$, (a). Donc $(j_{!}j^K_\liow)_\lidI$ en est un sur $X$, \[th.f!\\](a). \[pr.dec1\], \[eq.decK\] K\_= j\_[!\]{}j\^\K\_+ i\_\K”\_, où $K''_\liow\in K_w(Y,\Qlb)$. Donc $K_\lio=\sum_{\wiZ}j_{!}j^K_\liow + i_\sum_{\wiZ}K''_\liow$, d’où la de $(\sum_\wiZ K''_\liow)_\lidI$, qui implique la de $(K''_\liow)_\lidI$ par l’de , $\forall \wiZ$. La de $(K_\liow)_\lidI$ suit alors de (\[eq.decK\]). Indépendance de $l$ et intégralité pour la cohomologie d’intersection ===================================================================== Soit $X$ un schéma propre sur $k$ purement de dimension $d$. On définit la cohomologie (le complexe) d’intersection par $$IH^i(X_\kb,\Qlb) = H^i(X_\kb,(IC_X)_\kb), \text{(\resp $IC_X = IC(X,\Qlb) = (j_{!}(\Qlb [d]))[-d],$)}$$ où $j: U\har X$ est une immersion ouverte dominante telle que $U_\red$ soit lisse sur $k$. Notons que la normalisation ici diffère de celle dans [@BBD 0]. D’après \[th.Gabber\], $IC_X$ est pur de poids $0$. \[th.IH\] Soient $X$ un schéma propre équidimensionnel sur $k=\F_q$, $l$ un nombre premier $\nmid q$. Alors pour chaque $i$, $P_i(t) = \det(1-t\Fr, IH^i(X_\kb,\Qlb))$ est dans $\Z[t]$ et indépendant de $l$. D’après \[th.f!\\] et \[th.6op\], $$L(IC_X, t) = \det(1-t\Fr,R\Gamma(X_\kb,(IC_X)_\kb))^{-1} \in \Q(t)$$ et indépendant de $l$. Le morphisme $a: X\to \Spec k$ est propre, donc $Ra_IC_X$ est pur de poids 0. Il en résulte que $IH^i(X_\kb,\Qlb)$ est pur de poids $i$, donc les $P_i(t)$ peuvent être extraits de $L(IC_X,t)$ de manière indépendante de $l$, et par suite les $P_i(t)$ sont dans $\Q[t]$ et indépendants de $l$. Il reste à démontrer l’. $X$ réduit. Soit $f: X'\to X$ une normalisation. Prenons $j: U\har X$ comme plus haut. Alors $j=fj'$ où $j': U\har X'$ est une immersion ouverte. Donc $IC_X = f_(IC_{X'}) = f_(\oplus IC_{X_i})$, où $X_i$ sont les composantes connexes de $X'$. Donc $X$ intègre. D’après [@deJong 4.1], on a une altération $\pi: Y\to X$ génériquement étale telle que $Y$ soit , lisse et sur $\Spec k$. Prenons $V\subset U$ un ouvert non vide $\pi_V: Y\times_X V\to V$ soit un fini étale. Alors $\Qlb$ sur $V$ est un facteur direct de $R\pi_{V}\pi_V^* \Qlb = j_V^* K$, où $j_V: V\har X$, $K= R\pi_* \Qlb$. Donc $\Qlb [d]$ est facteur direct de $j_V^* (\pH^d K)$, où $d= \dim X$. Le complexe $K$ étant pur, $\pH^d K$ l’est aussi, \[th.pHi\]. Donc $IC_X[d]=j_{V!}(\Qlb [d])$ est facteur direct de $\pH^d K$, d’après \[pr.dec1\]. Il en résulte que $P_i(t)$ est facteur de $\det(1-t\Fr, H^i(Y_\kb,\Qlb))$, \[cor.dec\].[^2] Donc l’intégralité pour $X$ découle de celle pour $Y$, qui est vraie d’après [@WeilI] ou [@int 5.2.2]. On donnera une autre de l’au §\[sec.int\]. Appendice. Théorème d’ {#sec.int} ====================== Dans cet appendice, on fixe un nombre premier $l\nmid q$. Soit $T$ un ensemble de nombres premiers. Un de $\Qlb$ est dit $T$-entier* s’il est sur $\Q$ et entier sur $\Z[(1/t)_{t\in T}]$. Soit $X$ un schéma séparé sur $k= \F_q$. Un $\Qlb$-$\cF$ sur $X$ est dit $T$-entier si pour tout $x\in \|X\|$, les valeurs propres de l’action de $\Fr_x$ sur $\cF_\xb$ sont $T$-entières. Des $T$-entiers sont stables par sous-quotients et extensions. Un objet $K\in \Dbc(X,\Qlb)$ est dit $T$-entier si tous ses de cohomologie sont $T$-entiers. Cette notion est stable par $\Lt, f^, Rf_!$ [@int 5.2.2]. \[th.int\] Soient $f:X\to Y$ un morphisme de schémas séparés sur $\Spec k$ et $K\in \Dbc(X,\Qlb)$ $T$-entier. Alors $Rf_ K$ est $T$-entier. Ce résultat est démontré dans [@int 5.6] en supposant la résolution des singularités. On peut adapter la pour éliminer l’hypothèse de résolution. En fait, le premier usage de l’de (l. 4 de la ) peut être remplacé par [@SGA4d Finitude]. Pour le deuxième usage (bas de [@int p. 397]), rappelons que l’on est dans le cas suivant $$\begin{gathered} \label{eq.cond} \text{$X$ normal et $K\simeq (\cG\otimes_R E)\otimes_E \Qlb$, où $\cG$ est un $R$-faisceau lisse}\\ \text{\tq $\cG\otimes_R R/\fm$ soit constant sur chaque composante connexe de $X$,}\end{gathered}$$ où $E$ est une extension finie de $\Q_l$ convenable, $R$ son anneau des entiers, $\fm$ l’idéal maximal. Soit $f:X\to Y$ un morphisme de schémas séparés sur $S=\Spec F$, où $F$ est un corps parfait. Alors il existe un diagramme commutatif $$\xymatrix{X'\s \ar@{^{(}->}[r]^{j} \ar[d]_{\eps} & Z\s\ar[ddl]^{g} & D\s \ar@{_{(}->}[l] \\ X\ar[d]_{f}\\ Y}$$ où $\eps$ est un hyper-recouvrement propre et pour tout $n$, $j_n$ est une immersion ouverte, $g_n$ propre, $Z_n$ lisse sur $S$, $D_n$ un diviseur à croisements normaux dans $Z_n$ de complémentaire $X_n$. Conséquence facile de [@deJong 4.1] et de [@Hodge3 6.2]. Voir [@Orgogozo 2.6]. Alors $$Rf_* K = Rf_* R\eps_\eps^ K = Rg_* Rj_* \eps^K,$$ donc on a une suite spectrale $$E_1^{pq} = \fH^q(Rg_{p} Rj_{p} \eps_p^ K) \Rightarrow R^{p+q}f_* K.$$ $X'_p$ et $\eps_p^K$ satisfont encore (\[eq.cond\]), donc $\eps_p^ K$ est lisse sur $X'_p$ et modérément ramifié le long de $D_p$. [@int 5.6.1], la de $\eps_p^* K$ implique la de $Rj_{p}\eps_p^K$, qui donne alors la de $Rg_{p} Rj_{p} \eps_p^* K$ car $g_p$ est propre. Il en suit que $Rf_* K$ est . Soient $f:Z\to X$ un morphisme quasi-fini de schémas séparés sur $k$ et $K\in\Dbc(Z,\Qlb)$ un pervers $T$-entier. Alors $f_{!}K$ est . Grâce à \[th.int\], que $f$ soit une immersion ouverte $j : U \har X$. On applique \[cor.j!\\] pour voir que $$j_{!}K = \tYle[n]{-d_n -1}Rj_{n} \cdots \tYle[1]{-d_1 -1} Rj_{1} K.$$ La conclusion découle alors du \[th.int\]. En prenant $T=\emptyset$, on obtient que pour $a: X\to\Spec k$ vérifiant l’du \[th.IH\], $Ra_ IC(X,\Qlb)$ est $T$-entier, les valeurs propres de $\Fr$ sur $IH^i(X_\kb,\Qlb)$ sont entières sur $\Z$, ce qui donne une autre de l’dans le \[th.IH\]. [^1]: Ceci découle de la factorisation du dans $\Per(X,\Qlb)$ $$\xymatrix{{}^pj_!K_1 \ar[d]_{{}^pj_! f} \ar[r] & {}^pRj_* K_1\ar[d]^{{}^pRj_* f}\\ {}^pj_!K_2 \ar[r] &{}^pRj_* K_2}$$ en le diagramme $$\xymatrix{{}^pj_!K_1 \ar[d]_{{}^pj_! f} \ar@{>>}[r] & \ptYle{-1} Rj_* K_1\ar[d]_{\ptYle{-1} Rj_* f} \ar@{>->}[r] & {}^pRj_* K_1\ar[d]^{{}^pRj_* f}\\ {}^pj_!K_2 \ar@{>>}[r] & \ptYle{-1} Rj_* K_1 \ar@{>->}[r] &{}^pRj_* K_2}$$ dont le carré à gauche est commutatif car celui à droite l’est. [^2]: Pour notre $K$ ici, $\pH^dK$ est en fait facteur direct de $K[d]$ ([@BBD 5.4.10] et [@dec]).
--- abstract: 'In the derivation of synchrotron radiation formulas it has been assumed that the pitch angle of a charge remains constant during the radiation process. However from the radiation reaction formula, while the component of the velocity vector perpendicular to the magnetic field reduces in magnitude due to radiative losses, the parallel component does not undergo any change during radiation. Therefore there is a change in the ratio of the two components, implying a change in the pitch angle. We derive the exact formula for the change in energy of radiating electrons by taking into account the change of the pitch angle due to radiative losses. From this we derive the characteristic decay time of synchrotron electrons over which they turn from highly relativistic into mildly relativistic ones.' author: - \| Ashok K. Singal\ [Astronomy and Astrophysics Division, Physical Research Laboratory, Navrangpura, Ahmedabad - 380 009, India]{}\ \ E-mail: asingal@prl.res.in date: 'Accepted . Received ; in original form ' title: Radiation reaction and the pitch angle changes for a charge undergoing synchrotron losses --- galaxies: active - radiation mechanisms: non-thermal - radio continuum: general - relativistic processes INTRODUCTION ============ Formulas for synchrotron radiative losses were derived more than about 50 years back and have been in use ever since for calculating radiative losses in a variety of radio sources. Since the radiation is beamed with its angular distribution confined to a narrow cone around the direction of the instantaneous velocity vector of the charge, the momentum loss of the radiating charge is expected to be along the direction of motion of the charge. Therefore though the kinetic energy of the radiating charge would be reducing due to the radiation losses, its direction of motion should not undergo any changes and there ought not be any changes in the pitch angle of the charge during the radiation process. Thus in these formulas, it has always been assumed that the pitch angle of the radiating charged particle remains constant and the dynamics and the life-time of radiating electrons are accordingly derived (Kardashev 1962). This formulation is now a standard text-book material (Pacholczyk 1970; Rybicki & Lightman 1979). In this formulation, the power loss rate is written as (Kardashev 1962; Pacholczyk 1970; Rybicki & Lightman 1979), $$\label{eq:30.1} \frac{{\rm d}{\cal E}}{{\rm d}t} = -\zeta \sin^2 \theta\; {\cal E}^2\;,$$ where $$\begin{aligned} \label{eq:30.1a} %\nonumber \zeta = \frac{2e^4\, B^2} {3m_{o}^4c^7}=2.37 \times 10^{-3}B^2\; {\rm erg}^{-1} {\rm s}^{-1}\;.\end{aligned}$$ Let ${\cal E}_{\rm o}$ be the initial energy at $t=0$, then from Eq. (\[eq:30.1\]) the energy of the radiating charge at $t=\tau$ is calculated to be, $$\label{eq:30.4} {\cal E}= \frac{{\cal E}_{\rm o}}{1+\zeta\sin^2 \theta\; \tau\; {\cal E}_{\rm o}}\;.$$ Here pitch angle $\theta$ is treated as a constant of motion. From Eq. (\[eq:30.4\]), radiative losses have been worked out to calculate the life-time of the radiating electrons (Kardashev 1962; Kellermann 1964; van der Laan & Perola 1969; Pacholczyk 1970; Miley 1980). For instance, from Eq. (\[eq:30.4\]) it follows that the electron loses half of its energy in a time $\tau_{1/2} = 1/(\zeta\sin^2 \theta \;{\cal E}_{\rm o})$, and the synchrotron source that has been radiating for a time, say $\tau$, will have no electrons above an energy ${\cal E}_\tau=1/(\zeta\sin^2 \theta\;\tau)$, even if we started with electrons with an infinite energy to begin with. This in turn also implies that there will be a break in the spectrum at a frequency $\nu_\tau \propto \tau^{-2}$ for such a source. Such observed breaks in the spectra have been used to estimate the ages of extragalactic radio sources (van der Laan & Perola 1969) and supernovae remnants (Chandra et al. 2004). Evolution in the spectra of such radio sources have been studied for different scenarios of fresh particle injections (Kardashev 1962; Pacholczyk 1970). These formulas have appeared in a number of books and review articles, and a large number of papers in the literature have made use of these formulas. Actually it is not exactly true that the pitch angle remains constant in spite of radiation losses. But it is still a very good approximation for highly relativistic particles, like synchrortron electrons in radio galaxies. However, as it will be shown, this approximation does not remain valid in trans- and sub-relativistic regimes. From radiation reaction it has been shown (Petrosian 1985) that in the case of synchrotron losses while the component of the velocity vector perpendicular to the magnetic field reduces in magnitude, the parallel component does not undergo any change due to radiation. Thus a change in the ratio of the two components takes place, implying a change in the pitch angle. Therefore one must consider this change in pitch angle while calculating radiation losses and the consequent life times of synchrotron electrons. Moreover, Eq. (\[eq:30.1\]) is an approximate expression, again valid only in highly relativistic cases. The exact equation for radiative energy loss rate is (Melrose 1971; Jackson 1975; Longair 2011), $$\begin{aligned} \label{eq:30.1b} \frac{{\rm d}{\cal E}}{{\rm d}t} = -\zeta \beta ^2\sin^2 \theta\;{\cal E}^2,\end{aligned}$$ which also changes the solution in Eq. (\[eq:30.4\]). We shall derive the exact formulation for radiative losses, applicable even to non-relativistic or mildly relativistic cases, simultaneously taking into account the pitch angle changes. RADIATION REACTION ================== Radiation reaction was first derived by Lorentz (1909) from the self-force of a small charged sphere, and the detailed derivation is available in various forms in many text-books (Schott, 1912; Jackson, 1975; Panofsky & Phillips 1962; Heitler 1954; Griffiths 1999; Yaghjian 2006). But recently it has been derived even for a “point” charge by examining the electromagnetic fields in the neighbourhood of the point charge and then from momentum conservation employing Maxwell-stress tensor (Singal 2015a,b). Of course the two independent methods do lead to the same result, $$\label{eq:1d6} {\bf {f}}=\frac{2e^{2}}{3 c^{2}}\ddot{\mbox{\boldmath $\beta$}}\;.$$ This is the self-force on the charge in its instantaneous rest frame. The force due to the radiation reaction on a charge with relativistic motion (a generalization of Eq. (\[eq:1d6\])) for a synchrotron case (i.e., where $\dot{\mbox{\boldmath $\beta$}}\cdot{\mbox{\boldmath $\beta$}}=0$) is given by (Heitler 1954; Singal 2016), $$\label{eq:10.1} {\bf F}=\frac{2e^{2}}{3c^{2}}\gamma ^{4}\left[\ddot{\mbox{\boldmath $\beta$}}+ {\mbox{\boldmath $\beta$}}\times({\mbox{\boldmath $\beta$}}\times\ddot{\mbox{\boldmath $\beta$}})\right]\;,$$ where $$\label{eq:10.1.1} \ddot{\mbox{\boldmath $\beta$}}=\frac{e}{m_{\rm o}\gamma c}\dot{\mbox{\boldmath $\beta$}}\times{\bf B} =-\frac{e^2}{m_{\rm o}^2\gamma^2 c^2}B^2{\mbox{\boldmath $\beta$}}_\perp\;.$$ We then get force components parallel and perpendicular to the magnetic field direction (Petrosian 1985) as, $$\label{eq:10.1.2} {F}_\parallel=\frac{2e^2}{3c^2}\gamma ^{4} \ddot{\beta} \beta_\parallel {\beta}_\perp =-\frac{2e^4}{3m_{\rm o}^2c^4}\gamma ^{2} B^2 {\beta}_\perp^2 \beta_\parallel \;,$$ $$\label{eq:10.1.3} {F}_\perp=\frac{2e^{2}}{3c^{2}}\frac{\gamma ^{4}\ddot{\beta}}{\gamma_\parallel^2} =-\frac{2e^4}{3m_{\rm o}^2c^4}\frac{\gamma ^{2 }B^2 {\beta}_\perp}{\gamma_\parallel^2}\;,$$ where $\gamma_\parallel=\surd(1- \beta_\parallel^2)$. The negative sign indicates that the force component in each case is in a direction opposite to the direction of the corresponding velocity component of the charge. However, the ratio of the force components is not equal to that of the velocity components, $$\label{eq:10.1.3a} {F}_\perp/{F}_\parallel=1/(\gamma_\parallel^2 \beta_\parallel \beta_\perp)\ne \beta_\perp / \beta_\parallel\;.$$ which implies that the force vector is not parallel to the velocity vector. Also, in spite of a finite ${F}_\parallel$, there is no acceleration $\dot{\beta}_\parallel$ along the $z$-axis. It can be recalled that in special relativity, force and acceleration vectors are not always parallel, e.g., in a case where force is not parallel to the velocity vector, the acceleration need not be along the direction of the force. When the applied force is either parallel to or perpendicular to the velocity vector, it is only then that the acceleration is along the direction of force (see e.g., Tolman 1934). It has to be further kept in mind that the acceleration we are talking about here is not that due to the force by the magnetic field on the moving charge (which is perpendicular to the instantaneously velocity of the charge), but the acceleration (or rather a deceleration) caused on the charge due to the radiation reaction force. RADIATIVE LIFE TIMES ==================== The radiation drag will reduce only the perpendicular velocity component, ${\beta}_\perp$, of the charge (i.e., $\dot{\beta}_\parallel=0$). The rate of change of the parallel component of the momentum is, $$\begin{aligned} \label{eq:10.1.4} \frac{{\rm d}(m_{\rm o} \gamma\beta_\parallel c)}{{\rm d}t}=m_{\rm o} c \frac{{\rm d}{\gamma}}{{\rm d}t} \beta_\parallel =m_{\rm o} c\gamma^3 \dot{\beta}_\perp {\beta}_\perp \beta_\parallel\;.\end{aligned}$$ Equating it to $F_\parallel$ (Eq. (\[eq:10.1.2\])), the force due to radiation reaction, we can write, $$\begin{aligned} \label{eq:10.1.5} m_{\rm o} c\gamma^3 \dot{\beta}_\perp {\beta}_\perp \beta_\parallel =-\frac{2e^4}{3m_{\rm o}^2c^4}\gamma ^{2} B^2 {\beta}_\perp^2 \beta_\parallel\;,\end{aligned}$$ which simplifies to, $$\label{eq:30.6c} {\dot{\beta}_\perp} = \frac{-{\eta}{\beta_\perp}}{\gamma}\;.$$ Here ${\eta}=2e^4 B^2/(3m_{\rm o}^3c^5)=1.94 \times 10^{-9}B^2\; {\rm s}^{-1}$. On the other hand, from the perpendicular component of the rate of change of momentum (Eq. (\[eq:10.1.3\])), we have, $$\begin{aligned} \label{eq:10.1.6} \frac{{\rm d}{\gamma}}{{\rm d}t} {\beta}_\perp + \gamma \dot{\beta}_\perp =-{\eta}\frac{\gamma ^{2 }{\beta}_\perp}{\gamma_\parallel^2}\;,\end{aligned}$$ Eliminating $\dot{\beta}_\perp$ with the help of Eq. (\[eq:30.6c\]) we get (also see, Petrosian 1985), $$\label{eq:30.2.1} \frac{{\rm d}{\gamma}}{{\rm d}t} = -{\eta}\left(\frac{\gamma^2}{\gamma_\parallel^2}-1\right)\;.$$ Equation (\[eq:30.2.1\]) has a solution, $$\label{eq:30.2c} \tanh^{-1}\frac{\gamma_\parallel}{\gamma}= \frac{\eta t}{\gamma_\parallel} + a\;.$$ where $a$ is a constant of integration. Let ${\gamma_{\rm o}}$ correspond to the initial energy of the charge at $t=0$, implying ${\gamma_\parallel/\gamma_{\rm o}}=\tanh a$, then at $t=\tau$ we have, $$\label{eq:30.2e1} \tanh^{-1}\frac{\gamma_\parallel}{\gamma} = \tanh^{-1}\frac{\gamma_\parallel}{{\gamma_{\rm o}}}+\frac{{\eta} \tau}{\gamma_\parallel}\;.$$ This is a general solution for all values of $\gamma$. We can rewrite it as, $$\label{eq:30.2e2} \gamma = \gamma_\parallel\; \frac{{\gamma_{\rm o}}+\gamma_\parallel\tanh({\eta} \tau/\gamma_\parallel)} {{\gamma_{\rm o}}\tanh({\eta} \tau/\gamma_\parallel)+\gamma_\parallel}\;.$$ For an initially ultra-relativistic charge ($\gamma_{\rm o}\gg1, \beta_{\rm o} \approx 1$), we have $1/\gamma_\parallel =\surd(1-\beta_{\rm o}^{2}\cos^2 \theta_{\rm o})\approx \sin\theta_{\rm o}$. That also implies that (except for initially small pitch angle cases) $\sin \theta_{\rm o} \gg 1/\gamma_{\rm o}$ or $\gamma_\parallel / \gamma_{\rm o} \ll 1$, and from Eq. (\[eq:30.2e1\]) we could write, $$\label{eq:30.2e1a} \tanh^{-1}\frac{1}{\gamma \sin\theta_{\rm o}} = {{\eta} \tau}{\sin\theta_{\rm o}}\;.$$ This implies that for $\tau = \gamma_\parallel/ \eta \approx 1/ (\eta \sin \theta_{\rm o})$, we have $\gamma \approx 1.3/\sin \theta_{\rm o}$. Thus even if an electron had started with an almost infinite energy, it loses most of its kinetic energy in a time interval of the order of $1/ \eta$, reducing to perhaps a mildly relativistic status (for not too small an initial pitch angle). For instance let us consider $\gamma_{\rm o}=10^3$ and $\theta_{\rm o}=\pi/4$, then $\gamma_\parallel \approx 1/\sin \theta_{\rm o}=\surd{2}$, then from Eq. (\[eq:30.2e1\]) or Eq. (\[eq:30.2e1a\]) we get for $\tau = 1/\eta$, $\gamma=2.3$. In another example, taking $\gamma_{\rm o}=10^4$ and $\theta_{\rm o}=\pi/3$, for $\tau = 1/\eta$ we get $\gamma_\parallel \approx 1/\sin \theta_{\rm o}=2/\surd{3}$ and $\gamma=1.7$. Thus $1/ \eta$ represents the characteristic decay time of synchrotron electrons over which they turn from ultra-relativistic into mildly relativistic ones. On a shorter time scale where $\gamma$ is still highly relativistic, we could write $$\label{eq:30.2f} \frac{1}{\gamma} = \frac{1}{{\gamma_{\rm o}}}+\frac{{\eta} \tau}{\gamma_\parallel^2}\;,$$ Then we have, $$\label{eq:30.2g} \gamma = \frac{\gamma_{\rm o}}{1+{\eta}\tau \gamma_{\rm o} / \gamma^2_\parallel} =\frac{{\gamma_{\rm o}}}{1+{\eta}\tau\gamma_{\rm o}\sin^2 \theta_{\rm o}}\;.$$ Equation (\[eq:30.2g\]) is the same relation as in Eq. (\[eq:30.4\]), which tells us that there is an upper energy-cutoff for highly relativistic electrons in a synchrotron source, $\gamma_{\tau}=\gamma^2_\parallel/({\eta} \tau)=1/({\eta} \tau \sin^2 \theta_{\rm o})$. Also, while $\tau_{1/2} = 1/(\eta\sin^2 \theta_{\rm o} \;{\gamma}_{\rm o})$ gives half-life time of the synchrotron electrons (Kardashev 1962; Pacholczyk 1970; Rybicki & Lightman 1979), $\tau \sim 1/\eta$ gives decay time over which radiating synchrotron electrons turn from ultra-relativistic into mildly relativistic ones. This followed from Eq. (\[eq:30.2e1\]) which gives an exact formula, true for all $\gamma$, including mildly relativistic electrons and implicitly includes the effects of the pitch angle changes. An explicit expression for the pitch angle changes is derived in the next section. PITCH ANGLE CHANGES =================== Both $\beta$ and $\theta$ in $\beta_\perp=\beta \sin \theta$ are functions of time. Therefore we can rewrite Eq. (\[eq:30.6c\]) as, $$\label{eq:30.6c1} {\beta \cos \theta \frac{{\rm d}\theta}{{\rm d}t} + \dot{\beta} \sin \theta} = \frac{-{\eta}\beta \sin \theta} {\gamma}\;.$$ Also since $\dot{\beta}_\parallel=0$, we have, $$\label{eq:30.6d} \beta \sin \theta \frac{{\rm d}\theta}{{\rm d}t} = \dot{\beta} \cos \theta\;.$$ Eliminating $\dot{\beta}$ from Eqs. (\[eq:30.6c1\]) and (\[eq:30.6d\]), we get (also see, Petrosian 1985), $$\label{eq:30.6e} \frac{{\rm d}\theta}{{\rm d}t}= \frac{-{\eta}\sin \theta \cos \theta}{\gamma}=\frac{-{\eta}\sin 2\theta}{2\gamma}\;.$$ This is the relation for the rate of change of the pitch angle of a charge undergoing synchrotron radiative losses. The negative sign implies that the pitch angle decreases with time and aligns with the magnetic field. The rate of alignment is very slow for low pitch angles ($\theta \approx 0$) as well as for high pitch angles ($\theta \approx \pi/2$), and the highest rate of change of the pitch angle is for $\theta=\pi/4$. With the help of Eq. (\[eq:30.2c\]), we can integrate Eq. (\[eq:30.6e\]) to get, $$\label{eq:30.6i} {\tan \theta} = \frac{{\tan \theta_{\rm o}}}{\cosh \left(\frac{{\eta} \tau}{\gamma_\parallel}\right) + \frac{\gamma_\parallel}{\gamma_{\rm o}} \sinh \left(\frac{{\eta} \tau}{\gamma_\parallel} \right)}\;.$$ In Eq. (\[eq:30.6i\]), $\theta < \theta_{\rm o}$, because pitch angle always reduces with time. There are many notable points. If $\theta_{\rm o} =\pi/2$, then $\theta =\pi/2$ also, which is because if the pitch angle is $\pi/2$, then the radiating electron always moves in a circular path in the plane perpendicular to the magnetic field. And if $\theta_{\rm o} =0$, then $\theta=0$ too as there is no more reduction in the pitch angle. For any $0<\theta_{\rm o} <\pi/2$, $\theta \rightarrow 0$ as $\tau \rightarrow \infty$. For large $\gamma_{\rm o}$ values, $$\label{eq:30.6j} {\tan \theta} = \frac{{\tan \theta_{\rm o}}}{\cosh \left({{\eta} \tau}{\sin \theta_{\rm o}}\right)}\;,$$ which can be used to estimate change in pitch angle with time. For example for say, $\theta_{\rm o} =\pi/3$, and $\theta =\pi/6$, $\cosh ({\eta} \tau \sin \theta_{\rm o})=3$, which gives $\tau \approx 2 /\eta$ for this change in the pitch angle. Thus there are appreciable pitch angle changes in time $\tau \sim 1/\eta$ (except for in the vicinity of very small pitch angles). All charges of a given energy and pitch angles, directed towards the observer in a narrow angle $1/\gamma$ around the line of sight not only lose energy but will even get shifted outside the angle $1/\gamma$ around the line of sight towards the observer, in a time $\tau \sim 1/\eta$. Thus in a mono-energetic and a narrow pitch angle distribution, the pitch angle changes might be quite relevant. But it may be of less importance when there is a wide angular (isotropic!) distribution of pitch angles. In case of compact radio sources with ${\bf B} \sim 10^{-3}$G the time periods for pitch angle alignments may be $\sim 10^{8}$ years, but in extended radio sources where ${\bf B} \sim 10^{-5}$G, the pitch angle alignment would be much slower. However in mildly relativistic cases (see e.g. Petrosian 1981) pitch angle changes may be of sufficient importance and might not be ignored. Also in cases where magnetic field is much stronger, e.g., in case of curvature radiation models in pulsars, where magnetic fields could vary from ${\bf B} \sim 10^{12}$G near the neutron star surface to $\sim 10^{6}$G in the outer magnetosphere, the pitch angle changes will be rather fast and that is why the electrons will stream along the curved magnetic field lines, giving rise to curvature radiation. It may be mentioned that our discussion is well within the classical synchrotron radiation regime where energy loss time scales are much larger than the gyro period and the quantum effects do not become applicable (see e.g., Brainerd 1987; Brainerd & Petrosian 1987). CONCLUSIONS =========== We showed that the argument used in the standard formulation of the synchrotron radition that the radiation is beamed along the direction of motion and therefore radiation losses should not cause any change in the pitch angle of the charge is not correct and that the pitch angle in general varies. It is because from the radiation reaction formula while the component of the velocity vector perpendicular to the magnetic field reduces in magnitude due to radiative losses, the parallel component does not undergo any change during radiation, which implies a change in the pitch angle. We derived the exact formula for the change in energy of radiating electrons by taking into account the changes in the pitch angle due to radiative losses. This way we derived the characteristic decay time of synchrotron electrons over which they turn from highly relativistic into mildly relativistic ones. Further, an explicit formula for the pitch angle changes with time was also derived. [20]{} Brainerd, J. J., 1987, ApJ, 320, 714 Brainerd, J. J., & Petrosian, V. 1987, ApJ, 320, 703 Chandra, P., Ray, A., & Bhatnagar, S., 2004, ApJ, 604, L97 Griffiths, D. J., 1999, Introduction to electrodynamics, 3rd ed., Prentice, New Jersey Heitler, W., 1954, The quantum theory of radiation, Clarendon, Oxford Jackson, J. D., 1975, Classical electrodynamics, 2nd ed., Wiley, New York Kardashev, N. S., 1962, Sov. Astr. - AJ, 6, 317 Kellermann, K. I., 1964, ApJ, 140. 969 Longair, M. S., 2011, High energy astrophysics, Cambridge Univ. Press, Cambridge Lorentz, H. A., 1909 The theory of electron, Teubner, Leipzig. Reprinted 1952, 2nd ed., Dover, New York Melrose, D. B., 1971, ApL, 8, 35 Miley, G., 1980, ARA&A, 18, 165 Mould, R. A., 1994, Basic relativity, Springer, New York Pacholczyk, A. G., 1970, Radio astrophysics, Freeman, San Francisco Panofsky, W. K. 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--- author: - 'V.V. Syshchenko,' - 'A.I. Tarnovsky' title: 'Eikonal Approximation and the Conical Effect in Smith-Purcell Radiation' --- Introduction {#sec:intro} ============ Transition radiation permits precise description only in the case of the incident particle’s interaction with the simplest target that is the uniform infinite plate. All other cases need some approximate methods. Some of that methods are briefly described in the sections \[Born.method\]–\[eikonal.method\]. The diffraction and transition radiation (DR and TR) under normal incidence of a particle to the semi-infinite dielectric plate had been considered in [@jof.2010] using the eikonal approximation in the transition radiation theory [@arsa.2006; @pov.2007]. This approach is valid for the high radiation frequencies domain (where the dielectric permittivity of the plate material is close to unit), including X-ray domain. In the present paper this method is applied to the case of oblique incidence of the particle on the plate (but parallel to the plate’s edge) as well as to the periodic set of such plates (Smith–Purcell radiation). The oblique incidence of the particle (both in the case of the single plate and in the case of the periodic grating of such plates) leads to the so called conical effect in the radiation angular distribution recently observed in [@Naumenko]. We outline that the origin of this effect lies in the superluminal motion along the plate’s edge of the disturbance produced by the incident particle’s field in the plates. The analogous effects arise in various physical situations [@Bolot.Ginz; @Davydov; @Syshch12] (section \[interpretation\]). The TR spectral-anguylar density on such plate is calculated in the section \[TR.section\]. Also we compare our results to ones of [@Tishch.2004; @Tishch.2] obtained using different technique (section \[discussion\]). Eikonal and other approximations in TR and DR theory {#sec:method} ==================================================== The spectral-angular density of the diffraction or transition radiation can be written in the form [@arsa.2006; @pov.2007] $$\label{spectral.angular.0} \frac{d\mathcal E}{d\omega d\Omega} = \frac{\omega^2}{(8\pi^2)^2 c} \left\|\mathbf k\times\mathbf I\right\|^2,$$ where $$\label{I.determination} \mathbf I= \int \left(1-\varepsilon_\omega(\mathbf r) \right) \mathbf E_\omega(\mathbf r) \, e^{-i\mathbf k\mathbf r} \, d^3r \,,$$ $\mathbf k$ is the wave vector of the radiated wave, $\|\mathbf k\| = \omega /c$, $$\mathbf E_\omega (\mathbf r) = \int_{-\infty}^\infty \mathbf E (\mathbf r,t)e^{i\omega t} dt$$ is the Fourier component by time of the electric field produced by the moving particle in the substance of the target with the dielectric permittivity $\varepsilon_\omega(\mathbf r)$. Born approximation {#Born.method} ------------------ If $\left\| 1-\varepsilon_\omega(\mathbf r) \right\| \ll 1$, the precise value of the field in the target in (\[I.determination\]) could be replaced in the first approximation by non-disturbed Coulomb field of the uniformly moving particle in vacuum $$\label{Coulomb.Fourier} \mathbf E^{(Coulomb)}_\omega (\boldsymbol\rho , z) = \frac{2e\omega}{v^2 \gamma}\, e^{i(\omega/ v)z} \left\{ \frac{\boldsymbol\rho}{\rho} K_1 \left(\frac{\omega \rho}{v \gamma} \right) - i \frac{\mathbf v}{v \gamma} K_0 \left(\frac{\omega \rho}{v \gamma} \right) \right\} \,,$$ where $K_n(x)$ is the modified Bessel function of the second kind (McDonald function), $\boldsymbol\rho$ is the component of $\mathbf r$ perpendicular to $\mathbf v$, $z$ is the component parallal to $\mathbf v$, $\gamma = \sqrt{1-v^2/c^2}$ is the particle’s Lorentz factor. The resulting approximation is analogous to Born approximation in the quantum theory of scattering. Although this approximation permits to calculate the TR characteristics for the targets with complex geometry [@Syshch12], its applicability is restricted by the range of extremely high frequencies, $$\label{Born.condition} \omega \gg \gamma\omega_p \,.$$ For investigation of radiation in the range of more soft photons, some different approximate method is needed. Durand approximation {#Durand.method} -------------------- The simple modification of the above approach was proposed by L. Durand in [@Durand]; it consists in the substitution $\mathbf k \to \sqrt{\varepsilon_\omega}\, \mathbf k$ in (\[spectral.angular.0\]) and (\[I.determination\]) $$\label{spectral.angular.D} \frac{d\mathcal E}{d\omega d\Omega} = \frac{\omega^2}{(8\pi^2)^2 c} \left\|\sqrt{\varepsilon_\omega}\, \mathbf k\times\mathbf I^{(Durand)} \right\|^2,$$ $$\label{I.D} \mathbf I^{(Durand)}= \int \left(1-\varepsilon_\omega(\mathbf r) \right) \mathbf E^{(Coulomb)}_\omega(\mathbf r) \, e^{-i\sqrt{\varepsilon_\omega}\, \mathbf k\mathbf r} \, d^3r \,.$$ Eikonal approximation {#eikonal.method} --------------------- A simple variant of eikonal approximation in TR theory was developed in [@arsa.2006; @pov.2007]. In that approach the Coulomb field of the incident particle (\[Coulomb.Fourier\]) is approximated by the packet of the transverse waves $$\label{paket} \mathbf E_\omega (\mathbf r) = \frac{2e\omega}{v^2\gamma} \, e^{i\omega z/c} \, \frac{\boldsymbol\rho}{\rho} \, K_1\left( \frac{\omega\rho}{v\gamma}\right) \,,$$ and account of the gradual evolution of this field inside the target leads to the following formula for the $\mathbf I$ (\[I.determination\]) component that perpendicular to the particle’s velocity $\mathbf v$: $$\label{I.eik} \mathbf I^{(eik)}_\perp = \frac{2e\omega}{v^2\gamma} \int d^3 r\, e^{i(\omega/c -k_z)z} \, e^{-i\mathbf k_\perp\boldsymbol\rho}\, \frac{\boldsymbol\rho}{\rho} K_1\left( \frac{\omega\rho}{v\gamma}\right) \exp\left[-i \frac{\omega}{2c} \int_{-\infty}^z (1-\varepsilon_\omega(\mathbf r))\,dz\right] ,$$ The eikonal approximation in DR and TR theory is valid for $$\omega\gg\omega_p$$ (compare with the condition (\[Born.condition\])). For small angles between $\mathbf k$ and $\mathbf v$, $$\label{angle.restriction.0} \theta\ll \sqrt{c/L\omega},$$ where $L$ is the thickness of the target along the particle’s velocity $\mathbf v$, the first exponent in (\[I.eik\]) can be replaced by the unit, and the integration over $z$ in (\[I.eik\]) could be performed in general form (that had been done in [@jof.2010]). However, in the problem under consideration the integration over $z$ in (\[I.eik\]) can be performed without the restriction (\[angle.restriction.0\]). Eq. (\[I.eik\]) had been used in [@jof.2010] for description of the radiation arising under normal incidence of the particle on the semi-infinite uniform plate (as well as the set of parallel plates). Here we consider the case of oblique incidence. Let the particle is incident on the semi-infinite dielectric plate with the thickness $a$ and uniform dielectric permittivity $\varepsilon_\omega$. The particle’s velocity is parallel to the edge of the plate and makes the angle $\psi$ with the normal to the plate’s surface (figure \[geometry.1\]). It is convenient to choose the $z$ axis along the particle’s velocity $\mathbf v$, and then the particle’s flight above the plate’s edge corresponds to negative values of the impact parameter $x_0$ while the flight through the plate to positive ones. ![\[geometry.1\] Geometry of the problem. The arising radiation is concentrated near the gray cone with the semi-apex angle $\pi/2 - \psi$ and the axis along the plate’s rib. The parts of the cones that (bound border restrict) the domains (\[poln.vnut\]) are pictured in green.](sceme.png){width="80.00000%"} Inegration over $z$ in (\[I.eik\]) gives $$\label{I.eik.plate.0} \mathbf I^{(eik)}_\perp = \frac{2e\omega}{v^2\gamma} (1-\varepsilon_\omega) \frac{\exp\left\{ i\frac{a}{\cos\psi} \left( \frac{\omega}{c} - k_z - \frac{\omega}{2c} (1-\varepsilon_\omega) \right) \right\} - 1}{i \left( \frac{\omega}{c} - k_z - \frac{\omega}{2c} (1-\varepsilon_\omega) \right)}$$ $$\times \int \Theta(x_0-x) \, \underbrace{ \frac{x\mathbf e_x + y \mathbf e_y}{\sqrt{x^2 + y^2}} K_1\left( \frac{\omega}{v\gamma} \sqrt{x^2 + y^2} \right) } \, \exp\left\{i\left[ \left( \frac{\omega}{c} -k_z \right) \tan\psi - k_y \right] y\right\} \, \exp ( -ik_x x) \, dxdy \,,$$ where $\Theta (\xi)$ is the Heaviside’s step function ($\Theta (\xi) = 0$ for $\xi < 0$ and $\Theta (\xi) = 1$ for $\xi\geq 0$). Interpretation of the conical effect {#interpretation} ------------------------------------ Already at this stage we can see and discuss some features of the arising radiation. The factor marked by the underbrace disappears from (\[I.eik.plate.0\]) in the case of incidence of the plane wave $\mathbf E_\omega (\mathbf r) = \mathbf E_0 e^{i\,\mathbf k^{(i)} \,\mathbf r} = \mathbf E_0 e^{i \omega z/c}$ instead of the wave packet (\[paket\]), and the integration over $y$ would give the $\delta$-function $$\label{delta.function.0} \delta \left[ \left( \frac{\omega}{c} -k_z \right) \tan\psi - k_y \right] \,.$$ The argument of this $\delta$-function becomes zero when $$\label{delta.function.1} k_z \sin\psi + k_y \cos\psi = \frac{\omega}{c} \sin\psi \,.$$ The value on the left is the projection of the radiated wave vector $\mathbf k$ on the axis $y'$ parallel to the plate’s edge (figure \[geometry.1\]): $$\label{delta.function.2} k_z \sin\psi + k_y \cos\psi = k_{y'} \,,$$ so the equation (\[delta.function.1\]) means $$\label{delta.function.3} k_{y'} = \frac{\omega}{c} \sin\psi \,.$$ But the value $(\omega/c) \sin\psi$ can be interpreted as the projection of the incient wave vector $\mathbf k^{(i)}$ (parallel to $\mathbf v$) on the same axis $y'$: $$\label{delta.function.4} k_{y'} = k^{(i)}_{y'} \,.$$ The last equality together with the equality of the lengths of the wave vectors, $\|\mathbf k\| = \omega/c = \|\mathbf k^{(i)}\|$, leads to the conclusion that the arising radiation would be distributed over the conical surface with the main axis $y'$ and the half-opening angle $\psi$ (see figure \[geometry.1\]). ![The intersection of the wave front (red vertical line) travelling along the $z$ axis with the velocity $c$ with the target will move along the last one with the velocity $c/\sin\psi > c$.[]{data-label="superluminal"}](superluminal_2019.png){width="60.00000%"} This conical character of the angular distribution of the radiation under oblique incidence on the plate permits clear interpretation as the manifestation of Cherenkov mechanism. Consider the plane wave incident on our plate under the angle $\pi/2-\psi$ (figure \[superluminal\]). This wave creates in the plate the perturbation that travels along the plate’s edge with the superluminal speed $$\label{25} v_{\mbox{\small perturbation}} = \frac{\omega}{k^{(i)}_\parallel} = \frac{c}{\cos(\pi/2-\psi)} = \frac{c}{\sin\psi} > c \,.$$ Indeed, while the incident wavefront travels the distance $c\,dt$ in the $z$ direction, the point of intersection of the wave front and the plate’s edge travels the distance $$\label{26} \frac{z + c\,dt}{\sin\psi} - \frac{z}{\sin\psi} = \frac{c\,dt}{\sin\psi}$$ along the $y'$ axis, so the perturbation moves along the edge with the velociy $c/\sin\psi > c$. This superluminal motion leads to generation of the radiation analogous to Cherencov one. The half-opening angle of the Cherenkov cone determined by the relation $\cos\theta_{\mbox{\footnotesize Ch}} = c/v_{\mbox{\small perturbation}}$ is just equal to $\pi/2-\psi$: $$\label{27} \cos\theta_{\mbox{\footnotesize Ch}} = \frac{c}{v_{\mbox{\small perturbation}}} =\cos(\pi/2-\psi) \,.$$ This interpretation clearly demonstrates why the cone opening angle depends neither on the radiation frequency nor on the dielectric properties of the plate’s material. The analogous cone effects had been discussed in a large variety of physical situations (see, e.g., [@Bolot.Ginz; @Davydov; @Syshch12]). The presence of the underbraced factor in (\[I.eik.plate.0\]) will lead to the blur of the cone, see below. Diffraction radiation ===================== Integrating (\[I.eik.plate.0\]) over $y$ we obtain $$\label{I.eik.plate.2} \mathbf I^{(eik)}_\perp = \frac{2\pi e}{v} (1-\varepsilon_\omega) \frac{\exp\left\{ i\frac{a}{\cos\psi} \left( \frac{\omega}{c} - k_z - \frac{\omega}{2c} (1-\varepsilon_\omega) \right) \right\} - 1}{i \left( \frac{\omega}{c} - k_z - \frac{\omega}{2c} (1-\varepsilon_\omega) \right)} \, e^{ -i k_x x_0}$$ $$\times \,\, \frac{ \exp \left\{x_0 \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + \left[ \left( \frac{\omega}{c} -k_z \right) \tan\psi - k_y \right]^2} \right\} } {-ik_x + \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + \left[ \left( \frac{\omega}{c} -k_z \right)\tan\psi - k_y \right]^2}} % \times %\end{equation} %\begin{equation} % \times \left\{ -\mathbf e_x + i \frac{ \left( \frac{\omega}{c} -k_z \right) \tan\psi - k_y } { \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + \left[ \left( \frac{\omega}{c} -k_z \right) \tan\psi - k_y \right]^2}} \, \mathbf e_y \right\} .$$ The substitution of (\[I.eik.plate.2\]) into (\[spectral.angular.0\]) in the case $x_0 < 0$ leads to the spectral-angular density of DR: $$\label{spectral.angular.1} \frac{d\mathcal E}{d\omega d\Omega} = \frac{e^2}{8\pi^2 v^2} \, (1-\varepsilon_\omega)^2 \, \frac{1-\cos\left[ \frac{a\omega}{c\cos\psi} \left( \frac{1+\varepsilon_\omega}{2} - \varkappa_z \right) \right]}{\left( \frac{1+\varepsilon_\omega}{2} - \varkappa_z \right)^2} F(\theta , \varphi) \,,$$ where $\boldsymbol{\varkappa} = \mathbf k / \|\mathbf k\|$ is the unit vector in the $\mathbf k$ direction, $\varkappa_x = \sin\theta\cos\varphi$, $\varkappa_y = \sin\theta\sin\varphi$, $\varkappa_z = \cos\theta$, $$\label{spectral.angular.2} F(\theta , \varphi) = \frac{ \exp \left\{2x_0 \frac{\omega}{c} \sqrt{ \frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2} \right\} } {\varkappa_x^2 + \frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2} \, \frac{ \frac{1-\varkappa_x^2}{v^2\gamma^2/c^2} + (2-\varkappa_\perp^2) [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2} {\frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2} \,,$$ $\varkappa_\perp^2 = \varkappa_x^2 + \varkappa_y^2$. The value $\gamma^{-2} F(\theta, \varphi)$ that describe the angular distribution of radiation is presented in figures \[angular.1\] and \[angular.2\]. ![\[angular.1\] The value $F(\theta ,\varphi)$ that decribes the anguar distribution of DR acccording to (\[spectral.angular.2\]) under $\gamma = 12$, $x_0 \omega / c\gamma = -3$, $\psi = 75^\circ$, presented as a surface plot (a), as an isolevel contour plot ((b) and (d)), and as a directional diagram (c).](figure_3.png){width="\textwidth"} ![\[angular.2\] The same as in figure \[angular.1\] for the case $x_0 \omega / c \gamma = -0.5$.](figure_4.png){width="\textwidth"} The formulae (\[I.eik.plate.2\]) and (\[spectral.angular.2\]) demonstrate that the radiation is concentrated near the cone that is determined by the condition (\[delta.function.0\])–(\[delta.function.1\]): the argument of the $\delta$-function (\[delta.function.0\]) is present in (\[I.eik.plate.2\]) and (\[spectral.angular.2\]) in the square brackets. The main contribution to the radiation is made by the range of values $$\left\| (1 -\varkappa_z)\tan\psi - \varkappa_y \right\| \lesssim \frac{c}{v\gamma} \,,$$ so the characteristic “thickness” of the cone is $\sim 2c/v\gamma$. The exponential factor demonstrates that the increase of $\|x_0\|$ leads to the decrease of the total intensity, but, however, to the increase of the cone’s sharpness (compare figures \[angular.1\] and \[angular.2\]). On the other hand, the denominator in (\[I.eik.plate.2\]) and (\[spectral.angular.2\]) leads to supression of the radiation for large $\|\varkappa_x\|$, with the same effective values $$\left\| \varkappa_x \right\| \lesssim \frac{c}{v\gamma} \,,$$ that leads to the persence of two “horns” on the angular distribution. DR from the grating ------------------- Consider now the radiation arising under the incidence on the stack of the parallel plates with the period $b$. The contribution into $\mathbf I^{(eik)}_\perp$ from the every next plate will distinct from the contribution from the previous one by the phase factor $$\exp \left[ i\,\omega \frac{b}{\cos\psi} \left( \frac{1}{v} - \frac{\cos\theta}{c} \right) \right] .$$ The radiation in this case will be described (neglecting the emitted radiation refraction in the following plates) by the formula (\[spectral.angular.1\]) with the additional factor $$\label{Smith.Purcell.1} 2\pi N \sum_{j=-\infty}^\infty \delta \left\{ \omega \frac{b}{\cos\psi} \left( \frac{1}{v} - \frac{\cos\theta}{c} \right) -2\pi j \right\},$$ where $N$ is the total number of the plates, $N\gg 1$. These $\delta$-functions mean that the radiation under the angle $\theta$ takes the place only for the frequencies that satisfy the condition $$\label{Smith.Purcell.2} \omega_j = \frac{v}{\displaystyle 1 - \frac{v}{c} \cos\theta} \cos\psi \frac{2\pi j}{b}$$ ($j$ is the positive integer). This is well known Smith–Purcell condition [@Smith.Purcell]. Transition radiation {#TR.section} ==================== When the particle’s trajectory crosses the plate ($x_0 > 0$) the integration over $x$ in (\[spectral.angular.0\]) gives $$\mathbf I^{(eik)}_\perp = \frac{2\pi e}{v} (1-\varepsilon_\omega) \frac{\exp\left\{ i\frac{a}{\cos\psi} \left( \frac{\omega}{c} - k_z - \frac{\omega}{2c} (1-\varepsilon_\omega) \right) \right\} - 1}{i \left( \frac{\omega}{c} - k_z - \frac{\omega}{2c} (1-\varepsilon_\omega) \right)} \, \frac{1} {k_x^2 + \left(\frac{\omega}{v\gamma}\right)^2 + [( \frac{\omega}{c} -k_z)\tan\psi - k_y]^2}$$ $$\times \left\{ \mathbf e_x \left[ -2ik_x - \left( -ik_x + \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + [( \frac{\omega}{c} -k_z)\tan\psi - k_y]^2} \right) e^{-ik_xx_0} \, e^{-x_0 \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + [( \frac{\omega}{c} -k_z)\tan\psi - k_y]^2} } \right] \right.$$ $$\label{I.eik.plate.TR} \left. \phantom{a} + \mathbf e_y \left[ 2i \left[ \left( \frac{\omega}{c} -k_z \right) \tan\psi - k_y \right] \vphantom{\sqrt{\left(\frac{v}{v}\right)^2}}\right.\right.$$ $$\phantom{a} -i \frac{ ( \frac{\omega}{c} -k_z)\tan\psi - k_y } { \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + [( \frac{\omega}{c} -k_z)\tan\psi - k_y]^2}} \left( -ik_x + \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + [( \frac{\omega}{c} -k_z)\tan\psi - k_y]^2} \right)$$ $$\left. \phantom{a} \times e^{-ik_xx_0} \, e^{-x_0 \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + [( \frac{\omega}{c} -k_z)\tan\psi - k_y]^2} } \right\} \,.$$ The spectral-angular density of TR is described by the formula (\[spectral.angular.1\]), however the value $F(\theta , \varphi)$ now is equal to $$\label{spectral.angular.3} F(\theta , \varphi) = \frac{1} {\left( \varkappa_x^2 + \frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2 \right)^2}$$ $$\times \left\{ 4 \left( \varkappa_x^2 + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2 - \left(-\varkappa_x^2 + \varkappa_y [(1 -\varkappa_z)\tan\psi - \varkappa_y] \right)^2 \right) \vphantom{\sqrt{\left(\frac{v}{v}\right)^2}} \right.$$ $$+\exp \left\{-2x_0 \frac{\omega}{c} \sqrt{ \frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2} \right\}$$ $$\times \left( \varkappa_x^2 + \frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2 \right) \left( 1 - \varkappa_x^2 +(1-\varkappa_y^2) \frac{[(1 -\varkappa_z)\tan\psi - \varkappa_y]^2}{\frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2} \right)$$ $$\phantom{a} + 4 \exp \left\{-x_0 \frac{\omega}{c} \sqrt{ \frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2} \right\}$$ $$\times \left( - \varkappa_x^2 (1-\varkappa_x^2) \cos \left(\frac{\omega}{c}\varkappa_x x_0 \right) - \varkappa_x \sqrt{ \frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2} \, (1-\varkappa_x^2) \sin\left(\frac{\omega}{c}\varkappa_x x_0 \right) \right.$$ $$\phantom{a} - (1-\varkappa_y^2) \, [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2 \cos\left(\frac{\omega}{c}\varkappa_x x_0 \right)$$ $$\phantom{a} - \varkappa_x (1-\varkappa_y^2) \frac{[(1 -\varkappa_z)\tan\psi - \varkappa_y]^2}{\sqrt{\frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2}} \sin\left(\frac{\omega}{c}\varkappa_x x_0 \right)$$ $$- 2\varkappa_x^2\varkappa_y [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2 \cos\left(\frac{\omega}{c}\varkappa_x x_0 \right) + \varkappa_x^3 \varkappa_y \frac{[(1 -\varkappa_z)\tan\psi - \varkappa_y]^2}{\sqrt{\frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2}} \sin\left(\frac{\omega}{c}\varkappa_x x_0 \right)$$ $$\left.\left. \phantom{a} - \varkappa_x \varkappa_y \, [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2 \, \sqrt{ \frac{c^2}{v^2\gamma^2} + [(1 -\varkappa_z)\tan\psi - \varkappa_y]^2} \, \sin\left(\frac{\omega}{c}\varkappa_x x_0 \right) \right) \right\} \,.$$ Under $x_0\to\infty$ the second and third groups of terms (that contain exponents) disappear. Hence the first group of terms can be interpreted as describing the transition radiation on the infinite slanted plate. Then the second goup of terms describes the influence from the plate’s edge, and the third one describes the interference of these two contributions. Plots of the function $\gamma^{-2} F(\theta , \varphi)$ for different values $x_0$ are presented in figure \[DR.and.TR\]. ![\[DR.and.TR\] The value $\gamma^{-2} F(\theta ,\varphi)$ that describes the angular distribution of radiation according to (\[spectral.angular.2\]) for $x_0 \omega / c\gamma = -0.5$ and according to (\[spectral.angular.3\]) for $x_0 \omega / c\gamma = 0$, 0.5, 3. In all cases $\gamma = 12$, $\psi = 75^\circ$.](DR_and_TR.png){width="\textwidth"} Discussion ========== Compare our result for DR under oblique incidence on the semi-infinite plate (\[I.eik.plate.2\])–(\[spectral.angular.2\]) with the result obtained in [@Tishch.2004; @Tishch.2] for the same problem using Durand approach (see section \[Durand.method\]). Equation (\[I.D\]) gives us $$\mathbf I^{(Durand)}_\perp = \frac{2\pi e}{v} (1-\varepsilon_\omega) \frac{\exp\left\{ i\frac{a}{\cos\psi} \left( \frac{\omega}{v} - \sqrt{\varepsilon_\omega} \, k_z \right) \right\} - 1}{i \left( \frac{\omega}{c} - \sqrt{\varepsilon_\omega} \, k_z \right)} \, e^{ -i \sqrt{\varepsilon_\omega} \, k_x x_0}$$ $$\label{I.plate.Durand} \times \frac{ \exp \left\{x_0 \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + \left[\left( \frac{\omega}{v} - \sqrt{\varepsilon_\omega} \, k_z \right)\tan\psi - \sqrt{\varepsilon_\omega} \, k_y \right]^2} \right\} } {-ik_x + \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 + \left[\left( \frac{\omega}{v} - \sqrt{\varepsilon_\omega} \, k_z \right)\tan\psi - \sqrt{\varepsilon_\omega} \, k_y \right]^2}}$$ $$\times \left\{ -\mathbf e_x + i \frac{\left( \frac{\omega}{v} - \sqrt{\varepsilon_\omega} \, k_z \right)\tan\psi - \sqrt{\varepsilon_\omega} \, k_y} { \sqrt{ \left(\frac{\omega}{v\gamma}\right)^2 +\left[\left( \frac{\omega}{v} - \sqrt{\varepsilon_\omega} \, k_z \right)\tan\psi - \sqrt{\varepsilon_\omega} \, k_y \right]^2}} \, \mathbf e_y \right\} \,.$$ Substitution of (\[I.plate.Durand\]) into (\[spectral.angular.D\]) leads to the result that coinsides with (13) in [@Tishch.2004]. The radiation cone, like in the case (\[I.eik.plate.0\]), (\[delta.function.0\]), (\[I.eik.plate.2\]), is determined by the equality to zero of the value in the square brackets, which, however, is distinct from the same in (\[I.eik.plate.2\]), and leads to the condition $$\label{delta.function.5} k_{y'} = \frac{1}{\sqrt{\varepsilon_\omega}} \frac{\omega}{v} \sin\psi$$ different from (\[delta.function.3\]). The comparison of the spectral-angular density of DR computed using equations (\[I.eik.plate.2\]) and (\[I.plate.Durand\]) is presented in the figure \[our.vs.Tishch\] (a) and (b) for the case $\varepsilon_\omega = 0.95$. We see that the cone’s opening angle is very sensitive to the value $\varepsilon_\omega$. However, it is in controversy to Cherenkov interpretation of the conical effect discussed in the section \[interpretation\]. This difficulty is fixed after account of the radiation refraction on the plate’s boundary, as it has been made in [@Tishch.2]. To this goal we have to substitute (in our denominations) $$\sqrt{\varepsilon_\omega} \varkappa_x \to \varkappa_x \,,$$ $$\label{zamena} \sqrt{\varepsilon_\omega} \, (\varkappa_z \sin\psi + \varkappa_y \cos\psi) \to \varkappa_z \sin\psi + \varkappa_y \cos\psi \quad \mbox{(that is} \ \ \sqrt{\varepsilon_\omega} \, \varkappa_{y'} \to \varkappa_{y'} ) \,,$$ $$\sqrt{\varepsilon_\omega} \, (\varkappa_z \cos\psi - \varkappa_y \sin\psi) \to \sqrt{\varepsilon_\omega - 1 + (\varkappa_z \cos\psi - \varkappa_y \sin\psi)^2}$$ for the radiation that exits the target through the forward and backward faces. Under that, only the waves meeting the condition $$\label{poln.vnut} (\varkappa_z \cos\psi - \varkappa_y \sin\psi)^2 > 1- \varepsilon_\omega$$ can leave the target (condition (27) in [@Tishch.2]). The ranges permitted by (\[poln.vnut\]) are located in figure \[geometry.1\] on the left and right sides out of the green surfaces. After the change (\[zamena\]) the cone condition (\[delta.function.5\]) transforms into (\[delta.function.3\]), and the change in (\[I.plate.Durand\]) with the following substitution into (\[spectral.angular.D\]) leads to the result (30) in [@Tishch.2]. The last result in the range permitted by the condition (\[poln.vnut\]) is very close to our (\[spectral.angular.1\])–(\[spectral.angular.2\]), see figure \[our.vs.Tishch\] (d). ![\[our.vs.Tishch\] The value $F(\theta ,\varphi)$ that describes the angular distribution of DR for $x_0 \omega / c\gamma = -3$, $\gamma = 12$, $\psi = 75^\circ$, $\varepsilon_\omega = 0.95$ according to (\[I.eik.plate.0\]), (\[spectral.angular.2\]), that is the same as in figure \[angular.1\] (a), (\[I.plate.Durand\]) (b), (30) from [@Tishch.2] (c). The results (a) and (c) are compared in the panel (d).](sravnenie.png){width="\textwidth"} The range prohibited by the condition (\[poln.vnut\]) could be filled with the radiadiation refracted by the upper face of the plate. The description of such radiation could be found in [@Tishch.1], however, the authors outline that their results are valid for a thin wide plate (the width $a$ is much larger than the plate’s thickness along the $x$ axis). Conclusion {#sec:conclusion} ========== The unified description of diffraction and transition radiation arising in the X-ray domain under oblique incidence of relativistic particle on the semi-infinite plate parallel to its upper face is presented. The calculations are based on the simple variant of eikonal approximation developed earlier [@arsa.2006; @pov.2007]. This approach accounts the evolution of the electric field created in the target by the propagating particle. The comparison with the other authors’ results [@Tishch.2004; @Tishch.2] demonstrates that our approach automatically takes into account the radiation refraction on the target boundary, and also describes the radiation leaving the target through the forward and through the upper faces in a unified way. [99]{} N.F. Shul’ga, V.V. Syshchenko, Transition radiation on semi-infinite plate and Smith–Purcell effect, Journal of Physics: Conference Series* [236]{} (2010) 012010. N.F. Shul’ga, V.V. Syshchenko, Transition radiation by relativistic electrons in inhomogeneous substance, Advanced Radiation Sources and Applications, Springer Netherlands (2006) 129. V.V. Syshchenko, N.F. Shul’ga, On the eikonal approximation in the theory of transition radiation, Journal of Surface Investigation. X-ray, Synchrotron and Neutron Techniques [1]{}, No. 2 (2007) 217. G.A. Naumenko, A.P. Potylitsyn, D.Yu. Sergeeva, A.A. Tishchenko, M.N. Strikhanov, V.V. Bleko, First experimental observation of the conical effect in Smith–Purcell radiation, JETP Letters [105]{} (2017) 553. B.M. Bolotovskii, V.L. Ginzburg, The Vavilov-Cerenkov effect and the Doppler effect in the motion of sources with superluminal velocity in vacuum, Sov. Phys. Usp. [15]{} (1972) 184. V.A. Davydov, Yu.V. Korobkin, I.V. Romanov, Theory and experimental detection of transition radiation from a charged filament, JETP [78]{}, No. 6 (1994) 844. N.F. Shul’ga, V.V. Syshchenko, Transition radiation of high energy particles on fiber-like targets, Phys. Lett. A [313]{} (2003) 307. A.A. Tishchenko, A.P. Potylitsyn, M.N. Strikhanov Diffraction radiation from an ultrarelativistic charge in the plasma frequency limit, Phys. Rev. E [70]{} (2004) 066501. D.Y. Sergeeva, A.A. Tishchenko, M.N. Strikhanov, Conical effect in diffraction radiation, NIMB [355]{} (2015) 155. D.Y. Sergeeva, A.A. Tishchenko, M.N. Strikhanov, Conical diffraction effect in optical and x-ray Smith–Purcell radiation, Phys. Rev. ST Accel. Beams [18]{} (2015) 052801. L. Durand, Transition radiation from ultrarelativistic particles, [Phys. Rev.]{} D [11]{} (1975) 89. S.J. Smith, E.M. Purcell, Visible light fom localized surface charges moving across a grating, Phys. Rev. [92]{} (1953) 1069.
--- abstract: 'We use the continuum shell model approach to explore the resonance width distribution in unstable many-body systems. The single-particle nature of a decay, the few-body character of the interaction Hamiltonian, and collectivity that emerges in non-stationary systems due to the coupling to the continuum of reaction states are discussed. Correlations between structures of the parent and daughter nuclear systems in the common Fock space are found to result in deviations of decay width statistics from the Porter-Thomas distribution.' author: - Alexander Volya bibliography: - 'ptd.bib' title: 'Porter-Thomas distribution in unstable many-body systems ' --- The Porter-Thomas distribution (PTD) [@Porter:1956] of transition strengths is a central aspect of complex systems. This statistical law was noted by many authors [@Brody:1981] to be valid more generally than other predictions of the Random Matrix Theory from which it originates. The PTD emerges under the assumption that the relative orientation of the two states involved in the overlap describing a transition covers the $\Omega$-dimensional sphere in the Hilbert space uniformly. While this is true by definition for the Gaussian Orthogonal Ensemble (GOE), the validity of the PTD extends much farther as it constitutes the central limit theorem (CLT). Being a sum of a large number of uncorrelated components, the transitional amplitude is indeed expected to have a Gaussian distribution. There is a large volume of work on this subject; see reviews [@Brody:1981; @Zelevinsky:2004; @Papenbrock:2007; @Mitchell:2010; @RevModPhys.81.539] and references therein. Generally, there is a consensus among authors that while the specifics of an ensemble and the physics of transitions do matter for certain observables, any deviations from the PTD are quickly defeated by even small stochastic components due to the robust nature of the CLT; see for example Ref. [@Grimes:1983]. Any claims to the contrary, either experimental [@PhysRevLett.105.072502] or theoretical [@Celardo:2010ru; @Whitehead:1978], have always ignited debates and discussions [@Reich:2010]. Here we will not consider the data handling procedures, which on many occasions, have been deemed to be the most likely reasons for the deviations observed experimentally [@RevModPhys.81.539]. Instead, we focus on the possible reasons for high correlation between the transitioning states from the theoretical perspective. We analyze the feasible scenarios with the help of the continuum shell model approach [@Volya:2009; @Volya:2003PRC] which is one of the most equipped methods to address the structure-reaction physics of interest microscopically. In a unified picture we review the superradiance (SR) effects [@Celardo:2010ru; @Muller:1988], the wave-function localization effects in the two-, three-, and four-body ensembles [@Kaplan:2000], the role of rotational symmetry, and other parent-daughter structural correlations that emerge in a decay [@Grimes:1983]. The dynamics of an unstable many-body system projected onto the intrinsic space spanned by the bound (shell-model) states is generated by the effective, energy-dependent Hamiltonian [@Feshbach:1991; @Mahaux:1969; @Volya:2009] $${\cal H}=H-i\sum_{c({\rm open})}{\wp}^{c}\,\|c\rangle\langle c\|.\label{eq:HAM-1}$$ Here $H$ is the Hermitian part that is identified with the traditional shell model Hamiltonian. The second, imaginary term reflects the irreversible decays into the continuum of states excluded by the Feshbach projection. This factorized operator contains the kinematic penetrability factor ${\wp}^{c}$ and the set of channel vectors $\|c\rangle.$ For simplicity we omit the angular momentum, isospin, and other labelings; detailed notations are found in Ref. [@Volya:2009]. The problem is non-stationary; the Hamiltonian is understood as a component of the propagator and is dependent on the scattering energy. The Hermitian component includes the coupling to the continuum of reaction states via virtual excitations; the penetrability also depends on energy through the kinematics of the decay process. Away from thresholds the energy dependence is smooth, and its exact form mainly pertains questions of the experimental data analysis. We ignore this dependence here, and further assume that the penetrability is the same for all channels. The eigenvalues of the effective Hamiltonian are complex, ${\cal E}=E-i\Gamma/2,$ and represent the poles of the scattering matrix in the complex energy plane. These complex energies are associated with resonances and their widths. Let us first consider weak decays, for which the imaginary component in can be treated perturbatively. In this case the shell model eigenstate $\|I\rangle$ defined by $H\|I\rangle=E_{I}\|I\rangle$ is not modified by the decay instability, and the corresponding decay width is $$\Gamma_{I}=2{\wp}\,\gamma_{I},\quad{\rm where}\quad\gamma_{I}=\sum_{c({\rm open})}\left\|\langle I\|c\rangle\right\|^{2}$$ is the reduced width. The PTD of reduced widths $$P_{\nu}(\gamma)=\frac{1}{\gamma}\left(\frac{\nu\gamma}{2\overline{\gamma}}\right)^{\nu/2}\frac{1}{\Gamma(\nu/2)}\,\exp\left(-\frac{\nu\gamma}{2\overline{\gamma}}\right)\label{eq:PTD}$$ emerges under the uniform Hilbert space coverage assumption for $\|I\rangle.$ Here $\nu$ is the dimension spanned by the channel vectors, and $\overline{\gamma}$ is the average reduced width. For the orthogonal and normalized channels $\overline{\gamma}=\nu/\Omega$. In this work we examine situations with only one open channel. The strength of the continuum coupling is defined via the average decay width relative to the level spacing. Here we express this coupling using the parameter $\kappa={\wp}/\lambda,$ where $\lambda^{2}=\Omega^{-1}{\rm Tr}\left(H^{2}\right)$ is the variance of the density of states distribution of $H$. In Fig. \[fig:sr\] we consider an example of GOE+SR, where $H$ in (\[eq:HAM-1\]) is represented by the GOE. Here the PTD is reproduced numerically in the limit ${\wp}\rightarrow0.$ The density of states in this limit has a semicircular distribution bound by a radius $2\lambda.$ The imaginary component in (\[eq:HAM-1\]) is factorized, which is pertinent to the unitarity of the scattering matrix. The non-Hermitian component, when large, gives rise to the collectivity often referred to as superradiance (SR). A similar collectivity due to the factorized Hermitian interaction describes giant resonances. The resulting deformed random ensembles are discussed in Refs. [@RefWorks:1468; @Brody:1981; @Sokolov:1988]. As coupling to the continuum increases, and the average decay width relative to the level spacing $\kappa$ becomes large, the resonances start to overlap, thus reorienting the intrinsic structure. This could be hypothesized to result in the PTD being violated [@Celardo:2010ru]. An in-depth examination, however, shows that the SR mechanism alone is unlikely to cause a significant change to the PTD. Indeed, for $\kappa\ll1,$ the PTD simply follows from the definition of GOE. For $\kappa\gg1,$ in full mathematical equivalence to the deformed ensembles, the Hilbert space is separated into the SR channel space, which is one-dimensional here with a single eigenstate ${\cal E}=-i{\wp},$ and the orthogonal statistical (compound resonance) space of dimension $\Omega-1$. Because $H$ is orthogonally invariant the reduced-space dynamics is represented by the GOE. With the perturbation theory built in this limit one finds that the reduced widths for the compound resonances follow the PTD with $\overline{\gamma}=1/(\Omega\kappa^{2}).$ For large $\Omega$ the single SR state with a reduced width $\gamma_{SR}=1-\kappa^{-2}$ has no effect on the PTD. As seen in Fig. \[fig:sr\], numerical study confirms the PTD for both small and large values of $\kappa.$ Moreover, the slight deviation from the PTD for couplings $\kappa$ between around 0.4 and 2 is due to a small fraction of very broad states with $\gamma>10\overline{\gamma}$. This localized effect is shown in the inset of Fig. \[fig:sr\]. This subset of an exceptionally broad states is difficult to identify experimentally. Investigations [@RefWorks:1468; @Brody:1981; @Sokolov:1988] of deformed ensembles further confirmed a good agreement of the decay width distribution with the PTD and of the density of states distribution with the semicircular shape. ![(Color online) The normalized distribution of probabilities of the reduced widths for the GOE+SR ensemble with $\Omega=10^{4}.$ The curves for different continuum couplings $\kappa=0.5,$ 1, 2, and 5 are compared with the PTD. The corresponding average widths $\Omega\overline{\gamma}=0.76,$ 0.40, 0.20, and 0.05, are computed after the very broad state(s) are removed from consideration. For $\kappa>2$ this amounts to exclusion of a single superradiant state in which case $\Omega\overline{\gamma}=\kappa^{-2}.$ With the exception of very broad states, there is perfect agreement with PTD, the differences being magnified in the inset where the region of $\gamma>4\overline{\gamma}$ is shown. The largest deviation is observed for the transitional coupling strength $\kappa=1$. For $\kappa=5$ the distribution is already indistinguishable from the PTD.\[fig:sr\]](srx){width="7cm"} Experience with the realistic nuclear structure and some theoretical arguments [@Brody:1981; @Zelevinsky:2004; @Papenbrock:2007; @Mitchell:2010; @RevModPhys.81.539] suggest that the effective Hamiltonian involves only few-nucleon interactions, thus, the two-body random ensembles (TBRE) appear to be more appropriate. Many features of these ensembles are different from those of GOE, nevertheless the eigenvectors form a uniform coverage of the Hilbert space. Numerical studies confirm that, in agreement with the CLT, this leads to the PTD of transition strengths toward an uncorrelated channel vector [@Grimes:1983]. This logic, however, does not take into account the correlations that exist in the variable particle-number Fock space. Strong parent-daughter correlations emerge due to the microscopic physics of decay. Indeed, in the single-particle reaction processes all nucleons, except for one, are spectators, and the decay channels for the $N-$particle system $\|c;N\rangle$ are built from the $(N-1)$-particle eigenstates of the daughter nucleus $\|F;N-1\rangle$ that follow from the same two-body Hamiltonian. Thus, $\|c;N\rangle=\left\{ a^{\dagger}\|F;N-1\rangle\right\} ,$ where $a^{\dagger}$ is a single-particle creation operator corresponding to the decaying nucleon, and brackets {...} indicate normalization to unity and appropriate symmetry coupling. The correlation between eigenstates and channels leads to the violation of PTD. Single-particle removal amplitudes are related to the independent-particle basis, where departure from the PTD has been demonstrated in the past [@Whitehead:1978]. In Fig. \[fig:eax\] the EGOE+SR ensemble is considered, where $H$ in Eq. (\[eq:HAM-1\]) is given by an embedded two-body GOE (EGOE), and by definition does not include any symmetries. The previous example shows that it is important to draw attention to the statistics of the narrow states. In this work we do not present our fits of the observed distributions to the PTD (\[eq:PTD\]) treating $\nu$ as a parameter. We found that attempts to do so could produce misleading results. The effective $\nu$ is directly related to the normalization $\overline{\gamma}$ which is disproportionately influenced by statistically unimportant collective state(s). In an attempt to avoid any potential misjudgements, here we show the distribution of absolute values of amplitudes $x=\sqrt{\gamma/\overline{\gamma}}$. Then the Gaussian distribution $P_{G}(x)=\sqrt{2/\pi}\exp\left(-x^{2}/2\right)$ that corresponds to the PTD is easy to separate from a distribution provided by the Bessel function $P_{B}(x)=(2/\pi)K_{0}(x).$ In contrast to the CLT, the latter distribution emerges when the transitional overlap is possible only due to a single component in the wave-function along some direction in the Hilbert space given by a vector $\|1\rangle$; so that $\langle I\|c\rangle=\langle I\|1\rangle\langle1\|c\rangle,$ where both $\langle I\|1\rangle$ and $\langle1\|c\rangle$ are distributed normally (agree with PTD). ![(Color online) The normalized distribution of probabilities of decay amplitudes in the EGOE+SR ensemble. The decay of a 7-particle system to the ground state of a 6-particle system is considered. There are 16 single-particle states, thus $\Omega=11440$. The curves for different continuum couplings $\kappa=0,$ 1, and 2 are compared with the Gaussian and Bessel distributions. The curves are normalized so that the average widths do not include the single SR state. $\Omega\overline{\gamma}=1,\,0.19,$ and 0.06 for $\kappa=0,\,1$ and 2 respectively. The region of very narrow widths is shown in the inset using a log-log scale. While all observed distributions for very narrow states seem to approach a constant, they are still not described by Gaussian distribution of different variances (or $\overline{\gamma}$’s). The inset includes two Gaussian curves with variances 1 and 0.02 as labeled.\[fig:eax\]](amp_eax){width="7cm"} From Fig. \[fig:eax\] we find that none of the EGOE+SR results follow the PTD. In contrast to the Gaussian curve $P_{G}(x)$ the distributions have sharp peaks at low amplitudes and an extended exponential tail. In the SR limit of large $\kappa$ the distributions appear to approach the one given by the Bessel function $P_{B}(x)$. The rank of the force beyond the two-body interaction, and symmetries, such as rotational, may have additional influences. To examine this we consider a model where $N$ identical fermions occupy a single-$j$ level. This has been a popular model for exploring the properties of TBRE [@Johnson:1998; @Zelevinsky:2004; @Papenbrock:2007]. The resemblance of the low-lying spectra to those observed in realistic nuclei is the most intriguing feature. For our demonstration we select $j=19/2$ and discuss widths of the decay of many-body states in $9$-particle systems. The final state is the ground state of the system with $8$ nucleons. All states, in both parent and daughter nuclei, are eigenstates of the same Hamiltonian given by the $n-$body Random Ensemble ($n$-BRE) [@Volya:2008]. In this work we restrict our consideration to the two-, three-, and four-body forces, $n=2,3,$ and 4, respectively. We select only those realizations where the daughter system has ground state spin $F=0;$ and thus the channel spin is $I=19/2$. The fractions of such realizations are 42%, 64%, and 83% for $n=2,3,$ and 4 respectively. The resemblance between random ensembles with symmetries and realistic nuclei extends to parentage relations. The low-lying states in the odd-particle parent nucleus are predominantly of the single-particle nature. If $F=0$ for the even-particle core then the ground state of a system with an extra nucleon is likely to carry the single-particle quantum numbers $j=I=19/2.$ This is indeed observed, and the corresponding probabilities are 21%, 47%, and 37% for $n=2,3,$ and 4 respectively. The correlation between the parent and daughter ground states is demonstrated in Fig. \[fig:Identical-nucleons\] which shows the distribution of reduced widths for the decay from ground state to ground state when $F=0$ and $I=19/2$. Both parent and daughter systems have correlated structures because they are eigenstates of the same Hamiltonian for different number of particles. In the distribution of spectroscopic factors this correlation is seen as a peak near $\gamma=1.$ As the rank of interaction $n$ becomes higher, more remote configurations can be admixed, which reduces the ground state to ground state transitional collectivity. For totally uncorrelated systems the transitional strength due to the CLT is expected to follow the PTD. ![(Color online) Distribution of the reduced decay widths (spectroscopic factors) for the ground state $N=9$ spin $I=19/2$ parent decaying to the ground state $N=8$ spin $F=0$ daughter system. The $n$-BRE of identical nucleons in $j=19/2$ level is considered. Three curves correspond to two-, three-, and four-body ($n=2,3,$ and 4) random ensembles with rotational symmetry.\[fig:Identical-nucleons\]](CSN9){width="7cm"} Since most of the transitional strength is concentrated in a few states at the low end of the spectrum, here the PTD is not generally expected; however, there is also no agreement with this law for the widths of highly excited compound resonances. In Fig. \[fig:PT\] the distribution of reduced decay amplitudes is shown for the two-, three-, and four-body random ensembles with rotational symmetry. The curves are quite close to each other and are similar to the EGOE result ($\kappa=0$ in Fig. \[fig:eax\]). All findings indicate violation of the PTD. ![(Color online) Same model as in Fig. \[fig:Identical-nucleons\] The distribution of spectroscopic amplitudes for all 204 states with spin $I=19/2$ in the parent $9$-nucleon system is shown in the decay to the spin $F=0$ ground state of the daughter nucleus. The results are compared with the Gaussian and Bessel distributions. The curves do not change if states only in a certain energy region are considered.\[fig:PT\]](amp_pt){width="7cm"} This study is motivated by the long-standing debate in relation to the Porter-Thomas distribution and possibility of its violation [@Celardo:2010ru; @Grimes:1983; @PhysRevLett.105.072502; @Reich:2010; @Whitehead:1978]. The PTD is a robust prediction justified by the central limit theorem; it is easily confirmed for different random matrix ensembles. This, however, is a purely structural approach that does not take into account the microscopic physics of reactions. To address this we use a continuum shell model approach where violations of the PTD may result from one or a combination of the following: coherence in structure due to factorized nature of the effective Hamiltonian that reflects unitarity of the scattering matrix, the so called superradiance mechanism; parent-daughter relation between decaying systems in the common Fock space; few-body low rank interaction forces; and significant variations in the energy dependence of the effective Hamiltonian. We examine all of these possibilities, with the exception of the last, which is to be discussed elsewhere. In agreement with the studies of deformed ensembles we find that the SR mechanism by itself does not lead to violations of the PTD. However, the distribution, unambiguously different from PTD, is observed in random ensembles with a particle decay and with few-body intrinsic Hamiltonians. This remains true even when subsets of states are considered: restricted by a certain energy region, or by some reasonable limits on the value of the decay width itself. The latter allows for any collective or unmeasurably broad states to be excluded. The parent-daughter relation in the decay process appears to be central for this phenomenon, the results being only slightly influenced by SR, additional rotational symmetry or the rank of forces. The author is thankful to Kirby Kemper for motivating discussions. Support from the U. S. Department of Energy, grant DE-FG02-92ER40750 is acknowledged. The computing resources were provided by the Florida State University shared High-Performance Computing facility.
--- author: - 'Planck Collaboration: P. A. R. Ade' - 'N. Aghanim' - 'M. Arnaud' - 'M. Ashdown' - 'J. Aumont' - 'C. Baccigalupi' - 'A. Balbi' - 'A. J. Banday' - 'R. B. Barreiro' - 'J. G. Bartlett' - 'E. Battaner' - 'K. Benabed' - 'A. Benoît' - 'J.-P. Bernard' - 'M. Bersanelli' - 'R. Bhatia' - 'J. J. Bock' - 'A. Bonaldi' - 'J. R. Bond' - 'J. Borrill' - 'F. R. Bouchet' - 'F. Boulanger' - 'M. Bucher' - 'C. Burigana' - 'P. Cabella' - 'C. M. Cantalupo' - 'J.-F. Cardoso' - 'A. Catalano' - 'L. Cayón' - 'A. Challinor' - 'A. Chamballu' - 'R.-R. Chary' - 'L.-Y Chiang' - 'P. R. Christensen' - 'D. L. Clements' - 'S. Colombi' - 'F. Couchot' - 'A. Coulais' - 'B. P. Crill' - 'F. Cuttaia' - 'L. Danese' - 'R. D. Davies' - 'R. J. Davis' - 'P. de Bernardis' - 'G. de Gasperis' - 'A. de Rosa' - 'G. de Zotti' - 'J. Delabrouille' - 'J.-M. Delouis' - 'F.-X. Désert' - 'C. Dickinson' - 'K. Dobashi' - 'S. Donzelli' - 'O. Doré' - 'U. Dörl' - 'M. Douspis' - 'X. Dupac' - 'G. Efstathiou' - 'T. A. En[ß]{}lin' - 'E. Falgarone' - 'F. Finelli' - 'O. Forni' - 'M. Frailis' - 'E. Franceschi' - 'S. Galeotta' - 'K. Ganga' - 'M. Giard' - 'G. Giardino' - 'Y. Giraud-Héraud' - 'J. González-Nuevo' - 'K. M. Górski' - 'S. Gratton' - 'A. Gregorio' - 'A. Gruppuso' - 'F. K. Hansen' - 'D. Harrison' - 'G. Helou' - 'S. Henrot-Versillé' - 'D. Herranz' - 'S. R. Hildebrandt' - 'E. Hivon' - 'M. Hobson' - 'W. A. Holmes' - 'W. Hovest' - 'R. J. Hoyland' - 'K. M. Huffenberger' - 'A. H. Jaffe' - 'G. Joncas' - 'W. C. Jones' - 'M. Juvela' - 'E. Keihänen' - 'R. Keskitalo' - 'T. S. Kisner' - 'R. Kneissl' - 'L. Knox' - 'H. Kurki-Suonio' - 'G. Lagache' - 'J.-M. Lamarre' - 'A. Lasenby' - 'R. J. Laureijs' - 'C. R. Lawrence' - 'S. Leach' - 'R. Leonardi' - 'C. Leroy' - 'M. Linden-V[ø]{}rnle' - 'M. López-Caniego' - 'P. M. Lubin' - 'J. F. Macías-Pérez' - 'C. J. MacTavish' - 'B. Maffei' - 'N. Mandolesi' - 'R. Mann' - 'M. Maris' - 'D. J. Marshall' - 'P. Martin' - 'E. Martínez-González' - 'G. Marton' - 'S. Masi' - 'S. Matarrese' - 'F. Matthai' - 'P. Mazzotta' - 'P. McGehee' - 'A. Melchiorri' - 'L. Mendes' - 'A. Mennella' - 'S. Mitra' - 'M.-A. Miville-Deschênes' - 'A. Moneti' - 'L. Montier [^1]' - 'G. Morgante' - 'D. Mortlock' - 'D. Munshi' - 'A. Murphy' - 'P. Naselsky' - 'F. Nati' - 'P. Natoli' - 'C. B. Netterfield' - 'H. U. N[ø]{}rgaard-Nielsen' - 'F. Noviello' - 'D. Novikov' - 'I. Novikov' - 'S. Osborne' - 'F. Pajot' - 'R. Paladini' - 'F. Pasian' - 'G. Patanchon' - 'T. J. Pearson' - 'V.-M. Pelkonen' - 'O. Perdereau' - 'L. Perotto' - 'F. Perrotta' - 'F. Piacentini' - 'M. Piat' - 'S. Plaszczynski' - 'E. Pointecouteau' - 'G. Polenta' - 'N. Ponthieu' - 'T. Poutanen' - 'G. Prézeau' - 'S. Prunet' - 'J.-L. Puget' - 'W. T. Reach' - 'R. Rebolo' - 'M. Reinecke' - 'C. Renault' - 'S. Ricciardi' - 'T. Riller' - 'I. Ristorcelli' - 'G. Rocha' - 'C. Rosset' - 'M. Rowan-Robinson' - 'J. A. Rubiño-Martín' - 'B. Rusholme' - 'M. Sandri' - 'D. Santos' - 'G. Savini' - 'D. Scott' - 'M. D. Seiffert' - 'G. F. Smoot' - 'J.-L. Starck' - 'F. Stivoli' - 'V. Stolyarov' - 'R. Sudiwala' - 'J.-F. Sygnet' - 'J. A. Tauber' - 'L. Terenzi' - 'L. Toffolatti' - 'M. Tomasi' - 'J.-P. Torre' - 'V. Toth' - 'M. Tristram' - 'J. Tuovinen' - 'G. Umana' - 'L. Valenziano' - 'P. Vielva' - 'F. Villa' - 'N. Vittorio' - 'L. A. Wade' - 'B. D. Wandelt' - 'N. Ysard' - 'D. Yvon' - 'A. Zacchei' - 'S. Zahorecz' - 'A. Zonca' --- [^1]: Corresponding author = Ludovic.Montier@cesr.fr
--- abstract: 'In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev’s periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant ${\mathrm{K}}$-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane–Mele $\mathbb{Z}_2$-invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse Mayer-Vietoris exact sequence.' address: 'Graduate School of Mathematical Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan' author: - Yosuke Kubota bibliography: - 'bibABC.bib' - 'bibDEFG.bib' - 'bibHIJK.bib' - 'bibLMN.bib' - 'bibOPQR.bib' - 'bibSTUV.bib' - 'bibWXYZ.bib' - 'arXiv.bib' - 'bulkedge.bib' date: '9 February, 2016' title: 'Controlled topological phases and bulk-edge correspondence' --- Introduction ============ In condensed matter physics, it is known that some physical quantities are related to the topology of the Hamiltonian operator, which gives a reason for them to be quantized and robust. A classical example is the $2$-dimensional integer quantum Hall effect (IQHE). Here, the Hamiltonian operator with translation symmetry is regarded as a continuous family of self-adjoint operators on the Brillouin torus (the Pontrjagin dual of the translation group ${\mathbb Z}^2$) by the Fourier transform. If the Hamiltonian has a spectral gap at the Fermi level, the family of spectral subspaces corresponding to the gap forms a vector bundle on the Brillouin torus. The first Chern numbers of this vector bundle, which classifies vector bundles on the $2$-dimensional torus, is called their bulk index and related to the Hall conductance by the TKNN formula [@TKNN]. On the other hand, from the viewpoint of the edge channel picture, the Hamiltonian with a spectral gap at the Fermi level is insulated in the bulk and an electric flow occurs at the edge of the material. More precisely, the edge Hamiltonian obtained by cutting the bulk Hamiltonian by the projection onto the half-plane gives a continuous family of Toeplitz operators on the $1$-dimensional torus. Its spectral flow is called the edge index and regarded as the number of edge channels and hence the Hall conductance. The bulk-edge correspondence ensures that these two Hall conductances obtained from different presentations actually coincide. This is first proved by Hatsugai using the topology of a Riemann surface [@H1993PRB]. These arguments are understood systematically in terms of ${\mathrm{K}}$-theory and cyclic cohomology for ${\mathrm{C}^}$-algebras, which are basic tools in noncommutative geometry. We remark that the Brillouin torus is the same thing as (the Gelfand-Naimark dual of) the group ${\mathrm{C}^}$-algebra $C^_r ({\mathbb Z}^2)$. Actually, the bulk and edge Hamiltonian operators determine elements in ${\mathrm{K}}_0(C^_r({\mathbb Z}^2))$ and ${\mathrm{K}}_1(C^_r({\mathbb Z}))$ respectively. Moreover, the TKNN formula is translated into the language of noncommutative geometry as the pairing between these elements and the fundamental cyclic cocycle by Bellissard [@MR965981]. In [@MR1877916], a simple proof of the bulk-edge correspondence is given by using the boundary map of the six-term exact sequence of ${\mathrm{K}}$-groups associated to the Toeplitz exact sequence. A merit of this noncommutative geometric approach is that it can be applied for more general settings. Particularly, it is remarkable that it enables us to deal with the case that the translation symmetry is broken by disorder . Here, the study of the topology of quantum systems is divided into two problems: classification of topological phases (homotopy classes of gapped quantum systems) and calculation of the indices, which are related to physical quantities. Actually, in this case topological phases are no longer classified by the index but by the ${\mathrm{K}}$-group of the crossed product ${\mathbb Z}^d \ltimes C(\Omega)$, which plays the role of the noncommutative Brillouin torus. Here, $\Omega$ is a certain compactification of ${\mathbb Z}^d$ which is regarded as the probabilistic space of random Hamiltonians. The bulk index is defined in the same way as the translation-invariant case by using a cyclic cocycle. In [@MR1877916], the corresponding edge index is also defined and the bulk-edge correspondence is proved by using the Toeplitz extension for crossed products by ${\mathbb Z}$. As for bulk systems, they are generalized by Bellissard and his collaborators to materials without any translation symmetry on its distribution of atoms such as quasi-crystals. They adopt a certain groupoid ${\mathrm{C}^}$-algebra as the noncommutative Brillouin torus. In the recent study of topological insulators and topological superconductors, some of their topological invariants take values in ${\mathbb{Z}_2}$ instead of ${\mathbb Z}$. For example, consider $2$-dimensional spin quantum Hall systems [@PhysRevLett.95.226801], that is, a quantum system with time-reversal symmetry (an antilinear unitary operator) whose square is $-1$. Here, the vector bundle given by a spectral subspace of the bulk Hamiltonian determines an element in the ${\mathrm{KQ}}^0$-group of the Brillouin torus with the canonical involution, which is isomorphic to ${\mathbb Z}\oplus {\mathbb{Z}_2}$. The corresponding element in the second component ${\mathbb{Z}_2}$ is called the Kane–Mele invariant. In this case, the bulk-edge correspondence ensures that this invariant coincides with the parity of helical edge channels. It is formulated and proved by Avila–Schulz-Baldes–Villegas-Blas [@MR3063955] and Graf–Porta [@MR3123539] by means of an additional symmetry coming from spin operators and the topology of Riemann surfaces like [@H1993PRB] respectively. Moreover, a ${\mathbb{Z}_2}$-invariant is also defined for disordered systems in [@GrossmannSchulz-Baldes] and [@mathph150805485]. More generally, there are three fundamental operators in quantum mechanics: particle-hole symmetry $C$ (odd and antilinear), time-reversal symmetry $T$ (even and antilinear) and chiral (sub-lattice) symmetry $P$ (odd and linear). By Altland–Zirnbauer [@AZ1997] and Kitaev [@K2009], symmetries of quantum mechanics generated by some of these operators are classified in $10$ types, each of which corresponds to one of $2$ complex and $8$ real Clifford algebras. For each of them, classifications of topological phases (for both periodic and nonperiodic systems) are given by Thiang [@mathph14067366] and Kellendonk [@mathKT150906271] by using Karoubi’s ${\mathrm{K}}$-theory and van Daele’s ${\mathrm{K}}$-theory respectively. The notion of (bulk) index for them is introduced by Gro[ß]{}mann–Schulz-Baldes [@GrossmannSchulz-Baldes] as the pairing with a ${\mathrm{K}}$-cycle. For the bulk-edge correspondence, there is a ${\mathrm{KK}}$-theoretic approach by Bourne–Carey–Rennie [@mathKT150907210]. In addition to them, a more detailed classification for periodic systems is given by using homotopy theory in a series of works by Nittis–Gomi . In [@MR3119923], Freed and Moore studied quantum systems with symmetries given by finite extensions of translation groups such as topological crystalline insulators . A remarkable feature of their approach is that they developed the theory in an axiomatic way starting from a small number of general principles. According to Wigner’s theorem [@MR0106711] (see also Section 1 of [@MR3119923]), a symmetry of quantum mechanics is given by a twisted (i.e. even/odd projective unitary/antiunitary) representation of a group on a ${\mathbb{Z}_2}$-graded complex Hilbert space, which are classified by a triplet $(\phi,c,\tau)$ of group cocycles called twists (Definition \[defn:tw\]). From this viewpoint, a CT-symmetry (a symmetry of quantum mechanics generated by some of CPT operators) is characterized as a certain twist of a subgroup $\mathscr{A}$ of ${\mathbb{Z}_2}\times {\mathbb{Z}_2}$. Actually, the Altland–Zirnbauer classification is reproved in Proposition 6.4 of [@MR3119923] in terms of the group cohomology. They gave a classification of topological phases with the symmetry given by $(G,\phi,c,\tau)$ in terms of twisted equivariant ${\mathrm{K}}$-theory of groupoids when $G$ is a Galilean symmetry. In this paper, we introduce a new framework of topological phases, which is a relevant generalization of the one introduced in [@MR3119923] for systems with an arbitrary symmetry of quantum mechanics such as translations (not necessarily cocompact), rotations and their combinations. First, for simplicity we describe the case that the system does not have any symmetry. We focus on the property of tight-binding Hamiltonians that their propagations are controlled at the position space (or distribution of atoms) represented by a discrete metric space $X$. Actually, in previous researches such as [@MR1295473], Hamiltonian operators are assumed to be approximated in the norm topology by operators whose propagations (the maximum of $d(x,y)$ such that $H_{x,y}\neq 0$) are finite. This assumption corresponds to continuity of its Fourier transform in the case that $H$ is translation invariant. We say that such a Hamiltonian is a bulk Hamiltonian controlled at $X$ and two Hamiltonians are in the same controlled topological phase if they are homotopic (more precisely, see Definition \[defn:bulkTP\]). The norm closure $C^_u(X)$ of the algebra of bounded operators with finite propagation on $\ell ^2(X)$ is called the uniform Roe algebra of $X$. This ${\mathrm{C}^}$-algebra is similar to the noncommutative Brillouin torus. Actually, when $X={\mathbb Z}^d$ as a metric space, $C^_u(X)$ is isomorphic to the crossed product ${\mathbb Z}^d \ltimes C(\beta X)$ where $\beta X$ is the Stone–Čech compactification of $X$ (Example \[exmp:groupoid\]). Now, the set of controlled topological phases is nothing but ${\mathrm{K}}_0(C^_u(X))$ as in the noncommutative geometric approach for disordered systems (Proposition \[prp:bulk\]). The uniform Roe algebra of a metric space is one of central subjects in coarse geometry, a part of geometry studying “large-scale” properties of metric spaces and identifying two spaces that look the same from a large distance such as the Euclidean space and its lattice. Actually, the set of topological phases is flexible with respect to deformation of distribution of atoms. More precisely, it is independent of the choice of discrete subsets of the Euclidean space which is distributed uniformly (i.e. Delone sets, Definition \[defn:Delone\]). Unfortunately, the ${\mathrm{K}}$-group of the uniform Roe algebra is in general highly complicated and often has uncountably many basis. On the other hand, ${\mathrm{K}}$-theory of another ${\mathrm{C}^}$-algebra called the Roe algebra, which is originally introduced by Roe for the purpose of studying index theory of non-compact manifolds, is easy to handle and widely studied. A useful tool is the (equivariant) coarse Baum–Connes map , which gives an isomorphism between a certain ${\mathrm{K}}$-homology groups and ${\mathrm{K}}$-groups of Roe algebras (note that there is another use of the Baum-Connes isomorphism in connection with the T-duality studied in ). For example, when $X$ is a lattice of ${\mathbb{E}}^d$, the groups ${\mathrm{K}}_(C^(X))$ are isomorphic to ${\mathrm{K}}^{d-}({{\rm pt}})$. Here, we consider the canonical homomorphism of ${\mathrm{K}}$-groups induced from the inclusion $C^_u(X) \subset C^(X)$. Roughly speaking, it gives a rough classification of topological phases, which captures the primary part of the topology of quantum systems (see Remark \[remk:conc\]). Actually, it coincides with the Hall conductance defined by the pairing with the fundamental cyclic cocycle. Therefore, we call it the bulk index. Another powerful tool for the study of the ${\mathrm{K}}$-theory of (uniform) Roe algebras is the coarse Mayer–Vietoris exact sequence [@MR1219916]. Fortunately, the boundary map of the coarse Mayer–Vietoris exact sequence for a partition $X=Y_+ \cup Y_-$ provides a good explanation of the bulk-edge correspondence: it maps a bulk Hamiltonian $H$ to the corresponding edge Hamiltonian $\hat{H}$ obtained by cutting $H$ by the support projection onto $Y_1$. Following this, we give a definition of edge quantum systems controlled at a pair $Z \subset Y$ of metric spaces (corresponding to the edge and the bulk of a material) in Definition \[defn:edgeTP\]. The set of controlled edge topological phases is actually isomorphic to (a subgroup of) the ${\mathrm{K}}_{-1}$-group of the relative uniform Roe algebra $C^_u(Z \subset Y)$ (Definition \[defn:Roe\]). The edge index is also defined in the same way as the bulk case. Finally, the bulk-edge correspondence is a consequence of the fact that the Mayer–Vietoris boundary map is an isomorphism for good partitions. This formulation is more intuitive than the previous approach using Toeplitz extensions for ${\mathbb Z}$-crossed products because the edge algebra ${\mathbb Z}^{d-1} \ltimes C(\Omega)$ is a subalgebra of infinite direct product of $C^_u(Z)$ (see Corollary \[cor:Toep\] and its proof). The same arguments work for the study of quantum systems with symmetries of quantum mechanics. We deal with the situation that a discrete group $G$ acts in the position space properly and isometrically. Under an assumption on twists (Assumption \[assump:ext\]), we show that the bulk and edge topological phases are classified by a modification of the twisted equivariant ${\mathrm{K}}$-theory of invariant uniform Roe algebras (Definition \[defn:Roe\]). Here, we use the twisted equivariant ${\mathrm{K}}$-group for ${\mathrm{C}^}$-algebras defined in [@Kubota1] as a generalization of van Daele’s ${\mathrm{K}}$-theory. The bulk and edge indices take values in a modification of the twisted equivariant ${\mathrm{K}}$-group of the invariant Roe algebras, which are calculated by the equivariant coarse Baum-Connes isomorphism (Theorem \[thm:cBC\]). As long as considering about CT-symmetries, the twisted equivariant ${\mathrm{K}}$-group is the same thing as $DK$-groups in [@mathKT150906271] or the description of ${\mathrm{KR}}$-groups by Boersema–Loring [@mathOA150403284]. Moreover, our bulk index is the same thing as the index pairing in [@GrossmannSchulz-Baldes] (Remark \[remk:CTvsR\]). In particular, for type A and [[[AII]{}]{}]{} topological insulators, it coincides with the Hall-conductance and the Kane-Mele invariant respectively (Example \[exmp:2dIQHE\] and Example \[exmp:2dSQHE\]). Even in this case, our approach gives a generalization of the bulk-edge correspondence for materials whose distribution of atoms have no translation symmetry such as quasi-crystals. In particular, we obtain the bulk-edge correspondence for systems whose edge is not straight, which is studied from a different point of view for type A and [[[AII]{}]{}]{} insulators by Prodan . Moreover, in this framework we can deal with topological crystalline insulators with symmetry-preserving disorder. For example, when we consider the group $\mathscr{A} \times {\mathbb{Z}_2}$ where ${\mathbb{Z}_2}$ acts on ${\mathbb{E}}^d$ as a reflection, we obtain the same classification list as indicated in (Example \[exmp:cryst\]). Here, the use of twisted equivariant ${\mathrm{K}}$-theory is essential since we deal with the case that the reflection operator commutes or anticommutes with CPT operators. It is pointed out by Jean Bellissard and Johannes Kellendonk that, although the uniform Roe algebra is similar to the groupoid ${\mathrm{C}^}$-algebra of Bellissard, the (uniform) Roe algebras are not able to play the role of the noncommutative Brillouin zone since they are too large as the ${\mathrm{C}^}$-algebra of observables. Actually, they are not even separable. A reasonable interpretation is that the uniform Roe algebra of $X$ is the universal algebra containing every observable algebras controlled at $X$. Preliminaries {#section:2} ============= Twisted equivariant ${\mathrm{K}}$-theory ----------------------------------------- Throughout this paper, we use the letter ${\mathbb R}$ for the real number field as a real ${\mathrm{C}^}$-algebra (instead of the real line as a topological space) unless otherwise noted. We use the same letter for a real ${\mathrm{C}^}$-algebra (a real Banach $\ast$-algebra satisfying the ${\mathrm{C}^}$-condition) and the corresponding Real ${\mathrm{C}^}$-algebra (a ${\mathrm{C}^}$-algebra with an antilinear involution). For a complex ${\mathrm{C}^}$-algebra $A$, we write $A_{\mathbb R}$ for the underlying real ${\mathrm{C}^}$-algebra. For an element $a$ in a Real ${\mathrm{C}^}$-algebra $A$, we write ${{}^{0} \hspace{-0.1ex} } a :=a $ and ${{}^{1} \hspace{-0.1ex} } a:=\overline{a}$. ### Definitions First, we prepare terminologies on twists of groups in the sense of [@MR3119923]. For more details, see for example Section 2 of [@Kubota1]. \[defn:tw\] For a countable discrete group $G$, a twist* on $G$ is a triple $(\phi,c,\tau)$ where $\phi,c \in \operatorname{Hom}(G,{\mathbb{Z}_2})$ and $\tau \in H^2(G;\mathbb{T}_\phi)$, where $\mathbb{T}_\phi$ is the abelian group $\mathbb{T}$ with the $G$-action given by $g \cdot t:={{}^{\phi (g)} \hspace{-0.1ex} }t$. A $2$-cocycle $\tau \in H^2(G;\mathbb{T}_\phi)$ corresponds to a $\phi$-twisted central extension of $G$ by $\mathbb{T}$, that is, an extension $1 \to \mathbb{T} \to G^\tau \to G \to 1$ such that $gtg^{-1}={{}^{\phi(g)} \hspace{-0.1ex} } t $ for any $t \in \mathbb{T}$ and $g \in G^\tau$. For a twist $(\phi,c,\tau)$ of $G$, a $(\phi,c,\tau)$-twisted representation $({\mathscr{H}}, U)$ of $G$ (or a PUA representation [@mathph14067366]) is a unitary representation of $G^\tau$ on a ${\mathbb{Z}_2}$-graded Hilbert space ${\mathscr{H}}$ which is - $\phi$-linear ($U_g$ is complex linear if $\phi (g)=0$ and antilinear if $\phi (g)=1$), - $c$-graded ($U_g$ is even if $c(g)=0$ and odd if $c(g)=1$) and - $\tau$-projective ($U_t$ are the scalar multiplications for $t \in \mathbb{T}$). We write $\gamma _{{\mathscr{H}}}$ for the grading operator of ${\mathscr{H}}$. When $G$ is finite, we remark that ${{}^{\phi} \hspace{-0.1ex} } {\mathscr{H}}_G^{c,\tau}:= (\ell^2(G) \oplus \ell^2(G)^{\mathrm{op}})^\infty \operatorname{\hat{\otimes}}\mathscr{V}$ is independent of the choice of $(\phi,c,\tau)$-twisted representations $\mathscr{V}$ and contains all separable $(\phi,c,\tau)$-twisted representations of $G$ when $G$ is finite. We simply write ${{}^{\phi} \hspace{-0.1ex} } {\mathbb{K}}_G^{c,\tau}:={\mathbb{K}}({{}^{\phi} \hspace{-0.1ex} } {\mathscr{H}}_G^{c,\tau})$. Note that ${\mathbb{K}}_{G}^{c_1,\tau _1} \operatorname{\hat{\otimes}}{\mathbb{K}}_G^{c_2,\tau_2} \cong {\mathbb{K}}_G^{c,\tau}$ where $c:=c_1+c_2$ and $\tau:=\tau _1 +\tau _2 + \epsilon (c_1,c_2)$. Here, the $2$-cocycle $\epsilon (c_1,c_2)$ is determined by $\epsilon (c_1,c_2)(g,h):=(-1)^{c_1(h)c_2(g)}$. A $\phi$-twisted $G$-${\mathrm{C}^}$-algebra is a ${\mathrm{C}^}$-algebra $A$ together with a $\phi$-linear $G$-action $\alpha : G\times A \to A$. For example, any Real $G$-${\mathrm{C}^}$-algebra has a structure of $\phi$-twisted ${\mathrm{C}^}$-algebra for arbitrary $\phi$ by the action $\tilde{\alpha} _g(a):={{}^{\phi(g)} \hspace{-0.1ex} }\alpha _g(a)$. In particular, we call the action $\alpha _g(a):={{}^{\phi(g)} \hspace{-0.1ex} }a$ the trivial* $\phi$-twisted action of $G$ on a Real ${\mathrm{C}^}$-algebra $A$. For a $\phi$-twisted $G$-${\mathrm{C}^}$-algebra $A$, we say that $a \in A$ is $c$-twisted $G$-invariant if $\alpha _g(a)=(-1)^{c(g)}a$ and we write $\mathscr{F}^G_{c,\mathscr{V}}(A)$ for the space of $c$-twisted $G$-invariant symmetries (i.e. self-adjoint unitaries) $s \in A \operatorname{\hat{\otimes}}{\mathbb{K}}(\mathscr{V})$. For an inclusion $\mathscr{V} \subset \mathscr{W}$, we have $\mathscr{F}^G_{c,\mathscr{V}}(A) \to \mathscr{F}^G_{c, \mathscr{W}}(A)$ given by $s \mapsto s \oplus \gamma_{\mathscr{V}^\perp}$. \[defn:twK\] Let $G$ be a finite group and let $A$ be a $\phi$-twisted $G$-${\mathrm{C}^}$-algebra. We define the twisted equivariant ${\mathrm{K}}_0$-groups by $${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,c,\tau}^G(A):= \bigcup \nolimits_{\mathscr{V}} \mathscr{F}^G_{c,\mathscr{V}}(A)/\sim _{\mathrm{homotopy}}$$ where $\mathscr{V}$ runs over all finite dimensional $(\phi,c,\tau)$-twisted representations. In the same way as the usual ${\mathrm{K}}$-theory, we have an isomorphism $${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1,c,\tau}^G(A)\cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,c,\tau }^G ({\mathbb{B}}({\mathscr{H}}_{G,A})/ {\mathbb{K}}({\mathscr{H}}_{G,A}))$$ where ${\mathscr{H}}_{G,A}:=\ell^2(G)^\infty \otimes A$. In other words, $${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1,c,\tau}^G(A) \cong \mathrm{Fred}_{c}^{G} ({{}^{\phi} \hspace{-0.1ex} } {\mathscr{H}}_{G,A}^{c,\tau})/\sim _{\mathrm{homotopy}},$$ where $\mathrm{Fred}_{c}^G ({{}^{\phi} \hspace{-0.1ex} } {\mathscr{H}}_{G,A}^{c,\tau})$ is the set of bounded, $c$-twisted $G$-invariant and self-adjoint operators $F$ on ${{}^{\phi} \hspace{-0.1ex} } {\mathscr{H}}_{G,A}^{c,\tau}:={{}^{\phi} \hspace{-0.1ex} } {\mathscr{H}}_{G}^{c,\tau} \otimes A$ such that $F^2-1$ is compact. For an exact sequence $0 \to I \to A \to A/I \to 0$ of $\phi$-twisted $G$-${\mathrm{C}^}$-algebras, we define the homomorphism $$\partial : {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,\tau,c}^G(A/I)\to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1,\tau,c}^G(I), \ \partial [s]=[\sigma(\hat{s})]$$ where $$\sigma: A \operatorname{\hat{\otimes}}{\mathbb{K}}(\mathscr{V}) \subset A \operatorname{\hat{\otimes}}{\mathbb{K}}({{}^{\phi} \hspace{-0.1ex} } {\mathscr{H}}_G^{c,\tau}) \to \mathcal{M}(I \otimes {\mathbb{K}}({{}^{\phi} \hspace{-0.1ex} } {\mathscr{H}}_{G}^{c,\tau})) \cong {\mathbb{B}}({{}^{\phi} \hspace{-0.1ex} } {\mathscr{H}}_{G,I}^{c,\tau})$$ is the $\ast$-homomorphism as in Proposition 2.1 of [@MR1325694] (for a ${\mathrm{C}^}$-algebra $A$, we write $\mathcal{M}(A)$ for its multiplier algebra) and $\hat{s}$ is a $c$-twisted $G$-invariant self-adjoint lift of $s$ in $A \operatorname{\hat{\otimes}}{\mathbb{K}}(\mathscr{V})$. Theorem 5.14 of [@Kubota1] asserts that $$\cdots \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,\tau,c}^G(A) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,\tau,c}^G(A/I) {\xrightarrow}{\partial} {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1,\tau,c}^G(I) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1,\tau,c}^G(A) \to \cdots$$ is a long exact sequence. \[remk:detw\] In [@Kubota1], the twisted equivariant ${\mathrm{K}}$-groups ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{,c,\tau}(A)$ are in general defined for ${\mathbb{Z}_2}$-graded $\phi$-twisted $G$-${\mathrm{C}^}$-algebras. We remark that [$$\begin{aligned} {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{,c,\tau}(A) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_^G(A \operatorname{\hat{\otimes}}{\mathbb{K}}_G^{c,\overline{\tau}}) \cong {\mathrm{KR}}^G_((A \operatorname{\hat{\otimes}}{\mathbb{K}}_G^{c,\overline{\tau}})_{\mathbb R}) \label{form:detw} \end{aligned}$$]{} where $\overline{\tau}:=\tau + \epsilon (c,c)$ (Proposition 3.12 of [@Kubota1]). For a $\phi$-twisted ${\mathbb{Z}_2}$-graded $G$-${\mathrm{C}^}$-algebra $A$, the twisted crossed product $G \ltimes ^\phi _{c,\tau}A$ is the ${\mathbb{Z}_2}$-graded real ${\mathrm{C}^}$-algebra generated by unitaries $\{ u_g \}_{g \in G}$ and $A_{\mathbb R}$ with the relations $$u_gau_g^=(-1)^{c(g)\|a\|}\alpha _g(a), u_gu_h=\tau(g,h)u_{gh},$$ which has a canonical ${\mathbb{Z}_2}$-grading determined from that of $A$ and $\gamma(u_g):=(-1)^{c(g)}u_g$ (for more precise definition, see Definition 4.4 of [@Kubota1]). The twisted Green-Julg theorem (Theorem 4.10 of [@Kubota1]) asserts that there is an isomorphism [$$\begin{aligned} {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{,c,\tau}(A) \cong {\mathrm{KR}}_(G \ltimes^\phi_{c,-\overline{\tau}} A ). \label{form:GJ} \end{aligned}$$]{} \[remk:KK\] In Section 3 of [@Kubota1], the twisted analogue of the ${\mathrm{KK}}$-group, in particular the ${\mathrm{K}}$-homology group, is also defined. An element of ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{KK}}^G_{c,\tau}(A,{\mathbb R})$ is represented by a ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_G^{0,c,\tau}$-cycle* , that is, a triple $({\mathscr{H}}, \varphi, F)$ where ${\mathscr{H}}$ is a $(\phi,c,\tau)$-twisted representation of $G$, $\varphi : A \to {\mathbb{B}}({\mathscr{H}})$ is a graded $\ast$-homomorphism, and $F \in {\mathbb{B}}({\mathscr{H}})$ is an odd self-adjoint operator such that $[F,\varphi (a)], \varphi (a)(F^2-1), \varphi (a)[U_g,F] \in {\mathbb{K}}$ and $[U_g,\varphi (a)]=0$ for any $a \in A$ and $g \in G$. The twisted Kasparov product gives a pairing $$\langle {\text{\textvisiblespace}}, {\text{\textvisiblespace}}\rangle : {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{c_1,\tau_1} (A) \otimes {{}^{\phi} \hspace{-0.1ex} } {\mathrm{KK}}^G_{c_2,\tau_2}(A, {\mathbb R}) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{c,\tau}({\mathbb R}),$$ where $c=c_1+c_2$ and $\tau =\tau_1+\tau_2 +\epsilon (c_1,c_2)$. \[remk:lind\] For a subgroup $H \leq G$, there is a canonical isomorphism $${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{,c,\tau}(A \otimes C(G/H)) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^H_{,c,\tau}(A)$$ given by $[s] \mapsto [s_{eH}]$, where $s=\bigoplus s_{gH} \in (A\operatorname{\hat{\otimes}}{\mathbb{K}}(\mathscr{V})) \otimes C(G/H)$ (Theorem 20.5.5 of [@MR1656031]). Hence we obtain the $\ell ^2$-induction functor $${\ell ^2\mathchar`-\mathrm{Ind}}_H^G: {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^H_{,c,\tau}(A) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{,c,\tau}(A \otimes C(G/H)) {\xrightarrow}{l_} {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{,c,\tau}(A),$$ where $l:C(G/H) \to {\mathbb{K}}(\ell^2(G/H))$ is the inclusion. For example, in the case of topological ${\mathrm{K}}$-theory, $\operatorname{Ind}_G^{G \times \mathscr{T}} : {\mathrm{K}}_G^0(X) \to {\mathrm{KR}}_G^0(X)$ maps a complex vector bundle $V$ to the underlying real vector bundle $V_{\mathbb R}$. Passing through the Green–Julg isomorphism (\[form:GJ\]), ${\ell ^2\mathchar`-\mathrm{Ind}}_H^G$ is the same thing as the homomorphism of ${\mathrm{KR}}$-groups induced from the inclusion of real ${\mathrm{C}^}$-algebras $H \ltimes ^\phi _{c,-\overline{\tau}}A \to G \ltimes ^\phi_{c,-\overline{\tau}}A$. Finally, we remark that the notion of twists and twisted equivariant ${\mathrm{K}}$-theory are in general defined for groupoids. For the definition, see Section 7 of [@MR3119923]. In particular, for an action groupoid $G \ltimes X$, a twist $\nu \in H^2_G(X, \mathbb{T}_\phi)$ corresponds to a $(\phi ,c,\tau)$-twisted $G$-equivariant Hilbert bundle $\mathscr{E}$ (Theorem 2.1 of [@Kubota1]) and ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{,c,\nu}_G(X) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{}_G({\mathbb{K}}(\mathscr{E}))$. \[exmp:ext\] Let $1 \to\Pi \to G \to Q \to 1$ be an extension of discrete groups such that $Q$ is finite and $\Pi$ is free abelian and let $(\phi,c,\tau)$ be a twist of $Q$. We write $B_\Pi$ for the Pontrjagin dual of $\Pi$, on which $q \in Q$ acts as $$(q \cdot {\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}})(a)={{}^{\phi(q)} \hspace{-0.1ex} }{\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}(\tilde{q}a\tilde{q}^{-1})$$ where $\tilde{q}$ is a lift of $q$ on $G$. As in Section 2 of [@MR1909514], we write $\ell ^2_\Pi(G)$ for the completion of $c_c(G)$ by the $C^_r(\Pi)$-valued inner product $$\langle f_1,f_2 \rangle= \sum\nolimits_{h \in \Pi} \overline{f_1(gh)}f_2(h) \lambda _h.$$ In other words, $\ell^2_\Pi(G)$ is the section space of a Hilbert bundle over $B_\Pi$. Hence the $\phi$-twisted $G$-action $U_g(f)(h):={{}^{\phi(g)} \hspace{-0.1ex} } f(gh)$ on $\ell^2_\Pi(G)$ gives a $(\phi,\nu)$-twisted action of $Q$ for some $\nu$. By Theorem 4.1 and Corollary 3.7 of [@MR1002543], we can check that $$G \ltimes B \cong Q \ltimes _\nu (C^_r(\Pi) \otimes B) \sim_{\mathrm{Morita}} Q \ltimes ({\mathbb{K}}(\ell^2_\Pi(G)) \otimes B)$$ for any $Q$-${\mathrm{C}^}$-algebra $B$ and hence we obtain [$$\begin{aligned} {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_Q^{,c,\tau+\nu}(B_\Pi) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^Q_{,c,\tau}({\mathbb{K}}(\ell ^2_\Pi(G))) \cong {\mathrm{KR}}_(G \ltimes ({{}^{\phi} \hspace{-0.1ex} } {\mathbb{K}}_Q^{c,\overline{\tau}})_{\mathbb R}) \label{form:extK} \end{aligned}$$]{} by (\[form:detw\]) and the Green–Julg theorem. When $\nu =0$ (that is, $G \cong Q \ltimes \Pi$), the collapsing map $B_\Pi \to {{\rm pt}}$ and the inclusion ${{\rm pt}}\to B_\Pi$ into the origin induce a direct sum decomposition $${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_Q^{,c,\tau}(B_\Pi) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{,c,\tau} _Q(B_\Pi, {{\rm pt}}) \oplus {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{,c,\tau} _Q({{\rm pt}}),$$ where ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{,c,\tau} _Q(B_\Pi, {{\rm pt}})$ is the reduced ${\mathrm{K}}$-group and ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{,c,\tau} _Q({{\rm pt}})$ is the trivial part. On the other hand, there is no such decomposition when $\nu \neq 0$. In this case, we characterize “trivial” elements as following. For each finite subgroup $F$ of $G$, we identify it with the image $q(F) \subset Q$ and set $H:=q^{-1}(F)$. Then, $H \cong F \ltimes \Pi$ since there is a splitting. Now we consider ${\ell ^2\mathchar`-\mathrm{Ind}}_F^Q: {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{,c,\tau} _F(B_\Pi) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{,c,\tau+\nu} _Q(B_\Pi)$ and we write $$\mathrm{Triv} := \mathop{\mathrm{span}} \{ {\ell ^2\mathchar`-\mathrm{Ind}}_{F}^Q ({{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{,c,\tau} _F({{\rm pt}})) \mid \text{$F \leq G$ is finite} \} \subset {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{,c,\tau +\nu}_Q(B_\Pi).$$ ### CT-type symmetries For a quadruple $(G, \phi , c, \tau)$, we write $G_0:=\operatorname{Ker}(\phi,c)$ and $\mathscr{A}:=\operatorname{\mathrm{Im}}(\phi,c) \subset {\mathbb{Z}_2}\times {\mathbb{Z}_2}$. We say that a quadruple $(G, \phi,c,\tau)$ is - a CT-type symmetry* if there is a splitting $s: \mathscr{A} \to G$ and $\tau$ is the pull-back of a $2$-cocycle on $\mathscr{A}$, - a product CT-type symmetry if $G=\mathscr{A} \times G_0$ and $\tau$ is a $2$-cocycle on $\mathscr{A}$, - a CT-symmetry if $G_0$ is trivial. For a CT-type symmetry, we say that the pair $(\mathscr{A}, \tau)$ is its CT-type. It is proved in Proposition 6.4 of [@MR3119923] that there are $10$ choices of pairs $(\mathscr{A}, \tau)$ where $\mathscr{A}$ of $\mathscr{G}:={\mathbb{Z}_2}\times {\mathbb{Z}_2}$ and $\tau$ is a $2$-cocycle of $\mathscr{A}$, each of which corresponds to a Cartan labels as indicated in Table \[table:CT\]. Here, we write $\mathscr{C}$, $\mathscr{T}$, $\mathscr{P}$ for subgroups of $\mathscr{G}$ generated by $\underline{C}:=(1,1)$, $\underline{T}:=(1,0)$, $\underline{P}:=(0,1)$ respectively. In fact, $(\mathscr{A},\tau)$ is classified by $C^2=\pm 1$ and $T^2=\pm 1$ where $C,T,P$ are lifts of $\underline{C}, \underline{T},\underline{P}$ in $\mathscr{A}^\tau$ such that $P^2=1$ and $CT=P$. \[lem:KCT\] Let $(G \times \mathscr{A},\phi,c,\tau)$ be a product CT-type symmetry and let $A$ be a Real $G$-${\mathrm{C}^}$-algebra on which $\mathscr{A}$ acts trivially. Then, ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,c,\tau}^{G \times \mathscr{A}}(A)$ is isomorphic to one of ${\mathrm{K}}_^G(A)$ or ${\mathrm{KR}}_^G(A)$ as indicated in Table \[table:CT\] below. It follows from the fact that the twisted group algebras of CT-symmetries are isomorphic to Clifford algebras and (\[form:GJ\]). In this paper, we simply write as ${\mathrm{KL}}_ ^ G(A):={{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^{G \times \mathscr{A}}(A)$ by using its Cartan label ${\mathrm{L}}$. For example, ${\mathrm{KA\hspace{-.08em}I\hspace{-.08em}I}}^G_(A) \cong {\mathrm{KQ}}_^G(A)$ and ${\mathrm{KB\hspace{-.08em}D\hspace{-.08em}I}}^G_(A) \cong {\mathrm{KR}}_{+1}^G(A)$. \[0.92\] $\mathscr{A}$ $1$ $\mathscr{P}$ ---------------------- ---------------------- ---------------------- --------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- $C^2$ $1$ $-1$ $1$ $1$ $-1$ $-1$ $T^2$ $1$ $-1$ $1$ $-1$ $1$ $-1$ Cartan [[A]{}]{} [[AIII]{}]{} [[AI]{}]{} [[AII]{}]{} [[D]{}]{} [[C]{}]{} [[BDI]{}]{} [[DIII]{}]{} [[CI]{}]{} [[CII]{}]{} ${\mathrm{KL}}_0^G $ ${\mathrm{K}}_{0}^G$ ${\mathrm{K}}_{1}^G$ ${\mathrm{KR}}_0^G$ ${\mathrm{KR}}_{4}^G$ ${\mathrm{KR}}_{2}^G$ ${\mathrm{KR}}_{6}^G$ ${\mathrm{KR}}_{1}^G$ ${\mathrm{KR}}_{3}^G$ ${\mathrm{KR}}_{7}^G$ ${\mathrm{KR}}_{5}^G$ : The 10-fold way for twisted equivariant ${\mathrm{K}}$-groups[]{data-label="table:CT"} \[remk:CTvsR\] The groups ${\mathrm{KL}}_0(A)$ are essentially the same thing as the $DK$-groups studied in [@mathKT150906271]. Moreover, for real Cartan labels, the isomorphisms ${\mathrm{KL}}_0(A) \cong {\mathrm{KR}}_j(A)$ are given in another way by Boersema–Loring [@mathOA150403284]. Actually, when $P \not \in \mathscr{A}$, $U^{(j)}_\infty (A)$ in [@mathOA150403284] is the same thing as $\bigcup {{}^{\phi} \hspace{-0.1ex} } {\mathscr{F}}_{c,\mathscr{V}}^{\mathscr{A}}(A)$. Similarly, when $P \in \mathscr{A}$, $s \mapsto \Phi(s\gamma_\mathscr{V})\Phi$ (where $\Phi:=(1+P)/2$) gives a one-to-one correspondence between $\bigcup {{}^{\phi} \hspace{-0.1ex} } {\mathscr{F}}_{c,\mathscr{V}}^{\mathscr{A}}(A)$ and $U^{(j)}_\infty (A)$. Set $\mathscr{V}':=\mathscr{V}$ if $P \not \in \mathscr{A}$ and $\mathscr{V}':=\Phi\mathscr{V}$ if $P \in \mathscr{A}$. They introduce Real ${\mathrm{C}^}$-algebras $A_j$ which are ${\mathrm{KK}}$-equivalent to ${\mathrm{C}\ell}_{0,j}$ and ${\mathrm{KKR}}(A_j, A) \cong [A_j, A \otimes {\mathbb{K}}]$. By functoriality, for any $s \in {{}^{\phi} \hspace{-0.1ex} } {\mathscr{F}}_{c,\mathscr{V}}^{\mathscr{A}}(A)$ there is a $\ast$-homomorphism $\psi _s:A_j \to A \otimes {\mathbb{K}}(\mathscr{V}')$ such that $[s]=[\psi _s(u_0)]$ under the above identification, where $u _0\in U_\infty ^{(j)}(\tilde{A}_j)$ is a distinguished element such that $[u_0] \in {\mathrm{KR}}_j(A_j) \cong {\mathbb Z}$ is the generator. This identification is useful for studying the Kasparov product. Let $(\mathscr{A},\phi,c,\tau)$ be the CT-symmetry corresponding to ${\mathrm{KR}}_d$-theory and set ${\mathscr{H}}_d:={\mathscr{H}}_{\mathscr{A}}^{c,\tau}$. Set $$\mathrm{FS}_{j,d}:=\{ (F,s) \in {{}^{\phi} \hspace{-0.1ex} } \mathrm{Fred}_c^{\mathscr{A}}({\mathscr{H}}_d)^{\mathrm{odd}} \times U^{(j)}({\mathbb{B}}({\mathscr{H}}_d))^{\mathrm{even}} \mid [F,s] \in {\mathbb{K}}\}$$ if $j=-1,0,1,2$ and $\mathrm{FS}_{j,d}:=\mathrm{FS}_{j-4,d+4}$ if $j=3,4,5,6$. Applying the above identification for the ${\mathrm{C}^}$-algebra $D:=\{ T \in {\mathbb{B}}({\mathscr{H}}_d)^{\mathrm{even}} \mid [T,F] \in {\mathbb{K}}\}$, we obtain an isomorphism ${\mathrm{KR}}^d(A_j) \cong \pi_0(\mathrm{FS}_{j,d})$ (note that ${\mathrm{KR}}^d(A_{j+4})={\mathrm{KR}}^d(A_j \otimes \mathbb{H}) \cong {\mathrm{KR}}^{d+4}(A_j)$). Let $s \in {{}^{\phi} \hspace{-0.1ex} } {\mathscr{F}}^\mathscr{A}_{c,\mathscr{V}}(A)$ corresponding to $\psi _s : A_j \to A \otimes {\mathbb{K}}(\mathscr{V}')$. Then, by the above identification, the Kasparov product $${\mathrm{KKR}}(A_j,A) \otimes {\mathrm{KKR}}^d(A,{\mathbb R}) \to {\mathrm{KKR}}^d(A_j,{\mathbb R})$$ is given by $$[\psi _s] \otimes_A [{\mathscr{H}}, \varphi , F] = [{\mathscr{H}}\otimes \mathscr{V}', \psi _s \circ \varphi, F \otimes 1],$$ which corresponds to $[F \otimes 1, (\varphi \otimes {\mathrm{id}}_{{\mathbb{K}}(\mathscr{V}')})(s)] \in \pi_0(\mathrm{FS}_{j,d})$ under an arbitrary embedding of ${\mathscr{H}}\otimes \mathscr{V}'$ into ${\mathscr{H}}_d$ ($j=-1,0,1,2$) or ${\mathscr{H}}_{d+4}$ ($j=3,4,5,6$). Explicit isomorphisms of $\pi _0(\mathrm{FS}_{j,d})$ with ${\mathbb Z}$ or ${\mathbb{Z}_2}$ are given by the index pairings investigated by Gro[ß]{}mann–Schulz-Baldes [@GrossmannSchulz-Baldes]. In other words, their index pairing is the same thing as the Kasparov product. \[exmp:rep\] The groups ${\mathrm{KR}}_j({\mathbb R})$ for $j=0,1,2,3,4,5,6,7$ are isomorphic to ${\mathbb Z}, {\mathbb Z}_2, {\mathbb Z}_2, 0 , {\mathbb Z}, 0,0,0$ (the $8$-fold Bott periodicity). Hence the twisted equivariant ${\mathrm{K}}$-group ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{\mathscr{A}}_{0,c,\tau}({\mathbb R})$ for CT-symmetries are isomorphic to one of ${\mathbb Z}$, ${\mathbb{Z}_2}$ or $0$ as indicated in Table \[table:Kitaev\]. \[0.90\] $d$ A [[[AIII]{}]{}]{} [[[AI]{}]{}]{} [[[BDI]{}]{}]{} D [[[DIII]{}]{}]{} [[[AII]{}]{}]{} [[[CII]{}]{}]{} C [[[CI]{}]{}]{} ----- --------------- ------------------ ---------------- ------------------ ------------------ ------------------ ------------------ ------------------ --------------- ---------------- $0$ ${\mathbb Z}$ $0$ ${\mathbb Z}$ ${\mathbb{Z}_2}$ ${\mathbb{Z}_2}$ $0$ ${\mathbb Z}$ $0$ $0$ $0$ $1$ $0$ ${\mathbb Z}$ $0$ ${\mathbb Z}$ ${\mathbb{Z}_2}$ ${\mathbb{Z}_2}$ $0$ ${\mathbb Z}$ $0$ $0$ $2$ ${\mathbb Z}$ $0$ $0$ $0$ ${\mathbb Z}$ ${\mathbb{Z}_2}$ ${\mathbb{Z}_2}$ $0$ ${\mathbb Z}$ $0$ $3$ $0$ ${\mathbb Z}$ $0$ $0$ $0$ ${\mathbb Z}$ ${\mathbb{Z}_2}$ ${\mathbb{Z}_2}$ $0$ ${\mathbb Z}$ : the group ${\mathrm{KL}}_d({\mathbb R})$[]{data-label="table:Kitaev"} The group ${\mathrm{KR}}^{G}_{d}({\mathbb R})$ for $d=0,\dots, 7$ is written in terms of the representation ring as $$R_{\mathbb R}(G), R_{\mathbb R}(G)/\rho R(G), R(G)/jR_\mathbb{H}(G), 0, R_\mathbb{H}(G), R_{\mathbb{H}}(G)/\eta R(G), R(G)/iR_{\mathbb R}(G), 0$$ (for the definition of $i$, $j$, $\rho$ and $\eta$, see Section 6 and Section 8 of [@MR0259946]). Hence, we can write down ${\mathrm{KL}}^G_({\mathbb R})$ in terms of the representation rings. These groups are also understood in terms of the Frobenius–Schur indicator (Chapter 13.2 of [@MR0450380]). For any finite group $G$, $$C^_r(G) \cong \bigoplus _{\pi \in \operatorname{Irr}(G)_1} {\mathbb{M}}_{l_\pi}({\mathbb R}) \oplus \bigoplus _{[\pi] \in \operatorname{Irr}(G)_0/\sim} {\mathbb{M}}_{l_\pi}({\mathbb C}) \oplus \bigoplus _{\pi \in \operatorname{Irr}(G)_{-1}} {\mathbb{M}}_{l_\pi/2}({\mathbb Q}),$$ where $l_\pi:=\dim_{\mathbb C}\pi$, $\operatorname{Irr}(G)_{i}$ is the set of irreducible representations whose Frobenius–Schur indicator are $i=1,0,-1$ and $\pi \sim \pi'$ in $\operatorname{Irr}(G)_0$ if and only if $\pi'=\pi$ or $\pi^$. Hence we obtain [$$\begin{aligned} {\mathrm{KR}}_{}^G({\mathbb R}) \cong {\mathrm{KR}}_{}({\mathbb R})^{n_1} \oplus {\mathrm{K}}_{}({\mathbb C})^{n_0/2} \oplus {\mathrm{KQ}}_{}({\mathbb R})^{n_{-1}} \label{form:group} \end{aligned}$$]{} where $n_i:=\|\operatorname{Irr}(G)_i\|$ for $i=1,0,-1$. For example, consider the case that $G={\mathbb Z}/k{\mathbb Z}$. By the Pontrjagin duality, we obtain $C^_r(C_{k}) \cong C(\hat{G})$ with the involution ${\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}\mapsto \overline{{\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}}$ (here $\hat{G}$ is identified with the group of roots of unity). Hence we obtain [$$\begin{aligned} {\mathrm{KL}}^{C_k}_({\mathbb R}) \cong \begin{cases} {\mathrm{K}}_{j+}({\mathbb C})^k & {\mathrm{KL}}_ \cong {\mathrm{K}}_{j+}, \\ {\mathrm{KR}}_{j+} ( {\mathbb R}) \oplus {\mathrm{K}}_{j+} ({\mathbb C}) ^{\frac{k-1}{2}} & \text{${\mathrm{KL}}_ \cong {\mathrm{KR}}_{j+}$, $k$ is odd}\\ {\mathrm{KR}}_{j+} ({\mathbb R}) ^2 \oplus {\mathrm{K}}_{j+}({\mathbb C}) ^{\frac{k-2}{2}} & \text{${\mathrm{KL}}_ \cong {\mathrm{KR}}_{j+}$, $k$ is even}. \end{cases} \label{form:rot} \end{aligned}$$]{} \[exmp:ind\] As in Example \[exmp:rep\], ${\mathrm{KA\hspace{-.08em}I\hspace{-.08em}I}}^2_G({\mathbb R})$, ${\mathrm{KD\hspace{-.08em}I\hspace{-.08em}I\hspace{-.08em}I}}^1_G({\mathbb R})$ and ${\mathrm{KC\hspace{-.08em}I\hspace{-.08em}I}}^3_G({\mathbb R})$ are isomorphic to $R(G)/R_{\mathbb{H}}(G)$. In fact, the canonical surjection from $R(G)$ to these groups is given by the $\ell ^2$-induction from $G':=\operatorname{Ker}\phi$. To see this, first we reduce the problem to ${\ell ^2\mathchar`-\mathrm{Ind}}_e^{\mathscr{T}}: {\mathrm{K}}_0^G({\mathbb{C}\ell }_2) \to {\mathrm{KR}}_0^G({\mathrm{C}\ell}_{2,0})$ by the Green–Julg isomorphism (\[form:GJ\]) and show that ${\ell ^2\mathchar`-\mathrm{Ind}}_e^{\mathscr{T}}:{\mathrm{K}}^{-2}_G({{\rm pt}}) \to {\mathrm{KR}}^{-2}_G({{\rm pt}})$ is the desired surjection. It follows from the fact that the surjection $R(G) \cong M^{\mathbb R}_2(G) \to {\mathrm{KR}}_G^0(D^2,S^1) $ in Section 8 of [@MR0259946] maps $[V]$ to $$[ D ^2 \times V_{\mathbb R}, D^2 \times V_{\mathbb R}, x+iy ] \in {\mathrm{KR}}_G^0 (D^2,S^1),$$ which is the same thing as ${\ell ^2\mathchar`-\mathrm{Ind}}_e^{\mathscr{T}} (\beta \otimes [V])$ where $\beta$ is the Bott generator. \[exmp:reflect\] Let $\mathscr{R}:={\mathbb{Z}_2}$ and consider an arbitrary twist on $G:=\mathscr{R} \times \mathscr{A}$ such that $\phi$ and $c$ come from that of $\mathscr{A}$. We write the generator of ${\mathbb{Z}_2}$ as $\underline{R}$. In the same way as Lemma 6.17 of [@MR3119923], lifts $C,T,R$ of $\underline{C}$, $\underline{T}$ and $\underline{R}$ satisfies $(CT)^2=1$ and $R^2=1$. Hence, $\tau$ is classified by $C^2=\pm 1$, $T^2=\pm 1$, $CR=\pm RC$ and $TR=\pm RT$. The classification list is given in Table I of [@PhysRevB.88.075142]. Let us consider the $\mathscr{R}$-action on ${\mathrm{C}\ell}_{0,1}$ induced from the reflection of ${\mathbb R}$, that is, the $\ast$-automorphism given by $\underline{R}\cdot 1=1$ and $\underline{R} \cdot e=-e$. For the latter use, we give a calculation of the twisted equivariant ${\mathrm{K}}$-group ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{0,c,\tau}({\mathrm{C}\ell}_{0,1})$. First, we consider the case that $P \not \in \mathscr{A}$ or $PR=RP$. By definition of the twisted crossed product, we have the isomorphism $$G \ltimes ^\phi _{c,\overline{\tau}} {\mathrm{C}\ell}_{0,1} \cong \mathscr{A} \ltimes ^\phi _{c,\overline{\tau}} (\mathscr{R} \ltimes {\mathrm{C}\ell}_{0,1}).$$ Here, $\mathscr{A}$ acts on $\mathscr{R} \ltimes {\mathrm{C}\ell}_{0,1}$ as $e_1 \mapsto e_1$ and $R \mapsto \eta(g)R$, where $\eta(g) \in \{\pm 1\}$ is determined by $Rg=\eta(g)gR$. Now, the underlying complex ${\mathrm{C}^}$-algebra $\mathscr{R} \ltimes {\mathbb{C}\ell }_1$ is isomorphic to ${\mathbb{C}\ell }_{2}$ generated by $e$ and $iRe$. Hence the fixed point subalgebra of the $\mathscr{A}$-action on it is ${\mathrm{C}\ell}_{1,1}$ when $TR=RT$ and ${\mathrm{C}\ell}_{0,2}$ when $TR=-RT$. In other words, the $\mathscr{A}$-action on $R \ltimes {\mathrm{C}\ell}_{0,1}$ is the same thing as the trivial action on ${\mathrm{C}\ell}_{1,1}$ (resp. ${\mathrm{C}\ell}_{0,2}$) when $TR=RT$ (resp. $TR=-RT$). For the case that $PR=-RP$, apply the same argument for $G=\mathscr{A} \times \mathscr{R}'$, where $\mathscr{R}'$ is the subgroup generated by $RP$. It is directly checked that the underlying complex ${\mathrm{C}^}$-algebra $\mathscr{R}' \ltimes_c {\mathbb{C}\ell }_1$ is isomorphic to ${\mathbb{C}\ell }_1 \oplus {\mathbb{C}\ell }_1$ (generated by $(1+RP)e/2$ and $(1-RP)e/2$) and the $\mathscr{A}$-action on it is the same thing as the trivial action on ${\mathrm{C}\ell}_{0,1} \oplus {\mathrm{C}\ell}_{0,1}$ (resp. $({\mathbb{C}\ell }_1)_{\mathbb R}$)when $TRP=RPT$ (resp. $TRP=-RPT$). Finally, by the Green-Julg isomorphism (\[form:GJ\]) we obtain $${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{d-1,c,\tau}({\mathrm{C}\ell}_{0,1}) \cong \begin{cases}{{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{\mathscr{A}}_{d-1,c,\tau}({\mathbb R}) & \text{if $PR=RP$ and $TR=RT$},\\ {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{\mathscr{A}}_{d+1,c,\tau}({\mathbb R}) & \text{if $PR=RP$ and $TR=-RT$}.\\ {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{\mathscr{A}}_{d,c,\tau}({\mathbb R})^2 & \text{if $PR=-RP$ and $TRP=RPT$},\\ {\mathrm{K}}_{d,c,\tau}({\mathbb R}) & \text{if $PR=-RP$ and $TRP=-RPT$}.\\ \end{cases}$$ Coarse ${\mathrm{C}^}$-algebras -------------------------------- Next, we review definitions and basic properties of ${\mathrm{C}^}$-algebras arising in coarse geometry and set up our notation. The reader can find more details in . Let $X$ be a proper (i.e. all bounded closed subsets are compact) discrete metric space with a proper isometric action of $G$. A triplet $({\mathscr{H}}, U, \pi)$ is a (Real) $(X,G)$-module if $({\mathscr{H}},U)$ is a (Real) unitary representation of $G$ and $\pi:c_0(X) \to {\mathbb{B}}({\mathscr{H}})$ is a $G$-equivariant (Real) $\ast$-homomorphism (equivalently, a decomposition ${\mathscr{H}}=\bigoplus _{x \in X} {\mathscr{H}}_x$ compatible with the $G$-action). We say that a (Real) $(X,G)$-module is finite or uniformly finite if the rank of projections $\pi (\delta _x)$ ($x \in X$) are finite or uniformly finite respectively. Hereafter, we make a technical assumption on $G$: the order of finite subgroups of $G$ is uniformly bounded. For example, any discrete group acting on ${\mathbb{E}}^d$ satisfies this condition. For such $G$ acting properly and isometrically on $X$, set $\tilde{X}:=\{ (g,x) \in G \times X \mid gx=x \}$, which is a free $G$-space by the $G$-action $g \cdot (h,x):=(hg^{-1}, gx)$. Then, we obtain a uniformly finite Real $(X,G)$-module $${\mathscr{H}}_{X,G}:= \ell ^2 (\tilde{X})=\bigoplus\nolimits_{x\in X} \ell ^2 (G_x)$$ with the Real structure given by $\overline{f}(x):=\overline{f(x)}$. \[defn:Roe\] Let $X$, $G$ be as above and let $({\mathscr{H}}, U, \pi)$ be a (Real) $(X,G)$-module. We say that $T \in {\mathbb{B}}({\mathscr{H}})$ is - controlled (or $T$ has finite propagation) if there is $R>0$ such that $T_{xy}:=\pi (\delta_y)T\pi(\delta_x)=0$ if $d(x,y) >R$, - locally compact if $\pi (\delta _x)T$ and $T\pi(\delta _x)$ are compact operators for any $x \in X$, - supported near $A$ if there is $R>0$ such that $T_{xy}=0$ if $d(x , A)>R$ or $d(y,A)>R$. We write $C^_{\mathscr{H}}(X)^G$ (resp. $C^_{\mathscr{H}}(A \subset X)^G$) for the closure of the (Real) $\ast$-algebra of $G$-invariant, controlled and locally compact operators on ${\mathscr{H}}$ (resp. supported near $A$). In particular, we simply write $$\begin{array}{l} C^_u(X)^G := C^_{{\mathscr{H}}_{X,G}}(X)^G,\\ C^(X)^G := C^_{{\mathscr{H}}_{X,G}^\infty}(X)^G, \end{array} \ \begin{array}{l} C^_u(A \subset X)^G:=C^_{{\mathscr{H}}_{X,G}}(A \subset X)^G, \\ C^(A \subset X)^G := C^_{{\mathscr{H}}_{X,G}^\infty}(A \subset X )^G, \end{array}$$ and call them the $G$-invariant (relative) uniform Roe algebra and $G$-invariant (relative) Roe algebra respectively. The ${\mathrm{C}^}$-subalgebra $C^_{\mathscr{H}}(A \subset X)^G$ of $C^_{\mathscr{H}}(X)^G$ is an ideal. In this paper, we simply write $$C^_{\mathscr{H}}(X, A )^G:=C^_{{\mathscr{H}}} (X)^G/C^_{{\mathscr{H}}} (A \subset X)^G$$ and define $C^(X,A)^G$, $C^_u(X,A)^G$ in the same way. \[exmp:eqRoe\] When $G$ acts on $X$ cocompactly, $C^_u(X)^G$ is isomorphic to ${\mathbb{K}}(\ell ^2_G(X))$ (Lemma 2.3 of [@MR1909514]). For example, for the Cayley metric space $\|G\|$ on which $G$ acts by the left multiplication, we have $$C^_u(\|G\|)^G \cong C^_r(G) , \ C^(\|G\|)^G \cong C^_r(G) \otimes {\mathbb{K}},$$ where $C^_r(G)$ is the reduced group ${\mathrm{C}^}$-algebra. Similarly, for an extension $1 \to \Pi \to G \to Q \to 1$ as in Example \[exmp:ext\], the $Q$-${\mathrm{C}^}$-algebra $C^_{\ell ^2 (G)}(\|G\|)^\Pi$ is $Q$-equivariantly isomorphic to ${\mathbb{K}}(\ell^2_\Pi (G))$. \[exmp:groupoid\] It is known that these ${\mathrm{C}^}$-algebras are related to certain groupoids when $G$ is trivial. Let ${\mathcal{G}}_X$ be the translation groupoid defined in Section 3.2 of [@MR1905840]. Then, by the same argument as Lemma 4.4 of [@MR1905840], we obtain an isomorphism $$C^_{\mathscr{H}}(X) \cong {\mathcal{G}}_X \ltimes c_b(X; {\mathbb{K}}({\mathscr{H}}))$$ where $c_b(X;{\mathbb{K}}({\mathscr{H}})):=\prod _{x \in X} {\mathbb{K}}({\mathscr{H}}_x)$. For example, $C^_u(X) \cong C^_r (\mathcal{G}_X)$ and $C^(X) \cong \mathcal{G}_X \ltimes c_b(X,{\mathbb{K}})$. In particular, for the Cayley metric space $\|G\|$, we have $$C^_u(\|G\|) \cong G \ltimes c_b(G), C^(\|G\|) \cong G \ltimes c_b(G ,{\mathbb{K}}).$$ Example \[exmp:groupoid\] immediately implies that the algebra of observables ${\mathbb Z}^d \ltimes C(\Omega )$ introduced in is a subalgebra of $C^_u(X)$. The same holds for the observable algebra of aperiodic solids. Let $X$ be a Delone subset of ${\mathbb{E}}^d$ and let $\Gamma _X$ be the groupoid of the transversal of the hull of $X$ (Definition 1.9 of [@MR1798994]). Note that the target space $\Gamma _X^0$ is regarded as a compactification of $X$. \[lem:groupoid\] There is a surjective groupoid homomorphism ${\mathcal{G}}_X \to \Gamma _X$ which induces the inclusion $C^_r(\Gamma _X) \subset C^_u(X)$. Since $\Gamma _X^0$ is a compactification of $X$, we have a unique surjection $f: \beta X \to \Gamma _X^0$. Recall that the morphism set ${\mathcal{G}}_X^1$ is given by the union of all subspaces of $\beta (X \times X)$ which are closures of controlled (i.e. $d(x,y)$ is uniformly bounded) subspaces of $X \times X$. Let $Z$ be a controlled subset of $X \times X$. Then, the bounded function $v: Z \to {\mathbb R}^d$ determined by $v(x,y) \cdot x=y$ extends to the function on $\beta Z=\overline{Z} \subset \beta (X \times X)$. Now we obtain the desired groupoid morphism $z \mapsto (f(x),v(z))$. Next, we prepare some notions on inclusions of metric spaces related to the tight-binding approximation of the position space. For a subspace $Y$ of $X$ and $R>0$, we write ${\mathrm{Pen}}(Y,R):=\{ x \in X \mid d(x,Y)<R\}$. \[defn:Delone\] A discrete subspace $X$ of a metric space $M$ is - uniformly discrete* if there is $r>0$ such that any closed ball of radius $r$ in $M$ contains at most one point in $X$, - relatively dense if there is $R>0$ such that any closed ball of radius $R$ in $M$ contains at least one point in $X$, - a Delone subset if $X$ is uniformly discrete and relatively dense. For s closed subspace $N$ of $M$, we say that a pair of discrete subspaces $Z \subset Y$ of $M$ is a Delone pair of $N \subset M$ if $Y$ is a Delone subset of $M$ and the Hausdorff distance $d(N,Z)$ is finite (in other words, $Z$ is a Delone subset of ${\mathrm{Pen}}(N,R)$ for some $R$). For a partition $M=M_+ \cup M_-$, $X =Y_+ \cup Y_-$ is a Delone partition of $M=M_+ \cup M_-$ if $X$ is a Delone subset of $M$ and $d(Y_\pm , M_\pm) <\infty$, $d(Z,N)<\infty$ holds, where $N:=M_+ \cap M_-$ and $Z:=Y_+ \cap Y_-$. A discrete subspace $X$ is relatively dense in $M$ if and only if the inclusion is a coarse equivalence. If $M$ has bounded geometry, so does $X$ if and only if $X$ is uniformly discrete. If a partition $M=M_+ \cup M_-$ is $\omega$-excisive (i.e. for any $R>0$ there is $S>0$ such that ${\mathrm{Pen}}(Y_1,R) \cap {\mathrm{Pen}}(Y_2,R) \subset {\mathrm{Pen}}(Y_1 \cap Y_2,S))$, then so are its Delone partitions. Finally, we remark on coarse invariance of uniform Roe algebras. We say that two $G$-spaces are $G$-equivariantly coarsely equivalent if there are $G$-maps $f : \tilde{X} \to \tilde{Y}$ and $g: \tilde{Y} \to \tilde{X} $ such that there is a nondecreasing function $\rho : {\mathbb R}_{\geq 0} \to {\mathbb R}_{\geq 0}$ and $R>0$ satisfying - $d_Y(f(x),f(x'))) \leq \rho (d_X(x,x'))$ and $d_X(g(y),g(y')) \leq \rho (d_Y(y,y'))$, - $d_X(gf(x),x)<R, d_Y(fg(y),y)<R$, for any $x,x' \in \tilde{X}$ and $y,y' \in \tilde{Y}$, where $d_X$ and $d_Y$ are metric functions on $X$ and $Y$ lifted to $\tilde{X}$ and $\tilde{Y}$. For example, two $G$-invariant Delone subsets of a $G$-metric space $M$ are $G$-equivariantly coarsely equivalent. \[lem:Morita\] Let $X$ and $Y$ be uniformly discrete $G$-metric spaces with bounded geometry which are $G$-equivariantly coarse equivalent. Then, for any normal subgroup $\Pi$ of $G$, $C^_{{\mathscr{H}}_{X,G}}(X)^\Pi$ and $C^_{{\mathscr{H}}_{Y,G}}(Y)^\Pi$ are $G/\Pi$-equivariantly Morita equivalent [@MR0419677]. The proof is given in the same way as Theorem 4 of [@MR2363428] by using $\tilde{X}$ and $\tilde{Y}$ instead of $X$ and $Y$. Twisted equivariant ${\mathrm{K}}$-theory of coarse ${\mathrm{C}^}$-algebras ----------------------------------------------------------------------------- Our main subject in this paper is the topology of $c$-twisted $G$-invariant controlled symmetries on $(\phi ,c ,\tau)$-twisted $(X,G)$-modules* (a variation of $(X,G)$-modules using $(\phi,c,\tau)$-twisted representations as $({\mathscr{H}}, U)$). In order to reduce the problem to calculation of twisted equivariant ${\mathrm{K}}$-groups, we make the following assumption (note that equivariant ${\mathrm{K}}$-groups are not represented by $G$-invariant operators when $G$ is not compact). \[assump:ext\] There is an extension $1 \to \Pi \to G \to Q \to 1$ such that $Q$ is finite and $(\phi,c,\tau)$ is the pull-back of a twist of $Q$. For example, CT-type symmetries satisfy this assumption. In addition to that, we often assume that $\Pi$ is free abelian. \[exmp:extRoe\] Let $G:=G' \times N$, where $N$ is a finite group acting on ${\mathbb{E}}^d$ trivially and $G'$ is a discrete subgroup of the Euclidean group $\operatorname{Aut}({\mathbb{E}}^d)$. By Theorem 1 and Theorem 2 of [@MR574501], we obtain - a group extension $1 \to \Pi \to G' \to Q' \to 1$ such that $\Pi$ is a cocompact free abelian normal subgroup of rank $n$, - a decomposition ${\mathbb{E}}^d \cong {\mathbb{E}}^n \times {\mathbb{E}}^{d-n}$ such that $G'$ acts on ${\mathbb{E}}^n$ cocompactly and on ${\mathbb{E}}^{d-n}$ as a $Q'$-action. We remark that the $Q'$-action on ${\mathbb{E}}^n$ has a fixed point $x_0 \in {\mathbb{E}}$ and hence it is identified with the induced linear action on $V:=T_{x_0}{\mathbb{E}}^n$ by the exponential map. Set $Q:=Q' \times N$ and let $(\phi,c,\tau)$ be a twist of $Q$. Then, $(G,\phi,c,\tau)$ satisfies Assumption \[assump:ext\] Let $(G,\phi,c,\tau)$ be as Assumption \[assump:ext\] and let $X$ be a proper metric space on which $G$ acts properly and isometrically. For a $(X,G)$-module ${\mathscr{H}}$, $C^_{\mathscr{H}}(X)^\Pi$ has a canonical Real $Q$-${\mathrm{C}^}$-algebra structure. Hereafter, we simply write $$C^_{u,G}(X)^\Pi:=C^_{{\mathscr{H}}_{X,G}}(X)^\Pi, \ C^_G(X) ^\Pi:= C^_{{\mathscr{H}}_{X,G}^\infty}(X)^\Pi$$ respectively. Since any uniformly finite $(\phi,c,\tau)$-twisted $(X,G)$-module ${\mathscr{H}}$ is contained in ${\mathscr{H}}_{X,G} \operatorname{\hat{\otimes}}\mathscr{V}$ for some finite dimensional $(\phi,c,\tau)$-twisted representation $\mathscr{V}$, a controlled $c$-twisted $G$-invariant symmetry on ${\mathscr{H}}$ determines an element of the group ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^Q_{0,c,\tau}(C^_{u,G}(X)^\Pi )$. ### Direct calculations First of all, we give a calculation of the ${\mathrm{K}}$-group of uniform Roe algebras and Roe algebras of ${\mathbb Z}^d$ by using the Pimsner-Voiculescu exact sequence [@MR587369]. For example, since ${\mathrm{K}}_0(c_b({\mathbb Z})) \cong B({\mathbb Z})$ (where $B({\mathbb Z}) \subset {\mathbb Z}^{\mathbb Z}$ is the group of bounded sequences of integers), ${\mathrm{K}}_0(c_b({\mathbb Z}, {\mathbb{K}})) \cong {\mathbb Z}^{\mathbb Z}$ and ${\mathrm{K}}_1(c_b({\mathbb Z}, {\mathbb{K}})) \cong 0 \cong {\mathrm{K}}_1(c_b({\mathbb Z}, {\mathbb{K}}))$, we have exact sequences [$$\begin{aligned} 0 \to {\mathrm{K}}_{1}(C^_u(\|{\mathbb Z}\|)) \to B({\mathbb Z}) &{\xrightarrow}{\mathrm{shift}} B({\mathbb Z}) \to {\mathrm{K}}_0 (C^_u(\|{\mathbb Z}\|)) \to 0, \\ 0 \to {\mathrm{K}}_{1}(C^(\|{\mathbb Z}\|)) \to {\mathbb Z}^{\mathbb Z}&{\xrightarrow}{\mathrm{shift}} {\mathbb Z}^{\mathbb Z}\to {\mathrm{K}}_0 (C^(\|{\mathbb Z}\|)) \to 0. \end{aligned}$$]{} In particular, ${\mathrm{K}}_0(C^_u(\|{\mathbb Z}\|)) =0$ although ${\mathrm{K}}_0(C^_u(\|{\mathbb Z}\|))$ has uncountably many basis (Example 3.4 of [@Spakula]). By the same argument, we can calculate several ${\mathrm{KR}}_$-groups of uniform Roe algebras and compare them with that of Roe algebras. \[lem:uRoe\] The canonical map $${\mathrm{KR}}_j(C^_u(\|{\mathbb Z}^d\|)) \to {\mathrm{KR}}_j(C^(\|{\mathbb Z}^d\|))$$ is an isomorphism for $j=1,2,3,5,6,7$ when $d=1$, $j=1,2,5,6$ when $d=2$ and $j=1,5$ when $d=3$. Moreover, for $j=0,4$, $${\mathrm{KR}}_j(C^_u(\|{\mathbb Z}^d\|))/\operatorname{\mathrm{Im}}(\iota_u) _* \to {\mathrm{KR}}_j(C^(\|{\mathbb Z}^d\|)),$$ where $\iota _u : c_b({\mathbb Z}^d) \to C^_u(\|{\mathbb Z}^d\|)$ is the inclusion, is an isomorphism when $d=0,1,2,3$. They are direct consequences of the Pimsner-Voiculescu exact sequence [@MR587369] for the crossed product $C^_u(\|{\mathbb Z}^d\|) \cong {\mathbb Z}^d \ltimes c_b({\mathbb Z}^d)$. Note that ${\mathrm{KR}}_(c_b({\mathbb Z}^d)) \cong ({\mathbb{Z}_2})^{{\mathbb Z}^d}$ for $\ast=1,2$ and hence the shift homomorphisms are surjective. ### Coarse Baum-Connes isomorphism Let $M$ be a uniformly contractible metric space with a proper isometric $G$-action and let $X$ be a $G$-invariant Delone subset of $M$. Following to [@MR1451759] and [@MR2565716], we consider the twisted equivariant coarse assembly map $$\mu_{X,G}: {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^G(M) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^Q(C^_G(X)^\Pi)$$ defined as following. Let $[{\mathscr{H}}, \pi, F]$ be a ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^G$-cycle of $C_0(M)$. As in Lemma 9 of [@MR2565716], we can replace $F$ with $F'$ which is $G$-invariant and controlled. Moreover, as in Proposition 6.3.12 of [@MR1817560], we regard ${\mathscr{H}}$ as a $(\phi,c,\tau)$-twisted $(X,G)$-module by the $\ast$-representation $\pi'$ given by $\pi' (\delta _x):= \pi ({\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}_{F_x})$, where $M=\bigcup _{x \in X} F_x$ is a uniformly bounded Borel partition such that $F_x \cap X=\{x\}$ and $g(F_x)=F_{gx}$. By taking direct sum with a degenerate cycle, we may assume that ${\mathscr{H}}\cong {\mathscr{H}}_{X,G}^\infty \operatorname{\hat{\otimes}}\mathscr{V} \operatorname{\hat{\otimes}}\hat{{\mathbb R}} ^{2,2}_c$ as $(\phi,c,\tau)$-twisted $(X,G)$-modules, where $\mathscr{V}$ is a finite dimensional $(\phi,c,\tau)$-twisted representation. Therefore, we get $$[F'] \in {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{1,c,\tau}^Q(\mathcal{M}(C^_G (X)^\Pi)/C^_G(X)^\Pi)_{\mathrm{vD}}.$$ Here, for the definition of $\hat{{\mathbb R}} ^{2,2}_c$ and ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{1,c,\tau}^Q({\text{\textvisiblespace}})_{\mathrm{vD}}$, see Definition 5.4 of [@Kubota1]. Finally, we define the coarse assembly map as $$\mu_{X,G} [{\mathscr{H}}, \pi , F] := \partial [F'] \in {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,c,\tau}^Q(C^_G(X)^\Pi).$$ \[thm:cBC\] When $M={\mathbb{E}}^d$, $\mu _{X,G}$ is an isomorphism. In the proof, we use the Roe algebra for general (not necessarily discrete) metric spaces. For the definition, see for example Definition 6.3.8 of [@MR1817560]. We only remark that $C^_G(X)^\Pi$ and $C^_G({\mathbb{E}}^d)^\Pi$ have the same (twisted equivariant) ${\mathrm{K}}$-theory (Proposition 6.3.12 of [@MR1817560]). First, consider the case that the $G$-action on ${\mathbb{E}}^d$ is cocompact. By (\[form:detw\]), (\[form:extK\]) and the same argument as in [@MR1909514], $\mu_{X,G}$ is isomorphic to the Real Baum–Connes assembly map with coefficient in ${\mathbb{K}}(\mathscr{V})_{\mathbb R}$, which is an isomorphism since $G$ has the Haagerup property [@MR1487204]. For general $G$, the proof is given in a similar way as in [@MR1451759]. Let us choose a decomposition ${\mathbb{E}}^d \cong {\mathbb{E}}^{d-n} \times {\mathbb{E}}^n$ as in Example \[exmp:extRoe\]. For any $G$-subspace $D$ of ${\mathbb{E}}^d$, consider the “partial” localization algebra $C^_{G,L'}(D)$, that is, the closure of the $\ast$-algebra of elements $f \in C([0,\infty), C^(D))$ such that there is $R \in C_0[0,\infty)$ such that $\xi f(t)\eta =0$ if the distance of $p (\operatorname{supp}\xi)$ and $p (\operatorname{supp}\eta)$ is larger than $R(t)$, where $p : {\mathbb{E}}^d \to {\mathbb{E}}^n$ is the projection. In the same way as , we get $$\mu _{D,G,L'} : {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{,c,\tau}(D) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^Q_{,c,\tau}(C^_{G,L'}(D))$$ such that $\mu_{{\mathbb{E}}^d,G}=({\mathrm{ev}}_0)_* \circ \mu _{{\mathbb{E}}^d, G,L'}$, where ${\mathrm{ev}}_0$ is the evaluation map. It suffices to show that $\mu _{{\mathbb{E}}^d,G,L'}$ and $(ev_0)_$ are isomorphisms. Choose a simplicial decomposition $\tilde{K}$ of ${\mathbb{E}}^{d-n}$ compatible with the $G$-action and let $K^{(k)}:=\tilde{K}^{(k)} \times {\mathbb{E}}^{n}$. By a Mayer-Vietoris argument as in the proof of Theorem 3.2 of [@MR1451759],$\mu _{K^{(k)},G,L'}$ are isomorphisms for any $k$. Note that the isomorphism for $\mu _{K^{(0)},G,L'}$ follows from the Real Baum–Connes isomorphism discussed above. We can check that $({\mathrm{ev}}_0)_$ is an isomorphism in the same way as in Section 4 of [@MR1451759] by using the scaling map on ${\mathbb{E}}^{d-n}$. \[lem:cliff\] Let $G$ be a finite group and let $V$ be a $d$-dimensional real representation of $G$. Then, there is $c'' \in \operatorname{Hom}(G,{\mathbb{Z}_2})$ and $\tau'' \in H^2(G,{\mathbb{Z}_2})$ such that the twisted equivariant ${\mathrm{K}}$-homology group ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{,c,\tau}(V)$ is isomorphic to ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_G^{\dim V-,c',\tau'}({{\rm pt}})$ where $c':=c+c''$ and $\tau'=\tau + \tau''+\epsilon (c,c'')$. Moreover, $c''=0$ if and only if $\pi$ preserves the orientation of $V$ and $\tau''=0$ if and only if $G$ is a Pin representation (i.e. $G \to O(V)$ gives rise to $G \to \mathop{\mathrm{Pin}}(V)$). By taking direct sum with trivial representations, we may assume that $\dim V=8k$. Then, the $\phi$-twisted $G$-action on ${\mathrm{C}\ell}(V) \cong {\mathbb{K}}(\Delta _{V})$, where $\Delta _{V}$ is the unique ${\mathbb{Z}_2}$-graded representation of ${\mathrm{C}\ell}(V)$, gives rise to a $(\phi,c'',\tau'')$-twisted representation of $G$ on $\Delta _{V}$. It is actually induced from $G^{\tau''} \to \mathop{\mathrm{Pin}}(V)$ where the action of $g \in \mathop{\mathrm{Pin}}(V)$ on $\Delta _V$ is even if and only if $g \in \mathop{\mathrm{Spin}}(V)$ (see Section 1.2 of [@LawsonMichelsohn1989]). By Theorem 7 of [@MR582160], we have the isomorphism ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{KK}}^G_{c,\tau}(C_0(V),{\mathbb R}) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{KK}}^G_{c,\tau}({\mathbb R}, {\mathrm{C}\ell}(V))$. Now we obtain the desired isomorphism by (\[form:detw\]). \[cor:cBC\] Assume that $\Pi$ is free abelian. Let $n:=\operatorname{rank}\Pi$, $\nu \in H^2_Q(B_\Pi, \mathbb{T}_\phi)$ as in Example \[exmp:ext\], $V:=T_{x_0}{\mathbb{E}}^{d-n}$, $(c'',\tau'')$ the twist corresponding to ${\mathrm{C}\ell}(V)$ and $c':=c+c''$, $\tau':=\tau +\tau'' +\epsilon (c,c'')$. Then, there is an isomorphism $$\Psi_{X,G} :{{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^Q(C^_G(X)^\Pi) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{-+n,c',\tau'+\nu}_Q(B_\Pi).$$ In particular, for a product CT-type symmetry $(G \times \mathscr{A},\phi,c,\tau)$ such that $G$ is finite and acts on ${\mathbb{E}}^d$ by a Spin representation, ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{G \times \mathscr{A}}_{,c,\tau}(C^_G(X))$ is written in terms of the representation ring of $G$ as indicated in Example \[exmp:rep\]. By Theorem \[thm:cBC\] and Lemma \[lem:cliff\], we obtain the isomorphism $${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^Q(C^(X)^\Pi ) \xleftarrow{\mu_{X,G}} {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^G ({\mathbb{E}}^d) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-n,c',\tau'}^G ({\mathbb{E}}^{d-n}) \xrightarrow{\mu_{X',G}} {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c',\tau'}^Q(C^(X')^\Pi ),$$ where $X'$ is a $G$-invariant Delone subset of ${\mathbb{E}}^{d-n}$. It is the desired isomorphism by Example \[exmp:eqRoe\]. ### Coarse Mayer-Vietoris exact sequence Here we review the coarse analogue of the Mayer-Vietoris exact sequence introduced by Higson–Roe–Yu [@MR1219916] (see also [@mathKT12120241]). \[thm:MV\] There is an exact sequence $$\begin{array}{l} \cdots \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^{Q}(C^_{\mathscr{H}}(Z \subset X)^\Pi) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^{Q}(C^_{\mathscr{H}}(Y_1 \subset X)^\Pi) \oplus {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^{Q}(C^_{\mathscr{H}}(Y_2 \subset X)^\Pi) \to \\ \hspace{1.5em} \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^{Q}(C^_{\mathscr{H}}(X)^\Pi) {\xrightarrow}{\partial_{\mathrm{MV}}}{{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1,c,\tau}^{Q}(C^_{\mathscr{H}}(Z \subset X)^\Pi) \to \cdots. \end{array}$$ The boundary map $\partial _{\mathrm{MV}}: {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,c,\tau}^{Q}(C^_{\mathscr{H}}(X)^\Pi) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1,c,\tau}^{Q} (C^_{\mathscr{H}}(Z)^\Pi)$ is given by $$\partial _{\mathrm{MV}}[s]=[\sigma (\hat{s})] \in {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1,c,\tau}^{Q} (C^_{\mathscr{H}}(Z \subset X)^\Pi)$$ where $\hat{s}:=\pi ({\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}_{Y_1})s \pi ({\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}_{Y_1}) \in C^_{\mathscr{H}}(Y_1)$. Note that the inclusion $C^_{{\mathscr{H}}}(A)^\Pi \to C^_{{\mathscr{H}}}(A \subset X)^\Pi$ induces the isomorphism of twisted equivariant ${\mathrm{K}}$-groups (cf. Lemma 1 of [@MR1219916]). Set $A:=C^_{\mathscr{H}}(X)^\Pi$, $I_k:=C^_{\mathscr{H}}(Y_k \subset X)^\Pi$ for $k=1,2$ and $J:=I_1 \cap I_2$. Then, by assumption we have $J=C^_{\mathscr{H}}(Z \subset X)^\Pi$ and $I_1+I_2=A$. Note that $A/J\cong I_1 / J \oplus I_2 / J$. We write $q_k: A_k \to A_k/J$ and $\theta: A \to A/J$ for the quotient and $p _k$ for the projection $A/J \to I_k/J$. Now, the coarse Mayer-Vietoris exact sequence is obtained as the long exact sequence associated to the short exact sequence $$0 \to SA \to \Omega(A;I_1,I_2) \to I_1 \oplus I_2 \to 0$$ where $$\Omega (A;I_1,I_2) := \{ f\in C([0,1],A) \mid f(0) \in I_1 , \ f(1) \in I_2 \}.$$ Actually, since $\Omega(A;I_1,I_2) /C([0,1],J) \cong C_0([0,1),I/J_1) \oplus C_0((0,1],I/J_2)$ is contractible, we have ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^{Q}(\Omega (A;I_1,I_2) ) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^Q(J)$. Let $\psi$ be the $\ast$-homomorphism from $\Omega(A;I_1,I_2)$ to $${\mathrm{cone}}(q_1) := \{ (a,f) \in I_1 \oplus C([0,1), I_1/J) \mid f(0)=q_1(a) \}$$ given by $\psi (f)= (f(0), p_1 \circ \theta (f))$. Then, the diagram $$\xymatrix{ SA \ar[r]^{b} \ar[d]^{p_1 \circ \theta} &\Omega (A;I_1,I_2) \ar[d]^\psi & J \ar[l]^i \ar[ld]^{i'} \\ S(I_1/J) \ar[r]^{b'} & {\mathrm{cone}}(q_1) & }$$ commutes. Now, by definition $$\partial _{\mathrm{MV}}=(i_)^{-1}\circ b_: {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^{Q} (SA) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^{Q}(\Omega (A:I_1,I_2)) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{,c,\tau}^{Q}(J),$$ and hence we obtain $$\partial _{\mathrm{MV}}=(i_')^{-1}\circ b'_* \circ (p_1 \circ \theta)_* =\partial \circ (p _1 \circ \theta)_,$$ where $\partial$ is the boundary map of the exact sequence $0 \to J \to I_1 \to I_1/J \to 0$. \[cor:Toep\] Let $\partial _{\mathrm{Toep}}$ be the boundary map of the Toeplitz extension $$0 \to A \otimes {\mathbb{K}}\to \mathcal{T}(A) \to {\mathbb Z}\ltimes A \to 0$$ where $A:={\mathbb Z}^{d-1}\ltimes c_b({\mathbb Z}^d)$. Then, we have $\partial _{\mathrm{MV}}=f_ \circ \partial _{\mathrm{Toep}}$, where $f: {\mathbb Z}^{d-1} \ltimes c_b({\mathbb Z}^d) \to {\mathbb Z}^{d-1} \ltimes c_b({\mathbb Z}^{d-1})$ be the $\ast$-homomorphism of crossed products induced from the restriction. In particular, $\partial _{\mathrm{MV}} \circ \mathop{\mathrm{Res}} =\mathop{\mathrm{Res}} \circ \partial_{\mathrm{Toep}}$ on ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,c,\tau}^G(C^_r({\mathbb Z}^d))$, where $\partial_{\mathrm{Toep}}$ is the boundary map of the Toeplitz exact sequence $$0 \to C^_r({\mathbb Z}^{d-1}) \otimes {\mathbb{K}}\to C^_r({\mathbb Z}^{d}) \otimes \mathcal{T} \to C^_r({\mathbb Z}^d) \to 0$$ and $\mathrm{Res}$ is the map induced from the inclusion $C^_u(\|{\mathbb Z}^d\|)^{{\mathbb Z}^d}\to C^_u(\|{\mathbb Z}^d\|)$. We identify ${\mathbb Z}\ltimes A$ with the subalgebra of ${\mathbb{B}}(\ell ^2 ({\mathbb Z}^d)) \otimes {\mathbb{B}}(\ell ^2 ({\mathbb Z}))$ generated by $A \otimes 1$ and $u \otimes v$ where $u$ is a generator of $C^_r{\mathbb Z}\subset {\mathbb Z}\ltimes A$ and $v$ is the birateral shift operator on $\ell ^2 ({\mathbb Z})$. Then, $\mathcal{T}(A)$ is the ${\mathrm{C}^}$-algebra generated by $(1 \otimes P)({\mathbb Z}\ltimes A)(1 \otimes P)$ where $P$ is the projection onto $\ell ^2({\mathbb Z}_{\geq 0})$. Let $p$ be the projection onto $\ell ^2 ({\mathbb Z}^{d-1} \times D_{{\mathbb Z}} )$ where $D _{\mathbb Z}$ is the diagonal subset of ${\mathbb Z}\times {\mathbb Z}$. Then, $p$ commutes with $\mathcal{T}(A)$ and the exact sequence $$0 \to p(A \otimes {\mathbb{K}})p \to p(\mathcal{T}(A))p \to p({\mathbb Z}\ltimes A)p \to 0$$ is the same thing as $0 \to C^_u(Z \subset Y_1) \to C^_u(Y_1) \to C^_u(Z,Y_1) \to 0$ and the projection $A \otimes {\mathbb{K}}\to p(A \otimes {\mathbb{K}})p$ is the same thing as $f$. ### ${\mathrm{K}}$-cycles and cyclic cocycles At the end of this section, we study fundamental cycles of the coarse ${\mathrm{C}^}$-algebras for $G$-invariant Delone subsets of ${\mathbb{E}}^d$. For simplicity, let $G$ be a finite group. Let $V:=T_x{\mathbb{E}}^d$ as in Example \[exmp:ext\], $V^-:=V$ with the negative definite inner product $-{\left< {\text{\textvisiblespace}},{\text{\textvisiblespace}}\right>}$ and let $\tilde{V}:=V \oplus V^-$. Then, ${\mathrm{C}\ell}(\tilde{V})$ has a unique irreducible ${\mathbb{Z}_2}$-graded representation $\Delta _{\tilde{V}}$. For a finite $(X,G)$-module $({\mathscr{H}}, U, \pi)$, we obtain a twisted equivariant ${\mathrm{K}}$-cycle $$\Xi _{X,{\mathscr{H}}}:=[{\mathscr{H}}\operatorname{\hat{\otimes}}\Delta _{\tilde{V}},{\mathrm{id}}, F] \in {{}^{\phi} \hspace{-0.1ex} } {\mathrm{KK}}^G (C^_{\mathscr{H}}(X)^\Pi \operatorname{\hat{\otimes}}{\mathrm{C}\ell}(V^-) , {\mathbb R}).$$ where ${\mathrm{id}}$ means the canonical inclusion $C^_{\mathscr{H}}(X)^\Pi \operatorname{\hat{\otimes}}{\mathrm{C}\ell}(V^-) \subset {\mathbb{B}}({\mathscr{H}}) \operatorname{\hat{\otimes}}{\mathbb{B}}(\Delta _{\tilde{V}})$ and $$F:=D(1+D^2)^{-1/2}, D:= \sum \nolimits_{j=1}^{d} \pi (x_j) \operatorname{\hat{\otimes}}e_j \in {\mathbb{B}}({\mathscr{H}}),$$ where $x_j$ are coordinate functions on $X \subset {\mathbb{E}}^d$ and $e_j$ are Clifford generators of ${\mathrm{C}\ell}(V)$. Note that $C^_{\mathscr{H}}(X)$ are unital and in particular $\sigma$-unital. Therefore, as in Remark \[remk:KK\], the twisted Kasparov product induces the group homomorphism $$\Xi _{X,{\mathscr{H}}}^:={\text{\textvisiblespace}}\otimes \Xi _{X,{\mathscr{H}}}: {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{0,c,\tau}(C^_{\mathscr{H}}(X)) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{KK}}^G_{0,c,\tau}({\mathrm{C}\ell}(V^-),{\mathbb R}) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{0,c,\tau}({\mathrm{C}\ell}(V)).$$ Here, ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{0,c,\tau}({\mathrm{C}\ell}(V))$ is isomorphic to ${{}^{\phi} \hspace{-0.1ex} }{\mathrm{K}}_G^{0,c',\tau'}({{\rm pt}})$ by Lemma \[lem:cliff\]. For example, for a CT-symmetry, $\Xi _X^$ takes value in the group as in Table \[table:Kitaev\]. In particular, we simply write $\Xi _X:=\Xi _{X,{\mathscr{H}}_{X,G}} $. \[prp:index\] The diagram $$\xymatrix@R=1.2em{ {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{0,c,\tau} (C^_{u,G}(X)) \ar[d] \ar[r]^{\Xi _X^} & {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_G^{d,c',\tau'}({{\rm pt}}) \\ {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{0,c,\tau}(C^_G(X)) \ar[ru] _{\Psi_{X,G}} & }$$ commutes, where ${\Psi _{X,G}}$ is the isomorphism given in Corollary \[cor:cBC\]. Since $C^_G(X)=\overline{\bigcup _{{\mathscr{H}}} C^_{\mathscr{H}}(X)}$ where ${\mathscr{H}}$ runs over all finite $(X,G)$-submodules of ${\mathscr{H}}_{X,G}^\infty$, we have the isomorphism $$\varinjlim {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,c,\tau}^{Q}(C^_{\mathscr{H}}(X) ) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,c,\tau}^{Q}(C^(X) )$$ in the same as continuity of ${\mathrm{K}}$-groups. Since $\Xi _{X,{\mathscr{H}}}^$ commutes with the inclusion $C^_{{\mathscr{H}}}(X) \subset C^_{{\mathscr{H}}'}(X)$ for ${\mathscr{H}}\subset {\mathscr{H}}'$, $\Xi _X^$ factors through ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^Q_{0,c,\tau}(C^_G(X))$. By Corollary \[cor:cBC\], any element in ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^G_{0,c,\tau}(C^_G(X))$ is of the form $\mu(\alpha) \otimes \xi$ for some $\xi \in {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-d,c,\tau}^G({\mathbb R})$, where $\alpha \in {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{d}^Q({\mathbb{E}}^d)$ is the element represented by the de Rham operator on ${\mathbb{E}}^d$ (see Section 5.II of [@MR582160]). Therefore, it suffices to show that $\Xi _X^(\mu (\alpha))=1 \in {{}^{\phi} \hspace{-0.1ex} } {\mathrm{KK}}^Q({\mathbb R}, {\mathbb R})$. Actually, it is proved in the same way as Theorem 1 of [@MR1435703] that $\Xi _X^(\mu (\alpha))$ coincides with $\beta \otimes _{C_0({\mathbb{E}}^d) \otimes {\mathrm{C}\ell}_{0,d}} \alpha =1$, where $\beta$ is the Bott element. \[remk:genind\] For a general $(G,\phi,c,\tau)$ satisfying Assumption \[assump:ext\], we obtain a similar index pairing. Recall that we have a decomposition ${\mathbb{E}}^d \cong {\mathbb{E}}^{n-d} \times {\mathbb{E}}^{n}$ as in Example \[exmp:extRoe\]. By Lemma \[lem:Morita\], we may assume that $X=X' \times X''$ where $X' \subset {\mathbb{E}}^{d-n}$ and $X'':=G \cdot x'' \subset {\mathbb{E}}^n$ are $G$-invariant Delone subsets. Then, we have an isomorphism $$C^_{G}(X)^\Pi \cong C^_{Q}(X') \otimes C^_{G}(X'')^\Pi \cong C^_{Q}(X') \otimes {\mathbb{K}}(\ell^2_\Pi(G)).$$ Therefore, by using $\Xi _{X'}$, we obtain an index pairing which takes value in ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^Q_{-d,c',\tau'}({\mathbb{K}}(\ell^2_\Pi(G))) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{d-n,c',\tau' +\nu}_Q(B_\Pi)$. By the same proof as Proposition \[prp:index\], we obtain $\Xi _{X'}^=\Psi_{X,G}$ for this case. \[cor:iota\] Let $G$ be a discrete group as in Assumption \[assump:ext\] such that $\Pi$ is free abelian ($n:=\operatorname{rank}\Pi$) and let $X$ be a $G$-invariant Delone subset of ${\mathbb{E}}^d$. Let $\iota: c_b(X; {\mathbb{K}}({\mathscr{H}}_{X,G}^\infty)) \to C^_G(X)^\Pi$ be the inclusion. Under the isomorphism in Corollary \[cor:cBC\], - $\operatorname{\mathrm{Im}}\iota_=0$ when $n<d$. - $\operatorname{\mathrm{Im}}\iota_$ is spanned by ${\ell ^2\mathchar`-\mathrm{Ind}}_{G_x}^Q ({{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{G_x}^{,c,\tau}({{\rm pt}}))$ when $n=d$. In particular, $\operatorname{\mathrm{Im}}\iota _ \subset \mathrm{Triv}$. When $n<d$, it follows from $\Xi _X^* (\operatorname{\mathrm{Im}}\iota_)=0$. When $n=d$, it follows from the definition of the $\ell^2$-induction. Next, we relate it with the coarse Mayer–Vietoris exact sequence. For this, let $G$ be a finite group acting on ${\mathbb{E}}^{d-1}$. We choose a decomposition ${\mathbb{E}}^1={\mathbb{E}}^1_+ \cup {\mathbb{E}}^1_-$ such that ${\mathbb{E}}^1_+ \cap {\mathbb{E}}^1_- ={{\rm pt}}$ and set ${\mathbb{E}}^d_\pm:= {\mathbb{E}}^{d-1} \times {\mathbb{E}}^1_{\pm}$. \[cor:MVind\] Let $X=Y_+ \cup Y_-$ be a Delone partition of ${\mathbb{E}}^d={\mathbb{E}}^d_+ \cup {\mathbb{E}}^d_-$. Set $Z:=Y_+ \cap Y_-$. Then, we have $$\Xi _X^[s] = \Xi _Z^* \partial _{\mathrm{MV}}[s] \in {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_G^{d,c',\tau'}({{\rm pt}}).$$ In the same way as Remark \[remk:genind\], we obtain the same result when $G$ is a general discrete group. By Proposition \[prp:index\], it suffices to show that $\Psi_{Z,G} \circ \partial _{\mathrm{MV}}=\Psi _{X,G}$. It follows from the fact that the coarse Baum-Connes isomorphism is compatible with the Mayer-Vietoris exact sequences (see for example Section 3 of [@mathKT12120241]). For the case of type [[[A]{}]{}]{} or [[[AIII]{}]{}]{} CT-symmetry, this pairing is also related to the pairing with the fundamental cyclic cocycle. Let $\omega \in \beta {\mathbb N}\setminus {\mathbb N}$ be a free ultrafilter and let $\mathbf{P} _\omega \in c_b(X)^$ be the corresponding probability measure on $\beta X$, that is, $$\int f(x)d\mathbf{P} _\omega (x) =\lim _{n \to \omega } \|\Lambda _n \|^{-1} \sum\nolimits _{x \in \Lambda _n} f(x)$$ where $\Lambda _n:=[-n,n] \cap X$. As in the proof of Lemma 4.7 of [@MR2007488], the state $$\mathcal{T}^\omega (a):=\int d\mathbf{P} _\omega (x) (\operatorname{Tr}(a_{xx}))$$ on $C^_uX$ is tracial since $\Lambda _n$ is a F[ø]{}lner set of an amenable uniformly discrete metric space $X$. Hence, we obtain a cyclic $d$-cocycle [$$\begin{aligned} \mathcal{T} _d^\omega (a_0,a_1,\cdots , a_d):=\sum\nolimits_{\sigma \in\mathfrak{S}_n} (-1)^{\|\sigma\|}\mathcal{T} ^\omega(a _0 \nabla _{\sigma (1)}(a_1) \cdots \nabla _{\sigma (d)}(a_d)) \label{form:cyc} \end{aligned}$$]{} on the dense $\ast$-subalgebra $\mathcal{B}_u(X):=\{ a \in C^_u(X) \mid \nabla_j (a) \in C^_u(X) \}$ of $C^_u(X)$, where $\nabla _j(a):=i[\pi(x_j),a]$. Since $\mathcal{B}_u(X)$ has the same ${\mathrm{K}}$-theory as $C^_u(X)$ (see Appendix 3 of [@MR823176]), this pairing induces the group homomorphism ${\left< {\text{\textvisiblespace}}, \mathcal{T}^\omega_d \right>}:{\mathrm{K}}_d(C^_uX) \to {\mathbb R}$. \[lem:pair\] We have $\langle \xi , \mathcal{T}_d^\omega \rangle =\Xi_X^ \xi$ for any $\xi \in {\mathrm{K}}_d(C^_u(X))$. In particular, $\langle {\text{\textvisiblespace}}, \mathcal{T}_d^\omega \rangle $ is independent of the choice of $\omega \in \beta {\mathbb N}\setminus {\mathbb N}$. It suffices to show that the Chern-Connes character of $\Xi_X$ coincides with $\mathcal{T}^\omega_d$ as cyclic cocycles. The proof is given in the same way as Theorem 10 of [@MR1295473]. Let $X=Y_+ \cup Y_-$ be a Delone partition of ${\mathbb{E}}^d ={\mathbb{E}}^d_+ \cup {\mathbb{E}}_-^d$. In the same way as above, we obtain a cyclic $(d-1)$-cocycle $$\hat{\mathcal{T}} _{d-1}^\omega (a_0,a_1,\cdots , a_{d-1}):=\sum_{\sigma \in \mathfrak{S}_{d-1}} (-1)^{\|\sigma\|}\mathcal{T}^\omega (a _0 \nabla _{\sigma (1)}(a_1) \cdots \nabla _{\sigma (d-1)}(a_{d-1}))$$ on a dense subalgebra of $C^_u(Z \subset Y_1)$ where $Z:=Y_1 \cap Y_2$. By Corollary \[cor:MVind\] and Lemma \[lem:pair\], we have [$$\begin{aligned} \langle \xi, \mathcal{T}^\omega_d\rangle =\langle \partial_{\mathrm{MV}}(\xi), \hat{\mathcal{T}}^\omega_{d-1}\rangle. \label{form:MVcyc} \end{aligned}$$]{} Controlled topological phases {#section:3} ============================= In this section, we introduce the notion of controlled topological phase for bulk and edge quantum systems with an arbitrary symmetry of quantum mechanics in the sense of [@MR3119923]. Let $(G,\phi,c,\tau)$ be as Assumption \[assump:ext\] and let $X$ be a proper metric space with proper isometric $G$-action. \[defn:bulkTP\] We say that a bulk quantum system controlled at $X$ with the symmetry class $(G,\phi , c,\tau )$ is a quadruple $({\mathscr{H}}, U, \pi , H)$ where - $({\mathscr{H}},U,\pi)$ is a uniformly finite $(\phi,c,\tau)$-twisted $(X,G)$-module, - $H \in {\mathbb{B}}({\mathscr{H}})$ is a $c$-twisted $G$-invariant controlled self-adjoint operator with a spectral gap at $[-\varepsilon, \varepsilon]$ for some $\varepsilon >0$. A controlled bulk topological phase is a equivalence class of bulk quantum systems controlled at $X$ with the symmetry class $(G,\phi,c,\tau)$ with respect to the equivalence relation generated by the following: - $({\mathscr{H}}, U, \pi ,H) \sim ({\mathscr{H}}', U', \pi' ,H')$ if there is an isomorphism $V: {\mathscr{H}}\to {\mathscr{H}}'$ of $(\phi,c,\tau)$-twisted $(X,G)$-modules such that $VHV^=H'$, - $({\mathscr{H}}, U, \pi ,H_0) \sim ({\mathscr{H}}, U, \pi ,H_1)$ if there is a continuous path $H_t \in {\mathbb{B}}({\mathscr{H}})$ of $c$-twisted $G$-invariant controlled self-adjoint operators with a spectral gap at $[-\varepsilon, \varepsilon]$ for some $\varepsilon >0$, - $({\mathscr{H}}, U, \pi ,H) \sim ({\mathscr{H}}\oplus {\mathscr{H}}', U \oplus U', \pi \oplus \pi ' ,H \oplus H')$ if $H'$ commutes with $\pi (c_0(X))$. We write ${\mathscr{TP}}(X;G,\phi , c,\tau )$ for the set of bulk topological phases. For a CT-type $(\mathscr{A},\tau)$ with the Cartan label $\mathrm{L}$, we simply write ${\mathscr{TP}}(X;\mathrm{L}):={\mathscr{TP}}(X;\mathscr{A},\phi,c,\tau)$ and ${\mathscr{TP}}(X;G,\mathrm{L}):={\mathscr{TP}}(X;G \times \mathscr{A},\phi,c,\tau)$. In this framework, we can deal with a Hamiltonian $H$ with a long range term such that ${\left\\| H_{xy} \right\\|}$ decays as $d(x,y) \to \infty$ uniformly. Actually, such $H$ is approximated by the controlled operators $H_R$ given by $$(H_R) _{xy}= \begin{cases}H_{xy} & d(x,y)<R, \\ 0 & d(x,y)\geq R, \end{cases}$$ in the norm topology. For a sufficiently large $R>0$, ${\left\\| H-H_R \right\\|}$ is small enough that $H_R$ has a spectral gap at $0$ and all $H_{R'}$ with $R'>R$ are in the same topological phase. Here, we assume that the Hilbert space ${\mathscr{H}}$ of quantum states is equipped with an a priori ${\mathbb{Z}_2}$-grading. In the case of [[[AIII]{}]{}]{} systems, it corresponds to an a priori choice of reference isomorphisms $h_{\mathrm{ref}}$ in [@mathph150404863]. This assumption is not artificial because ${\mathscr{H}}$ has a canonical ${\mathbb{Z}_2}$-grading for a large class of examples. For example, when the system has the particle-hole symmetry such as Bogoliubov–de-Gennes Hamiltonians, ${\mathscr{H}}$ is decomposed as the direct sum of the particle and hole components. When ${\mathscr{H}}$ does not have a canonical ${\mathbb{Z}_2}$-grading, we only have to fix an arbitrary choice of such ${\mathbb{Z}_2}$-gradings. For example, when we consider type A, AI or [[[AII]{}]{}]{} symmetry class, we can use the trivial grading $\gamma_{{\mathscr{H}}} =1$. In fact, we give the relation (3) in order for the topological phase of a system to be independent of the choice of ${\mathbb{Z}_2}$-gradings of ${\mathscr{H}}$ which is compatible with $U$ and $\pi$. Actually, let ${\mathscr{H}}'$ be the Hilbert space ${\mathscr{H}}$ with another ${\mathbb{Z}_2}$-grading $\gamma$ compatible with $U$ and $\pi$. Then, [$$\begin{aligned} ({\mathscr{H}}, U,\pi,H) &\sim ({\mathscr{H}}, U, \pi ,H) \oplus ({\mathscr{H}}', U, \pi,\gamma) \sim ({\mathscr{H}}, U,\pi,\gamma ) \oplus ({\mathscr{H}}', U,\pi,H)\\ & \sim ({\mathscr{H}}', U,\pi,H), \end{aligned}$$]{} where the second equivalence is given by the homotopy $R_t(H \oplus \gamma)R_t^$ where $R_t$ are the $2$-dimensional rotation matrices. On the other hand, a general quantum system with sublattice symmetry does not admit any compatible ${\mathbb{Z}_2}$-grading. For example, let $X_1$ and $X_2$ be mutually disjoint Delone subsets of ${\mathbb{E}}^d$ and let $X:=X_1 \sqcup X_2$. If $P$ acts on each ${\mathscr{H}}_x$ as $+1$ for $x \in X_1$ and $-1$ for $x \in X_2$, there is no compatible ${\mathbb{Z}_2}$-grading on ${\mathscr{H}}$. However, even in this case, sometimes we can get a controlled bulk quantum system by deforming $X$. More precisely, if there is another $G$-invariant Delone subset $X'$ of ${\mathbb{E}}^d$ and a $G$-map $f : X \to X'$ such that $d(x,f(x))$ is uniormly bounded and $\sum_{x \in f^{-1}(x')} \dim {\mathscr{H}}_{x}^0=\sum_{x \in f^{-1}(x')} \dim {\mathscr{H}}_{x}^1$ for any $x' \in X'$, we can define a bulk quantum system $({\mathscr{H}}, U, \pi \circ f^, H)$ controlled at $X'$. For example, in the case of a honeycomb lattice, we get a map $X \to X_1$ by shifting $X_2$ as in Section 3.1 of [@MR3123539]. \[prp:bulk\] We have the isomorphism $${\mathscr{TP}}(X;G,\phi,\tau , c) \cong {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,\tau,c}^Q (C^_{u,G}(X)^\Pi )/\operatorname{\mathrm{Im}}(\iota_u) _,$$ where $\iota_u : c_b(X,{\mathbb{K}}({\mathscr{H}}_{X,G}))^\Pi \to C^_{u,G}(X)^\Pi$ is the inclusion. In particular, it is independent of the choice of full $G$-invariant Delone subsets $X$ of $M$ up to $\operatorname{\mathrm{Im}}(\iota _u)_$ by Lemma \[lem:Morita\]. We have a well-defined map $$\Theta : {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,\tau,c}^Q (C^_{u,G}(X)^\Pi ) \to {\mathscr{TP}}(X;G,\phi,c,\tau), [s] \mapsto ({\mathscr{H}}_{X,G}\otimes \mathscr{V} , U,\pi , s ).$$ It is suejective because any ${\mathscr{H}}$ is contained in some ${\mathscr{H}}_{X,G} \otimes \mathscr{V}$ and $$({\mathscr{H}}, U,\pi , H) \sim ({\mathscr{H}}_{X,G}\otimes \mathscr{V} , U,\pi , H \oplus \gamma _{{\mathscr{H}}^\perp} )=\Theta [H \oplus \gamma _{{\mathscr{H}}^\perp}].$$ By the relation (3), $\operatorname{\mathrm{Im}}\iota_* \subset \operatorname{Ker}\Theta $. To see that $\operatorname{Ker}\Theta = \operatorname{\mathrm{Im}}\iota _$, it suffices to check $[usu^]=[s]$ for any even $G$-invariant unitary $u \in c_b(X;{\mathbb{K}}({\mathscr{H}}_{X,G} \otimes \mathscr{V}))$. Set $$v_t:=(u \oplus 1)R_t(u^* \oplus 1) R_t^* \in c_b(X, {\mathbb{K}}(({\mathscr{H}}_{X,G} \otimes \mathscr{V})^2)).$$ Then, since $v_0 =1 \oplus 1$ and $v_1=u \oplus u^$, we get a path $v_t(s \oplus \gamma)v_t^$ connecting $s \oplus \gamma$ and $usu^* \oplus \gamma $, which shows $[s]=[usu^]$. \[rmk:twist\] Even if Assumption \[assump:ext\] is not satisfied such as the case that the translation symmetry is twisted by the magetic field, we can consider cotroled topological phases in the same way. In this case, controlled topological phases are classified by the twisted equivariant ${\mathrm{K}}$-group of “twisted” invariant uniform Roe algebras defined by using twisted $(X,G)$-modules. For example, when $G={\mathbb Z}^d$, $X=\|{\mathbb Z}^d\|$ and $\tau$ is nontrivial, this ${\mathrm{C}^}$-algebra is the same thing as the noncommutative torus $C^_{-\tau, r}({\mathbb Z}^d)$ used in the previous researches . When $\Pi={\mathbb Z}^d$ and $X=\|{\mathbb Z}^d\|$, then ${\mathscr{TP}}(X;\Pi,{\mathrm{L}})$ is isomrophic to the reduced ${\mathrm{KL}}^0$-group of the Brillouin torus $B_\Pi$ because $C^_u(X)^\Pi \cong C^_r(\Pi)$ as is described in Example \[exmp:eqRoe\]. More generally, when $G$ is a cocompact discrete Euclidean group, ${\mathscr{TP}}(X;G,{\mathrm{L}})$ is isomorphic to ${\mathrm{KL}}^{0,\nu}_Q (B_\Pi)/\mathrm{Triv}$, which is essentially the same thing as the set of reduced topological phases $\mathscr{RTP}_F(G,\phi,\tau,c)$ of type $F$ in [@MR3119923] (note that they deal with Galilean spacetimes). For a Delone subset of ${\mathbb{E}}^d$, the group ${\mathscr{TP}}(X;{\mathrm{L}})$ is isomorphic to ${\mathrm{KL}}_0(C^_u(X))/\operatorname{\mathrm{Im}}(\iota_u)_$. By Lemma \[lem:Morita\], they are isomorphic to the groups calculated in Lemma \[lem:uRoe\] when $\mathrm{L}$ is a real Cartan label. \[remk:compare\] The set ${\mathscr{TP}}(X;L)$ is related to the classification of topological phases via ${\mathrm{K}}$-groups of noncommutative Brillouin torus. Recall that the noncommutative Brillouin torus $A:={\mathbb Z}^d \ltimes C(\Omega)$ introduced in [@MR1295473] is a subalgebra of $C^_u(X)$ (Example \[exmp:groupoid\] and Lemma \[lem:groupoid\]). For any gapped Hamiltonian $H \in A\operatorname{\hat{\otimes}}{\mathbb{K}}(\mathscr{V})$ as in , the inclusion $j: A \to C^_u(X)$ induces a homomorphism $${\mathrm{KL}}_0(A) \to {\mathrm{KL}}_0 (C^_{u}(X) ) \to {\mathscr{TP}}(X;L)$$ mapping $[H]$ to $[\ell^2(X)\operatorname{\hat{\otimes}}\mathscr{V},U,\pi,H]$. We also remark that $j_* \Xi _X \in {\mathrm{KR}}^d(A)$ is the same thing as the ${\mathrm{KR}}$-cycle used in [@GrossmannSchulz-Baldes] and [@mathKT150907210]. Let $G$ be a discrete group acting on the hyperbolic space $M=SO^+(1,d)/SO(d)$ properly and isometrically and let $X$ be a $G$-invariant Delone subset of $M$. Then, the Hamiltonian studied in determines an element in ${\mathscr{TP}}(X;G, \mathrm{A})$ when the $2$-cocycle is trivial. Next we introduce the notion of edge topological phase. In this paper, we use the hat symbol for objects in edge quantum systems instead of objects related to ${\mathbb{Z}_2}$-gradings. Let $Z \subset Y$ be a pair of metric spaces with a proper isometric $G$-action. Here we write $q: C^_{\mathscr{H}}(Y) \to C^_{\mathscr{H}}(Y,Z)$ for the quotient. \[defn:edgeTP\] We say that an edge quantum system controlled at $Z \subset Y$ with the symmetry class $(G,\phi , c, \tau )$ is a quadruple $(\hat{{\mathscr{H}}}, \hat{U}, \hat{\pi} ,\hat{H})$ where - $(\hat{{\mathscr{H}}}, \hat{U}, \hat{\pi})$ is a uniformly finite $(\phi,c,\tau)$-twisted $(Y,G)$-module, - $\hat{H} \in {\mathbb{B}}(\hat{{\mathscr{H}}})$ is a $c$-twisted $G$-invariant controlled self-adjoint operator such that $q (\hat{H}) \in C^_{{\mathscr{H}}}(Y,Z)$ has a spectral gap at $[-\varepsilon,\varepsilon]$ for some $\varepsilon >0$. A controlled edge topological phase* is a equivalence class of edge quantum systems controlled at $Z \subset Y$ with the symmetry class $(G,\phi,c,\tau)$ with respect to the equivalence relation generated by the following: - $(\hat{{\mathscr{H}}}, \hat{U}, \hat{\pi} ,\hat{H}) \sim (\hat{{\mathscr{H}}} ',\hat{ U}', \hat{\pi}' ,\hat{H}')$ if there is a unitary $V: \hat{{\mathscr{H}}} \to \hat{{\mathscr{H}}} '$ such that $V\hat{U}_gV^* =\hat{U}_g'$, $V\hat{\pi} (f)V^=\hat{\pi} '(f)$ and $V\hat{H}V^=\hat{H}'$, - $(\hat{{\mathscr{H}}}, \hat{U}, \hat{\pi} ,\hat{H}_0) \sim (\hat{{\mathscr{H}}}, \hat{U}, \hat{\pi} ,\hat{H}_1)$ if there is a continuous path $\hat{H}_t \in {\mathbb{B}}(\hat{{\mathscr{H}}})$ of $c$-twisted $G$-invariant controlled self-adjoint operator such that $q (\hat{H}) \in C^_u(Y,Z)$ has a spectral gap at $[-\varepsilon,\varepsilon]$ for some $\varepsilon >0$, - $(\hat{{\mathscr{H}}}, \hat{U}, \hat{\pi},\hat{H}) \sim (\hat{{\mathscr{H}}} \oplus \hat{{\mathscr{H}}} ', \hat{U} \oplus \hat{U}', \hat{\pi} \oplus \hat{\pi} ' ,\hat{H} \oplus \hat{H}')$ if $\hat{H}'$ is invertible. We write $\hat{{\mathscr{TP}}} (Z \subset Y;G,\varphi , \tau ,c)$ for the set of controlled edge topological phases. \[prp:edge\] We have the isomorphism $$\hat{\mathscr{TP}}(Z \subset Y;G,\phi,\tau , c) \cong \operatorname{\mathrm{Im}}\partial \subset {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1, \tau,c}^{Q} (C^_{u,G}(Z \subset Y)^\Pi)$$ where $\partial $ is the boundary map of the long exact sequence associated to $$0 \to C^_{u,G}(Z \subset Y)^\Pi \to C^_{u,G}(Y)^\Pi \to C^_{u,G}(Y,Z)^\Pi \to 0.$$ In particular, it is independent of the choice of $G$-invariant Delone pairs $Z \subset Y$ of $N \subset M$ by Lemma \[lem:Morita\]. We have a well-defined map $$\hat{\Theta } : {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{-1, \tau,c}^{Q} (C^_{u,G}(Y,Z)^\Pi) \to \hat{{\mathscr{TP}}}(Z \subset Y;G,\phi,\tau , c), [s] \mapsto ({\mathscr{H}}_{Y,G}\otimes \mathscr{V}, U,\pi, \hat{s})$$ where $\hat{s}$ is a self-adjoint $c$-twisted $Q$-invariant lift of $s$ in $C^_{u,G}(Y)^\Pi$. In the same way as Proposition \[prp:bulk\], we obtain that $\operatorname{Ker}\hat \Theta =\operatorname{\mathrm{Im}}q_$ and $\hat \Theta$ is surjective. Finally, we obtain the conclusion since ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_{0,\tau,c}^Q(C^_{u,G}(Y,Z)^\Pi ) / \operatorname{\mathrm{Im}}q_* \cong \operatorname{\mathrm{Im}}\partial$. Bulk-edge correspondence {#section:4} ======================== In this section, we give the definition of the bulk and edge indices of topological phases controlled at Delone subsets of the Euclidean space ${\mathbb{E}}^d$ and formulate the bulk-edge correspondence for them. Let $(G,\phi,c,\tau)$ be as in Assumption \[assump:ext\] such that $\Pi$ is free abelian (particularly we take notice of the case as in Example \[exmp:extRoe\]) and let $X$ be a $G$-invariant Delone subset of ${\mathbb{E}}^d$. \[defn:bulk\] We write $\operatorname{ind_{\mathit{b}}}$ for the group homomorphism $$\operatorname{ind_{\mathit{b}}}: \mathscr{TP}(X;G,\phi,c, \tau) \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_Q^{n-d,c',\tau'+\nu }(B_\Pi)/\mathrm{Triv}_{d-n}$$ induced from the inclusion $C^* _{u,G}(X)^\Pi \to C^_G(X)^{\Pi}$ and the isomorphisms in Proposition \[prp:bulk\], Corollary \[cor:cBC\] and Corollary \[cor:iota\]. Here, we write $\mathrm{Triv}_0:=\mathrm{Triv}$ and $\mathrm{Triv}_k:=0$ for $k \neq 0$. When $n=d$, it is merely the quotient map $${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_Q^{n-d,c',\tau'+\nu }(B_\Pi)/\operatorname{\mathrm{Im}}(\iota_u)_ \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_Q^{n-d,c',\tau'+\nu }(B_\Pi)/\mathrm{Triv}$$ since $C^_G(X)^\Pi \cong {\mathbb{K}}(\ell^2_\Pi(G)) \otimes {\mathbb{K}}$ by Example \[exmp:extRoe\]. On the other hand, when $n=0$, $\operatorname{ind_{\mathit{b}}}=\Xi _X^$ by Proposition \[prp:index\]. In particular, for CT-symmetries, $\operatorname{ind_{\mathit{b}}}$ is the same thing as the index pairing introduced in [@GrossmannSchulz-Baldes] by Remark \[remk:CTvsR\]. Next, we give a definition of the edge index. Let ${\mathbb{E}}^d={\mathbb{E}}^d_+ \cup {\mathbb{E}}^d_-$ be as in Corollary \[cor:MVind\] and let $Z \subset Y$ be a $G$-invariant Delone pair (Definition \[defn:Delone\]) in ${\mathbb{E}}^{d-1} \subset {\mathbb{E}}^{d}_+$. \[defn:edge\] We write $\operatorname{ind_{\mathit{e}}}$ for the group homomorphism $$\operatorname{ind_{\mathit{e}}}: \hat{\mathscr{TP}}(Z \subset Y;G,\phi,c', \tau') \to {{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}_Q^{n-d,c',\tau'+\nu }(B_\Pi)$$ induced from the inclusion $C^* _{u,G}(Z \subset Y)^{\Pi} \to C^_G(Z \subset Y)^{\Pi}$, Proposition \[prp:edge\] and Corollary \[cor:cBC\]. Finally, we show the bulk-edge correspondence for these indices. Let $X=Y_+ \cup Y_-$ be a Delone partition of ${\mathbb{E}}^d={\mathbb{E}}^d_+ \cup {\mathbb{E}}^d_-$ and $Z:=Y_1 \cap Y_2$. Since $\operatorname{rank}\Pi <d$, the bulk and edge indices take values in the same group ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{n-d,c',\tau'}_Q(B_\Pi)$. For a bulk quantum system $({\mathscr{H}}, U, \pi , H)$ controlled at $X$ with the symmetry class $(G,\phi,c,\tau)$, we obtain the edge system of the same material as $$\hat{{\mathscr{H}}}:=\pi ({\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}_{Y_+}) {\mathscr{H}}, \ \hat{U}_g:= U_g \|_{\hat{{\mathscr{H}}}} , \ \hat{\pi} (f):=\pi (f)\|_{\hat{{\mathscr{H}}}}, \hat{H}:=\pi({\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}_{Y_+}) H \pi({\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}_{Y_+})$$ where ${\@ifnextchar_{ \@ifnextchar^{{ \latexchi_{}^{}}}{\latexchi^{}_{}}}}_{Y_+}$ is the characteristic function on $Y_+$. Let $X, Y _\pm , Z$ and $(G,\phi,c,\tau)$ be as above. Then, we have $$\operatorname{ind_{\mathit{b}}}[{\mathscr{H}}, U, \pi , H]= \operatorname{ind_{\mathit{e}}}[\hat{{\mathscr{H}}}, \hat{U} , \hat{\pi} , \hat{H}].$$ It follows from the definition of $\Psi_{X,G}$ in Corollary \[cor:cBC\] and the fact that $\mu _{X,G}$ is compatible with the Mayer-Vietoris exact sequences. \[exmp:2dIQHE\] Consider the $2$-dimensional IQHE without translation symmetry. As indicated in Table \[table:Kitaev\], the bulk and edge indices take values in ${\mathbb Z}$. Now we relate them with the Hall conductance. Let $\mathcal{T}_d^\omega$ be as in (\[form:cyc\]). In the same way as [@MR1295473] and [@MR1877916], we obtain the cohomological presentation for the bulk and edge Hall conductance as [$$\begin{aligned} \sigma_{\perp}^b &:= \frac{e^2}{h}2\pi i \mathcal{T}^\omega (P_- [\nabla_1(P_-), \nabla_2(P_-)])=\frac{e^2}{h} \langle [P_-], \mathcal{T} _2^\omega \rangle \\ \sigma _{\perp}^e &:= -\frac{e^2}{h}\lim _{\Delta \to \{ 0 \}}\frac{1}{\|\Delta\|} \hat{\mathcal{T}} (\hat{P}_{\Delta} \nabla _1(\hat{H})) =\frac{e^2}{h}\langle [-\exp (\pi i \hat{H})], \hat{\mathcal{T}} _1^\omega \rangle . \end{aligned}$$]{} Here note that $P_- \in {\mathbb{M}}_n(\mathcal{B}_u(X))$ since $P_-$ is obtained by a holomorphic functional calculus of the controlled operator $H$. By Lemma \[lem:pair\], they coincides with $\operatorname{ind_{\mathit{b}}}[{\mathscr{H}}, U,\pi,H]$ and $\operatorname{ind_{\mathit{e}}}[\hat{{\mathscr{H}}}, \hat{U}, \hat{\pi}, \hat{H}]$ respectively. By (\[form:MVcyc\]), we obtain $\sigma _{\perp}^b=\sigma _{\perp}^e$. \[exmp:oddchiral\] In the same way as above, the odd cyclic cocycle $\mathcal{T}_d^\omega $ gives a homomorphism ${\mathscr{TP}}(X; {{{\rm A\hspace{-.08em}I\hspace{-.08em}I\hspace{-.08em}I}}} ) \to {\mathbb R}$, which is explicitly written by Prodan–Schulz-Baldes [@mathph14025002] as $$\langle [H], \mathcal{T}_d^\omega \rangle := \frac{i(i\pi)^{(d-1)/2}}{d!!}\sum_{\sigma \in \mathfrak{S}_d} (-1)^{\|\sigma\|} \mathcal{T}^\omega (U^(\nabla_{\sigma(1)}U) \cdots U^(\nabla _{\sigma (d)}U))$$ where $\Phi:=(1+P)/2$ and $U:=\Phi H\gamma \Phi$ is a unitary on $\Phi {\mathscr{H}}$. By Lemma \[lem:pair\], it coincides with the index pairing $\Xi _X^* [H] \in {\mathbb Z}$. It is also shown in [@mathph14025002] and related to the index of limit operators. When $d=1$, the bulk-edge correspondence implies that $\langle [H], \mathcal{T}_1^\omega \rangle$ coincides with the Fredholm index of $\hat{H}_0$ where $\hat{H}={\begin{pmatrix} 0 & \hat{H}_0^* \\ \hat{H}_0 & 0 \end{pmatrix}}$ with respect to the ${\mathbb{Z}_2}$-grading given by $P$. \[exmp:2dSQHE\] For a $2$-dimensional Quantum Spin-Hall system (quantum systems with type [[[AII]{}]{}]{} symmetry) without any other symmetry, the bulk and edge indices take value in ${\mathrm{KA\hspace{-.08em}I\hspace{-.08em}I}}_2({\mathbb R}) ={\mathrm{KQ}}_2({\mathbb R}) \cong {\mathbb{Z}_2}$. We show that it coincides with the Kane–Mele ${\mathbb{Z}_2}$-invariant or parity of the Spin-Hall conductance. Assume that a bulk quantum system $({\mathscr{H}}, \pi, \varphi , H)$ has another symmetry, that is, ${\mathscr{H}}={\mathscr{H}}_+ \oplus {\mathscr{H}}_-$ such that $T{\mathscr{H}}_{\pm}={\mathscr{H}}_{\mp}$ and $H{\mathscr{H}}_\pm={\mathscr{H}}_\pm$ (given by spectral subspaces of the spin operator $s_z$). In other words, $[H_+] \in {\mathrm{K}}_0 (C^_u(X)) \cong {\mathbb Z}$, where $H_+:=H\|_{{\mathscr{H}}_+}$, satisfies $${\ell ^2\mathchar`-\mathrm{Ind}}_e^{\mathscr{T}} [H_+]=[H] \in {\mathrm{KA\hspace{-.08em}I\hspace{-.08em}I}}_0 (C^_u(X)) \cong {\mathrm{KQ}}_2({\mathbb R}) \cong {\mathbb{Z}_2}.$$ By Example \[exmp:ind\], we obtain $$\operatorname{ind_{\mathit{b}}}({\mathscr{H}}, \pi ,\varphi , H) = 2\pi i \mathcal{T}^\omega (P_- [\nabla_x P_1, \nabla_2 P_-]) \text{ mod $2$},$$ which coincides with the definition of Kane-Mele invariant in [@MR3063955]. In the same way, we obtain the edge index $$\operatorname{ind_{\mathit{e}}}(\hat{{\mathscr{H}}} , \hat{\pi} ,\hat{\varphi }, \hat{H}) = \lim _{\Delta \to \{ 0 \}}\frac{1}{\|\Delta\|} \hat{\mathcal{T}} (\hat{P}_{\Delta} \nabla _y\hat{H}) \text{ mod $2$},$$ which is the Spin-Hall conductance introduced by Prodan [@MR2525473] modulo $2$. We remark that the ${\mathbb Z}$-valued index is a topological invariant if the additional symmetry is preserved. In the same way, we obtain ${\mathbb{Z}_2}$-invariants for $1$-dimensional type [[[DIII]{}]{}]{} and $3$-dimensional type [[[CII]{}]{}]{} topological insulators with additional symmetry by using with odd cocycles. By Remark \[remk:CTvsR\] and Proposition \[prp:index\], these invariants are the same thing as the index pairing in [@GrossmannSchulz-Baldes] and hence the ${\mathbb{Z}_2}$-invariant in [@mathph150805485] as is noted in Remark (ii) of [@mathph150805485]. Consider $3$-dimensional type [[[AII]{}]{}]{} topological insulators. Set $\Pi;={\mathbb Z}^3$ and $X=\|{\mathbb Z}^3\|$. The bulk indices for $\Pi$-invariant and non-invariant systems take values in [$$\begin{aligned} {\mathrm{KA\hspace{-.08em}I\hspace{-.08em}I}}^0(B_\Pi)/\mathrm{Triv} &\cong {\mathrm{KQ}}^0(B_\Pi, {{\rm pt}}) \cong {\mathrm{KQ}}^2({{\rm pt}})^3 \oplus {\mathrm{KQ}}^3({{\rm pt}}) \cong ({\mathbb Z}_2 )^3 \oplus {\mathbb Z}_2, \\ {\mathrm{KA\hspace{-.08em}I\hspace{-.08em}I}}^{3}({{\rm pt}})&\cong {\mathrm{KQ}}^3({{\rm pt}}) \cong {\mathbb Z}_2 \end{aligned}$$]{} respectively. Moreover, the restriction map $$\operatorname{Res}:{\mathrm{KA\hspace{-.08em}I\hspace{-.08em}I}}_0(C^(X)^\Pi) \to {\mathrm{KA\hspace{-.08em}I\hspace{-.08em}I}}_0(C^(X))$$ is given by the projection onto ${\mathrm{KQ}}^3({{\rm pt}}) \cong {\mathbb{Z}_2}$ since the pairing of $\Xi _X$ with elements ${\mathrm{KQ}}^2({{\rm pt}}) ^3$ is zero. Therefore, we obtain one “strong” (i.e. robust when the translation symmetry is broken) and three “weak” (i.e. not preserved when the translation symmetry is broken) ${\mathbb{Z}_2}$-valued topological invariants as is studied by Moore–Balents [@MB2007] and Fu–Kane–Mele [@FKM2007]. Recall that the bulk and edge index maps are isomorphisms by Lemma \[lem:uRoe\]. This mean that the edge (and hence the bulk) index is nontrivial if and only if there is a topologically robust “topological metal” surface state in the sense of [@FKM2007]. Moreover, two $3$-dimensional [[[AII]{}]{}]{} systems with the same strong invariant are in the same topological phase when we the translation symmetry is broken. We remark that a model with nontrivial index is found in Section 3.6.3 of [@GrossmannSchulz-Baldes]. \[exmp:cryst\] Consider a topological crystalline insulator with disorder which is symmetric under the action of a point group $G$ coming from a spin representation such as rotations. Then, the bulk and edge indices take values in the group as in (\[form:group\]). In particular, when we consider the group $C_k:={\mathbb Z}/k{\mathbb Z}$ acting on ${\mathbb{E}}^2$ by rotations, the indices take values in the group as in (\[form:rot\]). \[exmp:refl\] Consider a topological crystalline insulator with reflection symmetry, that is, $\mathscr{R}$ acts on ${\mathbb{E}}^1 \times {\mathbb{E}}^{d-1}$ as $(x_1,x_2,\dots, x_n) \mapsto (-x_1, x_2, \dots x_n)$. Let $\tau$ be an arbitrary $2$-cocycle of $\mathscr{A} \times \mathscr{R}$ such that $(\phi,c,\tau)$ is a twist of $\mathscr{A} \times \mathscr{R}$. Then, by Corollary \[cor:cBC\], the indices take values in ${{}^{\phi} \hspace{-0.1ex} } {\mathrm{K}}^{\mathscr{A} \times \mathscr{R}}_{d-1,c,\tau}({\mathrm{C}\ell}_1)$, which is calculated in Example \[exmp:reflect\]. This conclusion is the same thing as the one indicated in table VI of [@PhysRevB.88.125129]. We remark that it is exposed in Section IV.B of [@PhysRevB.88.125129] that the case of $PR=-RP$ and $TRP=-RPT$ has two components. \[remk:conc\] As a concluding remark, we discuss on the meaning of these indices. In Example \[exmp:2dIQHE\] and Example \[exmp:2dSQHE\], they are related to the conductance by the TKNN formula. Moreover, by Lemma \[lem:uRoe\], the edge index is an isomorphism for - arbitrary types when $d=1$ (i.e. the dimension of the edge is $0$), - arbitrary types other than type [[[A]{}]{}]{} when $d=2$, - type [[[AI]{}]{}]{}, [[[AII]{}]{}]{}, [[[D]{}]{}]{}, [[[C]{}]{}]{}, [[[BDI]{}]{}]{} and [[[CII]{}]{}]{} when $d=3$. In other words, the edge index is trivial if and only if the edge Hamiltonian is homotopic to an invertible operator, that is, does not have topologically robust edge channels. On the other hand, Prodan [@mathph150103479] shows that weak invariants, which is not captured by our index, are protected in ${\mathscr{TP}}(G,\phi,c,\tau)$ in the case of type A and [[[AIII]{}]{}]{} insulators by using lower cyclic cocycles. Let us consider a variation of the quantum systems using finite $(X,G)$-modules instead of uniformly finite $(X,G)$-modules. Then, the equivalence relation of topological phases is also weakened. Actually, in the same way as Proposition \[prp:bulk\] and Proposition \[prp:edge\], they are isomorphic to a modification of twisted equivariant ${\mathrm{K}}$-groups of Roe algebras, that is, the range of the bulk and edge indices. In other words, two topological phases have the same index if and only if they are equivalent in this rough classification. In particular, the lower cocycle of Prodan [@mathph150103479] are not invariants of this rough equivalence relation. Acknowledgement {#acknowledgement .unnumbered} --------------- The author would like to thank his supervisor Yasuyuki Kawahigashi for his support and encouragement. This work is motivated by his collaborative research with Mikio Furuta, Shin Hayashi, Motoko Kotani, Shinichiro Matsuo and Koji Sato. He would like to thank them for suggesting the problem and many stimulating conversations. He also would like to thank Jean Bellissard, Peter Bouwknegt, Johannes Kellendonk, Yoshiko Ogata, Mathai Varghese, Emil V. Prodan and Guo Chuan Thiang for their helpful conversation. He is greatly indebted to referees for their many helpful advices. This work was supported by Research Fellow of the JSPS (No. 26-7081) and the Program for Leading Graduate Schools, MEXT, Japan.
--- abstract: 'This paper concerns representations of the integral general linear group. The extension groups $Ext^2$ between any pair of hook Weyl modules are determined via a detailed study of cyclic generators and relations associated to certain extensions. As a corollary, the modular extension groups $Ext^1$ between such modules are determined.' address: 'Department of Mathematics, University of Athens' author: - 'Dimitra-Dionysia Stergiopoulou' date: 'June 5, 2020' title: Integral $Ext^2$ between hook Weyl modules --- Introduction ============ In the study of polynomial representations of the integral general linear group $GL_n$, the Weyl modules $\Delta(\lambda)$, indexed by partitions $\lambda$, play a central role. These modules have an explicit construction, enjoy a standard basis theorem and their characters are the classical Schur functions [@Gr]. Moreover, the heads of the modules $K\otimes \Delta(\lambda)$ form a complete set of inequivalent polynomial simple modules for $GL_n(K)$, where $K$ is an infinite field. However, the structure of the modules $\Delta(\lambda)$ is not well understood. One of the important problems in the area is to determine the extension groups $Ext^i(\Delta(\lambda),\Delta(\mu))$. Relatively few explicit results are known. Modular extension groups for $GL_n(K)$ were studied in [@A], when $\lambda$ consists of a single column and $\mu$ of a single row. For $SL_2(K)$, all modular extension groups between Weyl modules were described in [@Pa], generalizing [@Er] and [@CE]. The modular extension groups $Ext^1(K\otimes\Delta(\lambda), K\otimes\Delta(\mu))$ for $GL_n(K)$ were determined in [@DG], when $\mu$ consists of a single row. Concerning the integral extension groups, $Ext^1(\Delta(\lambda),\Delta(\mu))$ for $GL_3$ was computed in [@BF], when $\lambda$ and $\mu$ differ by a multiple of a positive root. The groups $Ext^1(\Delta(\lambda),\Delta(\mu))$ for $GL_n$ was determined in [@Ku2], when $\lambda$ and $\mu$ differ by a single positive root. The groups $Ext^i(\Delta(h), \Delta (h(k)))$ were computed in [@MS] for $i=1,k$ and any pair of hooks $h =(a,1^b)$ and $h(k)=(a+k, 1^{b-k})$. The main method there was to determine cyclic generators of certain extension groups of the form $Ext^i(\Delta(h),D_{a+k}\otimes\Lambda^{b-k})$, where $D_$ and $\Lambda^$ denote the divided power algebra and the exterior algebra, respectively, of the natural $GL_n$-module. Except for some extreme cases, these generators usually have an involved form. The next step in our method was to calculate the images of these under canonical maps. The idea of using the ‘skew’ extensions $Ext^i(\Delta(h),D_{a+k}\otimes\Lambda^{b-k})$, in place of the usual extensions $Ext^i(\Delta(h),\Delta(h(k)))$, comes from the observation that the former seem to have more manageable presentation matrices. The reason for this is that there is no straightening law involved. The purpose of the present paper is to study $Ext^2$ between any two hook Weyl modules using the above strategy. The main result is the following. Consider hook partitions $h=(a,1^b)$ and $h(k)=(a-k,1^{b-k})$ for $k \ge 0$. Suppose $n\ge b+1$. Then, $Ext^2(\Delta(h), \Delta(h(k)))=0$, except possibly when $k\in\{2,3,4\},$ in which cases $$Ext^2(\Delta(h), \Delta(h(k))) = \begin{cases} \mathbb{Z}_s, & \mbox {if}\;k=2, \\ \mathbb{Z}_{3/t}, & \mbox {if}\;k=3, \\ \mathbb{Z}_{t}, & \mbox {if}\;k=4, \end{cases}$$ where $s= {(a+b)}/{gcd(2,a+b)}$, $t={gcd(3,a+b)}$. As a corollary, we determine the modular $Ext^1$ groups between any hook partitions. In Section 2 we gather necessary preliminaries. In Section 3 we consider the cases $k\neq 4$. In Section 4, which costitutes the main part of the paper, we complete the proof of the theorem. As a corollary, we determine the dimensions of the modular $Ext^1$ groups between any hook Weyl modules. Preliminaries ============= We will use the notation of [@MS] but, for completeness, we recall the main points here and supplement with additional material. Notation. --------- Let $F$ be a free abelian group of finite rank $n$. Fixing a basis of $F$ yields an identification of general linear groups $GL(F)=GL_n(\mathbb{Z})$. We will be working with homogeneous polynomial representations of $GL_n(\mathbb{Z})$ of degree $r$, or equivalently, with modules over the Schur algebra $S_\mathbb{Z}(n,r)$ [@Gr]. We will write $S(n,r)$ in place of $S_\mathbb{Z}(n,r)$. By $DF=\sum_{i\geq 0}D_iF$ and $\Lambda F=\sum_{i\geq 0}\Lambda^{i}F$ we denote the divided power algebra of $F$ and the exterior algebra of $F$ respectively. We will usually omit $F$ and write $D_i$ and $\Lambda^i$. Throughout this paper all tensor products are over the integers. For a partition $\lambda$ of $r$ with at most $n$ parts, we denote by $\Delta(\lambda)$ the corresponding Weyl module for $S(n,r)$. A hook $h$ is a partition of the form $h=(a,1^b)$. The following complex of $S(n,r)$-modules (which is the dual of the usual Koszul complex) is exact $$0 \rightarrow D_{a+b} \rightarrow ... \rightarrow D_{a+1} \otimes \Lambda^{b-1} \xrightarrow{\Theta_{a}} D_{a} \otimes \Lambda^{b} \rightarrow ... \rightarrow \Lambda^{a+b} \rightarrow 0,$$ where $\Theta_{a}$ is the composition $\ D_{a+1}\otimes \Lambda^{b-1} \xrightarrow{\triangle \otimes 1} D_{a}\otimes D_1 \otimes \Lambda^{b-1}\xrightarrow{1 \otimes m} D_{a} \otimes \Lambda^{b},$ where $\triangle$ (respectively, $m$) is the indicated component of the comultiplication (resp., multiplication) map of the Hopf algebra $DF$ (resp., $\Lambda F$). It is well known that if $h=(a,1^b)$ is a hook, $b \geq 1$, then $\Delta (h) \simeq cok(\Theta_{a}) \simeq ker(\Theta_{a-1}).$ Throughout this paper we use the notation $h=(a,1^b),h(k)=(a+k,1^{b-k})$ and $r=a+b.$ Straightening law and standard basis. ------------------------------------- We recall the straightening law and the standard basis theorem for $\Delta(h)$ [@ABW]. Fix an ordered basis $e_1,\dots,e_n$ of $F$. For simplicity, we denote the element $e_i$ by $i$ and accordingly the element $e_{i_1}^{(a_1)} \cdots e_{i_t}^{(a_t)} \otimes e_{j_1} \wedge \dots \wedge e_{j_b} \in D_a \otimes \Lambda^b$ by ${i_1}^{(a_1)} \cdots {i_t}^{(a_t)} \otimes {j_1} \cdots {j_b}$. The image of this element under the identification $\Delta (h) \simeq cok(\Theta_{a})$ will be denoted by ${i_1}^{(a_1)} \cdots {i_t}^{(a_t)} / {j_1} \cdots {j_b}$. Now suppose $i_1<i_2< \dots <i_t$ and $j_1 \le i_1$. Then in $\Delta(h)$ we have $${i_1}^{(a_1)}\cdots {i_t}^{(a_t)} / {j_1} \cdots {j_b} = \begin{cases} -\sum\limits_{s\geq 2}{i_1}^{(a_1+1)} \cdots {i_s}^{(a_s-1)}\cdots{i_t}^{(a_t)} / {i_s}{j_2} \cdots {j_b}, & \mbox{if}\;j_1=i_1 \\ -\sum\limits_{s\geq 1}j_1{i_1}^{(a_1)} \cdots {i_s}^{(a_s-1)}\cdots {i_t}^{(a_t)} / {i_s}{j_2} \cdots {j_b}, & \mbox {if}\;j_1<i_1. \end{cases}$$ A $\mathbb{Z}$ - basis of $\Delta(h)$ is the set of elements ${i_1}^{(a_1)} \cdots {i_t}^{(a_t)} / {j_1} \cdots {j_b} $, where $a_1+\cdots+a_t=a$, $i_1<\dots<i_t$ and $i_1 <j_1< \dots <j_b.$ Resolutions of hooks. --------------------- From [@Gr] or [@AB] we recall that for each sequence $a_1,\dots,a_n$ of non negative integers $a_i$ that sum to $r$, the $S(n,r)$-module $D_{a_1} \otimes \cdots \otimes D_{a_n}$ is projective. We will use the explicit finite projective resolution $P_{}(a,b)$ of $ \Delta (h)$, $$0 \longrightarrow \cdots \longrightarrow P_2(a,b) \xrightarrow{\Theta_{2}(a,b)} P_1(a,b) \xrightarrow{\Theta_{1}(a,b)} P_0(a,b)$$ of [@Ma] which we now recall. For short, we denote the tensor product $D_{a_1} \otimes \cdots \otimes D_{a_m}$ of divided powers by $D(a_1,\dots,a_m)$. We have $P_i(a,b)=\sum D(a_1,\dots,a_{b+1-i}) $ where the sum ranges over all sequences $(a_1,\dots,a_{b+1-i})$ of positive integers of length $b+1-i$ such that $a_1+\cdots+a_{b+1-i}=a+b$ and $a \le a_1 \le a+i$. The differential $\Theta_{i}(a,b)$ is defined be sending $x_1 \otimes \cdots \otimes x_{b+1-i} \in D(a_1,\dots,a_{b+1-i})$ to $$\sum_{j=1}^{b+1-i} (-1)^{j+1}x_1 \otimes \cdots \otimes \triangle(x_j) \otimes \cdots \otimes x_{b+1-i} \in D(a_1,\dots,u,v,\dots,a_{b+1-i}),$$ where $\triangle$ is the the two-fold diagonalization $D(a_j) \rightarrow \sum D(u,v)$, the sum ranges of all positive integers $u,v$ such that $u+v=a_j$ and $D(a_1,\dots,u,v,\dots,a_{b+1-i})$ is a summand of $P_{i-1}(a,b)$ with $u$ located in position $j$. Let $ \triangle_{u,v} : D(a_j) \rightarrow D(u,v)$ be the indicated component of the two-fold diagonalization $D(a_j) \rightarrow \sum D(u,v)$. If $A,B$ are $S(n,r)$ - modules, we write $Hom(A,B)$ and $Ext^i(A,B)$ in place of $Hom_{S(n,r)}(A,B)$ and $Ext^i_{S(n,r)}(A,B)$ respectively. We recall the recursions $$\begin{aligned} &P_0(a,b)=D(a) \otimes P_0(1,b-1), \\ &P_i(a,b)=P_{i-1}(a+1,b-1) \oplus D(a) \otimes P_i(1,b-1), i>0 \end{aligned}$$ and that under these identifications we have the following. The differential $Hom(\Theta_i(a,b), M)$ of the complex\ $Hom(P_(a,b),M)$ is given by Hom(P\_[i-2]{}(a+1,b-1),M) & Hom(P\_[i-1]{}(a+1,b-1),M)\ Hom(D(a) P\_[i-1]{}(1,b-1),M) &Hom(D(a) P\_i(1,b-1),M) where the top horizontal map is $Hom(\Theta_{i-1}(a+1,b-1),M)$, the bottom one is $-Hom(1 \otimes \Theta _i(1,b-1),M)$ and the restriction of the diagonal one on the summand $Hom(D(a,j,a_2,\dots,a_m),M)$ is $Hom(\triangle_{a,j} \otimes 1 \otimes \cdots \otimes 1), M) .$ We now consider weight spaces. For any $S(n,r)$-module $M$ and any sequence $a_1,\dots,a_m$ of nonnegative integers such that $a_1+\cdots +a_m=r$ and $m \le n$, we identify the $\mathbb{Z}$-module $Hom(D(a_1,\dots,a_m),M)$ with the $(a_1,\dots,a_m)$ weight subspace of $M$ (with respect to the action of $\mathbb{Z}^n$) [@AB]. We will use such identifications freely throughout this paper. In particular, suppose $M$ is a skew Weyl module for $S(n,r)$ (denoted be $K_{\lambda / \mu}$ in [@ABW]). Using the $\mathbb{Z}$-basis of $M$ given by standard tableaux of [@ABW], Thm. II.3.16, we see that the $\mathbb{Z}$-module $Hom(D(a_1,\dots,a_m),M)$ may be identified with the $\mathbb{Z}$-submodule of $M$ that has basis the standard tableaux of $M$ that contain the entry $i$ exactly $a_i$ times, $i\in\{1,\dots,m\}$. We call this the standard basis of $Hom(D(a_1,\dots,a_m),M)$. If $a$ is a non negative integer, we denote by $M_a$ the $\mathbb{Z}$-submodule of $M$ given by $$M_a=\sum Hom(D(a,a_2,\dots,a_m),M)$$ where the sum ranges over all nonnegative integers $a_2,\dots,a_m$ such that $a_2+\cdots+a_m$ $=r-a$ and $m \le n$. A $\mathbb{Z}$- basis of this is the set of standard tableaux of $M$ in which the entry 1 appears exactly $a$ times. $M_a$ is a $GL_{n-1}$ submodule of $M$ with $GL_{n-1}$ acting on the basis elements $2,\dots,n$. From the definition of $M_a$ it follows that we have the identification $$Hom_A(D(a_1,\dots,a_m),M)=Hom_B(D(a_2,\dots,a_m),M_{a_1})$$ and thus by summing, $Hom_A(D(a) \otimes P_i(1,b-1),M)=Hom_B(P_i(1,b-1),M_{a}),$ where $a=a_1$, $A=S(n,r), B=S(n-1,b)$. We refer to the standard basis of $Hom_A(D(a_1,\dots,a_m),M)$ as the standard basis of $Hom_B(D(a_2,\dots,a_m),M_a)$. Under the above identifications it follows from the definition of the complexes, that the maps $$\begin{aligned} &Hom(1 \otimes \Theta _i(1,b-1),M): \\ &Hom_A(D(a) \otimes P_{i-1}(1,b-1),M) \rightarrow Hom_A(D(a) \otimes P_i(1,b-1),M), \\ &Hom(\Theta _i(1,b-1),M_a): \\ &Hom_B(P_{i-1}(1,b-1),M_a) \rightarrow Hom_B(P_i(1,b-1),M_a) \end{aligned}$$ are equal. From now on we will drop from the above $Hom$ modules the subscripts $A,B$. Additional Preliminaries. ------------------------- We want to consider matrices of the differential of the complex $Hom(P_(a,b),M)$, where $M$ is a skew Weyl module. We consider the usual lexicographic ordering of the elements of the standard basis of $Hom(D(a_1,\dots,a_m),M)$ $=Hom(D(a_2,\dots,a_m),M_{a_1})$. We identify the sequence $(a_1,\dots,a_m)$ of nonnegative integers, where $m \le n$, with the sequence $(a_1,\dots,a_m,0,\dots,0) \in \mathbb{Z}^n $. Now if $(a_1,\dots,a_m)$ is greater than $(b_1,\dots,b_{m'})$ , where $m,m' \le n$, in the usual lexicographic ordering of sequences, me declare that each element of the standard basis of $Hom(D(a_1,\dots,a_m),M)$ is less than each element of the standard basis of $Hom(D(b_1,\dots,b_{m'}),M)$. For $M$ a skew Weyl module, let $E^{i}(\Delta(h),M)$ be the cokernel of the differential $Hom(\Theta_i (a,b),M)$ of the complex $Hom(P_{}(a,b),M)$. We know that the torsion part of this abelian group is isomorphic to $Ext^{i}(\Delta(h),M)$ [@AB]. Let $e^{(i)}(a,b,M)$ be the matrix of the differential $Hom(\Theta_i(a,b),M)$ with respect to the standard bases for the various spaces $Hom(D(a_1,\dots,a_m),M)$ and the above ordering. Likewise, let $e^{(i)}(1,b-1,M_a)$ be the matrix of the differential $Hom(\Theta_i(1,b-1),M_a)$ with respect to the standard bases for the various $Hom(D(a_2,\dots,a_m),M_a)$. From the previous discussion and Remark 2.1 we get the following statement. \[canonicalmatrix\] Suppose $n \ge b+1$ and $M$ is a skew Weyl module for $S(n,r)$. Then for $i=1$ $$e^{(1)}(a,b,M)= \left( \begin{array}{c} B^1(a,b,M)\\ \hline \; -e^{(1)}(1,b-1,M_a) \end{array} \right)$$ and for $i>1$ $$e^{(i)}(a,b,M)= \left( \begin{array}{c\|c} e^{(i-1)}(a+1,b-1,M) &B^i(a,b,M)\\ \hline \; & -e^{(i)}(1,b-1,M_a) \end{array} \right),$$ where in both cases $B^i(a,b,M)$ is the matrix of the diagonal map $Hom(D(a) \otimes P_{i-1}(1,b-1),M) \rightarrow Hom(P_{i-1}(a+1,b-1),M)$ of Remark 2.1. We will also need a lemma for the special case $M=D_{a+k}\otimes\Lambda^{b-k}$ which we now describe. We will need a different order of the basis elements. Let $B_i$ be the standard basis of $Hom(P_{i-1}(a+1,b-1,D_{a+k}\otimes\Lambda^{b-k}))$, $B_{i,1}$ the subset of the standard basis of $Hom(D(a) \otimes P_{i}(1,b-1),D_{a+k}\otimes\Lambda^{b-k}))$ consisting of the standard tableaux with one 1 in the $\Lambda^{b-d}$ part and $B_{i,0}$ the subset of the standard basis of $Hom(D(a) \otimes P_{i}(1,b-1),D_{a+k}\otimes\Lambda^{b-k}))$ consisting of the standard tableaux with no 1 in the $\Lambda^{b-k}$ part. We order each of the sets $B_i, B_{i,1}, B_{i,0}$ lexicographically and we declare that the elements of $B_i$ are less than the elements of $B_{i,1}$, and the elements of $B_{i,1}$ are less than the elements of $B_{i,0.}$ Hence we have a total order on the basis $B_{i-1} \cup B_{i-1,1} \cup B_{i-1,0}$ of the domain of $Hom(\Theta_i(a,b),D_{a+d}\otimes\Lambda^{b-k})$ and a total order on the basis $B_{i} \cup B_{i,1} \cup B_{i,0}$ of the codomain of $Hom(\Theta_i(a,b),D_{a+k}\otimes\Lambda^{b-k})$. \[skewmatrix\] With respect to the above ordered bases and for $i \ge2$, $$e^{(i)}(\Delta(h), D_{a+k}\otimes\Lambda^{b-k})= \left( \begin{array}{c\|c\|c} A &&\\ \hline \; &B& \\ \hline &&C \end{array} \right) ,$$ where $$\begin{aligned} A &=e^{(i-1)}(\Delta(a+1,1^{b-1}) ,D_{a+k}\otimes\Lambda^{b-k}),\\ B&=-e^{(i)}(\Delta(1,1^{b-1}),D_{1+k}\otimes\Lambda^{b-k-1}),\\ C&=-e^{(i)}(\Delta(1,1^{b-1}),D_{k}\otimes\Lambda^{b-k}). \end{aligned}$$ The claim for the matrix $A$ follows from Lemma 2.2. For $B$ consider the diagram of abelian groups spanB\_[i-1,1]{} &spanB\_[i,1]{}\ Hom(P\_[i-1]{}(1,b-1),D\_[1+k]{}\^[b-k-1]{}) &Hom(P\_[i]{}(1,b-1),D\_[1+k]{}\^[b-k-1]{}) where we regard the modules in the bottom row as $GL_{n-1}$ modules with $GL_{n-1}$ acting on the basis elements $1,\dots,n-1$. Here, $f_{i-1}$ is the isomorphism of abelian groups sending each standard basis element $1^{(a-1)}i_2^{(c_2)}\cdots i_q^{(c_q)}\otimes 1j_2\cdots j_{b-k} \in B_{i-1,1}$ to the standard basis element $(i_2-1)^{(c_2)}\cdots (i_q-1)^{(c_q)}\otimes (j_2-1)\cdots (j_{b-k}-1) \in Hom(P_{i-1}(1,b-1),D_{1+k}\otimes\Lambda^{b-k-1})$ and similarly for $f_i$, $\alpha$ is the restriction of $Hom(\Theta_i(a,b),D_{a+k}\otimes\Lambda^{b-k})$ to $spanB_{i-1,1}$ and $\beta = Hom(\Theta_i(1,b-1),D_{1+k}\otimes\Lambda^{b-k-1})$.We claim that the diagram commutes. Using Remark 2.1 and Remark 2.2 it suffices to show that $f_i\circ \theta_s = \theta_{s-1} \circ f_{i-1}$, $t \ge 2$. Indeed, if$$T=1^{(a-1)}i_2^{(c_2)}\cdots i_q^{(c_q)}\otimes 1j_2\cdots j_{b-k} \in B_{i-1,1},$$then $$\theta_s(T) = 1^{(a-1)}i_2^{(c_2)}\cdots i_u^{(c_u)}{i^{'}_{u+1}}^{(c_{u+1})}\cdots {i^{'}_{q}}^{(c_{q})}\otimes 1j_2\cdots j_vj^{'}_{v+1}\cdots j^{'}_{b-k},$$ where $i_u=max(\{i_2,\dots,i_q\} \cap (\{2,\dots,s\})$, $j_v=max(\{j_2,\dots,i_{b-k}\} \cap (\{2,\dots,s\})$, and $i^{'}=i-1$. We note that $\theta_s(T)$ is either 0 (if $j_v=j^{'}_{v+1}$) or a standard basis element (if $j_v \neq j^{'}_{v+1}$ and $i_u \neq i^{'}_{u+1}$ ) or a non-zero multiple of a standard basis element (if $j_v \neq j^{'}_{v+1}$ and $i_u = i^{'}_{u+1}$). Hence, in all cases we have $$\begin{aligned} f_i \circ \theta_s(T) =& (i_2-1)^{(c_2)}\cdots(i_u-1)^{(c_u)}({i^{'}_{u+1}-1})^{(c_{u+1})}\cdots({i^{'}_{q}-1})^{(c_{q})}\otimes (j_2-1)\\ &\cdots(j_v-1)(j^{'}_{v+1}-1)\cdots(j^{'}_{b-k}-1). \end{aligned}$$ Similarly, one verifies that $$\begin{aligned} \theta_{s-1} \circ f_{i-1} (T) =& (i_2-1)^{(c_2)}\cdots(i_u-1)^{(c_u)}({i^{'}_{u+1}-1})^{(c_{u+1})}\cdots({i^{'}_{q}-1})^{(c_{q})}\otimes (j_2-1)\\ &\cdots(j_v-1)(j^{'}_{v+1}-1)\cdots(j^{'}_{b-k}-1) \end{aligned}$$ and thus the diagram commutes. The proof for $C$ is similar and thus omitted. Finally, the middle block in the right column block is indeed 0, since for every $T \in B_{i-1,0}$, and every $s \ge 2$ we have $\theta_s(T) $ is of the form $x\otimes j_1\dots j_{b-k}$ with $1 \notin \{ j_1,\dots,j_{b-k} \}$. Remark. We will apply many times the isomorphism $f_2$ of the previous proof in section 4 in order to describe specific relations of $E^{2}(\Delta(h),D_{a+k}\otimes \Lambda^{b-k})$. The next lemma will be used several times. The first equality follows from the main result of [@Ku1]. \[basiclemma\] Suppose $n \ge {b+1}$ and $0 \le k < b$. Then $$Ext^i(\Delta (h),D_{a+k} \otimes \Lambda ^{b-k})=Ext^i(\Lambda ^{k+1}, D_{k+1}).$$ In particular, $\;$ 1. $Ext^1(\Delta (h),D_{a+k} \otimes \Lambda ^{b-k})= \mathbb{Z}_2, \; k \ge 1,$ 2. $Ext^2(\Delta (h),D_{a+k} \otimes \Lambda ^{b-k})= \begin{cases}\mathbb{Z}_3, \; k=2,3 \\ 0, \; \;\; k \ne 2,3.\end{cases}$ The first statement and (1) are Lemma 2.3 of [@MS]. In [@To], Ex. 11.9, $Ext^2(\Lambda ^{k+1}, D_{k+1})$ was determined and hence we have (2). We give below a different proof of Touzé’s result for the sake of self completeness. We calculate $Ext^2(\Lambda ^{k+1}, D_{k+1})$ using the projective resolution $P_(1,k)$ of $\Lambda ^{k+1}$ and the method of [@AB], Section 9. If $k=0,1$, the length $P_(1,k)$ is less than 2 and hence $Ext^2(\Lambda ^{k+1}, D_{k+1})=0$. Suppose $k \ge 2$. According to Remark 2.1, the map $Hom( \Theta_2(1,k),D_{k+1})$ looks like: Hom(P\_0(2,k-1),D\_[k+1]{}) & Hom(P\_1(2,k-1),D\_[k+1]{})\ Hom(D(1) P\_1(1,k-1),D\_[k+1]{}) &Hom(D(1) P\_2(1,k-1),D\_[k+1]{}) If $a_1,\dots,a_m$ are positive integers such that $a_1+\cdots+a_m=k+1,$ then we have $Hom(D(a_1,\dots,a_m),D_{k+1})= \mathbb{Z}$, a generator is simply the multiplication map, which will be denoted by $1^{(a_1)}\cdots m^{(a_m)}$. Thus, the map $$Hom(D(a_1,\dots,u,v,\dots,a_m),D_{k+1}) \rightarrow Hom(D(a_1,\dots,a_t,\dots,a_m),D_{k+1}),$$ where $u+v=a_t$, induced by $1 \otimes \cdots \otimes \triangle_{u,v} \otimes \cdots \otimes 1: D(a_1,\dots,a_t,\dots,a_m) \rightarrow D(a_1,\dots,u,v,\dots,a_m)$ is multiplication by the binomial coefficient $ \binom{a_t}{u}$. We have the following ordered $\mathbb{Z}$ - bases. - $Hom(P_0(2,k-1),D_{k+1}) : 1^{(2)}2\cdots k,$ - $Hom(D(1) \otimes P_1(1,k-1),D_{k+1}) :12^{(2)}3\cdots k, 123^{(2)}\cdots k, \dots,12\cdots k^{(2)},$ - $Hom(P_1(2,k-1),D_{k+1}) : 1^{(3)}2 \cdots (k-1), 1^{(2)}2^{(2)}4 \cdots (k-1),$\ $1^{(2)}23^{(2)}4 \cdots (k-1),\dots ,1^{(2)}23\cdots (k-1)^{(2)}.$ - $Hom(D(1) \otimes P_2(1,k-1),D_{k+1}) : 12^{(3)}3 \cdots (k-1), 12^{(2)}3^{(2)}4 \cdots (k-1),$\ $\dots, 12^{(2)}\cdots (k-1)^{(2)}, \dots ,123^{(3)}\cdots (k-1), 123^{(2)}4^{(2)}\cdots (k-1),$\ $\dots,123^{(2)}\cdots (k-1)^{(2)}, \dots , 12\cdots (k-1)^{(3)}.$ Now with the above ordered bases, a quick computation shows that the matrix of the top horizontal arrow of $Hom( \Theta_2(1,k),D_{k+1})$ is the $(k-1) \times 1$ matrix $$e^{(1)}(2,k-1,D_{k+1}) = (3 \; \; -2 \; \; \; 2 \; \; \; \dots \; \; (-1)^{k-2}2)^t$$ and the matrix of the diagonal arrow is the $(k-1) \times (k-1)$ matrix $$B^2(1,k,D_{k+1})=diag(3,2,\dots,2).$$ Also, it follows that the matrix $e^{(2)}(1,k,D_{k+1})$ is of size $ \binom{k}{2} \times k$. In the notation of Lemma 2.2, we have $M=D_{k+1}$ and thus $M_1=D_k$. Therefore the recursion for $e^{(2)}(1,k,D_{k+1})$ is $$e^{(2)}(1,k,D_{k+1})= \left( \begin{array}{c\|c} e^{(1)}(2,k-1,D_{k+1}) &B^2(1,k,D_{k+1})\\ \hline \; & -e^{(2)}(1,k-1,D_k) \end{array} \right)$$ where $k \ge 2$ and $e^{(2)}(1,2,D_3)=(3 \; \; 3)$ (this follows by an immediate computation). Hence, $$e^{(2)}(1,3,D_4)= \begin{pmatrix} 3 &3 & \\ -2 & & 2 \\ &-3 &-3 \end{pmatrix}.$$ The nonzero invariant factors of these last two matrices are 3 and 1,3 respectively and therefore $Ext^2( \Lambda ^3,D_3)=Ext^2( \Lambda ^4,D_4)= \mathbb{Z}_3$. Let $k \ge 4$. In order to show that $Ext^2(\Lambda ^{k+1}, D_{k+1})=0$, it suffices to show that the last nonzero invariant factor of $e^{(2)}(1,k,D_{k+1})$ is equal to 1. This matrix has rank at most $k-1$ since the matrix product $e^{(2)}(1,k,D_{k+1}) \cdot e^{(1)}(1,k,D_{k+1}) $ is zero, due to the fact that $Hom(P_(1,k),D_{k+1})$ is a complex. (We have $e^{(1)}(1,k,D_{k+1}) \ne 0 $ since $Ext^1(\Lambda^{k+1},D_{k+1}) \ne 0$. In fact here $e^{(1)}(1,k,D_{k+1})$ is the $k \times 1$ matrix $(2 \; \; -2 \; \; \; 2 \; \; \; \cdots \; \; (-1)^{k-1}2)^t)$. It is straightforward to verify that the minor of $e^{(2)}(1,k,D_{k+1})$ corresponding to columns $2,3,\dots,k$ and rows $2,3,\dots,k-1, k+1$ is equal to $\pm 2^{k-1}$ and an immediate induction shows that the minor corresponding to columns $1,2,\dots,k-1$ and rows $1, 1+(k-1), 1+(k-1)+(k-2),\dots, \binom{k}{2}$ is equal to $\pm 3^{k-1}$. Since there exist two relatively prime minors of size the rank of the matrix $e^{(2)}(1,k,D_{k+1})$, the last nonzero invariant factor of this matrix is equal to 1. Next we describe the differential of $Hom(\Theta_i (a,b),M)$ for $M$ a skew Weyl module. For $T \in Hom(D(a_1,\dots,a_m),M)$ a standard basis element, let $\theta_s (T)$, $1 \le s \le m$, be the element of $Hom(D(a_1,\dots,a_s+a_{s+1},\dots,a_m),M)$ obtained from $T$ by replacing each occurrence of $j>s$ by $j-1$. If $s>m$, let $\theta_s (T)=0$. By extending linearly, we obtain for each degree $i$ a map of $\mathbb{Z}$-modules $Hom(P_i(a,b),M) \rightarrow Hom(P_{i+1}(a,b),M)$ which is denoted by $\Theta_i$. It is clear that only a finite number of these maps are nonzero. From the definition of the differential of $P_(a,b)$, with the previous notation, we obtain the following fact. $Hom(\Theta_i(a,b),M) = \sum_{s \ge 1}(-1)^{s-1}\theta_s.$ The cases $k=3$ and $k\ge 5$ ============================ It is well known, for example by [@Ja], B.3 (4), and [@Ma] respectively, that $Ext^2(\Delta(h),\Delta(h))=$ $Ext^2(\Delta(h),\Delta(h(1)))=0$. Theorem 4.1 of [@MS] implies that $Ext^2(\Delta(h),h(2)))=\mathbb{Z}_{(a+b)/gcd(2,a+b)}$. In this Section we will determine the extension groups $Ext^2(\Delta(h),\Delta(h(k)))$ for $k=3$ and $k\geq 5$. We will use certain exact sequences that we now describe. Consider the short exact sequence $$\label{s1} 0\rightarrow\Delta(h(k+1))\xrightarrow{i_k} D_{a+k}\otimes \Lambda^{a-k}\xrightarrow{\pi_k} \Delta(h(k))\rightarrow 0$$ for every $k\in\mathbb{Z}_{>0}$, where $\pi_k$ is induced by the identity map on generators and $i_k$ is induced by the composition $$D_{a+k+1}\otimes \Lambda^{b-1} \to D_{a+k}\otimes D_1 \otimes \Lambda^{b-1} \to D_{a+k}\otimes \Lambda^{b}$$ of comultiplication in $ D_* $ and multiplication in $ \Lambda^$. We have $Hom_{S_{\mathbb{Q}}(n,r)}(\Delta(h),\Delta(h(k)))=0$, since $\Delta(h)$ and $\Delta(h(k))$ are distinct irreducible representations of $S_{\mathbb{Q}}(n,r)$. Thus, $Hom(\Delta(h),\Delta(h(k)))=0$. Applying then $Hom(\Delta(h),-)$ to (\[s1\]) we obtain the exact sequence $$\begin{aligned} \label{long1} 0&\longrightarrow Ext^1(\Delta(h),\Delta(h(k+1)))\xrightarrow{i_k^{(1)}} Ext^1(\Delta(h),D_{a+k}\otimes\Lambda^{b-k})\nonumber \\ &\xrightarrow{\pi_k^{(1)}}Ext^1(\Delta(h),\Delta(h(k))) \longrightarrow Ext^2(\Delta(h),\Delta(h(k+1)))\nonumber\\ &\xrightarrow{i_k^{(2)}}Ext^2(\Delta(h),D_{a+k}\otimes\Lambda^{b-k}) \xrightarrow{\pi_k^{(2)}} Ext^2(\Delta(h),\Delta(h(k))) \nonumber \\ &\longrightarrow Ext^3(\Delta(h),\Delta(h(k+1)))\xrightarrow{i_k^{(3)}}Ext^3(\Delta(h),D_{a+k}\otimes\Lambda^{b-k}) \end{aligned}$$ Using Theorem 3.5 of [@MS] and Lemma \[basiclemma\], we have the following remark. $\;$ 1. $Ext^1(\Delta(h),\Delta(h(k+1)))= Ext^1(\Delta(h),D_{a+k}\otimes\Lambda^{b-k})=\mathbb{Z}_2$ and\ $Ext^1(\Delta(h),\Delta(h(k)))=0$, or 2. $Ext^1(\Delta(h),D_{a+k}\otimes\Lambda^{b-k})=Ext^1(\Delta(h),\Delta(h(k)))=\mathbb{Z}_2$ and\ $Ext^1(\Delta(h),\Delta(h(k+1)))=0$. Thus (\[long1\]) takes the following form $$\begin{aligned} \label{long2} &0\rightarrow Ext^2(\Delta(h),\Delta(h(k+1)))\xrightarrow{i_k^{(2)}}Ext^2(\Delta(h),D_{a+k}\otimes\Lambda^{b-k})\nonumber \\ &\xrightarrow{\pi_k^{(2)}}Ext^2(\Delta(h),\Delta(h(k)))\longrightarrow Ext^3(\Delta(h),\Delta(h(k+1)))\nonumber \\ &\xrightarrow{i_k^{(3)}} Ext^3(\Delta(h),D_{a+k}\otimes\Lambda^{b-k}) \end{aligned}$$ Then, Theorem 4.1 by [@MS] and (3) yield the case $k=3$ of the theorem. We have $Ext^2(\Delta(h),\Delta(h(3)))=\mathbb{Z}_{3/gcd(3,a+b)}.$ We observe that $Ext^3(\Delta(h),D_{a+2}\otimes\Lambda^{b-2})=0$, as $Ext^3(\Delta(h),D_{a+2}\otimes\Lambda^{b-2})$ $=Ext^3(\Lambda^3, D_3)$, by Lemma 2.4, and $Ext^3(\Lambda^3, D_3)=0$, as $\Lambda^3$ has a projective resolution of length 2 by [@A]. So, for $k=2$, (\[long2\]) has the following form $$\begin{aligned} \label{glyko} &0\rightarrow Ext^2(\Delta(h),\Delta(h(3)))\xrightarrow{i_2^{(2)}}Ext^2(\Delta(h),D_{a+2}\otimes\Lambda^{b-2})\nonumber \\ &\xrightarrow{\pi_2^{(2)}}Ext^2(\Delta(h),\Delta(h(2)))\longrightarrow Ext^3(\Delta(h),\Delta(h(3)))\xrightarrow{i_2^{(3)}} 0. \end{aligned}$$ Using again Lemma 2.4 we have $Ext^2(\Delta(h),D_{a+2}\otimes\Lambda^{b-2})=\mathbb{Z}_3$, which implies that $Ext^2(\Delta(h),\Delta(h(3)))=\mathbb{Z}_3$ or $0$, as $i_2^{(2)}$ is a monomorphism. We also know that $Ext^2(\Delta(h),\Delta(h(2)))=\mathbb{Z}_{d_2}$ and $Ext^3(\Delta(h),\Delta(h(3)))=\mathbb{Z}_{d_3}$, by Theorem 4.1 of [@MS], where $d_2=gcd(a+b,\tbinom{a+b}{2})$ and $d_3=gcd(a+b,\tbinom{a+b}{2},\tbinom{a+b}{3})$. Simple calculations yield that $d_2=d_3$ if and only if $3\notdivides a+b$, so the exact sequence (\[glyko\]) implies that $i_2^{(2)}$ is an isomorphism if and only if $3\notdivides a+b$. It follows that $Ext^2(\Delta(h),\Delta(h(3)))=\mathbb{Z}_3$ if and only if $3\notdivides a+b$ and $Ext^2(\Delta(h),\Delta(h(3)))=0$ otherwise. Lemma \[basiclemma\] (b) also yields that $Ext^2(\Delta(h),D_{a+k-1}\otimes \Lambda^{b-k+1})=0$ for every $k\geq5$ and using again the exact sequence (\[long2\]) we obtain the following result. We have $Ext^2(\Delta(h),\Delta(h(k)))=0$ for every $5\leq k\leq b$. The case $k=4$ of the Theorem ============================= In this Section we show Theorem 1.1 for $k=4$, which constitutes the main part of the paper. This is done using a refinement of the main strategy of [@MS], namely by first determining a cyclic generator of $Ext^2(\Delta(h),D_{a+3}\otimes \Lambda^{b-3})$, Subsection 4.1, and then computing its image in $Ext^2(\Delta(h),D_{a+2}\otimes \Lambda^{b-2})$, Subsection 4.3, with the aid of the relations that we establish in Subsection 4.2. In order to compute $Ext^2(\Delta(h),\Delta(h(4)))$, we use the exact sequence (\[long2\]) for $k=3$, which takes the following form $$\begin{aligned} &0\longrightarrow Ext^2(\Delta(h),\Delta(h(4))) \xrightarrow{i_3^{(2)}}Ext^2(\Delta(h),D_{a+3}\otimes \Lambda^{b-3})\\ &\xrightarrow{\pi_3^{(2)}} Ext^2(\Delta(h),\Delta(h(3))). \end{aligned}$$ By Lemma \[basiclemma\] (b) we have $Ext^2(\Delta(h),D_{a+3}\otimes \Lambda^{b-3})=\mathbb{Z}_3$. In order to determine whether $Ext^2(\Delta(h),\Delta(h(4)))=0$ or $\mathbb{Z}_3$, we consider the following composition $$Ext^2(\Delta(h),D_{a+3}\otimes \Lambda^{b-3})\xrightarrow{\pi_3^{(2)}} Ext^2(\Delta(h),\Delta(h(3)))\xrightarrow{i_2^{(2)}} Ext^2(\Delta(h),D_{a+2}\otimes \Lambda^{b-2}).$$ Let $\phi=i_2^{(2)}\circ\pi_3^{(2)}$. Then, $\phi $ is a map from $\mathbb{Z}_3$ to $\mathbb{Z}_3$ by Lemma \[basiclemma\]. Our aim in this Section is to find a cyclic generator of $Ext^2(\Delta(h),D_{a+3}\otimes \Lambda^{b-3})$ and determine its image under $\phi$ as a multiple of the cyclic generator of $Ext^2(\Delta(h),D_{a+2}\otimes \Lambda^{b-2})$ described by [@MS] in Section 4.1. Cyclic generator of $Ext^2(\Delta (h),D_{a+3}\otimes \Lambda^{b-3})$. --------------------------------------------------------------------- In this subsection we will determine a cyclic generator of the group $Ext^2(\Delta (h),D_{a+3}\otimes \Lambda^{b-3})$. This is accomplished using the recursion of Lemma 2.3 for $i=2$, according to which the matrix $e^{(2)}(\Delta(h),D_{a+3}\otimes \Lambda^{b-3}V)$ of the differential $Hom(\Theta_2(a,b),D_{a+3}\otimes\Lambda^{b-3})$ has the following form $$\left( \begin{array}{c\|c\|c} A &&\\ \hline \; &B& \\ \hline &&C \end{array} \right) ,$$ where $A=e^{(1)}(\Delta(a+1,1^{b-1}) ,D_{a+3}\otimes\Lambda^{b-3})$, $B=-e^{(2)}(\Delta(1,1^{b-1}),D_{4}\otimes\Lambda^{b-4})$ and $C=-e^{(2)}(\Delta(1,1^{b-1}),D_{3}\otimes\Lambda^{b-3})$. Consider the following standard basis elements: - $T_{i,j}^1=1^{(a)}i^{(2)}j\otimes 2\cdots \hat{i} \cdots \hat{j} \cdots b$\ for $i\in\{2,\dots,b-1\}$, $j\in\{i+1,\dots,b\}$\ in $Hom(D_a\otimes {D_1}^{\otimes (i-2)}\otimes D_2 \otimes {D_1}^{\otimes (b-i)}, D_{a+3}\otimes \Lambda^{b-3})$, - $T_{i,j}^k=1^{(a-1)}ki^{(2)}j\otimes 1\cdots\hat{k}\cdots \hat{i} \cdots \hat{j} \cdots b$\ for $k\in\{2,\dots,b-2\}$, $i\in\{k+1,\dots,b-1\}$, $j\in\{i+1,\dots,b\}$\ in $Hom(D_a\otimes {D_1}^{\otimes (i-2)}\otimes D_2 \otimes {D_1}^{\otimes (b-i)},D_{a+3}\otimes \Lambda^{b-3})$, and the standard basis elements: - $B_{1,j}^{a,b}=1^{(a+2)}j\otimes 2\cdots \hat{j} \cdots (b-1)$ for $j\in\{2,\dots ,b-1\}$,\ in $Hom(D_{a+2}\otimes {D_1}^{\otimes (b-2)},D_{a+3}\otimes \Lambda^{b-3})$, , - $B_{i,1}^{a,b}=1^{(a)}i^{(3)}\otimes 2\cdots\hat{i}\cdots(b-1)$ for $i\in\{2,\dots,b-1\}$,\ in $Hom(D_a\otimes {D_1}^{\otimes (i-2)}\otimes D_3\otimes {D_1}^{\otimes (b-i-1)}, D_{a+3} \otimes \Lambda^{b-3})$, - $B_{i,j}^{a,b}=1^{(a-1)}i^{(3)}j\otimes 1\cdots \widehat{\{i,j\}} \cdots (b-1)$ for $i\in\{2,\dots,b-1\}$\ and $j\in\{2,\dots,i-1\}\cup\{i+1,\dots,b-1\}$,\ in $Hom(D_a\otimes {D_1}^{\otimes (i-2)}\otimes D_3\otimes {D_1}^{\otimes (b-i-1)}, D_{a+3} \otimes \Lambda^{b-3})$. Recall that the cokernel of the differential $Hom(\Theta_i (a,b),M)$ of the complex $Hom(P_{} (a,b),M)$ is denoted by $E^i (\Delta(h),M)$ (cf. 2.4). Let $\pi: Hom(P_i (a,b),M)\rightarrow E^{i}(\Delta(h),M)$ be the natural projection. Remark. In the following proposition and its proof, if the lower bound of the summation index is less than the upper bound, then we regard the sum as zero (empty sum). Let $$\begin{aligned} \Gamma_{a,b}=&\tbinom{a+2} {3}\sum_{j=2}^{b-1}{(-1)}^{j}B_{1,j}^{a,b} -\Big(aB_{2,1}^{a,b}+\sum_{j=3}^{b-1}{(-1)}^{j}B_{2,j}^{a,b}\Big)+ \\ &\sum_{i=3}^{b-2}{(-1)}^{i-1}\Big(aB_{i,1}^{a,b} -\sum_{j=2}^{i-1} {(-1)}^j B_{i,j}^{a,b}+\sum_{j=i+1}^{b-1} {(-1)}^{j} B_{i,j}^{a,b} \Big)+\\ &{(-1)}^{b}\Big( aB_{b-1,1}^{a,b}-\sum_{j=2}^{b-2}{(-1)}^jB_{b-1,j}^{a,b}\Big).\end{aligned}$$ Then, for $b\geq 3$, $3\pi(\Gamma_{a,b})=0$ in $E^{2}(\Delta(h),D_{a+3}\otimes \Lambda^{b-3})$ and thus $\pi(\Gamma_{a,b})$ is an element of $Ext^2(\Delta(h),D_{a+3}\otimes \Lambda^{b-3})$. Let $$A_{a,b}=a\sum_{i=2}^{b-1}\sum_{j=i+1}^{b} {(-1)}^{j-i+1}\sum_{s=1}^{b-1}{(-1)}^{s-1}\theta_s(T_{i,j}^1),$$ $$C_{a,b}=\sum_{k=2}^{b-2} \sum_{i=k+1}^{b-1} \sum_{j=i+1}^{b} {(-1)}^{j-i-k} \sum_{s=1}^{b-1}{(-1)}^{s-1}\theta_s (T_{i,j}^k).$$ We will use induction on $b\geq3$ to show that $$\begin{aligned} \label{eqnionia} A_{a,b}+C_{a,b}=3\Gamma_{a,b} \end{aligned}$$ For $b=3$ we observe that $C_{a,3}=0$ as it is an empty sum, and for $T_{2,3}^1=1^{(a)}2^{(2)}3$ in $Hom(D_a\otimes D_2\otimes D_1,D_{a+3})$ we have that $$\begin{aligned} &A_{a,3}+C_{a,3}=A_{a,b}=a(\theta_1(T_{2,3})-\theta_1(T_{2,3}))=a\big({\tbinom{a+2}{2}}\cdot1^{(a+2)}2-3\cdot 1^{(a)}2^{(3)}\big)\\ &=3{\tbinom{a+2}{3}}\cdot B_{1,2}^{a,3}-3a\cdot B_{2,1}^{a,3}=3\Gamma_{a,3}. \end{aligned}$$ Let $b>3$. For $i=2$ and $j\in\{3,\dots,b\}$ we observe that $$\theta_1 (T_{2,j}^1)=\tbinom{a+2}{2} 1^{(a+2)}(j-1)\otimes 2\cdots \widehat{(j-1)} \cdots (b-1)=\tbinom{a+2}{2}B_{1,j-1}^{a,b},$$ and so we have $$\begin{aligned} \label{a1} A_{a,b}&=a\sum_{i=2}^{b-1}\sum_{j=i+1}^{b} {(-1)}^{j-i+1}\theta_1(T_{i,j}^1) \nonumber \\ &\,\,\,\,\,\,\,+a\sum_{i=2}^{b-1}\sum_{j=i+1}^{b} {(-1)}^{j-i+1}\sum_{s=2}^{b-1}{(-1)}^{s-1}\theta_s(T_{i,j}^1)\nonumber\\ &=a\tbinom{a+2}{2}\sum_{j=3}^{b}{(-1)}^{j-1}B_{1,j-1}^{a,b}+ a\sum_{i=3}^{b-1}\sum_{j=i+1}^{b} {(-1)}^{j-i+1}\theta_1(T_{i,j}^1)\nonumber \\ &\,\,\,\,\,\,\,+a\sum_{i=2}^{b-1}\sum_{j=i+1}^{b} {(-1)}^{j-i+1}\sum_{s=2}^{b-1}{(-1)}^{s-1}\theta_s(T_{i,j}^1). \end{aligned}$$ Consider now the standard basis elements: - $t_{1,j-1}^0 = 1^{(2)}(j-1)\otimes 2\cdots \widehat{(j-1)} \cdots (b-1)$ in\ $Hom(D_2\otimes {D_1}^{\otimes (b-1)},D_3 \otimes \Lambda^{b-1})$, for $i=2$ and $j\in\{3,\dots,b\}$, - $t_{i-1,j-1}^0={(i-1)}^{(2)}(j-1)\otimes 1\cdots \widehat{(i-1)} \cdots \widehat{(j-1)} \cdots (b-1)$ in\ $Hom({D_1}^{\otimes (i-2)}$ $\otimes D_2\otimes {D_1}^{\otimes (b+2-i)},D_3 \otimes \Lambda^{b-1})$, for $i\in\{3,\dots,b-1\}$\ and $j\in\{i+1,\dots,b\}$. We observe that $\theta_{s-1}(t^0_{i-1,j-1})=f_2(\theta_s(T^1_{i,j}))$ for every $i\in\{2,\dots,b-1\}$, $j\in\{i+1,\dots,b\}$ and $s\in\{2,\dots,b-1\}$, where $f_2$ is the isomorphism described in Lemma \[skewmatrix\] for $i=2$. By [@MS], Lemma 4.4 for $k=2$, we have the following relation in $Ext^2(\Delta(1,1^{(b-1)}),D_3 \otimes \Lambda^{b-3})$ $$\label{eqms} Hom(\Theta_2(a,b),D_{a+2}\otimes \Lambda^{b-2})(\sum_{i=2,j>i}^{b-1,b}{(-1)}^{j-i+1}t_{i-1,j-1}^0)=3\gamma_{1,b-1},$$ where $\gamma_{1,b-1}=\sum_{i=2}^{b-1}{(-1)}^i\delta_{i-1}^{1,b-1}$, - $\delta_1^{1,b-1}=1^{(3)}\otimes 2\cdots (b-2)$ in $Hom(D_3\otimes {D_1}^{\otimes (b-3)},D_3\otimes\Lambda^{b-3})$, for $i=2$, - $\delta_{i-1}^{1,b-1}={(i-1)}^3\otimes 1\cdots \widehat{(i-1)} \cdots (b-2)$ in $Hom({D_1}^{\otimes (i-2)}\otimes D_3\otimes {D_1}^ {\otimes (b-i-1)},$ $D_3\otimes\Lambda^{b-3})$, for $i\in\{3,\dots,b-1\}$, and $\pi(\gamma_{1,b-1})$ is the generator of $Ext^2(\Delta(1,1^{b-1}),D_3\otimes \Lambda^{b-3})$ described by [@MS] in Subsection 4.1. Using Remark 2.5, (\[eqms\]) is equivalent to $$\label{rel} \sum_{i=2}^{b-1}\sum_{j=i+1}^{b} {(-1)}^{j-i+1}\sum_{s=2}^{b-1}{(-1)}^{s}\theta_{s-1}(t_{i-1,j-1}^0) =3\sum_{i=2}^{b-1}{(-1)}^i\delta_{i-1}^{1,b-1}.$$ By the proof of Lemma \[skewmatrix\], we have that $\delta_{i-1}^{1,b-1}=f_2(B_{i,1}^{a,b})$ where - $B_{2,1}^{a,b}=1^{(a)}2^{(3)}\otimes 3\cdots (b-1)$ for $i=2$, - $B_{i,1}^{a,b}=1^{(a)}i^{(3)}\otimes 2\cdots \hat{i} \cdots (b-1)$ for $i\in\{3,\dots,b-1\}$. and $f_2$ is the isomorphism described in Lemma \[skewmatrix\] for $i=2$. Applying the isomorphism $f_2^{-1}$ to Equation (\[rel\]) we have, $$\sum_{i=2}^{b-1}\sum_{j=i+1}^{b} {(-1)}^{j-i+1}\sum_{s=2}^{b-1}{(-1)}^{s}f_2^{-1}(\theta_{s-1}(t_{i-1,j-1}^0)) =3\sum_{i=2}^{b-1}{(-1)}^i f_2^{-1}(\delta_{i-1}^{1,b-1})$$ or, equivalently, $$\label{e14e} \sum_{i=2}^{b-1}\sum_{j=i+1}^{b} {(-1)}^{j-i+1}\sum_{s=2}^{b-1}{(-1)}^{s}\theta_s(T_{i,j}^1) =3\sum_{i=2}^{b-1}{(-1)}^iB_{i,1}^{a,b}.$$ Using (\[e14e\]), (\[a1\]) is equivalent to $$\begin{aligned} \label{a4} A_{a,b}&=a\tbinom{a+2}{2}\sum_{i=2}^{b-1}{(-1)}^{i}B_{1,i}^{a,b}+a\sum_{i=3}^{b-1}\sum_{j=i+1}^{b} {(-1)}^{j-i+1}\theta_1(T_{i,j}^1)\nonumber\\ &\,\,\,\,\,\,-3a\sum_{i=2}^{b-1}{(-1)}^i B_{i,1}^{a,b}.\end{aligned}$$ We also have $$\begin{aligned} \label{a8} C_{a,b}&=\sum_{i=3}^{b-1} \sum_{j=i+1}^{b} {(-1)}^{j-i}\theta_1 (T_{i,j}^2) + \sum_{k=3}^{b-2} \sum_{i=k+1}^{b-1} \sum_{j=i+1}^{b} {(-1)}^{j-i-k}\theta_1 (T_{i,j}^k)\nonumber \\ &\,\,\,\,\,\,+\sum_{k=2}^{b-2} \sum_{i=k+1}^{b-1} \sum_{j=i+1}^{b} {(-1)}^{j-i-k} \sum_{s=2}^{b-1}{(-1)}^{s-1}\theta_s (T_{i,j}^k)\nonumber \\ &=\sum_{i=3}^{b-1} \sum_{j=i+1}^{b} {(-1)}^{j-i}\theta_1 (T_{i,j}^2) +\sum_{k=2}^{b-2} \sum_{i=k+1}^{b-1} \sum_{j=i+1}^{b} {(-1)}^{j-i-k} \sum_{s=2}^{b-1}{(-1)}^{s-1}\theta_s (T_{i,j}^k),\end{aligned}$$ as $\theta_1 (T_{i,j}^k)=0$ for every $k\in\{3\dots,b-2\}$, because of the $12$ part of $T_{i,j}^k$ in $\Lambda^{b-3}$. Consider now the basis elements: - $t_{i-1,j-1}^1=1{(i-1)}^{(2)}(j-1)\otimes 2\cdots\widehat{(i-1)}\cdots\widehat{(j-1)}\cdots (b-1)$ in\ $Hom({D_1}^{\otimes (i-2)}\otimes D_2 \otimes {D_1}^{\otimes (b-i)},D_4\otimes\Lambda^{b-4})$, for $k=2$ and $i\in\{3,\dots,b-1\}$, $j\in\{i+1,\dots,b\}$, - $t_{i-1,j-1}^{k-1}=(k-1){(i-1)}^{(2)}(j-1)\otimes 1\cdots\widehat{(k-1)}\cdots\widehat{(i-1)}\cdots \widehat{(j-1)} \cdots (b-1)$ in $Hom({D_1}^{\otimes (i-2)}\otimes D_2 \otimes {D_1}^{\otimes (b-i)},D_4\otimes\Lambda^{b-4})$ , for $k\in\{3,\dots,b-2\}$, $i\in\{k+1,\dots,b-1\}$, $j\in\{i+1,\dots,b\}$. By induction on $b$, for $a=1$, we have $A_{1,b-1}+C_{1,b-1}=3\Gamma_{1,b-1},$ or, equivalently, $$\begin{aligned} \label{eq10} &\sum_{k=2}^{b-2} \sum_{i=k+1}^{b-1} \sum_{j=i+1}^{b} {(-1)}^{j-i-k+1} \sum_{s=2}^{b-1}{(-1)}^{s}\theta_{s-1} (t_{i-1,j-1}^{k-1})\nonumber \\ &=3\sum_{j=3}^{b-1}{(-1)}^{j-1}B_{1,j-1}^{1,b-1} -3\Big(B_{2,1}^{1,b-1}+\sum_{j=4}^{b-1}{(-1)}^{j-1}B_{2,j-1}^{1,b-1}\Big)\nonumber \\ &\,\,\,\,\,\,+3\sum_{i=4}^{b-1}{(-1)}^{i}\Big(B_{i-1,1}^{1,b-1} -\sum_{j=3}^{i-1} {(-1)}^{j-1} B_{i-1,j-1}^{1,b-1}+\sum_{j=i+1}^{b-1} {(-1)}^{j-1} B_{i-1,j-1}^{1,b-1} \Big)\nonumber \\ &\,\,\,\,\,\,+3{(-1)}^{b-1}\Big(B_{b-2,1}^{1,b-1}-\sum_{j=3}^{b-2}{(-1)}^{j-1}B_{b-2,j-1}^{1,b-1}\Big).\end{aligned}$$ Applying $f_2^{-1}$ to (\[eq10\]), using that $f_2(B_{i,j}^{a,b})=B_{i-1,j-1}^{1,b-1}$ and $f_2(\theta_s(T_{i,j})^k)=\theta_{s-1}(t_{i-1,j-1}^{k-1})$ for $i\geq 2$, $j\in\{3,\dots,i-1\}\cup\{i+1,\dots,b-2\}$, we obtain $$\begin{aligned} \label{eq11} &\sum_{k=2}^{b-2} \sum_{i=k+1}^{b-1} \sum_{j=i+1}^{b} {(-1)}^{j-i-k+1} \sum_{s=2}^{b-1}{(-1)}^{s}\theta_{s} (T_{i,j}^{k})\nonumber \\ &=3\sum_{j=3}^{b-1}{(-1)}^{j-1}B_{2,j}^{a,b} -3\Big(B_{3,2}^{a,b}+\sum_{j=4}^{b-1}{(-1)}^{j-1}B_{3,j}^{a,b}\Big)\nonumber \\ &\,\,\,\,\,\,+3\sum_{i=4}^{b-2}{(-1)}^{i}\Big(B_{i,2}^{a,b} -\sum_{j=3}^{i-1} {(-1)}^{j-1} B_{i,j}^{a,b}+\sum_{j=i+1}^{b-1} {(-1)}^{j-1} B_{i,j}^{a,b} \Big)\nonumber \\ &\,\,\,\,\,\,+3{(-1)}^{b-1}\Big(B_{b-1,2}^{a,b}-\sum_{j=3}^{b-2}{(-1)}^{j-1}B_{b-1,j}^{a,b}\Big).\end{aligned}$$ Equation (\[eq11\]) implies that (\[a8\]) is equivalent to $$\begin{aligned} \label{eq12} C_{a,b}&=\sum_{i=3}^{b-1} \sum_{j=i+1}^{b} {(-1)}^{j-i}\theta_1 (T_{i,j}^2) +3\sum_{j=3}^{b-1}{(-1)}^{j-1}B_{2,j}^{a,b} \nonumber \\ &\,\,\,\,\,\,-3\Big(B_{3,2}^{a,b}+\sum_{j=4}^{b-1}{(-1)}^{j-1}B_{3,j}^{a,b}\Big)\nonumber \\ &\,\,\,\,\,\,+3\sum_{i=3}^{b-2}{(-1)}^{i}\Big(B_{i,2}^{a,b} -\sum_{j=3}^{i-1} {(-1)}^{j-1} B_{i,j}^{a,b}+\sum_{j=i+1}^{b-1} {(-1)}^{j-1} B_{i,j}^{a,b} \Big)\nonumber \\ &\,\,\,\,\,\,+3{(-1)}^{b-1}\Big(B_{b-1,2}^{a,b}-\sum_{j=3}^{b-2}{(-1)}^{j-1}B_{b-1,j}^{a,b}\Big).\end{aligned}$$ We observe that $\theta_1(T_{i,j}^2)=a\theta_1(T_{i,j}^1)$ for $i\in\{3,\dots,b-1\}$, $j\in\{i+1,\dots,b\}$, as $\theta_1(T_{i,j}^1)=1^{(a)}{(i-1)}^2(j-1)\otimes 1\cdots\widehat{(i-1)}\cdots \widehat{(j-1)}\cdots (b-1)$, and $\theta_1(T_{i,j}^2)=a1^{(a)}{(i-1)}^2(j-1)\otimes 1\cdots\widehat{(i-1)}\cdots \widehat{(j-1)}\cdots (b-1)$. Using this observation and adding equations (\[a4\]) and (\[eq12\]), we obtain $A_{a,b}+C_{a,b}=3\Gamma_{a,b}$ and we have proved inductively equation (\[eqnionia\]). As we have the relations $\sum_{s=1}^{b-1}{(-1)}^{s-1}\pi(\theta_s(T_{i,j}^k))=0$ for every $k\in\{1,\dots,b-2\}$, $i\in\{k+1,\dots,b-1\}$, $j\in\{i+1,\dots,b\}$, it follows that $\pi(A_{a,b})=\pi(C_{a,b})=0$ and so $3\pi(\Gamma_{a,b})=0$. The element $\pi(\Gamma_{a,b})$ of Proposition 3.1 is a cyclic generator of $Ext^2(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})$. We will first show that $\pi(\Gamma_{a,b})$ is nonzero. Indeed, if $\pi(\Gamma_{a,b})=0$, then $\pi(\Gamma_{a,b})$ would be equal to a linear combination of the relations in the extension group $Ext^2(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})$ corresponding to the columns of the matrix $e^{(2)}(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})$. This implies that the coefficient $-1$ of $\pi(B_{b-1,b-2})$ in $\pi(\Gamma_{a,b})$ should be a linear combination of the nonzero entries of the last row of $e^{(2)}(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})$. These are the coefficients of $\pi(B_{b-1,b-2})$ resulting from the application of $\pi\circ\theta_{b-1}$ on the standard basis elements: $$1^{(a-1)} (b-2) {(b-1)}^{(2)} b\otimes 1\cdots(b-3), \, 1^{(a-1)}(b-2)(b-1)b^{(2)}\otimes 1\cdots(b-3).$$ Indeed, let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_b)$ be a sequence with $\lambda_1\in\{0,1\},\,\lambda_i\in\{1,2\},\, \sum_{i=1}^{b}\lambda_i$ $=b$, and a standard basis element $T_{\lambda}$ in $Hom(D_{a+\lambda_1}\otimes D_{\lambda_2}\otimes \cdots \otimes D_{\lambda_b}, D_{a+3}\otimes\Lambda^{b-3})$, such that $\theta_{b-1}(T_{\lambda})=c\cdot B_{b-1,b-2}^{a,b}$ for some $c\in\mathbb{Z}$. Then $\theta_{b-1}(T_{\lambda})$ has weight $(1^{a+\lambda_1},2^{\lambda_2},\dots,{(b-1)}^{\lambda_{b-1}+\lambda_b})$ which must be equal to $(1^{a},2,\dots,(b-2),{(b-1)}^3)$, the weight of $B_{b-1,b-2}^{a,b}$. This implies that $\lambda_1=0$, $(\lambda_{b-1},\lambda_{b})\in\{(1,2),(2,1)\}$ and $$T_{\lambda}\in\{1^{(a-1)} (b-2) {(b-1)}^{(2)} b\otimes 1\cdots(b-3), 1^{(a-1)}(b-2)(b-1)b^{(2)}\otimes 1\cdots(b-3)\}.$$ By straightforward calculations, we have, $$\begin{aligned} &\pi(\theta_{b-1}(1^{(a-1)} (b-2) {(b-1)}^{(2)} b\otimes 1\cdots(b-3)))\\ &=\pi(\theta_{b-1}(1^{(a-1)}(b-2)(b-1)b^{(2)}\otimes 1\cdots(b-3)))\\ &=3 \pi(B_{b-1,b-2}^{a,b}), \end{aligned}$$ which means that $-1$ should be a linear combination of $3$ and $3$, i.e. multiple of $3$. This implies that $\pi(\Gamma_{a,b})$ is nonzero. As $Ext^2(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})= \mathbb{Z}_3$, $3\pi(\Gamma_{a,b})=0$ (by Proposition 4.1) and $\pi(\Gamma_{a,b})\neq 0$, we conclude that $\pi(\Gamma_{a,b})$ is a generator of $Ext^2(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})$. Relations in $Ext^2(\Delta(h),D_{a+2}\otimes \Lambda^{b-2})$. ------------------------------------------------------------- In this Subsection we will prove certain relations in $E^2(\Delta(h),D_{a+2}\otimes \Lambda^{b-2})$ using the columns of $e^{(2)}(\Delta(h), D_{a+2}\otimes \Lambda^{b-2})$, i.e. using the equalities $\sum_{s=1}^{b-1}{(-1)}^{s-1}\pi(\theta_s(T))=0$, where $T$ is a standard basis element of the domain of the differential $\Theta_2 (a,b)$. These relations will be described in Lemmas 4.3 and 4.4. They will yield Proposition 4.5 which will be used in the proof of Proposition 4.6, in order to facilitate the computation of the image of the cyclic generator of Proposition 4.2. Proposition 4.6 will then yield the case $k=4$ of Theorem 1.1. Let $$\begin{aligned} {b_1}^{(1)}&=1^{(a+2)} \otimes 2\cdots (b-1),\\{b_j}^{(1)}&=1^{(a+1)}j \otimes 1\cdots \hat{j} \cdots (b-1)\end{aligned}$$ be the indicated standard basis elements in $Hom(D_{a+2}\otimes {D_1}^{\otimes (b-2)}, D_{a+2} \otimes \Lambda^{b-2})$ and $$\begin{aligned} {b_1}^{(i)}=&1^{(a)}i^{(2)} \otimes 2 \cdots i \cdots (b-1),\\{b_j}^{(i)}=&1^{(a-1)} i^{(2)}j \otimes 12\cdots \hat{j} \cdots (b-1)\end{aligned}$$ be the indicated standard basis elements in $Hom(D_a \otimes {D_1}^{\otimes (i-2)} \otimes D_3 \otimes {D_1}^{\otimes (b-i-1)},$ $ D_{a+2}\otimes \Lambda^{b-2})$ for $i\in\{2,\dots ,b-1\}$ and $j\in\{2,\dots,b-1\}$. \[bl1\] For every $i\in\{2,\dots,b-1\}$, $$a\pi(b_1^{(i+1)}) - \sum_{j=2}^{i}{(-1)}^j \pi(b_j^{(i+1)})= a\pi(b_1^{(i)})-\sum_{j=2}^{i-1}{(-1)}^{(j)}\pi(b_j^{(i)}) +3{(-1)}^{i+1}\pi(b_i^{(i)}).$$ We consider the basis elements $T_1^{(i+1)}=1^{(a)}{(i+1)}^{(2)}\otimes 2\cdots \widehat{(i+1)} \cdots b$ and $T_j^{(i+1)}=1^{(a-1)}j{(i+1)}^{(2)}\otimes 1\cdots \hat{j}\cdots \widehat{(i+1)}\cdots b$, for $j\in\{2,\dots,i\}$. Let $$D_{a,b}=a\sum_{s=1}^{b-1} {(-1)}^{s-1} \theta_s(T_1^{(i+1)})+\sum_{j=2}^i {(-1)}^j\sum_{s=1}^{b-1} {(-1)}^{s-1} \theta_s(T_j^{(i+1)}).$$ We know that $\sum_{s=1}^{b-1} {(-1)}^{s-1}\pi( \theta_s(T_j^{(i+1)}))=0$, for $j\in\{1,\dots,i\}$, which implies that $\pi(D_{a,b})=0$. We observe that $\theta_s (T_j^{(i+1)})=0$ for every $j\in\{1,\dots,i\}$ and $s\in\{i+2,\dots,b-1\}$ because of the $(i+2)(i+3)\cdots b$ part $T_j^{(i+1)}$ in $\Lambda^{b-2}$. So we obtain $$a\sum_{s=i+2}^{b-1} {(-1)}^{s-1} \theta_s(T_1^{(i+1)})+\sum_{j=2}^i {(-1)}^j\sum_{s=i+2}^{b-1} {(-1)}^{s-1} \theta_s(T_j^{(i+1)})=0$$ and $$\label{2e} D_{a,b}=a\sum_{s=1}^{i+1} {(-1)}^{s-1} \theta_s(T_1^{(i+1)})+\sum_{j=2}^i {(-1)}^j\sum_{s=1}^{i+1} {(-1)}^{s-1} \theta_s(T_j^{(i+1)}).$$ By immediate calculations we have $\theta_s(T_1^{(i+1)})=0$ for every $s\in\{2,\dots,i-1\}$, $\theta_s(T_j^{(i+1)})=0$ for every $s\in\{1,\dots,j-2\}\cup\{j+1,\dots,i-1\}$, $j\in\{2,\dots,i-1\}$ and $\theta_s(T_i^{(i+1)})=0$ for every $s\in\{1,\dots,i-2\}$. Hence $$\begin{aligned} D_{a,b}=&a\sum_{s=1}^{i-1} {(-1)}^{s-1} \theta_s(T_1^{(i+1)})+\sum_{j=2}^i {(-1)}^j\sum_{s=1}^{i-1} {(-1)}^{s-1} \theta_s(T_j^{(i+1)})\nonumber \\ &+a\sum_{s=i}^{i+1} {(-1)}^{s-1} \theta_s(T_1^{(i+1)})+\sum_{j=2}^i {(-1)}^j\sum_{s=i}^{i+1} {(-1)}^{s-1} \theta_s(T_j^{(i+1)})\nonumber \\ =& (a\theta_1(T_1^{(i+1)})-\theta_1(T_2^{(i+1)})) + \sum_{s=2}^{i-1}(\theta_s(T_s^{(i+1)})-\theta_s(T_{s+1}^{(i+1)}))\nonumber \\ &+a\sum_{s=i}^{i+1} {(-1)}^{s-1} \theta_s(T_1^{(i+1)})+\sum_{j=2}^i {(-1)}^j\sum_{s=i}^{i+1} {(-1)}^{s-1} \theta_s(T_j^{(i+1)})\nonumber\\ =&a\sum_{s=i}^{i+1} {(-1)}^{s-1} \theta_s(T_1^{(i+1)})+\sum_{j=2}^i {(-1)}^j\sum_{s=i}^{i+1} {(-1)}^{s-1} \theta_s(T_j^{(i+1)}), \end{aligned}$$ as $a\theta_1(T_1^{(i+1)})=\theta_1(T_2^{(i+1)})=a1^{(a)}i^{(2)}\otimes 1\cdots \hat{i} \cdots (b-1)$ and $\theta_s(T_s^{(i+1)})=\theta_s(T_{s+1}^{(i+1)})=1^{(a-1)}si^{(2)}\otimes 1\cdots\hat{i} \cdots (b-1)$. It follows that $$\pi(D_{a,b})=a\sum_{s=i}^{i+1} {(-1)}^{s-1} \pi(\theta_s(T_1^{(i+1)}))+\sum_{j=2}^i {(-1)}^j\sum_{s=i}^{i+1} {(-1)}^{s-1}\pi(\theta_s(T_j^{(i+1)}))=0,$$ which is equivalent to $$\label{6e} a\pi(\theta_i(T_1^{(i+1)}))-\sum_{j=2}^{i}{(-1)}^j\pi(\theta_i(T_j^{(i+1)})) = a\pi(\theta_{i+1}(T_1^{(i+1)}))-\sum_{j=2}^{i}{(-1)}^j\pi(\theta_{i+1}(T_j^{(i+1)})).$$ More straightforward calculations yield: - $\theta_i(T_1^{(i+1)})=1^{(a)}i^{(2)}\otimes 2\cdots (b-1)=b_1^{(i)}$, - $\theta_i(T_j^{(i+1)})=1^{(a-1)}ji^{(2)}\otimes 1\cdots \hat{j} \cdots (b-1)=b_j^{(i)}$ for every $j\in\{2,\dots ,i-1\}$, - $\theta_i(T_i^{(i+1)})=3\cdot 1^{(a-1)}i^{(3)}\otimes 1\cdots \hat{i} \cdots (b-1)=3 \cdot b_i^{(i)}$ - $\theta_{i+1}(T_1^{(i+1)})=1^{(a)}{(i+1)}^{(2)}\otimes 2\cdots (b-1) = b_1^{(i+1)}$, - $\theta_{i+1}(T_j^{(i+1)})=1^{(a-1)}j{(i+1)}^2\otimes 1 \cdots \hat{j} \cdots (b-1)=b_j^{(i+1)}$ for every\ $j\in\{2,\dots ,i\}$, so (\[6e\]) is equivalent to $$a\pi(b_1^{(i+1)}) - \sum_{j=2}^{i}{(-1)}^j \pi(b_j^{(i+1)})= a\pi(b_1^{(i)})-\sum_{j=2}^{i-1}{(-1)}^{j}\pi(b_j^{(i)}) +3{(-1)}^{i+1}\pi(b_i^{(i)}).$$ With the notation established before Lemma 4.3, we have the following statement. \[1q1q1\] For every $i=2,\dots ,b-2$, $$\sum_{j=i+2}^{b-1}{(-1)}^{j-i}\pi(b_j^{(i+1)})-3\pi(b_{i+1}^{(i+1)})=\sum_{j=i+1}^{b-1}{(-1)}^{j-i}\pi(b_j^{(i)}).$$ Consider the standard basis elements $$T_j^{(i+1)}=1^{(a-1)}(i+1)^{(2)}j\otimes 1\cdots \widehat{(i+1)} \cdots \hat{j} \cdots b$$ in $Hom(D_a \otimes {D_1}^{\otimes (i-1)}\otimes D_2 \otimes {D_1}^{\otimes (b-i-1)}, D_{a+2} \otimes \Lambda^{b-2})$ for $j\in\{i+2,\dots,b\}$. We know that $\sum_{s=1}^{b-1}{(-1)}^{s-1}\pi(\theta (T_j^{(i+1)}))=0$ for $j\in\{i+2,\dots,b\}$. This implies that $$\label{t1} \sum_{j=i+2}^b {(-1)}^{j}\sum_{s=1}^{b-1}{(-1)}^{s-1}\pi(\theta_s(T_j^{(i+1)}))=0.$$ But $\theta_s(T_j^{(i+1)})=0$ for every $s\in\{1,\dots , i-1\}$, for every $j\in\{i+2 ,\dots ,b\}$ because of the $1\cdots i$ part of $T_j^{(i+1)}$ in $\Lambda^{b-2}$. So (\[t1\]) is equivalent to $$\label{t2} \sum_{j=i+2}^b {(-1)}^{j}\sum_{s=i}^{i+1}{(-1)}^{s-1}\pi(\theta_s(T_j^{(i+1)}))+ \sum_{j=i+2}^b {(-1)}^{j}\sum_{s=i+2}^{b-1}{(-1)}^{s-1}\pi(\theta_s(T_j^{(i+1)}))=0.$$ By straightforward calculations we have $\theta_s(T_{i+2}^{(i+1)})=0$ for every $s\in\{i+3,\dots,b-1\}$, $\theta_s(T_j^{(i+1)})=0$ for every $s\in\{i+2,\dots ,j-2\}\cup \{j+1, \dots ,b-1\}$ for every $j\in\{i+3, \dots , b-1\}$ and $\theta_s(T_b^{(i+1)})=0$ for every $s\in\{i+2, \dots ,b-2\}$. So (\[t2\]) yields $$\begin{aligned} \label{t3} \sum_{j=i+2}^b {(-1)}^{j}\sum_{s=i}^{i+1}{(-1)}^{s-1}\pi(\theta_s(T_j^{(i+1)}))- \sum_{j=i+2}^{b-1}(\pi(\theta_j(T_j^{(i+1)}))-\pi(\theta_j(T_{j+1}^{(i+1)})))=0.\end{aligned}$$ We observe now that $\theta_j(T_j^{(i+1)})=\theta_j(T_{j+1}^{(i+1)})$, so (\[t3\]) is equivalent to $$\sum_{j=i+2}^b {(-1)}^{j-i}(\pi(\theta_{i+1}(T_j^{(i+1)}))-\pi(\theta_i(T_j^{(i+1)})))=0,$$ or $$\label{t6} \sum_{j=i+2}^{b-1}{(-1)}^{j-i} \pi(\theta_i(T_j^{(i+1)}))= \sum_{j=i+3}^{b-1}{(-1)}^{j-i} \pi(\theta_{i+1}(T_j^{(i+1)})) + \pi(\theta_{i+1}(T_{i+2})),$$ where - $\theta_i(T_j^{(i+1)})=b_{j-1}^{(i)}$ for every $j\in\{i+2,\dots ,b\}$, - $\theta_{i+1}(T_{i+2})= 3\cdot b_{i+1}^{(i+1)}$ and $\theta_{i+1}(T_j^{(i+1)})=b_{j-1}^{(i+1)}$ for every $j\in\{i+3, \dots ,b\}$. Equation (\[t6\]) is now equivalent to $$\sum_{j=i+2}^{b-1}{(-1)}^{j-i}\pi(b_j^{(i+1)})-3\pi(b_{i+1}^{(i+1)})=\sum_{j=i+1}^{b-1}{(-1)}^{j-i}\pi(b_j^{(i)})$$ for every $i\in\{2,\dots ,b-2\}$. We are now ready to prove the main result of the present subsection, which provides usefull relations that will be used in Subsection 4.3. \[l1\] $\;$ 1. For $i=1$, $$\sum_{j=2}^{b-1} (-1)^j \pi(b_j^{(1)})= (a+2)\pi(b_1^{(1)}).$$ 2. For every $i\in\{2,\dots, b-1\}$, $$a(-1)^i \pi(b_1^{(i)})+ (-1)^{i}\sum_{j=2}^{i-1} (-1)^{j-1} \pi(b_j^{(i)})+{(-1)}^i \sum_{j=i+1}^{b-1} (-1)^{j-1} \pi(b_j^{(i)}) = 3\pi(b_i^{(i)}).$$ For $i=1$ we consider the standard basis elements $T_j^{(1)}=1^{(a+1)}j\otimes 2\cdots\hat{j} \cdots b$ of $Hom(D_{a+1}\otimes {D_1}^{\otimes (b-1)}, D_{a+2} \otimes \Lambda^{b-2})$ for $j\in\{2,\dots , b\}$. Then we have the relations $\sum_{s=1}^{b-1} (-1)^{s-1}\pi(\theta_s (T_j^{(1)}))=0$, for every $j\in\{2,\dots ,b\}$, and so $$\label{e1} \sum_{j=2}^b (-1)^{j-1} \pi(\theta_1 (T_j^{(1)})) + \sum_{s=2}^{b-1} (-1)^{s-1} \sum_{j=2}^b (-1)^{j-1}\pi(\theta_s (T_j^{(1)})) =0.$$ But $\theta_2 (T_j^{(1)})=0$ for $j\geq 4$, $\theta_s (T_j^{(1)})=0$ for $j \in \{2,\dots,s-1\}\cup \{s+1,\dots b\}$, and $\theta_{b-1}(T_j^{(1)})=0$ for $j\leq b-2$. So $(\ref{e1})$ is equivalent to $$\label{e2} \sum_{j=2}^b (-1)^{j-1} \pi(\theta_1 (T_j^{(1)})) + \sum_{s=2}^{b-1} (\pi(\theta_s (T_s^{(1)})) -\pi(\theta_s (T_{s+1}^{(1)}))) =0.$$ We observe now that $\theta_s (T_s^{(1)}) = \theta_s (T_{s+1}^{(1)}) = 1^{(a+1)}s \otimes 2\cdots (b-1)$ and $\theta_1 (T_2^{(1)})= (a+2) b_1 ^{(1)}$ and $\theta_1(T_j^{(1)})=b_{j-1}^{(1)}$ for $j\in\{3,\dots , b\}$, so $(\ref{e2})$ is equivalent to $$\sum_{j=2}^{b-1} (-1)^{j} \pi(b_j^{(1)}) = (a+2)\pi(b_1^{(1)}).$$ For $i=2$, we consider the standard basis elements $T_1^{(2)}=1^{(a)}2^{(2)}\otimes 3\dots b$ and $T_j^{(2)}=1^{(a-1)}2j\otimes 1\hat{2}\dots \hat{j}\dots b$ in $Hom(D_a \otimes D_2 \otimes D_1 \otimes \dots \otimes D_1,D_{a+2}\otimes \Lambda^{b-2})$, for $j\in\{3,\dots ,b\}$. Then $\sum_{s=1}^{b-1} {(-1)}^{s-1}\pi(\theta_s (T_1^{(2)}))=0$ and $\sum_{s=1}^{b-1}{(-1)}^{s-1} \pi(\theta_s (T_j^{(2)}))=0$ for $j\in\{3,\dots ,b\}$. So we obtain $$\label{e3} \sum_{j=3}^{b}{(-1)}^{j-1}\sum_{s=1}^{b-1}{(-1)}^{s-1} \pi(\theta_s (T_j^{(2)}))=0,$$ which is equivalent to $$\begin{aligned} \label{e7} &\sum_{j=3}^{b} {(-1)}^{j-1} \pi(\theta_1 (T_j^{(2)})) - \sum_{j=3}^{b} {(-1)}^{j-1} \pi(\theta_2 (T_j^{(2)})) \nonumber \\ &+ \sum_{s=3}^{b-1}{(-1)}^{s-1} \sum_{j=3}^{b}{(-1)}^{j-1}\pi(\theta_s (T_j^{(2)}))=0.\end{aligned}$$ But $\theta_s (T_3^{(2)})=0$ for every $s\in\{4,\dots,b-1\}$, $\theta_s(T_b^{(2)})=0$ for every $s\in\{3,\dots b-2\}$, and $\theta_s (T_j^{(2)})=0$ for $s\in \{3,\dots,j-2\}\cup \{j+1,\dots,b-1\}$ for every $j\in\{4, \dots, b-1\}$ and also we have $\theta_s(T_s^{(2)})=\theta_s (T_{s+1}^{(2)})=1^{(a-1)}2^{(2)}s\otimes 1\hat{2}\dots \hat{s}\dots (b-1)$. As a result, $(\ref{e7})$ is equivalent to $$\label{e9} \sum_{j=3}^{b} {(-1)}^{j-1} \pi(\theta_1 (T_j^{(2)})) - \sum_{j=3}^{b} {(-1)}^{j-1} \pi(\theta_2 (T_j^{(2)})) =0.$$ But $\theta_1(T_j^{(2)})= \tbinom{a+1}{2}b_{j-1}^{(1)}$, for $j=3,\dots,b$, and thus $$\begin{aligned} \label{e10} &\sum_{j=3}^{b} {(-1)}^{j-1} \pi(\theta_1 (T_j^{(2)}))=\sum_{j=3}^{b} {(-1)}^{j-1}\tbinom{a+1}{2}\pi(b_{j-1}^{(1)}) = \tbinom{a+1}{2}\sum_{j=2}^{b-1}{(-1)}^j \pi(b_j^{(1)}) \nonumber \\ &=\tbinom{a+1}{2}(a+2)\pi(b_1^{(1)})=a\tbinom{a+2}{2}\pi(b_1^{(1)}),\end{aligned}$$ where we used part (a) of the Proposition 4.5 in the second equality. We observe that $\theta_2(T_3^{(2)})=3b_2^{(2)}$ and $\theta_2(T_j^{(2)})=b_{j-1}^{(2)}$ for $j\in\{4,\dots,b\}$, and so $$\label{e11} \sum_{j=3}^{b} {(-1)}^{j-1} \pi(\theta_2 (T_j^{(2)})) = 3\pi(b_2^{(2)}) + \sum_{j=4}^{b} {(-1)}^{j-1} \pi(b_{j-1}^{(2)}) =3\pi(b_2^{(2)}) + \sum_{j=3}^{b-1} {(-1)}^{j} \pi(b_{j}^{(2)}).$$ Combining $(\ref{e9})$, $(\ref{e10})$ and $(\ref{e11})$ we obtain $$\label{e12} a\tbinom{a+2}{2}\pi(b_1^{(1)})+\sum_{j=3}^{b-1} {(-1)}^{j-1} \pi(b_{j}^{(2)})=3\pi(b_2^{(2)}).$$ But $\sum_{s=1}^{b-1} {(-1)}^{s-1}\pi(\theta_s (T_1^{(2)}))=0$ and $\theta_s(T_1^{(2)})=0$ for every $s\in\{3,\dots,b-1\}$, because of the $3\cdots b$ part of $T_1^{(2)}$ in $\Lambda^{b-2}$, which implies that $\pi(\theta_1(T_1^{(2)}))=\pi(\theta_2(T_1^{(2)}))$ or equivalently $\tbinom{a+2}{2}\pi(b_1^{(1)})=\pi(b_1^{(2)}).$ Equation $(\ref{e12})$ is now equivalent to $$a\pi(b_1^{(2)})+\sum_{j=3}^{b-1} {(-1)}^{j-1} \pi(b_{j}^{(2)})=3\pi(b_2^{(2)}),$$ and we have proved the case $i=2$. The general case for $i\geq 2$ will be proved by induction on $i$. Suppose that for $i\geq 2$ we have $$\label{inductive} a(-1)^i \pi(b_1^{(i)})+ (-1)^{i}\sum_{j=2}^{i-1} (-1)^{j-1} \pi(b_j^{(i)})+ {(-1)}^i\sum_{j=i+1}^{b-1} (-1)^{j-1} \pi(b_j^{(i)}) = 3\pi(b_i^{(i)}).$$ Using (\[inductive\]), Lemma \[bl1\] yields $$\begin{aligned} \label{1q1q} a&(-1)^{i+1} \pi(b_1^{(i+1)})+ (-1)^{i+1}\sum_{j=2}^{i} (-1)^{j-1} \pi(b_j^{(i+1)}) \nonumber \\ &=a(-1)^{i+1} \pi(b_1^{(i)})+ (-1)^{i+1}\sum_{j=2}^{i-1} (-1)^{j-1} \pi(b_j^{(i)}) +3\pi(b_i^{(i)}) \nonumber \\ &=-\big(a(-1)^{i} \pi(b_1^{(i)})+ (-1)^{i}\sum_{j=2}^{i-1} (-1)^{j-1} \pi(b_j^{(i)})\big)+3\pi(b_i^{(i)}) \nonumber \\ &={(-1)}^i \sum_{j=i+1}^{b-1}{(-1)}^{j-1}\pi(b_j^{(i)}).\end{aligned}$$ By Lemma \[1q1q1\] we have $$\label{1q1q1q} \sum_{j=i+2}^{b-1}{(-1)}^{j-i}\pi(b_j^{(i+1)})=3\pi(b_{i+1}^{(i+1)})+\sum_{j=i+1}^{b-1}{(-1)}^{j-i}\pi(b_j^{(i)}),$$ and adding (\[1q1q\]) and (\[1q1q1q\]) we obtain $$a(-1)^{i+1} \pi(b_1^{(i+1)})+ (-1)^{i+1}\sum_{j=2}^{i} (-1)^{j-1} \pi(b_j^{(i+1)})+\sum_{j=i+2}^{b-1}{(-1)}^{j-i}\pi(b_j^{(i+1)})=$$ $${(-1)}^i\sum_{j=i+1}^{b-1}{(-1)}^{j-1}\pi(b_j^{(i)})+3\pi(b_{i+1}^{(i+1)})+\sum_{j=i+1}^{b-1}{(-1)}^{j-i}\pi(b_j^{(i)})=3\pi(b_{i+1}^{(i+1)}).$$ The image of the generator $\pi(\Gamma_{a,b})$ ---------------------------------------------- By straightforward calculations, one verifies that the composition $i_2 \circ \pi_3: D_{a+3}\otimes \Lambda^{b-3}\rightarrow \Delta(h(3))\rightarrow D_{a+2}\otimes\Lambda^{b-2}$ is the map $f:D_{a+3}\otimes \Lambda^{b-3}\xrightarrow{\triangle \otimes 1} D_{a+2}\otimes D_1 \otimes \Lambda^{b-3}\xrightarrow{1 \otimes m}D_{a+2}\otimes \Lambda^{b-2}$. Let $\phi: Ext^2(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})\rightarrow Ext^2(\Delta(h),D_{a+2}\otimes\Lambda^{b-2})$ be the map induced by $i_2\circ \pi_3: D_{a+3}\otimes \Lambda^{b-3}\rightarrow D_{a+2}\otimes \Lambda^{b-2}$, i.e. $\phi$ is the composition $i_2^{(2)}\circ \pi_3^{(2)}$, $$\begin{aligned} \phi:&Ext^2(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})\xrightarrow{\pi_3^{(2)}}Ext^2(\Delta(h),\Delta(h(3)))\\ &\xrightarrow{i_2^{(2)}}Ext^2(\Delta(h),D_{a+2}\otimes\Lambda^{b-2}).\end{aligned}$$ In this Subsection we will compute the image $\phi(\pi(\Gamma_{a,b}))$ of the cyclic generator $\pi(\Gamma_{a,b})$ that was defined in Proposition 4.1. More precisely, and using the notation established at the beginning of Subsection 4.2, we will prove the following. We have $$\phi(\pi(\Gamma_{a,b}))=(a+b)\pi(\gamma_{a,b}),$$ where $\gamma_{a,b}=\tbinom{a+2}{3} b_1^{(1)}+\sum_{i=2}^{b-1}{(-1)}^{i-1}b_i^{(i)}$ and $\pi(\gamma_{a,b})$ is a cyclic generator of $Ext^2(\Delta(h),D_{a+2}\otimes \Lambda^{b-2})$. We recall from [@MS], Lemma 4.4, that $\pi(\gamma_{a,b})$ is a cyclic generator of the group $Ext^2(\Delta(h),D_{a+2}\otimes \Lambda^{b-2}).$ As $\phi$ is a linear map, we obtain $$\begin{aligned} \phi(\pi(\Gamma_{a,b}))=&\sum_{i=3}^{b-2}{(-1)}^{i-1}\Big(a\phi(\pi(B_{i,1}^{a,b})) -\sum_{j=2}^{i-1} {(-1)}^j \phi(\pi(B_{i,j}^{a,b}))+\sum_{j=i+1}^{b-1} {(-1)}^{j} \phi(\pi(B_{i,j}^{a,b})) \Big)\\ &+\tbinom{a+2} {3}\sum_{j=2}^{b-1}{(-1)}^{j}\phi(\pi(B_{1,j}^{a,b})),\end{aligned}$$ where empty sums are equal to zero. Using the straightening law we have $$\begin{aligned} \phi(\pi(B_{1,j}^{a,b}))&={(-1)}^j\cdot \pi(1^{(a+2)}\otimes 2\dots (b-1)) + \pi(1^{(a+1)}j \otimes 1\dots \hat{j} \dots (b-1))\\&={(-1)}^j \pi(b_1^{(1)})+\pi(b_j^{(1)}).\end{aligned}$$ This implies that $$\begin{aligned} \label{eqgen1} &\tbinom{a+2}{3}\sum_{j=2}^{b-1}{(-1)}^{j}\phi(\pi(B_{1,j}^{a,b}))=\tbinom{a+2}{3}\sum_{j=2}^{b-1} \Big(\pi( b_1^{(1)})+ {(-1)}^j\pi( b_j^{(1)})\Big)\nonumber \\ &=\tbinom{a+2}{3}\Big((b-2)\pi(b_1^{(1)}) + \sum_{j=2}^{b-1} {(-1)}^j \pi(b_j^{(1)})\Big)=\tbinom{a+2}{3}(a+b)\pi(b_1^{(1)}),\end{aligned}$$ as $ \sum_{j=2}^{b-1} {(-1)}^j \pi(b_j^{(1)})=(a+2)\pi(b_1^{(1)})$, by part (a) of Proposition \[l1\]. Using again the straightening law, we observe that - $\phi(\pi(B_{i,j}^{a,b}))={(-1)}^{j-1}\pi(b_i^{(i)})+{(-1)}^i \pi(b_j^{(i)})$ for $1\leq j<i\leq b-1$, and - $\phi(\pi(B_{i,j}^{a,b}))={(-1)}^j \pi(b_i^{(i)})+{(-1)}^{i-1} \pi(b_j^{(i)})$ for $2\leq i<j\leq b-1$. Then, by Proposition \[l1\] for $i\in\{2,\dots,b-1\}$ and (\[eqgen1\]), we have $$\begin{aligned} \phi (\pi(\Gamma_{a,b}))&=\sum_{i=2}^{b-1}{(-1)}^{i-1}\Big(a\phi(\pi(B_{i,1}^{a,b})) -\sum_{j=2}^{i-1} {(-1)}^j \phi(\pi(B_{i,j}^{a,b}))+\sum_{j=i+1}^{b-1} {(-1)}^{j} \phi(\pi(B_{i,j}^{a,b})) \Big)\\ &\,\,\,\,\,\,+\tbinom{a+2}{3}(a+b)\pi(b_1^{(1)}) \\ &=\sum_{i=2}^{b-1}{(-1)}^{i-1}\Big((a+b-3)\pi(b_i^{(i)}) + a{(-1)}^i\pi(b_1^{(i)})+{(-1)}^i\sum_{j=2}^{i-1}{(-1)}^{j-1}\pi(b_j^{(i)})\\ &\,\,\,\,\,\,+{(-1)}^i\sum_{j=i+1}^{b-1}{(-1)}^{j-1}\pi(b_j^{(i)})\Big)+\tbinom{a+2}{3}(a+b)\pi(b_1^{(1)})\\ &=(a+b)\Big(\tbinom{a+2}{3} \pi(b_1^{(1)})+\sum_{i=2}^{b-1}{(-1)}^{i-1}\pi(b_i^{(i)})\Big)=(a+b)\pi(\gamma_{a,b}),\end{aligned}$$ Using Proposition 4.6, we will now prove the case $k=4$ of Theorem 1.1. We have $Ext^2(\Delta(h),\Delta(h(4)))=\mathbb{Z}_{gcd(3,a+b)}. $ We noted at the beginning of Section 4.3 that $\phi$ factors as follows $$\begin{aligned} \phi:&Ext^2(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})\xrightarrow{\pi_3^{(2)}}Ext^2(\Delta(h),\Delta(h(3)))\\ &\xrightarrow{i_2^{(2)}}Ext^2(\Delta(h),D_{a+2}\otimes\Lambda^{b-2}). \end{aligned}$$ We also saw in Proposition 4.6 that $$\phi(\pi(\Gamma_{a,b}))=(a+b)\pi(\gamma_{a,b}),$$ where $\pi(\Gamma_{a,b})$ is a generator of $Ext^2(\Delta(h),D_{a+3}\otimes \Lambda^{b-3})=\mathbb{Z}_3$ and $\pi(\gamma_{a,b})$ is a generator of $Ext^2(\Delta(h),D_{a+2}\otimes \Lambda^{b-2})=\mathbb{Z}_3$. Hence, $\phi$ is an isomorphism if and only if $3\notdivides a+b$ if and only if $\ker \phi =0$. Using (\[long2\]) for $k=3$, we obtain the exact sequence $$\begin{aligned} 0&\longrightarrow Ext^2(\Delta(h),\Delta(h(4)))\xrightarrow{i_3^{(2)}}Ext^2(\Delta(h),D_{a+3}\otimes\Lambda^{b-3})\\ &\xrightarrow{\pi_3^{(2)}}Ext^2(\Delta(h),\Delta(h(3))),\end{aligned}$$ and so $Ext^2(\Delta(h),\Delta(h(4)))=\ker \pi_3^{(2)}$. As $i_2^{(2)}$ is an injection, we obtain $\ker\phi = \ker \pi_3^{(2)}$ and so we have $Ext^2(\Delta(h),\Delta(h(4)))=\ker \phi$. We have $\ker \phi =0$ if and only if $3\notdivides a+b$ and, also, $\ker \phi=\mathbb{Z}_3$ if and only if $3\| a+b$. We conclude that $Ext^2(\Delta(h),\Delta(h(4)))=\mathbb{Z}_{gcd(3,a+b)}$. Modular $Ext^1$ --------------- Let $K$ be an infinite field of characteristic $p > 0$. The corollary below is a statement for $S_K(n,r)$-modules, where $S_K(n,r)$ is the Schur algebra for $GL_n(K)$ corresponding to homogeneous polynomial representations of $GL_n(K)$ of degree $r$ [@Gr]. By $\Delta_K(h)$ we denote the Weyl module of highest weight $h$ for $S_K(n,r)$. From [@Ma] it follows that $Ext^1(\Delta_K(h),\Delta_K(h(1)))$ is one dimensional if $p\|a+b$ and zero otherwise. Using the universal coefficient theorem ([@AB], Thm. 5.3), Theorem 3.5 by [@MS], and Theorem 1.1 we have the following complete picture. Let $h=(a,1^b)$ and $h(k)=(a+k,1^{b-k})$, where $ 1 \le k \le b$. Suppose $n \ge b+1$. Then $Ext^1(\Delta_K (h),\Delta_K (h(k)))$ is 1-dimensional in each of the following cases: - $k=1$ and $p$ divides $a+b$, - $k=2$ and $p$ divides $ 2(a+b)/{gcd(2,a+b)}^2$, - $k=3$ and $p$ divides $ 6/{gcd(2,a+b+1)gcd(3,a+b)}$, - $k=4$ and $p$ divides $ {2gcd(3,a+b)}/{gcd(2,a+b)}$, - $k\geq 5$ and $p=2$, $a+b+k$ is odd. In all other cases, $Ext^1(\Delta_K (h),\Delta_K (h(k)))=0.$ [99]{} Akin K., Extensions of symmetric tensors by alternating tensors, J. Algebra 121 (1989), 358-363. Akin K. and Buchsbaum D., Characteristic-free representation theory of the general linear group II: Homological considerations, Adv. Math. 72 (1988), 172-210. Akin K., Buchsbaum D. and Weyman J., Schur functors and Schur complexes, Adv. Math. 44 (1982), 207-278. Buchsbaum D. and Flores de Chela D., Intertwining numbers: the three rowed case, J. Algebra 183 (1996), 605-635. Cox A. and Erdmann K., $Ext^2$ between Weyl modules for quantum $GL_n$, Math. Proc. Cambridge Philos. Soc. 128 (2000), 441-463. Donkin S. and Geranios H., First degree cohomology of Specht modules and extensions of symmetric powers, Adv. Math. 345 (2019), 618-701. Erdmann, K., $Ext^1$ for Weyl modules of $SL_2(K)$, Math. Z. 218 (1995), 447-459. Green, J. A., Polynomial Representations of GLn, 2nd Edition, Springer (2007). Jantzen J. C., Representations of algebraic groups, volume 107 AMS, Providence, RI, 2nd Edition, 2003. Kulkarni U., Skew Weyl modules for $GL_n$ and degree reduction for Schur Algebras, J. Algebra 224 (2000), 248-262 Kulkarni U., On the Ext groups between Weyl modules for $GL_n$, J. Algebra 304 (2006), 510-542. Maliakas M., Resolutions, homological dimensions, and extension of hook representations, Commun. Algebra 19 (1991), 2195-2216. Maliakas M. and Stergiopoulou D.-D., On extensions of hook Weyl modules, arXiv:2005.00578. Parker A., Higher extensions between modules for $SL_2$, Adv. Math. 209 (2007), 381-405. Touzé, A., Bar complexes and extensions of classical exponential functors, Annales de L’Inst. Fourier 64 (2014), 2563-2637.
--- abstract: 'We consider a two-component scaling picture for the resistivity of two-dimensional (2D) weakly disordered interacting electron systems at low temperature with the aim of describing both the vicinity of the bifurcation and the low resistance metallic regime in the same framework. We contrast the essential features of one-component and two-component scaling theories. We discuss why the conventional lowest order renormalization group equations do not show a bifurcation in 2D, and a semi-empirical extension is proposed which does lead to bifurcation. Parameters, including the product $z\nu$, are determined by least squares fitting to experimental data. An excellent description is obtained for the temperature and density dependence of the resistance of silicon close to the separatrix. Implications of this two-component scaling picture for a quantum critical point are discussed.' author: - 'D.J.W. Geldart' - 'D. Neilson' title: 'Two-Component Scaling near the Metal-Insulator Bifurcation in Two-Dimensions' --- =10000 Introduction ============ The determination of the low temperature properties, and ultimately the ground state phase diagram, of two-dimensional (2D) systems of strongly interacting electrons at low carrier density in the presence of low levels of disorder remains a very challenging problem in spite of a great deal of experimental and theoretical effort. The longstanding view that all charge carrier states become localized at zero temperature ($T=0$) in the limit of a large 2D system has been called into question by experimental observations of a finite temperature “transition” from insulator-like ($\partial\rho/\partial T < 0$) to metal-like ($\partial\rho/\partial T > 0$) behavior as the electron density is increased in very high purity semiconductor MOSFETs and heterostructures [@Kravchenko; @Simonian; @Coleridge; @Simmons]. This bifurcation of the resistivity $\rho(T)$ into two families of curves occurs at a critical carrier density of the 2D electron (or hole) system. The critical carrier density depends on the specific disorder characteristics of the given sample. In the vicinity of the bifurcation, $\rho(T)$ has intriguing scaling properties as a function of density and temperature. The lower density insulating family of curves having $d\rho/dT<0$ can be collapsed onto a single curve when plotted as a function of a scaling variable $T_0(\delta)/T$. The new density dependent temperature scale $T_0(\delta)$ has a power law dependence on the magnitude of $\delta =(n-n^\star)/n^\star$ where $n^\star$ is the critical density at the bifurcation. Similarly, the metallic family of curves at higher density with $d\rho/dT>0$ collapses onto another unique curve with a power law dependence on $T_0(\delta)/T$ with precisely the same critical exponent. There is also scaling behavior with respect to density via $\|\delta\|$ and applied electric field, with a different characteristic critical exponent. A review has been given by Abrahams [et al]{} [@RMP]. A bifurcation with scaling behavior at finite $T$ is expected to be a generic feature associated with a $T=0$ quantum critical point (QCP)[@Sondhi]. Consequently, the demonstrated scaling for the resistivity can be taken as evidence for a metal-insulator transition (MIT). With this assumption, Dobrosavljević [et al]{} have given a phenomenological description of the observed scaling properties and power laws in terms of a one-parameter scaling theory [@DAMC]. In this picture, the bifurcation point lies on a separatrix which terminates at $T=0$ at a QCP which is a fixed point of a set of scaling equations. The fixed point is repulsive with respect to the temperature variable and separates the insulating and metallic regions of the low T phase diagram. There is a unique correspondence between the QCP and the bifurcation point and a two-phase ground state is implied. However, this correspondence provides only indirect evidence for a MIT. Of course the power law fits near the bifurcation point are based on linearization of a scaling flow equation and the linearization must eventually fail at sufficiently low temperature, for which $T_0(\delta)/T$ is no longer small, but this is not evidence against a QCP. A more crucial point is that a physical effect which is negligible for temperatures $T\sim 1$ K near the bifurcation, may become dominant in the low $T$ limit. Scaling properties characteristic of an imminent QCP would then still be observed near the bifurcation but the suggested QCP itself could be totally removed in the zero temperature limit. A possible example of this situation is the proposal by Simmons [et al]{} [@SimmonsWL] that in $p$-type GaAs the $\rho(T)$ curves of the metallic family will eventually turn upward to exhibit insulator-like behavior with $\partial \rho/\partial T < 0$ if the electron temperature $T$ could be made sufficiently low. A similar conclusion has recently been reached for Si MOSFETs. [@Prus] However the turn-up is actually only directly observed for metallic curves just above the critical density, and, as we discuss below, the magnetoresistance data cited as evidence of a universal weak localization in 2D in Refs. and may alternatively reflect different magnetic field dependencies of the competing localizing and delocalizing triplet spin state interactions. Irrespective of the mechanism responsible for the upturn, properties of the system at temperatures well below the bifurcation temperature are not correctly described by a one-component scaling theory. We will outline a semi-empirical two-component scaling theory that describes within a single framework the temperature dependent resistivity both in the vicinity of the bifurcation and in the low resistance metallic regime. Following early work by Altshuler [et al]{} [@AAL] further progress toward an understanding of disordered 2D electron systems in the strong coupling regime was made by Finkelstein [@Finkelstein1; @Finkelstein2] and by Castellani [et al]{} [@Castellani] who showed that electron-electron interactions can lead to $d\rho/dT > 0$ (metallic behavior) in the disordered 2D system at finite $T$. This is in contrast to the localizing insulating behavior found in the absence of electron-electron interactions [@Abrahams]. The physical properties of the system at low $T$ are determined by nonlinear interactions of low energy diffusive modes. A set of renormalization group (RG) equations was generated by means of a formal perturbation expansion in powers of a dimensionless temperature dependent resistivity ${\cal{R}}$ which is related to the physical resistance per square by ${\cal{R}} = (e^2/\pi h)R_{\fbox{\!}}$. These scaling results include electron-electron interactions to all orders and are valid in the diffusive regime $k_BT<\hbar/\tau$ provided ${\cal{R}}$ is small (in principle, ${\cal{R}}<<1$). Additional progress was made by Zala [et al]{} [@Zala] who considered interaction corrections to transport properties at intermediate temperatures in the ballistic regime $k_BT>\hbar/\tau$ to all orders in the electron-electron interaction using a Fermi-liquid approach. These results are also restricted to the perturbative regime, ${\cal{R}}<<1$. Comparisons with experiment for the temperature dependent resistivity of Si MOSFETs have been made for both the scaling theory [@Punn] and the intermediate temperature theory [@Vitkalov; @Pudalovinteraction]. The intermediate temperature theory gives consistent semi-quantitative agreement with experiment in the range of densities for which ${\cal{R}}$ is small. We emphasize that both these theories are perturbative in ${\cal{R}}$ and are limited to the regime of small ${\cal{R}}$. Hence they must fail to describe the regime near bifurcation where ${\cal{R}}\sim 1$ and they contain no bifurcation point. In addition, in the RG procedure the equations develop singularities at a nonzero temperature so that the approach to the ground state cannot be described. In contrast to the 2D case there is a qualitative understanding of the metal-insulator transition in 3D electron systems. A consistent qualitative picture of the transition has been obtained by RG methods in a space of dimension $2+\epsilon$ with $\epsilon>0$ taken as an expansion parameter. Physical quantities such as critical exponents are expressed as power series in $\epsilon$. While only the leading terms in $\epsilon$ have been obtained it is presumed that resummation methods would give good results in 3D if sufficient correction terms could be calculated. In the case of $\epsilon>0$ it is clear that the interplay of strong electron-electron interactions with disorder is an essential aspect of the bifurcation. It is remarkable that in spite of great effort a corresponding theory does not exist for 2D systems, that is for $\epsilon=0$. This lack casts doubt on the physical significance and relevance of the “universal” aspects of electron-electron interactions in disordered systems. In addition the lack of even an approximate scaling theory has prevented quantitative discussion of the low temperature limit and the approach to the ground state. We expect that the interplay of strong electron-electron interactions with disorder must be an essential aspect of the bifurcation in 2D irrespective of whether or not a true metal-insulator transition occurs. Of course in some systems material dependent effects also contribute. In this paper we discuss why the lowest order RG equations in 2D do not show a bifurcation and we give a semi-empirical extension which does describe a bifurcation region. We focus on the immediate vicinity of the bifurcation and on the metallic regime. The strong insulator limit will be discussed elsewhere. The proposed scaling equations have a physical low temperature limit and have no singularities at finite length or temperature scales. This scaling picture provides a theoretical framework for the interpretation of experimental results, more specifically the “universal” contributions due to electron-electron interactions and disorder. Renormalization Group Equations =============================== The RG equations established at one-loop level for a 2D system of electrons in the presence of disorder are based on four dimensionless scaling parameters ${\cal{R}}$, $\gamma_2$, $Z$ and $\gamma_c$, [@Finkelstein1; @Finkelstein2; @Castellani] where $\Gamma_2=Z\gamma_2$ is the electron-hole scattering amplitude for the triplet spin state, $\Gamma_c=Z\gamma_c$ is the singlet state particle-particle scattering amplitude and $Z$ is the dynamical energy rescaling function. Together with ${\cal{R}}$, these quantities are all functions of the variable $y=\ln\lambda^{-1}$ which describes rescaling of the momenta after integrating over the momentum shell specified by $\lambda k_0^2<k^2<k_0^2$. [@Castellani2] We consider only the case when the disorder is due to purely potential scattering. When a connection between length and temperature scaling is needed, we relate a thermal length $\ell_{th}$ to temperature by $\ell_{th}/\ell_{el}=(T_{el}/T)^{1/z}$, where $\ell_{el}$ and $T_{el}$ are the length and temperature scales for elastic scattering and $z$ is the dynamical critical exponent [@Sondhi]. Particle-particle scattering is not considered to play an essential role relative to the electron-hole scattering represented by $\gamma_2$ so we omit consideration of $\gamma_c$. The energy scaling function $Z(y$) is not followed explicitly but its effect is taken into account at the bifurcation fixed point by $y=\ln[(T_{el}/T)^{1/z}]$. These approximations permit discussion of a two-component scaling theory based on the scaling parameters ${\cal{R}}$ and $\gamma_2$. With these simplifications the RG equations are $$\begin{aligned} d{\cal{R}}/dy&=&\alpha(\gamma_2){\cal{R}}^2\\ d\gamma_2/dy&=&\frac{(1+\gamma_2)^2}{2} {\cal{R}} \label{RGEqns} \end{aligned}$$ where $$\alpha(\gamma_2)=1+\left[n_v+((2n_v)^2-1)\left\{1\!-\frac{1\!+\gamma_2}{\gamma_2} \ln(1\!+\gamma_2)\right\}\right] \ . \label{alpha}$$ $n_v$ is the number of valleys, so for Si $n_v=2$. Equations (1) and (2) are to be integrated upward in $y$, starting from initial bare values for the dependent variables at $y = y(0)$ for which $T_{el}/T$ is of order unity. Increasing $y$ corresponds to integrating out shorter wavelength and higher energy excitations. This amounts to increasing the length scale and decreasing the temperature scale. Our Eqs. (1) – (3) are essentially the same as Eqs. (1) and (2) of Ref. . The notational difference is that we use a running variable $y=-\ln(\lambda)$ as in Ref. , while Ref. uses $\xi=-\ln(k_B T \tau/\hbar)$. Prior to renormalization due to interacting diffusive modes, we can set $z=1$. After the RG flow has proceeded to the bifurcation fixed point, we set $y=\ln[(T_{el}/T)^{1/z}]$ where $z$ is the dynamical critical exponent of the fixed point. The first term (unity) on the right hand side of Eq. (\[alpha\]) contains no electron-electron interactions and arises from weak localization. On scaling this term enhances the resistivity. The other term, in the square brackets, is due to electron-electron interactions in the singlet and triplet particle-hole spin states and can be negative when $\gamma_2$ is large enough. When it is negative this term has the opposite trend of reducing the resistivity upon scaling. The bare value of $\gamma_2$ is given by $\gamma_2^{bare} = -F_0^a /(1 + F_0^a )$ where $F_0^a<0$ is the spin-antisymmetric Landau parameter. From Eq. (2), $\gamma_2$ increases with rescaling. With $n_v=2$, the expression for $\alpha(\gamma_2)$ changes sign when $\gamma_2$ reaches $0.46$. A change in sign of $\alpha(\gamma_2)$, corresponding to a net delocalizing effect in zero external magnetic field, has no particular [a priori]{} implication for magnetoresistance. The magnetoresistance data of Refs. can be consistent with our scaling conclusions if, for example, the localizing contribution in Eq. (\[alpha\]) has a stronger magnetic field dependence (at small fields) than the delocalizing contribution from the electron-electron interactions. Thus the appearance of a weak localization precursor signature in magnetoresistance data does not necessarily imply there will be an eventual turn-up in the zero field resistivity at very low temperatures. As we integrate Eq. (2), $\gamma_2$ increases from its initial bare value but diverges at a finite value of $y=y_{max}$ provided ${\cal{R}}$ remains finite at $y=y_{max}$. [@Finkelstein1; @Finkelstein2; @Castellani] It is easy to confirm that a solution of Eqs. (1) and (2) with finite ${\cal{R}}={\cal{R}}_{min}$ and $\gamma_2$ arbitrarily large is consistent. In this limit Eq. (2) can be written $$d\gamma_2/dy=({\cal{R}}_{min}/2)\gamma_2^2\ . \label{dgamma2dy}$$ It follows that $\gamma_2$ diverges at a finite $y_{max}$. To verify the consistency of a finite ${\cal{R}}_{min}$ when $\gamma_2$ diverges, we can divide Eq. (1) by Eq. (2) and rearrange obtaining, $$d{\cal{R}}/{\cal{R}}=-d\gamma_2(6\ln\gamma_2)/\gamma_2^2\ .$$ The integral of the left hand side is finite so it follows that ${\cal{R}}$ indeed reaches a finite lower limit ${\cal{R}}_{min}$ as $\gamma_2\rightarrow\infty$. The rescaling cannot be continued beyond the singularity at $y=y_{max}$ which means that the zero temperature limit cannot be reached. In principle, this singularity in the triplet state scattering amplitude might signal the onset of a magnetic instability in the system. On the other hand, the divergence might simply be an artefact of a low order perturbation expansion. The ultimate fate of this singularity at very low T is not completely clear. Castellani [et al]{} [@Castellani2] showed that a similar divergence in the energy rescaling function $Z(y)$ can cause the singular point to shift to extremely low $T$. The paramagnetic metallic regime would then extend over a wide temperature. More recently, Kirkpatrick and Belitz [@KB] and Chamon and Mucciolo [@CM] have indeed found a solution of the RG equations corresponding to a disordered ferromagnet. The question of how low the temperatures would be where such transitions might occur is open. In the absence of theoretical guidance on this point we turn to experiment for information on the triplet spin state scattering amplitude $\gamma_2$ in the temperature range of the bifurcation. References concluded from an analysis of magnetoresistance data that there would be a ferromagnetic instability very near the density of the bifurcation. However, measurements in Refs. found an enhancement of the effective g-factor $g^$, but no singularity for densities down to $r_s=8.4$, a range which includes the bifurcation. This is interpreted in terms of the spin-antisymmetric Landau parameter as $g^=2/(1+F_0^a)$. $F_0^a$ would be $-1$ at the onset of a ferromagnetic instability. For $r_s=8$ Ref. gives $F_0^a \simeq -0.5$, corresponding to a spin susceptibility enhancement factor of $2$. At temperatures well above the bifurcation the bare values of the electron-electron interaction amplitudes like $\gamma_2^{bare}$ have negligible diffusion corrections. $\gamma_2^{bare}$ is then related to $F_0^a$ by $\gamma_2^{bare} = -F_0^a /(1 + F_0^a )$. A value $F_0^a = -0.5$ corresponds to $\gamma_2^{bare} =1$. On the basis of the direct measurement of $g^$ we will assume that the triplet spin state scattering amplitude $\gamma_2$ is finite and well behaved throughout the density and temperature range of the experiments we consider. With this assumption, an explicit RG equation is not needed for $\gamma_2$. Nonmagnetic Fermi liquid behavior of the system is a sufficient condition for a smooth $\gamma_2$ but may not be necessary. Bifurcation in 2D ================= To discuss why Eqs. (1) and (2) fail to describe a bifurcation in 2D electron systems, we first recall the corresponding RG results in $d=2+\epsilon$ dimensions with $\epsilon$ small and positive. Making the same physical assumptions regarding the interacting diffusive modes and in the same one-loop approximation, Eq. (1) becomes $$d{\cal{R}}/dy = -(\epsilon/2){\cal{R}} + \alpha(\gamma_2) {\cal{R}}^2 \ . \label{dRdy}$$ The first term on the right hand side of Eq. (\[dRdy\]) is the consequence of the $L^{2-d}$ form factor when converting resistivity to resistance in a space of $d$ dimensions, and the coefficient $\alpha(\gamma_2)$ in the second term is the same as in Eq. (\[alpha\]). We identify the bifurcation point as the point at which $d{\cal{R}}/dy = 0$. The zero of Eq. (\[dRdy\]) occurs at the critical value ${\cal{R}}^ = \epsilon/(2 \alpha)$, provided $\alpha > 0$, where $\alpha$ is the value of $\alpha(\gamma_2)$ when the bifurcation occurs. In order for the bifurcation to be a precursor for a quantum critical point, the temperature must be a relevant variable. This requires that $\tau^{-1} > 0$ in the linearized flow equation $$d( {\cal{R}} -{\cal{R}}^)/dy = \tau^{-1} ( {\cal{R}} -{\cal{R}}^) \ . \label{dRdy1}$$ This procedure for identifying a critical point by a linearized flow equation is standard. From Eq. (\[dRdy\]), $\tau^{-1} \epsilon/2 > 0$. The condition that $\tau^{-1} > 0$ implies that the resistance has positive (negative) slope with respect to $T$ in the metallic (insulating) regime. It is clear that a non-trivial root of $d{\cal{R}}/dy=0$ requires at least two terms and that for the 2D case Eq. (1) as presented (that is, Eq.(\[dRdy\]) with $\epsilon=0$) will not be sufficient. Attempts have been made to discuss possible metal-insulator transitions in 2D using Eq. (1) by tuning the parameter $\alpha(\gamma_2)$ to zero. However, a critical point is a robust property of the entire $d{\cal{R}}/dy$ and cannot be described by the properties of only a single term $\alpha(\gamma_2) {\cal{R}}^2$ in a series. Such a procedure is not stable to the addition of higher order terms. The simplest modification of Eq. (1) that can show a bifurcation and is consistent with two-component scaling is $$d{\cal{R}}/dy = \alpha(\gamma_2) {\cal{R}}^2 + \beta(\gamma_2) {\cal{R}}^3+\ \dots \label{dRdy2}$$ The function $\beta(\gamma_2)$ is not known explicitly but its sign can be determined by the conditions that ${\cal{R}}^$ and $\tau^{-1}$ are both positive. Linearizing Eq. (\[dRdy2\]) about the zero of its right hand side gives the linearized flow equation (\[dRdy1\]) with ${\cal{R}}^= -\alpha/\beta$ and exponent $\tau = \beta/\alpha^2$. Here $\alpha$ and $\beta$ are the values of $\alpha(\gamma_2)$ and $\beta(\gamma_2)$ when the bifurcation occurs. ${\cal{R}}^$ and $\tau^{-1}$ are both positive provided $\alpha<0$ and $\beta>0$. There is no bifurcation in 2D if $\alpha(\gamma_2)$ is always positive. Integrating Eq. (\[dRdy1\]) starting from an initial ${\cal{R}}_0$ at $y=y_0$ gives two families of curves. A metallic regime and a bifurcation can thus be described. We conclude that the bifurcations in the 2D system and the $2+\epsilon$ system are controlled by different fixed points. The fixed point ${\cal{R}}^=\epsilon/\alpha$ for $2+\epsilon$ requires $\alpha>0$ which is in the range of weak electron-electron interactions. This fixed point becomes trivial (${\cal{R}}^=0$) in the $\epsilon\rightarrow0$ limit and plays no role in 2D. This allows a new fixed point ${\cal{R}}^ = -\alpha/\beta$ to become physical in 2D at a scale where electron-electron interactions have become strong enough to change the sign of $\alpha$. In each case there is only one fixed point and the physical picture of interacting diffusive modes is correct for the determination of the universal contributions. The strength of the electron-electron interactions in 2D is crucial for generating a bifurcation. At high densities where the electron-electron interactions are weak, $\gamma_2^{bare}$ is small and $\alpha(\gamma_2^{bare})$ is positive. As the density is lowered the initial $\gamma_2^{bare}$ increases. Using the data of Ref. for $g^$ in Si, the sign change in $\alpha(\gamma_2^{bare})$ occurs for a density corresponding to $r_s\simeq3$. This provides an upper limit to the density at which a bifurcation can occur in Si. For $r_s<3$ the electron-electron interactions are too weak. In order to describe quantitatively the resistivity near the bifurcation where $R\sim1$, as well as in the insulating regime where $R>>1$, the sum of the series implied in Eq. (\[dRdy2\]) must be adequately represented. If the series is truncated at an arbitrary finite order a spurious divergence at a finite $y_{max}$ can occur in the insulating region. While in the insulating limit ${\cal{R}}$ and $d{\cal{R}}/dy$ are expected to diverge as $y\rightarrow\infty$ (that is, at $T\rightarrow0$), the divergence in $d{\cal{R}}/dy$ must be sufficiently weak that a spurious divergence in ${\cal{R}}$ at a finite $y_{max}$ does not occur. A linear power law in ${\cal{R}}$ is the strongest growth of $d{\cal{R}}/dy$ at large ${\cal{R}}$ that permits this (with possible logarithmic corrections). For simplicity, we maintain the fixed point structure of the low order terms and introduce a denominator into Eq. (\[dRdy2\]) to represent the net effect of higher order terms including the linear growth at large ${\cal{R}}$, $$\frac{d{\cal{R}}}{dy} = \frac{\alpha(\gamma_2) {\cal{R}}^2 + \beta(\gamma_2) {\cal{R}}^3}{1+\kappa(\gamma_2){\cal{R}}^2}\ . \label{dRdy3}$$ Equation (\[dRdy3\]) satisfies the minimal conditions of having a bifurcation with two classes of well defined solutions (metallic and insulating) depending on the choice of initial conditions at $y=y(0)$. It may be regarded as a semi-empirical Padé approximation to the full series. Of course, these minimal conditions do not uniquely determine the functional form (see also Section IV. B). We focus in this paper on the metallic regime and the close vicinity of the separatrix. The known perturbative results are contained explicitly in Eq. (\[dRdy3\]) so the solution for $y\rightarrow\infty$ and ${\cal{R}}\rightarrow0$ is exact. The strongly insulating limit ${\cal{R}}\rightarrow\infty$ contains additional logarithmic corrections so Eq. (\[dRdy3\]) is incomplete in this limit, and a detailed discussion of the deep insulating regime will be given elsewhere. Close to the separatrix the functions $\alpha(\gamma_2$), $\beta(\gamma_2)$, and $\kappa(\gamma_2)$ are taken to be slowly varying and so are replaced by their constant values $\alpha$, $\beta$ and $\kappa$ for $\gamma_2$ near the start of the bifurcation. Information on these parameters is given by fitting to experimental data in the following section. Equation (\[dRdy3\]) can then be rewritten in the form $$\frac{1}{{\cal{R}}}\frac{d{\cal{R}}}{dy} = \frac{1}{\tau} \frac{\Delta+\Delta^2}{1+\phi(2\Delta+\Delta^2)}\ , \label{dRdyDelta}$$ where $$\begin{aligned} {\cal{R}}^&=&-\alpha/\beta \nonumber \\ {\cal{R}}/{\cal{R}}^&=&1+\Delta \nonumber \\ \tau^{-1}&=&\beta{{\cal{R}}^}^2/(1+\kappa{{\cal{R}}^}^2) \nonumber \\ \phi&=&\kappa{{\cal{R}}^}^2/(1+\kappa{{\cal{R}}^}^2)\ . \end{aligned}$$ Linearizing Eq. (\[dRdyDelta\]) in $\Delta$ we recover Eq. (\[dRdy1\]), with the solution $$\ln\{\|{\cal{R}}-{\cal{R}}^\star\|/\|{\cal{R}}(0)-{\cal{R}}^\star\|\}=\tau^{-1}(y-y(0))\ . \label{lnR}$$ Since $\tau$ describes the rescaling of an inverse length squared, it is related to the critical exponent $\nu$ of the correlation length by $\tau=2\nu$. The temperature is introduced by the thermal length $\ell_{th}\sim1/T^{1/z}$ giving $$\|{\cal{R}}-{\cal{R}}^\star\| = \|{\cal{R}}(0)-{\cal{R}}^\star\|\left(T_0/T\right)^{1/z\nu} , \label{R}$$ with the same exponent for both the metallic and the insulating branches. The prefactor defines a temperature scale $T_0 (\delta) \sim \|\delta\|^{z\nu} T_0$. Both of these features agree with the observed scaling and with the phenomenological scaling of Ref. . Results ======= Vicinity of separatrix ---------------------- We now compare our results based on Eq. (\[dRdyDelta\]) with experimental values of the resistivity $\rho(T)$ close to the separatrix obtained for a Si-MOSFET taken from Fig. 1(b) of Ref. (see also Ref.). Equation (\[dRdyDelta\]) can be integrated analytically. The parameters ${\cal{R}}^\star$, $z$, $\nu$, $\alpha$, $\beta$, and $\kappa$ in ${\cal{R}}(y)$ are chosen to give a best fit to the experimental data. Not all these parameters are independent of each other. The experimental data fix ${\cal{R}}^\star=2.8$. Since ${\cal{R}}^\star$=$-\alpha/\beta$, we can then consider $\alpha$ to be fixed with $\beta$ the free variable. The $z$ and $\nu=\tau/2$ enter together as a product via $\tau^{-1}(y-y(0)=\ln(T_0/T)^{1/z\nu}$. The temperature $T_0$ just prior to the bifurcation was taken to correspond to the temperature scale of elastic scattering $T_{el}=1.75$ K. From the definition of $\tau$, we have $\kappa=\tau\beta - 1/{{\cal{R}}^\star}^2$, which relates $\kappa$ to $\beta$. There are therefore three independent variables $z$, $\nu=\tau/2$ and $\beta$. From the combination of electric field scaling and temperature scaling of the resistivity the dynamical critical exponent $z$ is believed to be in the range $0.8$ to $1.2$.[@RMP] We have therefore made least squares fits to the experimental data with $b=z\nu$ and $\beta$ as free parameters for the fixed values of $z$ in the range $0.8\leq z\leq1.2$. The optimum values of the fitting parameters are determined by minimizing the root mean square relative deviation between theory and experiment $$D = \frac{1}{N}\left.\sum_{j=1}^{N} \frac{\sqrt{({\cal{R}}^{theory}_j-{\cal{R}}^{expt}_j)^2}} {{\cal{R}}^{expt}_j}\right.\ , \label{Rrmsq}$$ where $N$ is the total number of points included in the fit. We have also examined the sensitivity of the fitted parameters to the temperature and density range of the fits. This is essential because the values of the least squares fitted parameters can vary with the range of temperature and density considered. Eq. (\[dRdyDelta\]) has been established on the assumption of constant parameters for sufficiently small $T_0(\delta)/T$. Fits to data will be valid only if the derived parameters are stable with respect to reducing the maximum allowed $T_0(\delta)/T$, that is by restricting the temperature and density range of the fits. We first carried out fits including all of the data points in Fig. 1(b) of Ref. . To test sensitivity to the range of fit we successively restricted the allowed data points by the conditions $T_0(\delta)/T < 1.0$, $0.5$, and $0.25$. For $T_0(\delta)/T < 0.5$ and fixed $z$ the $D$ and the values of the fitted parameters become stable. The coefficients $\beta$ and $\kappa$ vary slowly with $z$. However the essential parameters from the point of view of fitting to a universal scaling form are constant throughout the range of $z$. These values are $D=0.036\pm 0.0005$, $b=z\nu=1.09\pm0.005$ and $\phi=0.80\pm0.005$, where the uncertainties reflect the small variations due to $z$. A constant $b$ and $\phi$ is consistent, since $(2\phi-1)/b$ is the universal coefficient of the first non-linear correction $\Delta^2$ in Eq. (\[dRdyDelta\]). ![Solid lines: measured resistivity $\rho(T)$ in units of $h/e^2$ from Ref. \[25\] in a Si-MOSFET close to separatrix for electron densities (from the top) $n=8.6$, $8.8$, $9.0$, $9.3$, $9.5$, $9.9$, $11.0\times10^{10}$ cm$^{-2}$. Dashed lines: our calculations using Eq. (10). The values of the parameters are given in the text. Dotted lines are in the small $T$ region $T_0(\delta)/T>0.5$ that is excluded from the fit. Small vertical arrows: see text. \[Fig.1\]](Fig1.eps){width="8.1cm"} Figure 1 compares our results (dashed lines) from Eq. (\[dRdyDelta\]) for the dimensionless resistivity in the form $\rho e^2/h={\cal{R}}\pi$ with the experimental measurements (solid lines). The agreement is excellent. The dotted lines indicate the data points excluded from the fit by the condition $T_0(\delta)/T < 0.5$, and the small arrows show the edge of the corresponding excluded data range for $T_0(\delta)/T < 0.25$. Exponential form in metallic region ----------------------------------- Previous fits to experimental data have shown that the temperature dependence of the resistivity is approximately exponential [@Pudalove; @Pudalovc]. This result was accounted for by Ref. by arguing that the beta-function of the conductivity was a logarithmic function of conductivity, even in the metallic range near the separatrix. This is equivalent to assuming that the sum of the full series in Eq. (\[dRdy2\]), again with the coefficients evaluated at the bifurcation, is approximately logarithmic. Then Eq. (\[dRdyDelta\]) is replaced by $$\frac{d \ln{\cal{R}}/{\cal{R}}^\star}{dy} \simeq \tau^{-1}\ln{\cal{R}}/{\cal{R}}^\star\ , \label{dlnRdy}$$ giving $${\cal{R}}(T)\simeq {\cal{R}}^\star\exp\left\{-C\left[\frac{T_0(\delta)}{T}\right]^{1/z\nu}\right\} \label{expR}$$ with $C$ a positive constant of order unity. For the purposes of least squares fitting to data for small $\|\Delta\|$, it is important to note that Eqs. (\[dRdyDelta\]) and (\[dlnRdy\]) are not inconsistent. Standard procedures for Padé approximations give the identity $$\ln(1+\Delta) = \frac{\Delta+\frac{1}{2}\Delta^2}{1+\Delta+\frac{1}{6}\Delta^2} +{\cal{O}}(\Delta^5)\ . \label{dln1+Delta}$$ The exact sum of the perturbation series near the bifurcation has the same form $$\frac{d \ln{\cal{R}}/{\cal{R}}^\star}{dy}= \frac{1}{\tau}\frac{\Delta+a \Delta^2}{1+b\Delta+c\Delta^2} +{\cal{O}}(\Delta^5)\ , \label{dlnRonR}$$ with the coefficients being of order unity. Equation (\[dRdyDelta\]) and Eq. (\[dlnRdy\]) with (\[dln1+Delta\]) can thus be regarded as different approximations to Eq. (\[dlnRonR\\]) and are therefore equivalent from the point of view of least squares fitting to data for small $\|\Delta\|$. It is interesting to note that the low order expansion of Eq. (\[dln1+Delta\]), $\ln(1+\Delta) =\Delta-\frac{1}{2}\Delta^2+{\cal{O}}(\Delta^3)$, gives a value $\phi=\frac{3}{4}$ which is comparable to our fitted value of $\phi=0.80$. Deep metallic limit ------------------- Due to its $y$ dependence $\gamma_2$ continues to rescale as the low temperature limit of the metallic regime is approached. If we assume that $\gamma_2$ approaches a finite limiting plateau value $\gamma_2^0$ as $T\rightarrow0$, then $\alpha(\gamma_2)$ and $\beta(\gamma_2)$ also approach finite limiting values $\alpha(\gamma_2^0)$ and $\beta(\gamma_2^0)$. Similarly, again due to the rescaling, the zero temperature limit $z^0\nu^0$ of the exponent $z\nu$ is expected to be different from the bare value and also from the value determined at the bifurcation. In this case for $T$ sufficiently small we can obtain a consistent solution of Eq. (\[dRdy2\]) with ${\cal{R}}<<1$ so that the perturbation expansion is well represented by the leading term, $$d{\cal{R}}/dy= \alpha(\gamma_2^0){\cal{R}}^2+{\cal{O}}({\cal{R}}^3) \ , \label{dRdy4}$$ with the solution $${\cal{R}}/{\cal{R}}_0=\left[1-\alpha(\gamma_2^0)(y-y_0)\right]^{-1} \ . \label{R2}$$ Using $(y-y_0)=\ln(T_0/T)^{1/z}$ and with the dimensionless conductivity $g={\cal{R}}^{-1}$, we find $$\frac{d\,g\ \ }{d\ln T}= z^0\alpha(\gamma_2^0) \ . \label{dgdlnT}$$ The dominant temperature dependence is then logarithmic as observed experimentally. References and express this logarithmic contribution to the conductivity $G$ as $\Delta G=(e^2/h)C(n)\ln T$. Using Eq.(\[dgdlnT\]) we identify $C(n)=\alpha(\gamma_2^0)/(\pi z^0)$. As the density increases $\gamma_2^0$ and $\alpha(\gamma_2^0)$ decrease in magnitude and $\alpha(\gamma_2^0)$ may even change sign. This dependence of $C(n)$ on density is in agreement with that observed experimentally. Conclusions =========== The low order perturbative RG equations of Refs. do not describe a bifurcation in 2D, but a proper description of a fixed point and a bifurcation in 2D can be obtained when higher order terms in the perturbation expansion for ${\cal{R}}$ are retained (see Eqs. (\[dRdy2\]), (\[dRdy3\])). The resulting 2D fixed point exists only for $\alpha(\gamma_2)$ negative and so is unrelated to the fixed point in dimensions $2+\epsilon$ for which $\alpha(\gamma_2)$ is positive. Our results provide a coherent semi-empirical two-component scaling description of the density and temperature regime near the observed bifurcation and throughout the metallic regime. Near the separatrix a least squares fit to experimental data using Eq. (\[dRdyDelta\]) gives an excellent description of the observed density and temperature dependence of the resistivity. A full discussion for the insulating range will be given elsewhere. The scaling results apply only to the “universal” contributions to the resistivity which are a generic consequences of the interplay between electron-electron interactions and disorder. There are also “non-universal” contributions to the resistivity which will be material-dependent. The present picture is based on a ${\cal{R}}$ which shows a bifurcation and on a $\gamma_2$ which scales smoothly with temperature. The bifurcation “point” ${\cal{R}}^=-\alpha/\beta$ therefore varies smoothly with temperature and the separatrix, at which $d{\cal{R}}/dT=0$, is “tilted” upwards. A tilted separatrix can lead in the metallic regime to a turn-up of the resistance at low temperatures. This cannot occur in a one-component scaling theory where the separatrix is flat and the bifurcation point is uniquely determined as a function of density. An exact solution of Eq. (\[dRdy4\]) in the very low temperature limit shows a logarithmic dependence of the conductance on temperature with a coefficient which is negative and decreases in magnitude as the density increases. The low temperature behavior is in agreement with experimental results [@Pudalovc; @Pudalovd]. The physical origin of this logarithmic electron-electron contribution to the conductivity is the same as that observed at high temperature. However the numerical value of the low $T$ coefficient can differ from the bare value at higher temperature. This work is supported by the Natural Sciences and Engineering Research Council of Canada and an Australian Research Council Grant. We thank Alex Hamilton and Michelle Simmons for very useful discussions. DJWG thanks Ian Campbell for hospitality at the Université de Montpellier II. [99]{} S.V. Kravchenko, W.E. Mason, G.E. Bowker, J.E. Furneaux, V.M. Pudalov, and M. D’Iorio, , 7038 (1995); S.V. Kravchenko, D. Simonian, M.P. Sarachik, W. 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--- abstract: 'The bright star 55 Cancri is known to host five planets, including a transiting super-Earth. We use the CHARA Array to directly determine the following of 55 Cnc’s stellar astrophysical parameters: $R=0.943 \pm 0.010 R_{\odot}$, $T_{\rm EFF} = 5196 \pm 24$ K. Planet 55 Cnc f ($M \sin i = 0.155 M_{Jupiter}$) spends the majority of the duration of its elliptical orbit in the circumstellar habitable zone (0.67–1.32 AU) where, with moderate greenhouse heating, it could harbor liquid water. Our determination of 55 Cancri’s stellar radius allows for a model-independent calculation of the physical diameter of the transiting super-Earth 55 Cnc e ($\simeq 2.1 R_{\earth}$), which, depending on the assumed literature value of planetary mass, implies a bulk density of 0.76 $\rho_{\earth}$ or 1.07 $\rho_{\earth}$.' author: - 'K. von Braun$^1$, T. S. Boyajian$^{2,3}$, T. A. ten Brummelaar$^6$, G. T. van Belle$^4$, S. R. Kane$^1$, D. R. Ciardi$^1$, M. Lopez-Morales$^5$, H. A. McAlister$^2$, G. Schaefer$^6$, S. T. Ridgway$^7$, L. Sturmann$^6$, J. Sturmann$^6$, R. White$^2$, N. H. Turner$^6$, C. Farrington$^6$, & P. J. Goldfinger$^6$' title: 'The 55 Cancri System: Fundamental Stellar Parameters, Habitable Zone Planet, and Super-Earth Diameter' --- Introduction ============ 55 Cancri (= HD 75732) is a late G / early K dwarf / subgiant currently known to host five extrasolar planets with periods between around 0.7 days and 14 years and minimum masses between 0.026 and 3.84 $M_{Jupiter}$. The super-Earth 55 Cnc e was recently discovered to transit (Winn et al. 2011, Demory et al. 2011), prompting a number of studies of the properties of this system (e.g., Kane et al. 2011, von Braun et al. 2011b). We used the CHARA Interferometric Array to directly measure the stellar angular diameter, which, when combined with Hipparcos parallax measurement and calculation of bolometric flux based on spectral templates and literature broad-band photometry, yields the physical $R_{star}$ and $T_{\rm EFF}$ (von Braun et al. 2011a). Properties of the 55 Cancri System ================================== Details of our observations and calculations of stellar properties are described in von Braun et al. (2011b). We give our results in Table 1. Of particular interest are the location and extent of the circumstellar habitable zone (HZ), based on the equations of Jones & Sleep (2010), and the physical radius of the transiting super-Earth, based on simply the measured flux decrement during transit and our calculated stellar radius value. Fig. 1 illustrates that planet f, with its elliptical orbit ($e\simeq0.4$), spends about 74% of its year inside the HZ, while its equilibrium temperature varies between 221K (apastron) and 302K (periastron) for the assumption of perfect efficiency in the redistribution of energy received from the star (von Braun et al. 2011b). Coupled with the recent literature values of assumed planetary mass and measured flux decrement during transit, our measured stellar radius implies $R_{p} = 2.007 \pm 0.136 R_{\earth}$ and $\rho_{p} =1.067 \pm 0.132 \rho_{\earth}$ (for $M_{p}$ = $8.63 \pm 0.35 M_{\earth}$; Winn et al. 2011), and $R_{p} = 2.193 \pm 0.146 R_{\earth}$ and $\rho_{p} = 0.757 \pm 0.109 \rho_{\earth}$ (for $M_{p}$ = $7.98 \pm 0.69 M_{\earth}$; Demory et al. 2011). ------------------------------------- ------------------- Parameter Value $\theta_{\rm UD}$ (milliarcseconds) $0.685 \pm 0.004$ $\theta_{\rm LD}$ (milliarcseconds) $0.711 \pm 0.004$ Radius ($R_{\rm \odot}$) $0.943 \pm 0.010$ Luminosity ($L_{\rm \odot}$) $0.582 \pm 0.014$ $T_{\rm EFF}$ (K) $5196 \pm 24$ HZ boundaries (AU) $0.67 - 1.32$ ------------------------------------- ------------------- : Stellar Properties of the 55 Cancri System. $\theta_{\rm UD}$ and $\theta_{\rm LD}$ correspond to the uniform disk and limb-darkening corrected angular stellar diameters, respectively. $\theta_{\rm LD}$ corresponds to the angular diameter of the Rosseland, or mean, radiating surface of the star.
--- abstract: 'Phase separation in salt-free symmetric mixtures of oppositely charged rodlike polyelectrolytes is studied using quasi-analytical calculations. Stability analyses for the isotropic-isotropic and the isotropic-nematic phase transitions in the mixtures are carried out and demonstrate that electrostatic interactions favor nematic ordering. Coexistence curves for the symmetric mixtures are also constructed and are used to examine the effects of linear charge density and electrostatic interaction strength on rodlike polyelectrolyte complexation. It is found that the counterions are uniformly distributed in the coexisting phases for low electrostatic interaction strengths dictated by the linear charge density of the polyelectrolytes and Bjerrum’s length. However, the counterions also partition along with the rodlike polyelectrolytes with an increase in the electrostatic interaction strength. It is shown that the number density of the counterions is higher in the concentrated (or “coacervate”) phase than in the dilute (or supernatant) phase. In contrast to such rodlike mixtures, flexible polyelectrolyte mixtures can undergo only isotropic-isotropic phase separation. A comparison of the coexistence curves for weakly-charged rodlike mixtures with those of analogous flexible polyelectrolyte mixtures reveals that the electrostatic driving force for the isotropic-isotropic phase separation is stronger in the flexible mixtures.' author: - '[Rajeev Kumar, Debra Audus and Glenn H. Fredrickson ]{}' title: Phase Separation in Symmetric Mixtures of Oppositely Charged Rodlike Polyelectrolytes --- Introduction ============ For numerous biological processes[@dubin_review; @pe_complex_reviews; @vlad_muthu; @shklovskii_virus] and emerging technologies[@waite; @bazan_biosensor], complexation between oppositely charged polyelectrolytes[@voorn_review; @veis_complexation; @kabanov_work1; @kabanov_work2; @dautzenberg_work1; @pogodina_work; @dautzenberg_work2] is the underlying fundamental phenomenon. However, our understanding of how the electrostatic attraction between opposite charges on the polyelectrolytes is coupled with the conformational characteristics of polyelectrolyte chains is very limited. Motivated by a plethora of relevant biological processes and the impetus for developing advanced technologies such as underwater adhesives[@waite] and biosensors[@bazan_biosensor], extensive experimental[@dubin_review; @pe_complex_reviews; @waite; @bazan_biosensor] and theoretical[@zhaoyang_complex; @voorn_review; @veis_complexation; @voorn_overbeek; @voorn_1; @voorn_2; @voorn_3; @voorn_4; @voorn_5; @borue_complexation; @monica_complexation; @monica_complexation_2; @joanny_01; @jay_yuri_1; @jay_yuri_2] work has been carried out to understand the mechanisms of polyelectrolyte charge complexation. A few insights into the origin of polyelectrolyte complexation have been provided by simulations. Langevin dynamics simulations[@zhaoyang_complex] of two oppositely charged polyelectrolyte chains have revealed that the entropy of counterion release is the predominant driving force for complexation in highly charged polyelectrolytes. In contrast, the simulations show that direct electrostatic attractions drive the association of two weakly charged polyelectrolytes whose counterions are not condensed onto the chains. Such simulations involving only a single pair of polymers are of course relevant only to extremely dilute solutions of polyelectrolytes and are only suggestive of complexation phenomena at finite concentrations. At such higher concentrations, complexation between oppositely charged polyelectrolytes has the character of a phase separation process in which a supernatant phase that is extremely dilute in polymer macroscopically separates from a polyelectrolyte rich phase that is either a fluid (“complex coacervate”) or a solid (“precipitate”). In this paper we shall be exclusively concerned with systems that produce fluid-like polyelectrolyte complexes properly classified as complex coacervates.[@dubin_review; @pe_complex_reviews; @voorn_review; @veis_complexation; @voorn_overbeek; @voorn_1; @voorn_2; @voorn_3; @voorn_4; @voorn_5]Extending conventional Langevin, Monte-Carlo, or molecular dynamics particle-based simulations to polyelectrolyte mixtures at finite concentrations has been hindered by the need for extremely large computational resources to address the long-range character of the electrostatic interactions and the inherently slow kinetics of the dense coacervate phase and phase separation process. Development of a theory to study the phase separation in a mixture of oppositely charged polyelectrolytes is a daunting problem due to the non-trivial coupling between long range effects such as electrostatics and chain connectivity, and short range excluded volume interactions. The theoretical treatment is further complicated by the inevitable need to include electrostatic screening and correlation effects in the theory, analogous to the classical Debye-Hückel theory of electrolyte solutions, to capture the electrostatic forces that drive phase separation. This is the reason behind the inability of the self-consistent field theory (SCFT), a type of mean-field theory, to capture even the signature of a phase separation[@borue_complexation; @monica_complexation; @monica_complexation_2; @joanny_01; @jay_yuri_1; @jay_yuri_2]. In other words, in a field-theoretic description of a mixed polyelectrolyte system, one needs to go beyond the mean-field or “saddle-point” approximation in order to study phase separation phenomena. Typically, the random phase approximation[@borue_complexation; @monica_complexation; @monica_complexation_2] is used to compute the leading correction to the saddle-point results. In the case of flexible polyelectrolytes[@muthu_double], it can be shown rigorously that the random phase approximation is valid in the dual limit of high monomer densities and low small ion densities. Physically, such a situation is realized in concentrated solutions of weakly charged polyelectrolytes. More sophisticated field-theoretic results beyond the random phase approximation have also been obtained (including a phase diagram) for a particular model of flexible polyelectrolyte mixtures using the self-consistent one-loop (Hartree) approximation[@joanny_01]. Most recently, direct numerical simulations of a related field theory model (so-called “field-theoretic simulations”) have been carried out[@jay_yuri_1; @jay_yuri_2] to capture the effects of fluctuations and correlations to all orders, without invoking any approximation. An important result that came out of these simulations of flexible polyelectrolyte mixtures is that the phase boundaries in the concentrated regime are accurately predicted by the random phase approximation. In this work, we consider solutions containing binary mixtures of oppositely charged rodlike polyelectrolytes and study their phase behavior using the random phase approximation, which, as noted above, has been previously shown to provide an accurate description of complexation phenomena in flexible polyelectrolyte mixtures. An important focus of our study is a comparison of the phase behavior of rodlike and flexible polyelectrolyte mixtures. The rodlike system is particularly relevant to a recently developed biosensor technology[@bazan_biosensor] involving complexation of cationic conjugated polyelectrolytes with anionic DNA. A fundamental question that we have tried to answer is: what is the role played by the flexibility of the polyelectrolytes in the phase separation processes that lead to a complex coacervate? Unlike flexible polyelectrolyte mixtures[@borue_complexation; @monica_complexation; @monica_complexation_2; @joanny_01; @jay_yuri_1; @jay_yuri_2], local enhancements in concentration of rodlike systems can produce orientationally ordered liquid crystalline phases (such as nematic, smectic and cholesteric phases).[@degennes_prost] Thus, a coacervate produced by complexing oppositely charged rodlike polymers may be a liquid crystal rather than an isotropic fluid. In this paper, where we consider “symmetric” achiral rodlike polymers with fore-aft symmetry, equal length, and equal but opposite charge, we have explored the competition between isotropic and nematic ordering in rodlike polyelectrolyte mixtures and elucidated how the phase boundaries depend on different parameters of the system. Before presenting our theory and numerical results on the binary mixtures, it is worthwhile to briefly review the theoretical literature aimed at understanding the phase separation of single component flexible[@degennes_pincus; @joanny_leibler; @khokhlov_nyrkova; @muthu_weak_polymer; @chilun_muthu] and rodlike[@onsager_paper; @odijk_pe_rods; @chen_koch; @carri_muthu99; @potemkin_1; @potemkin_2; @potemkin_3; @stell_rods] polyelectrolyte solutions. These studies have been primarily focused on poor solvent conditions. In the case of solutions containing a flexible polyelectrolyte, it has been shown that the dependence of the phase boundaries on the polyelectrolyte chain length is weak (referred to as the “weak polymer effect”[@muthu_weak_polymer]). Note that the condition of Donnan equilibrium was invoked in these calculations so that locally the system is electroneutral. This implies that a local increase in the charge on the polyelectrolyte leads to an increase in the local concentration of counterions. The localisation of the counterions is entropically unfavorable and hence, the region of two-phase coexistence shrinks with an increase in the charge on the polyelectrolyte chains[@joanny_leibler; @khokhlov_nyrkova; @muthu_weak_polymer]. Phase-separating mixtures of two oppositely charged polyelectrolytes are fundamentally different from these solutions containing a single kind of polyelectrolyte in two ways. Firstly, oppositely charged polyelectrolyte mixtures can phase separate even under good solvent conditions for the two individual polymer components. Secondly, it is clear in the binary mixture case that counterions are free to partition very differently upon phase separation. In particular, Donnan equilibrium can be satisfied by bringing the oppositely charged macromolecules together in the concentrated coacervate phase,[@voorn_1; @voorn_2; @voorn_3; @voorn_4; @voorn_5] while freeing the small ions to gain entropy by populating both dilute and concentrated phase in charge compensating proportions. Clearly, it is necessary to account for these extra degrees of freedom available to the counterions in studying phase separation in mixtures of oppositely charged polyelectrolytes. In the case of single component rodlike polyelectrolyte solutions, it has been shown recently[@potemkin_1; @potemkin_2; @potemkin_3] that electrostatic interactions among like-charged rods favors orientational ordering at concentrations lower than the overlap concentration. This non-trivial result is an outcome of long-range, multi-rod correlations present in these systems. A consequence is that, in addition to the lyotropic isotropic-nematic transition observed in neutral rodlike solutions, thermotropic isotropic-nematic as well as nematic-nematic phase transitions have been predicted for charged rodlike polyelectrolytes. Note that these predictions are a consequence of collective phenomena at higher concentrations that are not manifest in earlier theories of dilute rodlike polyelectrolytes, where the twisting effect[@onsager_paper; @parsegian] of pairwise electrostatic interactions between rods was shown to destroy nematic ordering[@onsager_paper; @odijk_pe_rods; @chen_koch; @carri_muthu99]. With this background on the phase separation in polyelectrolyte solutions, we present our theoretical model for the binary mixtures. This paper is organized as follows: the theory is presented in Section \[sec:theory\], the results are presented in Section \[sec:results\], and Section \[sec:conclusions\] contains our conclusions. Theory {#sec:theory} ======= We consider a symmetric binary mixture of rodlike polyelectrolytes bearing opposite charges on their backbone in the presence of the counterions originating from the polyelectrolyte chains. By symmetric mixtures, we mean mixtures containing equal number of polyelectrolyte chains, which are identical to each other in every aspect other than the sign of the charge they are bearing. For generality in the following, we maintain notation representing different polymeric species such as $n_+$ and $n_-$ representing the number of polycationic and polyanionic chains, resepectively. For the theoretical treatment, we consider a binary mixture of rods with the same diameters ($= d$) for each species but lengths $L_k$ for $k=+,-$. Furthermore, $\sigma_k$ is taken to be the linear charge density for polyelectrolytes of type $k=+,-$ so that there are $n_{ck} = \sigma_k L_k$ counterions released by each polyelectrolyte of type $k$. As the charge on the polymers is spread uniformly along their length, this description is often referred to as the “smeared charge” model. We specialize to the symmetric case only as needed after plugging $n_+ = n_- = n, L_+ = L_- = L$ and $\sigma_+ = \sigma_- = \sigma$ at the appropriate place. Also, we work in the canonical ensemble with the volume of the system denoted by $\Omega$. The solvent is treated implicitly and as a uniform dielectric background for the purpose of computing electrostatic interactions. The small ions are taken to be point like and we ignore short range structure, correlations, and polarization effects in the solvent. Qualitative Picture and Scaling Analysis {#sec:scaling_section} ---------------------------------------- Before presenting quantitative details of our theory for the symmetric rodlike polyelectrolyte mixtures, a qualitative picture can be drawn by using three important results available from the literature. First, the macrophase separation in the isotropic phase (by analogy with symmetric mixtures of flexible polylectrolytes) takes place at very low monomer densities[@borue_complexation; @monica_complexation; @monica_complexation_2; @joanny_01; @jay_yuri_1; @jay_yuri_2], when the electrostatic attractions between oppositely charged polyelectrolytes are strong enough to drive the phase separation. Second, the isotropic-nematic phase transition observed in neutral rodlike molecules and treated by Onsager and many others[@edwardsbook; @degennes_prost; @onsager_paper; @flory_semiflex; @flory_rods; @khokhlov_semiflex; @khokhlov_semenov] takes place at relatively high monomer densities above the overlap concentration and is driven primarily by entropic effects associated with the excluded volume of rodlike objects. Thirdly, we recall that addition of charges to rodlike polymers can induce thermotropic (temperature dependent) isotropic-nematic phase transitions, as opposed to the lyotropic (concentration dependent) transitions familiar in neutral rodlike polymer systems. The origin of this thermotropic behavior lies in the long-ranged electrostatic interactions, which depend on the charge on the polyelectrolytes and the temperature dependent Bjerrum’s length $l_B = e^2/(4\pi \epsilon_0\epsilon_r k_B T)$, which is the distance at which the Coulomb interaction energy between two elementary charges is of the order of thermal energy ($=k_BT, \; k_B$ being the Boltzmann’s constant and $T$ is the temperature). Here, $\epsilon_0$ and $\epsilon_r$ are the permittivities of vacuum and the medium, respectively. By integrating these three competing effects, a qualitative phase diagram of a symmetric rodlike polyelectrolyte mixture is sketched in Fig. \[fig:phase\_cartoon\] in the coordinates of $l_B /l$ versus $2 n L^2 d/\Omega$, where $l$ is a reference monomer length, $L$ is the overall rod length, $d$ is the rod diameter, $2 n$ is the total number of cationic and anionic rods, and $\Omega$ is the system volume. Broadly, we see two trends in the figure. At high temperatures, i.e. low values of $l_B /l$, non-interacting isotropic-isotropic and isotropic-nematic coexistence regions appear at low and high rod concentrations, respectively. As the temperature is lowered ($l_B /l$ raised), these features collide and ultimately at low temperature there is a broad region of two phase coexistence between a dilute isotropic supernatant phase and a semidilute or concentrated nematic “coacervate” phase. At intermediate values of $l_B /l$, more complex phase behavior is evident, including the possibility of three-phase coexistence between two isotropic phases and a nematic phase. With regard to the isotropic-isotropic coexistence region in Fig. \[fig:phase\_cartoon\], which is also present in flexible polyelectrolyte mixtures, a scaling analysis is helpful to understand the variables that control its extent and shape and differentiate the cases of rigid and flexible systems. For this purpose, we begin by considering a symmetric mixture containing an equal number $n$ of oppositely charged flexible polyelectrolytes in the absence of any counterions. Physically, such a situation is realized in a solution containing polyacids and polybases. Apart from electrostatic interactions, excluded volume effects can be simply accounted for (in an implicit solvent model) by an excluded volume parameter $w$. This particular system was recently studied using the field-theoretic simulation technique[@jay_yuri_1; @jay_yuri_2]. The osmotic pressure of such a system in the dilute regime[@degennes_pincus; @rubinstein_review], i.e. $\rho = 2nN/\Omega \ll \rho_f^\star \sim 1/(4\pi l_B \sigma^2 N^2)$, with $\rho, \;\rho_f^\star$ being the monomer density and the overlap concentration, respectively, is given by $$\begin{aligned} \frac{P_{f}^0}{k_BT} &=& \frac{\rho}{N} + \frac{w}{2} \rho^2 - \frac{\kappa_d^3}{24 \pi}, \label{eq:pressure_flex_dil}\end{aligned}$$ where $\kappa_d^2 = 4\pi l_B \sigma^2 N \rho$ is the Debye screening length in the asymptotic dilute regime where the polyelectrolytes behave as multivalent macroions. We note that although the polymers are assumed to be flexible, the expression that we use for $\rho_f^\star$ assumes that they adopt extended rodlike conformations at infinite dilution. In the above expression, $\sigma$ is the charge per monomer (in units of the fundamental charge $e$), which is the same for each kind of polyelectrolyte in the assumed symmetric mixture and $N$ is the total number of monomers. In Eq. \[eq:pressure\_flex\_dil\], the first and second terms correspond to the translational entropy of the polyelectrolyte chains and the effect of excluded volume interactions on the pressure, respectively. The third term containing $\kappa_d$, a Debye-Hückel contribution, originates from attractive electrostatic correlations among dilute, non-overlapping polyelectrolyte chains. In contrast, at higher polymer concentrations where chains overlap the internal structure of the polyelectrolytes plays a significant role. The dominant electrostatic contribution to the osmotic pressure in the high density, overlapping regime ($\rho \gg \rho_f^\star $) can be represented by the scaling expression $$\begin{aligned} \frac{P_{f}^{e}}{k_BT} &=& - \frac{\kappa_d^3}{24 \pi} \: \mathcal{F} \left(\rho / \rho_f^\star \right),\end{aligned}$$ where $\mathcal{F} (x)$ is a dimensionless scaling function that we assume to be a power law $\sim x^m$ for large argument. The exponent $m$ is determined by the requirement that in the concentrated regime, the local properties are not different for a solution having multiple chains containing $N$ monomers each or a single chain that fills space with the same overall segment density $\rho$ ($N\rightarrow \infty$). In other words, the electrostatic contribution to the osmotic pressure must be independent of $N$ at fixed $\rho$, i.e. $P_f^e \sim N^0$. This requirement, along with the fact that the overlap concentration scales as $\rho_f^\star \sim 1/(4\pi l_B \sigma^2 N^2)$, leads to the well-known result[@muthu_double; @kaji_kanaya] $$\begin{aligned} \frac{P_{f}^{e}}{k_B T} &\sim& - \kappa_p^{3/2}, \label{eq:pressure_flex}\end{aligned}$$ where $\kappa_p^2 = 4\pi l_B \sigma^2 \rho$. In the concentrated regime, the osmotic pressure of flexible, symmetric polyelectrolyte mixtures is thus dictated by a balance between excluded volume and this modified electrostatic correlation energy, i.e. $$\frac{P_f^c}{k_B T} \sim \frac{w}{2} \rho^2 - \kappa_p^{3/2}$$ We can now retrace these scaling arguments for the case of symmetric mixtures of rodlike polylectrolytes. The osmotic pressure in the dilute concentration regime ($\rho \ll \rho_r^\star \sim 2nL^2 d/\Omega$) is unchanged except that we replace the excluded volume coefficient $w$ with the Onsager expression $\pi d l^2 /2$, where $l \equiv L/N$ is a monomer segment length, $$\begin{aligned} \frac{P_{r}^0}{k_BT} &=& \frac{\rho}{N} + \frac{\pi d l^2}{4} \rho^2 - \frac{\kappa_d^3}{24 \pi}. \label{eq:pressure_rods_qual}\end{aligned}$$ In the concentrated regime, the electrostatic free energy of a system of rods can be approximated by the sum of the self-energies of individual rods experiencing the electrostatic potential induced by the other rods. For the purpose of estimation, these are placed on a periodic lattice. It is well-known that the self-energy of a charged cylindrical rod of length $L$ and linear charge density $\sigma$ is divergent[@odijk_self; @safran_pincus], and is given by $$\begin{aligned} \frac{U_{cyl}}{L k_BT} &\sim& l_B \sigma^2 \ln \frac{R}{\delta}, \label{eq:cyl_energy}\end{aligned}$$ where $\delta$ is a cut-off introduced to regularize the self-energy and $R$ is the radius of a unit cell (Wigner-Seitz) applied to each cylinder. Using the fact that $R \sim 1/\sqrt{\rho}$ in the concentrated regime and writing the electrostatic free energy of the solution containing $n$ rods as $F^e = n U_{cyl}$, the electrostatic contribution to the pressure is given by $$\begin{aligned} \frac{P_{r}^e}{k_BT} &\sim& - \kappa_p^2. \label{eq:pressure_rods_qual2}\end{aligned}$$ Comparing Eqs. \[eq:pressure\_flex\] and \[eq:pressure\_rods\_qual2\], it is clear that the qualitative difference between flexible and rodlike polyelectrolyte mixtures is in the functional form for the attractive electrostatic contribution to the osmotic pressure. In the case of rodlike polyelectrolytes, this term scales like $\kappa_p^2 \sim \rho$ in contrast to $\kappa_p^{3/2} \sim \rho^{3/4}$ for flexible coils. Thus, at the same set of electrostatic parameters and monomer densities, and for weakly charged polyelectrolytes where $\kappa_p \ll 1$, electrostatic correlation effects are weaker for rodlike polyelectrolytes in comparison with flexible polyelectrolyte mixtures. Since the electrostatic contribution to the pressure is responsible for polyelectrolyte complexation, it follows that rodlike polyelectrolyte mixtures are less prone to complex coacervation than analogous flexible polyelectrolyte mixtures. The regions of two-phase coexistence sketched in Fig. \[fig:phase\_cartoon\] should thus be narrower in rigid rod systems. This general observation will be born out by detailed numerical calculations in Section \[sec:results\] of the spinodals and binodals of each type of system based on the free energy expression presented below. Free Energy ----------- An expression for the Helmholtz free energy of the assumed mixture of rodlike polyelectrolytes, counterions, and implicit solvent is derived in Appendix A using Onsager’s treatment for the neutral interactions and the random-phase approximation for the electrostatic interactions. Explicitly, the free energy is given by $$\begin{aligned} F &=& F_{en} + F_w + F_e \label{eq:free_gen}\end{aligned}$$ where $$\begin{aligned} \frac{F_{en}}{k_BT} & = & \sum_{j=+,-}n_j \int d\mathbf{u}_{j} \: p_j(\mathbf{u}_{j},\nh) \ln \left[4\pi p_j(\mathbf{u}_{j},\nh)\right] + \sum_{\gamma=+,-,c+,c-} n_{\gamma} \left[\ln \frac{n_{\gamma}}{\Omega} - 1\right],\\ \frac{F_w}{k_BT} &=& \frac{1}{2\Omega} \sum_{j,k=+,-}^\prime n_j n_k \int d\mathbf{u} \int d\mathbf{u}' \: p_j(\mathbf{u},\nh) \left [2L_j L_k d \|\mathbf{u}\times\mathbf{u}'\| \right]p_k(\mathbf{u}',\nh), \\ \frac{F_e}{k_BT} &=& \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{\kappa^2 + \sum_{k=+,-}\kappa_k^2 t_k(q L_k)}{q^2}\right] - \frac{\kappa^2 + \sum_{k=+,-}\kappa_k^2 t_k(qL_k)}{q^2}\right\}\label{eq:free_rods}\end{aligned}$$ and $$\begin{aligned} t_k(qL_k) &=& \int d\mathbf{u} \: p_{k}(\mathbf{u},\nh)\left[\frac{\sin\left[\left(\mathbf{q}\cdot\mathbf{u}\right)L_k/2\right]}{\left[\left(\mathbf{q}\cdot\mathbf{u}\right)L_k/2\right]}\right]^2, \quad \mbox{for} \quad k = +,-.\end{aligned}$$ Here, $\kappa$ is the inverse of the Debye screening length for small ions given by $\kappa^2 = 4\pi l_B \sum_{\gamma=c+,c-} Z_{\gamma}^2 n_{\gamma}/\Omega$ so that $Z_\gamma$ is the valency (with sign) of the charged species of type $\gamma$. We define a similar object for the two polymer species: $\kappa_k^2 = 4\pi l_B Z_{k}^2 \sigma_{k}^2 L_k^2 n_k/\Omega, Z_k$ being the valency of the charged monomers of type $k$. The functions $p_{k}(\mathbf{u},\nh)$ represent the probability distribution function for finding a rod of type $k$ oriented along the unit vector $\mathbf{u}$ when the director is chosen to be the unit vector $\nh$. Each distribution function satisfies the normalization condition $\int d \mathbf{u} \: p_{k}(\mathbf{u},\nh) = 1 $. Finally, the primed superscript in the expression for $F_w$ indicates that the $j=k$ terms are omitted from the double sum. So far, the theory is quite general and the subsequent analysis depends on the functional form for the orientational probability distribution function for the rods. In this work, we focus on possibilities for phase separation that involve the isotropic and nematic phases only. For these two phases, $p_{k}(\mathbf{u},\nh) \equiv p_{k}(\mathbf{u} \cdot \nh) $ and the free energy for the each phase can be evaluated as described below. Isotropic Phase --------------- We begin by considering the isotropic phase, in which case the orientation distribution function for the rods is independent of the angle between the director $\nh$ and the unit vector along the axes of the rods ($\mathbf{u}$). In this case, $p_{k}(\mathbf{u}\cdot \nh) = 1/4\pi$ (obtained from the normalization condition) for $k=+,-$ so that the free energy in Eq. \[eq:free\_gen\] can be written as $$\begin{aligned} F^{iso} &=& F_{en}^{iso} + F_{w}^{iso} + F_{e}^{iso} \label{eq:free_iso}\end{aligned}$$ where $$\begin{aligned} \frac{F_{en}^{iso}}{k_BT} & =& \sum_{\gamma = +,-,c+,c-} n_{\gamma} \left[\ln \frac{n_{\gamma}}{\Omega} - 1\right],\\ \frac{F_w^{iso}}{k_BT} &=& \frac{\pi d}{4\Omega}(n_+ L_+ + n_- L_-)^2, \\ \frac{F_e^{iso} }{k_BT} &=& \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{\kappa^2 + \sum_{k=+,-} \kappa_k^2 t^{iso}(qL_k)}{q^2}\right] - \frac{\kappa^2 + \sum_{k=+,-} \kappa_k^2 t^{iso}(qL_k)}{q^2}\right\} \label{eq:feiso}\end{aligned}$$ and $$\begin{aligned} t^{iso}(qL_k) &=& \frac{2Si(qL_k)}{q L_k} - \left(\frac{2\sin \left(qL_k/2\right)}{qL_k}\right)^2, \label{eq:tkq_iso}\end{aligned}$$ where $Si(x) = \int_{0}^{x}dt \frac{\sin t}{t}$ is the sine integral. In the following, we rigorously show that the scaling argument presented in section \[sec:scaling\_section\] predicting a logarithmic correction to the free energy of isotropic phase in the mixtures of rods is indeed correct. For the analysis, we use the following asymptotic expressions for $t^{iso} (q L_k )$ : $$t^{iso}(qL_k) = \left \{ \begin{array}{ll} 1, & qL_k \rightarrow 0 \\ \pi/qL_k, & qL_k \rightarrow \infty. \end{array} \right . \label{eq:tkq_lim}$$ Also, we consider two limiting cases of short and long rods. For these cases, the electrostatic contribution to the free energy depends on how these limits are being approached. These limits can be approached by either fixing the linear charge density or the net charge per rod during the variation of length $L_k$. In the former case, $\kappa_k$ increases (decreases) with an increase (decrease) in $L_k$ and diverges (vanishes) strictly in the limit of $L_k\rightarrow \infty$ ($L_k \rightarrow 0$). On the other hand, for the latter approach, $\kappa_k$ is held fixed, while approaching the limits of $L_k \rightarrow \infty$ or $L_k \rightarrow 0$. Consider the latter case so that $\kappa_k$ is well-defined in approaching either limits of $L_k$. Defining a Debye-like parameter proportional to the total ionic strength, $\kappa_{eff}^2 = \kappa^2 + \sum_{k=+,-} \kappa_k^2 $ and approaching the limit of short rods, $L_k \rightarrow 0$ while keeping the charge per rod fixed, the free energy becomes $$\begin{aligned} \frac{F^{iso}\left\{L_k \rightarrow 0\right\}}{k_BT} &=& \sum_{\gamma = +,-,c+,c-} n_\gamma \left[\ln \frac{n_\gamma}{\Omega} - 1\right] + \frac{\pi d}{4\Omega}(n_+ L_+ + n_- L_-)^2 -\frac{\Omega \kappa_{eff}^3}{12 \pi} . \end{aligned}$$ Similarly, consider the limit of long rods, i.e. $L_k$ being large and approaching $\infty$. Approaching this limit while keeping the charge per rod fixed, the free energy becomes $$\begin{aligned} \frac{F^{iso}\left\{L_k \rightarrow \infty \right\}}{k_BT} &=& \sum_{\gamma = +,-,c+,c-} n_\gamma \left[\ln \frac{n_\gamma}{\Omega} - 1\right] + \frac{\pi d}{4\Omega}(n_+ L_+ + n_- L_-)^2 + \frac{F_e^{iso}\left\{L_k \rightarrow \infty \right\}}{k_BT},\nonumber \\ && \end{aligned}$$ where the electrostatic contribution to the free energy is given by $$\begin{aligned} \frac{F_e^{iso}\left\{L_k \rightarrow \infty \right\}}{k_BT} &=& \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{\kappa^2 }{q^2} + \frac{\pi \kappa_p^2}{q^3}\right] - \frac{\kappa^2 }{q^2} - \frac{\pi \kappa_p^2}{q^3}\right\}. \label{eq:fe_limit} \end{aligned}$$ Here, we have introduced $\kappa_p^2 = \left(\frac{\kappa_+^2}{L_+} + \frac{\kappa_-^2}{L_-}\right)$. Furthermore, note that in writing Eq. \[eq:fe\_limit\], we have used the aysmptotic form for $t^{iso}$ in the limit of $qL \rightarrow \infty$ for the entire range of $q$ (e.g., even in the case of $q = 0$). This causes the integral in Eq. \[eq:fe\_limit\] to diverge, while the original integral in Eq. \[eq:feiso\] is convergent. These divergences are mere artifacts of the approximation scheme. Despite these divergences, we show that the leading contribution to the free energy of the rods is of the form $- n \sigma^2 l_B L\ln n/\Omega$, as described using the scaling arguments. For weakly charged polyelectrolytes so that $\kappa_p^2 \ll 1$, Eq. \[eq:fe\_limit\] can be written as (see Appendix B) $$\begin{aligned} \frac{F_e^{iso}\left\{L_k \rightarrow \infty \right\}}{k_BT} &\simeq& -\frac{\Omega \kappa^3}{12 \pi} - \frac{\Omega \kappa_p^2}{4 \pi}\ln \left[\frac{\kappa}{q}\right]_{q\rightarrow 0} - \frac{\Omega \pi \kappa_p^4}{32 \kappa^3} + \cdots, \label{eq:fe_limit1} \end{aligned}$$ where the leading term in the free energy expression has the functional form similar to the one described in Eq. \[eq:cyl\_energy\]. On the other hand, for strongly charged rods in weakly screened solutions $\kappa_p^2 \gg 1$ and $\kappa \rightarrow 0$. In this limit, $F_e^{iso}$ becomes (see Appendix B) $$\begin{aligned} \frac{F_e^{iso}\left\{L_k \rightarrow \infty \right\}}{k_BT} &\simeq& -\frac{\Omega \kappa^3}{12 \pi} - \frac{\Omega \kappa_p^2}{6 \pi} \ln \left[\frac{\sqrt{\pi}\kappa_p}{q}\right]_{q\rightarrow 0} + \frac{\Omega \kappa^2 \kappa_p^{2/3}}{6\sqrt{3}\pi^{2/3}} \nonumber \\ && - \frac{\Omega \kappa^4}{6\sqrt{3}\pi^{4/3}\kappa_p^{2/3}}- \frac{\Omega \kappa^2}{4 \pi^2} \left[q\right]_{q\rightarrow \infty} + \cdots. \label{eq:fe_limit2} \end{aligned}$$ Like the weakly charged rods, the free energy of the strongly charged system has the same functional form as in Eq. \[eq:cyl\_energy\]. Using these approximate expressions for the free energy, the osmotic pressure can be readily computed using the thermodynamic relation $ P = - \left(\partial F^{iso}/\partial \Omega\right)_{n,T}$. For the limiting case of $L_k \rightarrow 0$, this gives $$\begin{aligned} \frac{P\left\{L_k \rightarrow 0 \right\}}{k_BT} &=& \sum_{k = +,-} \frac{\rho_k}{N_k} + \sum_{\gamma = c+,c-} \rho_\gamma -\frac{\kappa_{eff}^3}{24 \pi}, \label{eq:pressure_l0} \end{aligned}$$ which is the well-known Debye-Hückel limiting law for electrolyte solutions. Here, $\rho_k = n_k N_k/\Omega$ is the monomer number density of type $k = +,-$ and $\rho_\gamma = n_\gamma/\Omega$ is the number density of counterions of type $\gamma = c+,c-$. For the other limiting cases presented above, i.e., for $\kappa_p^2 \ll 1$ $$\begin{aligned} \frac{P\left\{L_k \rightarrow \infty \right\}}{k_BT} &=& \sum_{k = +,-} \frac{\rho_k}{N_k} + \sum_{\gamma = c+,c-} \rho_\gamma -\frac{\kappa^3}{24 \pi} + \frac{\pi d}{4} (\rho_+ l_+ + \rho_- l_-)^2 - \frac{\kappa_p^2}{8 \pi} + \frac{\pi \kappa_p^4}{64 \kappa^3}, \end{aligned}$$ where $l_k \equiv L_k/N_k$ is the monomer length. Similarly, for $\kappa_p^2 \gg 1$ and $\kappa \rightarrow 0$ $$\begin{aligned} \frac{P\left\{L_k \rightarrow \infty \right\}}{k_BT} &=& \sum_{k = +,-} \frac{\rho_k}{N_k} + \sum_{\gamma = c+,c-} \rho_\gamma -\frac{\kappa^3}{24 \pi} + \frac{\pi d}{4}(\rho_+ l_+ + \rho_- l_-)^2 - \frac{\kappa_p^2}{12 \pi} \nonumber \\ && + \frac{\kappa^2 \kappa_p^{2/3}}{18 \sqrt{3}\pi^{2/3}} - \frac{\kappa^4}{9\sqrt{3}\pi^{4/3}\kappa_p^{2/3}}. \end{aligned}$$ Note that the leading contribution to the pressure coming from the electrostatic correlations of the charged rods is negative and of the form $-\kappa_p^2$ as already anticipated from the simple scaling arguments in section \[sec:scaling\_section\]. This particular contribution can drive macrophase separation in isotropic solutions of rodlike polyelectrolytes, a theme that is considered later in the paper. Next, we turn to consider the weakly ordered nematic phase using the approximate free energy given in Eq. \[eq:free\_gen\]. Weakly Ordered Nematic Phase ---------------------------- For a nematic phase, the distribution function can be reduced to a function of the cosine of the angle between the director and rod orientation, i.e. $p_{k}(\mathbf{u},\nh) = p_{k}(\mathbf{u}\cdot \nh)$ for $k=+,-$. In order to study a fully formed nematic phase, the complete functional form for the probability distribution function is needed. However, a stability analysis for a weakly ordered nematic phase can be carried out without knowing the probability distribution function a priori by assuming a two-term Legendre expansion for $p_{k}(\mathbf{u}\cdot \nh)$ written as $$\begin{aligned} p_{k}(\mathbf{u}\cdot \nh) &\simeq& \frac{1}{4\pi}\left[1 + 5 S_k \frac{3(\mathbf{u}\cdot \nh)^2 - 1}{2}\right]. \label{eq:legend_expansion} \end{aligned}$$ Here, $S_k$ is the nematic order parameter, given by $$\begin{aligned} S_k &=& \int d\mathbf{u} \: p_{k}(\mathbf{u}\cdot \nh) \left[ \frac{3(\mathbf{u}\cdot \nh)^2 - 1}{2} \right] . \end{aligned}$$ Here, we have used the normalization condition $\int d\mathbf{u} \: p_{k}(\mathbf{u}\cdot \nh) = 1$ to write the Legendre expansion. Note that $S_k = 0$ for the isotropic phase and $1$ for the completely ordered nematic phase, for which $p_{k}(\mathbf{u}\cdot \nh) = \delta(\mathbf{u}-\nh)$. A similar analysis for the case of single component rodlike polyelectrolyte solutions has been carried out in Refs. [@potemkin_1; @potemkin_2; @potemkin_3]. Using Eq. \[eq:legend\_expansion\], $t_k(qL_k)$ for a weakly ordered nematic phase can be written as $t_k(qL_k) = t^{iso}(qL_k) + \delta t_k(qL_k)$, where $\delta t_k(qL_k)$ is given by $$\begin{aligned} \delta t_k(qL_k) &=& \frac{5 S_k}{4\pi}\int d\mathbf{u} \left[\frac{3(\mathbf{u}\cdot \nh)^2 - 1}{2}\right] \left[\frac{\sin(\mathbf{q}\cdot \mathbf{u} L_k/2)}{(\mathbf{q}\cdot \mathbf{u} L_k/2)}\right]^2 \\ &=& \frac{15 S_k}{2} \left[3\frac{(\mathbf{q}\cdot \nh)^2}{q^2} - 1\right] \left[ \frac{1}{q^2L_k^2}\left(1 - \frac{\sin qL_k}{qL_k}\right) - \frac{t^{iso}(qL_k)}{6}\right] \end{aligned}$$ Using these expressions for $p_k$ and $t_k(qL_k)$, the free energy of a weakly ordered nematic phase (expanded to second order in $S_k$) can be written as $$\begin{aligned} F & = & F^{iso} + \Delta F,\end{aligned}$$ where $$\begin{aligned} \Delta F & = & \Delta F_{en} + \Delta F_{w} + \Delta F_{e}\end{aligned}$$ so that $$\begin{aligned} \Delta F_{en} &=& \sum_{k=+,-}\frac{5}{2}n_k S_k^2, \\ \Delta F_w &=& -\frac{5\pi d}{32\Omega}\left(n_+ L_+ S_+ + n_- L_- S_-\right)^2, \\ \Delta F_e &=& - \frac{90\Omega}{\left(4 \pi\right)^2}\int_{0}^{\infty} dq q^2\frac{\left[\sum_{k=+,-} \kappa_k^2 S_k h(qL_k)\right]^2}{\left[q^2 + \kappa^2 + \sum_{k=+,-}\kappa_k^2 t^{iso}(qL_k)\right]^2}.\end{aligned}$$ Here, we have used the approximation $\sin \theta \simeq \left[1 - 5P_2(\cos\theta)/8\right]\pi/4$, $P_2$ being the Legendre polynomial of order $2$, in writing $\Delta F_w$, and $$\begin{aligned} h(qL_k) &=& \frac{1}{q^2 L_k^2}\left( 1 - \frac{\sin qL_k}{qL_k}\right) - \frac{t^{iso}(qL_k)}{6}. \end{aligned}$$ Note that the contribution to the free energy coming from electrostatic correlations (i.e., $\Delta F_e$) is negative. In other words, electrostatic correlations favor the nematic phase. This observation will prove important to the the stability of the weakly ordered nematic phase with respect to the isotropic phase. Some useful results can be inferred from the above expression for the free energy of a weakly ordered nematic phase. The spinodal of the isotropic-nematic transition (i.e. stability limit of the isotropic phase in the nematic region) is given by the condition $$\begin{aligned} \det \left[\begin{array}{cc} \frac{\partial^2 F}{\partial S_+ \partial S_+} & \frac{\partial^2 F}{\partial S_+ \partial S_-} \\ \frac{\partial^2 F}{\partial S_+ \partial S_-} & \frac{\partial^2 F}{\partial S_- \partial S_-} \end{array} \right ] &=& 0 \label{eq:spino_iso_nem}\end{aligned}$$ For the case of symmetric mixtures, i.e. equal length ($=L$), diameter ($=d$), and number density of the rods ($=n/\Omega$), Eq. \[eq:spino\_iso\_nem\] reduces to $$\begin{aligned} S_{IN} = \frac{25 n^2 }{4 }\left[1 - \frac{\pi}{16}c_p - \left(4 \pi \tilde l_B Z^2 \sigma^2 N\right)^2 \Gamma\left\{c_p\right\} \frac{L}{d}c_p \right] &=& 0 \label{eq:iso_nem_spino}\end{aligned}$$ where $$\begin{aligned} \Gamma\left\{c_p\right\} &=& \frac{9}{4 \pi^2 }\int_{0}^{\infty} d\tilde{q} \tilde{q}^2\frac{h^2(\tilde{q})}{\left [\tilde{q}^2 + \tilde \kappa^2 + \tilde \kappa_{sm}^2 t^{iso}(\tilde{q})\right ]^2}, \end{aligned}$$ and $c_p = 2nL^2 d/\Omega$ is the dimensionless parameter of the order of overlap concentration of the rods. For numerical purposes, dimensionless parameters are introduced after renormalizing different parameters with the length of the rods so that $\tilde \kappa^2 = \kappa^2 L^2 = 4\pi \tilde l_B \sigma c_p L l/d, \; \tilde \kappa_{sm}^2 = L^2(\kappa_+^2 + \kappa_-^2) = 4\pi \tilde l_B \sigma^2 c_p L^2 l/d$, and $\tilde q = qL$. Furthermore, in order to make a qualitative connection with the Manning’s theory of counterion condensation[@manning] later in this work, we have renormalized Bjerrum’s length using monomer length ($l \equiv L/N$) and defined $\tilde l_B = l_B/l$. Note that for the symmetric mixtures, the spinodal can also be obtained after putting $S_+ = S_- = S$ and then evaluating $\partial^2 F/\partial S^2 = 0$. Furthermore, $S_{IN}<0$ corresponds to the regime where the isotropic phase is unstable relative to the weakly ordered nematic phase. It is clear from Eq. \[eq:iso\_nem\_spino\] that the electrostatics and the steric effects (last two terms in square brackets) act against the orientational entropy to drive the system from the isotropic phase to the anisotropic nematic phase. In the absence of electrostatics[@onsager_paper; @edwardsbook], the nematic phase becomes stable when $c_p > 16/\pi \simeq 5.1$. It is also clear from Eq. \[eq:iso\_nem\_spino\] that the nematic phase in polyelectrolyte mixtures becomes stable at lower polymer concentrations compared to the neutral mixtures. Numerical solutions of Eq. \[eq:iso\_nem\_spino\] are presented in Fig. \[fig:compare\_spinod\_in\_rods\] for symmetric mixtures without and with the counterions (Fig. \[fig:compare\_spinod\_in\_rods\]a and \[fig:compare\_spinod\_in\_rods\]b, respectively) and for different linear charge densities. On the right hand side of these boundaries, the nematic phase is stable. It is clear from Fig. \[fig:compare\_spinod\_in\_rods\] that the nematic phase becomes stable at lower monomer densities with an increase in linear charge density at fixed Bjerrum length. Furthermore, comparing Fig. \[fig:compare\_spinod\_in\_rods\]a and \[fig:compare\_spinod\_in\_rods\]b, it is evident that the nematic phase becomes stable at higher monomer densities in the presence of counterions, i.e. the stable regime of the nematic phase is smaller in the presence of counterions. Note that these numerical results are in agreement with earlier theoretical predictions for one-component rodlike polyelectrolyte solutions that electrostatic interactions favor uniaxial ordering of the rods.[@potemkin_1; @potemkin_2; @potemkin_3] Nematic Phase for Symmetric Mixtures: Variational Treatment ----------------------------------------------------------- In order to study a nematic phase with an arbitrary magnitude of the orientational order parameter, we need to resolve the full probability distribution function. In principle, this can be done by a calculus of variations approach by minimizing the free energy of the system, which leads to an integral equation[@edwardsbook]. For the case of neutral polymers, the problem has been attacked by three different routes. The first is a variational treatment using Onsager’s trial function[@edwardsbook; @onsager_paper; @onsager_trial_func], where the variational parameter is determined by minimizing the free energy. A second route is through the use of a Legendre expansion[@lakatos_legendre; @lasher_legendre] and determining the coefficients in the Legendre series that minimize the free energy. A third scheme is to directly attack the integral equation in real space[@intgral_equation_real] using a non-linear equation solver. From a computational point of view, the last two routes are more demanding and become especially difficult for strongly ordered nematic phases ($S>0.9$). The first route is the easiest and readily describes a nematic phase with arbitrary order. However, it leads to a slight overprediction[@edwardsbook; @lasher_legendre] of the coexisting densities at the isotropic-nematic transition. In this work, we adopt Onsager’s variational approach to study nematic phases in the symmetric polyelectrolyte mixtures. Due to the assumed symmetry, the probability distribution functions for the two types of rods must be the same and only one of the distribution functions needs to be considered. In Onsager’s approach, the probability distribution function is taken to be of the form $$p(\mathbf{u}\cdot \nh) = \frac{\alpha \cosh \left[\alpha \left(\mathbf{u}\cdot \nh\right)\right]}{4 \pi \sinh \alpha},$$ where $\alpha$ is a variational parameter, which is determined by minimization of the free energy. The order parameter corresponding to this distribution function for a symmetric mixture is $$S = 1 + \frac{3}{\alpha^2}(1-\alpha \coth \alpha)$$ so that $S=0$ for $\alpha = 0$ corresponds to the isotropic phase and $S=1$ when $\alpha \rightarrow \infty$, corresponding to a perfectly ordered nematic phase. The free energy density can be written in terms of $\alpha$ so that the free energy density of the symmetric mixtures becomes $f = F/\Omega = f_{en} + f_{w} + f_e $, where $$\begin{aligned} \frac{f_{en}L^2 d}{k_BT} & = & c_{p}\left[\ln \left[\alpha \coth \alpha \right] - 1 + \frac{\tan^{-1}\left[\sinh \alpha\right]}{\sinh\alpha}\right] + c_{s} \left[\ln \frac{c_s}{2L^2d} - 1\right] \nonumber \\ && + c_{p} \left[\ln \frac{c_p}{2L^2d} - 1\right], \label{eq:free_en_nem}\\ \frac{f_w L^2 d}{k_BT} &=& \frac{\pi}{2} \frac{I_2(2\alpha)}{\left[\sinh\alpha\right]^2}c_p^2,\label{eq:free_w_nem}\\ \frac{f_e L^2 d}{k_BT} &=& \frac{d}{8\pi^2 L}\int_{0}^{\infty}d\tilde{q}\: \tilde{q}^2\int_{-1}^{1} dx\left\{\ln \left[1 + \frac{\tilde{\kappa}^2 + \tilde\kappa_p^2 t(\tilde{q},x)}{\tilde{q}^2}\right] - \frac{\tilde{\kappa}^2 + \tilde \kappa_p^2 t(\tilde{q},x)}{\tilde{q}^2}\right\}\label{eq:free_rods_2}\end{aligned}$$ and $$\begin{aligned} t(\tilde{q},x) &=& \int_{-1}^{1}dx' \: \frac{\alpha \cosh \left[\alpha x'\right]}{2 \pi \sinh \alpha}\int_{-1}^{1}\frac{dy}{\sqrt{1-y^2}}\left[\frac{\sin\left[\tilde{q}\left(xx' + \sqrt{1-x^2}\sqrt{1-x'^2} \:y\right)/2\right]}{\left[\tilde{q}\left(xx' + \sqrt{1-x^2}\sqrt{1-x'^2} \: y\right)/2\right]}\right]^2 \label{eq:tfunc_var_nem}\end{aligned}$$ where $c_s = \sigma L c_p$ is the dimensionless number density of small ions (both positive and negative) and $I_2$ is the modified Bessel function of order $2$. The integrals in the expression for $t(\tilde{q},x)$ can be readily evaluated using Gauss-Legendre and Gauss-Chebyshev quadratures for $x'$ and $y$ integrals, respectively. Also, the integral over $\tilde{q}$ ranging from $0$ to $\infty$ can be evaluated using Gauss-Legendre quadrature after using the transformation $q = (1-z)/(1+z)$. The free energy density can be optimized with respect to $\alpha$ so that $\frac{\partial f}{\partial \alpha} = 0$, which gives $$\begin{aligned} c_{p}\left[\frac{1}{\alpha} - \frac{\tan^{-1}\left[\sinh \alpha\right]\cosh \alpha}{\left[\sinh\alpha\right]^2}\right] + \frac{\pi c_p^2}{2\left[\sinh\alpha\right]^2} \left[I_1(2\alpha) + I_3(2\alpha) - 2\coth\alpha I_2(2\alpha)\right]&&\nonumber\\ - \frac{d}{8\pi^2 L}\int_{0}^{\infty}d\tilde{q} \int_{-1}^{1} dx \frac{\partial t(\tilde{q},x)}{\partial \alpha}\left\{\frac{\tilde \kappa_p^2\left[\tilde \kappa^2 + \tilde \kappa_p^2 t(\tilde{q},x)\right]}{ \tilde{q}^2 + \tilde \kappa^2 + \tilde \kappa_p^2 t(\tilde{q},x)}\right \} = 0 \qquad \qquad \qquad && \label{eq:order_equation}\end{aligned}$$ Note that for a root ($=\alpha^\star$) of Eq. \[eq:order\_equation\] to be a minimum of the free energy, $\frac{\partial^2 f}{\partial \alpha^2}\|_{\alpha = \alpha^\star} > 0 $ must be satisfied. In other words, some of the roots of Eq. \[eq:order\_equation\] may correspond to a local maximum in the free energy density rather than a local minimum. To ensure that we retain only the physical roots, we have conducted tandem numerical solutions of Eq. \[eq:order\_equation\] and direct minimization of the free energy. In the former, instead of solving for $\alpha$ using different values of $c_p$, it proves easier to solve Eq. \[eq:order\_equation\] for $c_p$ using different values of $\alpha$. In second set of calculations, we have carried out numerical minimization (using Brent’s method[@numerical_recipe]) of the free energy with respect to $\alpha$ for different values of $c_p$. Results of these two sets of calculations are presented in Fig. \[fig:in\_bet\_n1000\]. Fig. \[fig:in\_bet\_n1000\]a presents the results of the calculations without counterions and Fig. \[fig:in\_bet\_n1000\]b corresponds to results with counterions. Solid and dashed lines correspond to the roots of Eq. \[eq:order\_equation\] and the numerical minimization, respectively. In both the figures, the results of the two sets of calculations match perfectly well except in the transition regime, where there are multiple values of the order parameter $S$ for a given value of $c_p$. In fact, this is the metastable regime for the isotropic-nematic phase transition and such a diagram has already been mapped out for the neutral rods using bifurcation analysis[@bifurcation]. The jump in $S$ for the numerical minimization calculations (i.e., dashed lines) is a characteristic of the first order isotropic-nematic phase transition. From both the figures, it is clear that the isotropic-nematic phase transition takes place at lower monomer densities as the linear charge densities of the rodlike polyelectrolytes is increased. Also, in comparing the two figures, it is clear that the isotropic-nematic phase transition takes place at higher monomer densities in the presence of counterions. These results are consistent with the stability analysis of the weakly ordered nematic phase as presented in the previous section. We note that the isotropic-nematic transition in this solvated system is actually spanned by a region of two-phase coexistence in which a diluted (in polymer) isotropic phase coexists with an enriched nematic phase. Such two-phase regions are confined to within the regions of hysteresis shown in Fig. \[fig:in\_bet\_n1000\], but we have not taken the trouble to elaborate them in this figure. The two-phase regions will be revealed when we consider the full phase diagram. However, before getting to the full phase diagram, an important remark on the theoretical treatment considered in this paper is appropriate. During the computation of the full phase diagram, it is found that for some set of parameters a completely ordered nematic phase (i.e., the phase for which $\alpha \rightarrow \infty$ and hence, $S = 1$) becomes one of the competing phases. Physically, the completely ordered phase corresponds to perfectly aligned rods. In the next section, we show that Onsager’s approach (which is extended to polyelectrolyte mixtures in this work) is not able to describe such a coexistence between the completely ordered nematic phase and any other phase. Completely Ordered Nematic Phase for Symmetric Mixtures ------------------------------------------------------- For the completely ordered nematic phase, Onsager’s variational parameter, $\alpha$, diverges. In order to evaluate the free energy in this limit, we rewrite Eqs. \[eq:free\_en\_nem\]- \[eq:free\_w\_nem\] using asymptotic expansions for the Bessel and hyperbolic functions[@onsager_paper] leading to the result $$\begin{aligned} \frac{f_{en}^{h}L^2 d}{k_BT} & = & c_{p}\left[\ln \alpha - 1 \right] + c_{s} \left[\ln \frac{c_s}{2L^2d} - 1\right] + c_{p} \left[\ln \frac{c_p}{2L^2d} - 1\right], \label{eq:free_c_en_nem}\\ \frac{f_w^h L^2 d}{k_BT} &=& \sqrt{\frac{\pi}{\alpha}} c_p^2\left[1 - \frac{30}{32 \alpha} + \frac{210}{\left(32 \alpha\right)^2} + \frac{1260}{\left(32 \alpha\right)^3} + \cdots \right].\label{eq:free_c_w_nem}\end{aligned}$$ In the limit of $\alpha \rightarrow \infty$, the entropic contribution to the free energy of the completely ordered nematic phase diverges logarithmically (i.e., $f_{en}^c = f_{en}^{h}\left\{\alpha \rightarrow \infty \right \}\rightarrow \infty$) and the excluded volume contribution vanishes (i.e., $f_{w}^c = f_{w}^{h}\left\{\alpha \rightarrow \infty \right \} \rightarrow 0$). Furthermore, the limiting expression for Eq. \[eq:tfunc\_var\_nem\] becomes $$\begin{aligned} t^c(\tilde{q}x) &\equiv& t(\tilde{q},x)\|_{\alpha \rightarrow \infty} = \left[\frac{\sin\left[\tilde{q}x/2\right]}{\tilde{q}x/2}\right]^2, \label{eq:tfunc_c_var_nem}\end{aligned}$$ which allows us to rewrite Eq. \[eq:free\_rods\_2\] as $$\begin{aligned} \frac{f_e^c L^2 d}{k_BT} &=& \frac{d}{\pi^2 L}\int_{0}^{\infty}d\tilde{q}\: \tilde{q}^2\left\{\frac{\tilde{\kappa}^2 + \tilde\kappa_p^2 t^c(2\tilde{q})}{4\tilde{q}^2} - \left[1 + \frac{\tilde{\kappa}^2 + \tilde\kappa_p^2 t^c(2\tilde{q})}{4\tilde{q}^2}\right]\ln \left[1 + \frac{\tilde{\kappa}^2 + \tilde\kappa_p^2 t^c(2\tilde{q})}{4\tilde{q}^2}\right] \right\}.\nonumber \\ &&\label{eq:free_rods_2_c}\end{aligned}$$ This integral can be evaluated numerically and is found to be negative. Note that Eq. \[eq:free\_rods\_2\_c\] is the same as the electrostatic contribution to the free energy considered in Ref. [@stell_rods] in the context of phase separation of charged aligned needles. The divergence of $f_{en}^c$ complicates the evaluation of coexisting densities between the isotropic and the completely ordered phases. Also, the vanishing of $f_w^c$ for any arbitrary polymer density highlights the limitation of the Onsager’s virial approach to describe phase separation when one of the competing phases is fully ordered. A more suitable description of such highly ordered phases is presented in Ref. [@stell_rods], where the phase separation of a completely ordered parent phase into two completely ordered daughter phases having different polymer densities is considered. The phase separation calculations were performed using the Percus-Yevick equation of state for hard cylinders. This particular description takes into account the higher order terms in describing the excluded volume interactions and goes beyond the Onsager approach. However, this study did not consider the nematic phase as a candidate in the free energy competition. Such an analysis, while possibly relevant in an orientationally constrained situation, could produce unphysical results if the system can freely choose the orientation and concentration of both parent and daughter phases. In the following, in order to avoid the above complications associated with strongly ordered nematic phases, we limit our results to regimes where the nematic order parameter is not fully saturated at $S=1$. Results {#sec:results} ======= Using the theoretical approach and free energies presented in the previous section, we have investigated the phase behavior of symmetric mixtures of oppositely charged rodlike polyelectrolytes. In the following, we consider the possibility of macrophase separation in the isotropic phase due to attractive electrostatic correlations between oppositely charged polyelectrolytes in addition to phase coexistence between isotropic and the nematic phases. Unless mentioned, all lengths are normalized by the monomer length $l$ so that $L = Nl$, with $N$ the number of monomers. In order to keep the theoretical analysis simple and avoid the issue of counterion condensation[@manning], we have focused on the “weakly charged” regime corresponding to $l_B/l < 1$. Also, we restrict attention to salt-free symmetric mixtures here and leave the effect of added salt on the phase behavior for the future. In order to identify the role played by the counterions in the salt-free symmetric mixtures, we consider two comparison systems - one without any counterions from the polyelectrolytes and the other with counterions. As noted above, the counterion free situation could conceptually be realized in symmetric mixtures containing polyacids and polybases, where the charge on the polyelectrolyte backbones can be controlled by the pH of the solution. While the theory is quite general and a wide variety of phenomena can be investigated, we have chosen to further limit the parameter space of our study by fixing the rod length and aspect ratio to $N = 1000$ and $d/l = 1/50$. Prior to presenting results on the full phase diagram, it is illustrative to study macrophase separation in the isotropic phase, where we set aside the possibility of nematic order. Even for such a simpler situation, there are some key questions that need to be answered. For example, it is not clear how the counterions are partitioned between different phases and where the phase boundaries (spinodals and binodals) are located relative to the analogous phase boundaries in mixtures containing flexible polyelectrolytes. In the next subsection, we present our results for phase separation in the isotropic phase and address these key issues. In a subsequent subsection, we present the full phase diagram by considering the possibility of isotropic-nematic phase behavior in addition to the isotropic-isotropic transition. Isotropic-isotropic transition ------------------------------ A salt-free symmetric mixture of oppositely charged polyelectrolytes with counterions is a system with five components - two types of polymers, two types of counterions and the solvent molecules. However, in the simplified theoretical description presented here, the solvent molecules are treated implicitly and by restricting attention to symmetric mixtures only, the equations obtained by enforcing the equality of the chemical potentials of the polymers in the two phases are degenerate for the two types of polymers. The same is true for the counterions. As a consequence, we only have to analyze a pseudo two-component system, where the polymers and the counterions need to be partitioned among the coexisting phases. Using the same set of arguments, salt-free symmetric mixtures without counterions can be treated as a pseudo one-component system. ### Effect of counterions For the quasi-one and two component systems, we have carried out a direct minimization of the total free energy density, obtained by appropriate weighting of two isotropic phases and the use of the lever rule[@chandler_book]. This approach is equivalent to equating the chemical potential of each component in the two phases and equating the pressure of each phase, but replaces solving a nonlinear system by the numerically more robust procedure of minimization. Explicitly, for the quasi-one component system, we minimize the total free energy density ($f_{total}^{iso}$) of the phase segregated system with respect to the densities in each phase (i.e., two dimensional minimization) after writing $$\begin{aligned} f_{total}^{iso} &=& \frac{c_{p}^{\text{II}} - c_p}{c_{p}^{\text{II}} - c_{p}^{\text{I}}}f^{iso}\left\{c_{p}^{\text{I}}\right\} + \frac{c_p - c_{p}^{\text{I}}}{c_{p}^{\text{II}} - c_{p}^{\text{I}}}f^{iso}\left\{c_{p}^{\text{II}}\right\}, \label{eq:ftotal_iso}\end{aligned}$$ where $f^{iso}\left\{c_{p}^{\text{I}}\right\}$ and $f^{iso}\left\{c_{p}^{\text{II}}\right\}$ are the free energy densities of isotropic phases $\text{I}$ and $\text{II}$, respectively, given by Eq. \[eq:free\_iso\]. The parameters $c_p^{\text{I}}$, $c_p$, and $c_p^{\text{II}}$ ($c_{p}^{\text{I}} < c_p < c_{p}^{\text{II}}$) are the dimensionless number density of the polyelectrolytes in dilute phase $\text{I}$, overall number density, and the number density in the concentrated phase $\text{II}$, respectively. A similar equation can be written for the quasi-two component system (i.e. the system with counterions) where we note that the total number of counterions is related to the total number of monomers by $n_c^{total} = 2 \sigma n_t L$, where $n_t$ is the total number of polyelectrolyte chains of one type. For this system, three dimensional minimizations of the free energy with respect to the monomer densities in each phase along with the counterion density in one of the phases have been carried out to compute the coexistence curves. The counterion density in the second phase is computed using the lever rule for the counterions. All of the multi-dimensional minimizations of the free energy density have been carried out using the simplex method[@numerical_recipe]. In Fig. \[fig:ii\_bet\_n1000\], we present the results of these calculations for different total monomer densities. Coexisting monomer and counterion densities in the two phases are presented in Figs. \[fig:ii\_bet\_n1000\]a and \[fig:ii\_bet\_n1000\]b, respectively. In these calculations, the linear charge density $\sigma$ for the polyelectrolytes is kept the same so that different total monomer number densities correspond to different total counterion number densities. Also, for comparison purposes, the coexisting monomer densities in the counterion-free system are also presented in Fig. \[fig:ii\_bet\_n1000\]a for a total monomer density of $0.002$, and are denoted by $\star$. Note that in Fig. \[fig:ii\_bet\_n1000\]b the counterion densities on the right side of the dashed lines correspond to the densities in the phase with higher monomer density. It should be noted that these results are in qualitative agreement with similar calculations for flexible polyelectrolytes, where the system phase separates into a very low density (supernatant) phase and a dense (coacervate) phase. The asymmetric nature of the coexistence curve[@fisher_levin; @fisher] is a result of the long-range electrostatic interactions at play in these systems coupled with the reduced translational entropy of polymers relative to solvent and ions. Beyond these observations, two important results can be inferred from the figures. In Fig. \[fig:ii\_bet\_n1000\]a, it is clear that the coexistence regime shrinks continuously with the increase in the total counterion density. This implies that the counterions suppress the isotropic-isotropic phase transition. This is in agreement with the notion that the counterions need to be partitioned among the phases upon macrophase separation, which is entropically unfavorable. Furthermore, from Fig. \[fig:ii\_bet\_n1000\]b, it is found that indeed, the counterions get partitioned between the two phases and the counterion density is only slightly higher in the coacervate phase. Note that this result is in qualitative agreement with the theoretical and experimental results reported by Voorn[@voorn_5], although the treatment of electrostatics in Voorn’s theory is much more primitive than ours. An important remark regarding the phase diagrams presented in this work is due here. Typically, in a two-phase region the boundaries of the phase diagram are the same irrespective of the initial concentration; just the relative amounts of the two phases vary. The situation in the presence of counterions is very different. A change in the initial concentration of polyelectrolytes changes the concentration of counterions in the solution and the phase boundaries may shift as described in Fig. \[fig:ii\_bet\_n1000\]. As different concentrations of the polyelectrolytes and their counterions correspond to different states of the system, the diagrams presenting the coexisting phases for different initial states should be called “state diagrams” in general. However, in this work, we ignore this semantics issue and call these diagrams binodals (or coexistence curves). ### Comparison with flexible polyelectrolytes: counterion free symmetric mixtures In order to compare the phase boundaries of symmetric mixtures containing rodlike polyelectrolytes to those of flexible polyelectrolyte mixtures, we have computed the free energy of mixtures of flexible polyelectrolytes using an analogous random phase approximation to that employed in the rigid case (see Appendix C). Explicitly, the free energy of a flexible mixture is given by $$\begin{aligned} F_f &=& F_{en}^f + F_{w}^f + F_{e}^f \label{eq:free_flex} \end{aligned}$$ where $$\begin{aligned} \frac{F_{en}^{f}}{k_BT} &=& -\sum_{k=+,-} n_k \ln Q_k^0 + \sum_{\gamma = +,-,c+,c-} n_{\gamma} \left[\ln \frac{n_{\gamma}}{\Omega} - 1\right] \\ \frac{F_{w}^{f}}{k_BT} &=& \frac{w}{2 \Omega}\left( n_+ N_+ + n_- N_-\right)^2 \nonumber \\ && + \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{w}{\Omega}\sum_{k=+,-}N_{k}^2 n_k g\left(q^2N_k l_k^2/6\right)\right] - \frac{w}{\Omega}\sum_{k=+,-}N_{k}^2 n_k g\left(q^2N_k l_k^2/6\right)\right \}\nonumber \\ && \\ \frac{F_{e}^{f}}{k_BT} &=& \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{\kappa^2 + \sum_{k=+,-}\kappa_{k}^2 g\left(q^2N_k l_k^2/6\right)}{q^2}\right] - \frac{\kappa^2 + \sum_{k=+,-}\kappa_{k}^2 g\left(q^2N_k l_k^2/6\right)}{q^2} \right \} \nonumber \\ &&\end{aligned}$$ Here, $w$ is the excluded volume parameter[@edwardsbook] and $Q_k^0$ is the partition function for a noninteracting Gaussian chain of length $L_k = N_k l_k, k = +,-$ ($l_k$ being the Kuhn segment length and $N_k$ is the number of segments)[@edwardsbook]). Furthermore, $g(x) = 2 (e^{-x} - 1 + x)/x^2$ is the Debye function. We note that in the case of flexible polyelectrolyte mixtures the final term in $F_w^f$ (involving the integral) is the well-known Edwards’ screening contribution to the free energy[@edwardsbook] and is negative. For the comparison between the rodlike and the flexible systems, this contribution will be ignored and its effect captured by using a renormalized excluded volume parameter $w_r$ instead of the bare excluded volume parameter $w$. Since our focus here is on the effect of chain flexibility on the electrostatic contribution to the free energy, we can identify an appropriate value of $w_r$ by forcing agreement between the excluded volume contributions to the free energy for the rodlike and flexible symmetric mixtures (cf. Eqs. \[eq:free\_iso\] and \[eq:free\_flex\]). The comparison reveals that $w_r = \pi d l^2/2$ is the suitable choice, which makes all free energy contributions other than electrostatics identical for the rodlike and flexible systems. Using this value for the renormalized excluded volume parameter, the spinodals and the binodals for symmetric mixtures of flexible polyelectrolytes can be readily computed using the same approach as used for the rodlike system. To avoid any complications arising from the presence of counterions in comparing the phase boundaries for symmetric rodlike and flexible mixtures, we have considered a salt-free and counterion-free system. For this model flexible polyelectrolyte mixture, the spinodal for the isotropic-isotropic transition is given by $$\begin{aligned} S_f^{iso} = 1 + c_p \left[w_r - \left(4 \pi \tilde l_B Z^2 \sigma^2 N\right)^2 \frac{L}{d}D_e\left\{c_p\right\}\right] &=& 0 \label{eq:iso_iso_spino}\end{aligned}$$ where $$\begin{aligned} D_e\left\{c_p\right\} &=& \frac{1}{4 \pi^2 }\int_{0}^{\infty} d\tilde{q} \tilde{q}^2\frac{g^2\left(\tilde{q}^2/6N\right)}{\left [\tilde{q}^2 + \tilde \kappa_{sm}^2 g^2\left(\tilde{q}^2/6N\right)\right ]^2},\label{eq:iso_iso_de} \end{aligned}$$ and $c_p = 2nL^2 d/\Omega$ is the corresponding dimensionless monomer number density. Also, as before, we have defined $\tilde l_B = l_B/l, \; \tilde \kappa = \kappa L$, $\tilde \kappa_{sm}^2 = L^2(\kappa_+^2 + \kappa_-^2)$, and $\tilde q = qL$. The spinodal for the isotropic-isotropic phase transition in a corresponding rodlike symmetric polyelectrolyte mixture can be obtained from Eqs. (\[eq:iso\_iso\_spino\] - \[eq:iso\_iso\_de\]) by replacing the Debye function $g\left(\tilde{q}^2/6 N\right)$ by the function $t^{iso}(\tilde{q})$ given in Eq. \[eq:tkq\_iso\]. The set of parameters that lead to $S_f^{iso}<0$ corresponds to the regime of instability to macrophase separation in the isotropic phase. Note that the electrostatic term $D_e \left\{c_p\right\}$ is positive and hence, the electrostatics drives the macrophase separation in flexible as well as rodlike polyelectrolyte mixtures. Also, from Eq. \[eq:iso\_iso\_spino\] it is clear that the translational entropy (which appears as unity in the equation) and the excluded volume interactions oppose this driving force (assuming $w_r > 0$ for good solvents). Furthermore, we note the prefactor of $\left(4 \pi \tilde l_B Z^2\sigma^2 N\right)^2$ in front of $D_e \left\{c_p\right \}$. This implies that an increase in the polymer length, Bjerrum length or the linear charge density leads to a strengthening of the electrostatic driving force favoring the macrophase separation. However, note that an increase in monomer density leads to screening of the electrostatics, appearing through $\tilde \kappa_{sm}$ in the expression for $D_e$. This screening effect places an upper concentration bound on the unstable regime, ultimately stabilizing a single isotropic phase. In Fig. \[fig:rods\_flex\_bp1\], we have compared the spinodal phase boundaries for flexible and rodlike symmetric polyelectrolyte mixtures in a salt-free, counterion-free situation. It is clear that for these weakly charged polyelectrolyte systems (characterized by $4\pi l_B \sigma^2 \rho < 1$) the isotropic-isotropic coexistence regime is broader in the case of flexible symmetric mixtures in comparison with rodlike mixtures. As all the other contributions to the free energies of the two systems are the same except the electrostatic contributions, it is clear that the electrostatic driving force is stronger in the case of flexible polyelectrolytes than rodlike polyelectrolytes. We note that these numerical results are consistent with the simple scaling arguments presented in the introduction. Full Phase Diagram ------------------ From the results presented and discussions so far, it is clear that electrostatics drives macrophase separation in the isotropic phase and also stabilizes the nematic phase. So, in principle, isotropic-isotropic, isotropic-nematic and/or nematic-nematic phase transitions can take place in symmetric mixtures of rodlike polyelectrolytes. Similar to the isotropic-isotropic phase calculations considered above, we consider the extra possibilities of nematic-isotropic and nematic-nematic coexistence, where the orientational order parameter in each phase is determined during the minimization of the total free energy density of the phase separated system. For the quasi-one component system without counterions (cf. Eq. \[eq:ftotal\_iso\]), we minimize the total free energy density written as $$\begin{aligned} f_{total} &=& \frac{c_{p}^{\text{II}} - c_p}{c_{p}^{\text{II}} - c_{p}^{\text{I}}}f\left\{c_{p}^{\text{I}}, \alpha^{\text{I}} \right\} + \frac{c_p - c_{p}^{\text{I}}}{c_{p}^{\text{II}} - c_{p}^{\text{I}}}f\left\{c_{p}^{\text{II}}, \alpha^{\text{II}} \right\}, \label{eq:ftotal_nn}\end{aligned}$$ where the variational parameters $\alpha^{\text{I}}$ and $\alpha^{\text{II}}$ for the phases $\text{I}$ and $\text{II}$, respectively, are obtained by numerically minimizing (using Brent’s method[@numerical_recipe]) the free energy density of the phase ($f\left\{c_{p}^{\text{I}}\right\}$ and $f\left\{c_{p}^{\text{II}}\right\}$ for the phases $\text{I}$ and $\text{II}$, respectively, given by Eqs. \[eq:free\_en\_nem\] - \[eq:free\_rods\_2\]) at the given monomer densities ($c_{p}^{\text{I}}$ and $c_{p}^{\text{II}}$ for the phases $\text{I}$ and $\text{II}$, respectively). A similar approach can be taken in quasi two-dimensional systems that include counterions. The motivation behind carrying out such calculations is the fact that all the phase transitions that we have considered thus far (i.e., isotropic-isotropic and isotropic-nematic) are subsets of these more “general” calculations. Unfortunately, such high dimensional optimizations are computationally demanding, so we have looked for opportunities to accelerate the construction of full phase diagrams. In a complementary set of calculations, we have considered isotropic-nematic and isotropic-isotropic phase transitions separately. It was found that the results of the “general” calculations described above exactly match the results of our isotropic-nematic and the isotropic-isotropic calculations (data not presented here). Through such calculations, we have also established that there are no regions of nematic-nematic coexistence, so the various phase diagrams can be mapped out by tracking individual I-I and I-N boundaries. The phase diagrams presented in the remaining sections were obtained by the above technique and are presented in Figs. \[fig:phase\_diagram\_rods\] and \[fig:phase\_diagram\_rods\_count\], respectively in the absence and presence of counterions. Overall, the results of these calculations are in qualitative agreement with the picture presented in Fig. \[fig:phase\_cartoon\]. Some important features of the full phase diagram are discussed below. ### Phase diagram without counterions In Fig. \[fig:phase\_diagram\_rods\], we have plotted the phase diagram for symmetric mixtures of oppositely charged rodlike polyelectrolytes in the absence of counterions. In the figure, the linear charge density is varied to explore the effect of electrostatics on the phase boundaries. From Fig. \[fig:phase\_diagram\_rods\]a, it is clear that the coexisting densities in the two phases decrease from the uncharged case with an increase in the charge densities (compare the results for $\sigma = 0,0.02$ and $0.04$). However, a further increase in the charge density leads to three distinct regimes corresponding to the low, intermediate and high values of $l_B$. For very low values of the Bjerrum length ($l_B/l \rightarrow 0$), the coexisting phases are isotropic and nematic phases. Note that in this regime the coexisting densities for the isotropic and nematic phases for the polyelectrolyte systems are close to that for the neutral system. For high values of the Bjerrum length (close to unity) and high charge densities such as $\sigma = 0.08, 0.1$ in Fig. \[fig:phase\_diagram\_rods\]a, completely ordered nematic phase becomes one of the coexisting phases. However, we haven’t been able to compute the densities of the coexisting phases in this regime due to the numerical issues discussed in subsection F above. For intermediate values of $l_B$, there are regimes (e.g., $l_B/l \in 0.1-0.6$ for $\sigma = 0.1$ in Fig. \[fig:phase\_diagram\_rods\]a), where isotropic-isotropic coexistence and isotropic-nematic phase separation can be separately realized by varying the concentration of the rodlike polyelectrolytes in solution. With an increase in $l_B$ in this regime, three phase coexistence (isotropic-isotropic-completely ordered nematic) can be realized in these systems. Moreover, the value of $l_B$ at which the three phase coexistence takes place is dependent on the linear charge density of the polyelectrolytes. In fact, the Bjerrum length at which the three phase coexistence takes place decreases with an increase in the linear charge density of the rods (compare the results for $\sigma = 0.08$ and $\sigma = 0.1$ in Fig. \[fig:phase\_diagram\_rods\]a). Note that in this intermediate regime, the entropic effects driving the isotropic-nematic transition are comparable in strength with the energetic effects (coming from electrostatics), but the system quickly evolves into a broad isotropic-nematic phase envelope upon increasing $l_B$. This leads to a sharp increase in the density of the coexisting nematic phase (see the plots for $\sigma = 0.08$ and $\sigma = 0.1$ in Fig. \[fig:phase\_diagram\_rods\]a). A further increase in $l_B$ leads to the completely ordered nematic phase as a coexisting phase. In order to keep track of the degree of alignment of the rods in the nematic phase in Fig. \[fig:phase\_diagram\_rods\]a, we have plotted the order parameter in Fig. \[fig:phase\_diagram\_rods\]b. From Fig. \[fig:phase\_diagram\_rods\]b, it is clear that the order parameter increases with an increase in $l_B/l$, which is in agreement with the stability analysis carried out in this paper. Hence, the numerical results support the prediction that electrostatics favor orientational ordering. ### Effect of counterions In contrast to the phase diagram obtained in the absence of the counterions, the phase diagram in the presence of counterions depends on the total number density of the rodlike polyelectrolytes and correspondingly on the number of counterions in the system. In order to conduct a systematic study, we have carried out two sets of calculations. In the first set, we vary the linear charge density of the polyelectrolytes while keeping the total number density of the rodlike polyelectrolytes fixed at a particular value. In the second set, the total number density of the rods is varied keeping the linear charge density fixed at a particular value. So, in both the sets, the total number density of counterions is varied. Fig. \[fig:phase\_diagram\_rods\_count\] presents the results of the first set of calculations. Comparing Fig. \[fig:phase\_diagram\_rods\_count\] with Fig. \[fig:phase\_diagram\_rods\], we observe that the qualitative features of the phase diagram in the presence of counterions remain the same as in the absence of counterions. Furthermore, Fig. \[fig:counterions\_coexisting\] presents the number densities of the counterions in the coexisting phases. It is found that counterions are uniformly distributed between the two phases for these values of linear charge densities and the Bjerrum’s length. The results for the second set of calculations are shown in Figs. \[fig:den\_effect\_phase\] and \[fig:den\_effect\_phase2\], where the total number density of polyelectrolytes is changed for a particular value of the linear charge density. These results show the isotropic-isotropic and isotropic-nematic coexistence regimes just like the ones seen in the absence of counterions in Fig. \[fig:phase\_diagram\_rods\]. In these calculations, the change in the total number densities of the polyelectrolytes leads to a change in the total number density of the counterions. The effect of varying the total number density of the polyelectrolytes and counterions on the phase boundaries can be explained as follows. As already discussed, the origin of the isotropic-nematic transition for neutral rods near the overlap concentration or for very low values of $l_B$ ($l_B/l \rightarrow 0$) is entropic and is nearly independent of electrostatics. Hence, the presence of counterions does not affect these boundaries (compare the phase boundaries for $2n_t L^2 d/\Omega = 4$ and $4.7$). However, the origin of the isotropic-isotropic transition at low number density of polyelectrolytes is electrostatically driven and is strongly dependent on the presence of the counterions. The width of this particular coexistence regime can be tuned by changing the number of counterions. In particular, the coexistence regimes shrink with an increase in the number of counterions due to the screening of electrostatic interactions(compare $2n_t L^2 d/\Omega = 0.002, 0.01$ and $0.1$ in Fig. \[fig:den\_effect\_phase\]). Note that the results for the isotropic-isotropic phase transition in Fig. \[fig:den\_effect\_phase\] are the same as in Fig. \[fig:ii\_bet\_n1000\]. Figure \[fig:den\_effect\_phase2\] shows the counterion distribution in the coexisting phases. Consistent with our prior results, it is evident that the counterions also phase segregate at high Bjerrum length with a higher density in the concentrated (coacervate) phase. Conclusions {#sec:conclusions} =========== We have studied the phase behavior of salt-free symmetric mixtures of oppositely charged rodlike polyelectrolytes using the random phase approximation. In this work, we have focused on weak polyelectrolytes in the regime $l_B/l < 1$ to avoid complications arising from possible counterion condensation. For a variety of symmetric mixtures, we have computed the phase boundaries for regions of isotropic-isotropic and isotropic-nematic coexistence. We were not able to identify any regions of nematic-nematic coexistence in these symmetric systems. Our stability analysis and numerical results for coexistence curves reveal that electrostatic interactions favor nematic ordering of the rodlike components in solution. Nonetheless, the screening of these electrostatic interactions by higher concentrations of counterions weakens or destroys this ordering. It is shown that the phase boundaries for symmetric mixtures containing oppositely charged rodlike polyelectrolytes are dependent on the electrostatic interaction strengths characterized by the linear charge density, $\sigma$, and the Bjerrum length, $l_B$. In particular, it is demonstrated that at low electrostatic interaction strengths (i.e., $l_B/l \rightarrow 0, \: \sigma \rightarrow 0$), the densities in the coexisting isotropic and nematic phases decrease with an increase in the linear charge density of the polyelectrolytes in the absence of counterions. However, an increase in the electrostatic interaction strength by increasing $\sigma$ leads to three distinct regimes characterized by $l_B/l$. At low $l_B /l$ (close to zero), a narrow region of isotropic-nematic coexistence prevails, whose origin lies in the entropy of the system. On the other hand, at relatively high $l_B /l$ (close to unity), the completely ordered nematic phase becomes one of the coexisting phases. However, its origin lies in the electrostatic attraction between oppositely charged polyelectrolytes. At intermediate values of $l_B$, isotropic-isotropic or isotropic-nematic coexistence can prevail depending on the concentration of the polyelectrolytes. Also, in this regime, at a particular value of $l_B$, three phases (low density isotropic, moderate density isotropic, and high density nematic) coexist with each other. The value of $l_B$ at which the three phase coexistence takes place depends sensitively on the linear charge density of the polyelectrolytes. We have also investigated the effect of counterions on the phase coexistence boundaries (isotropic-isotropic and isotropic-nematic). Comparison of the results for the systems with and without counterions reveals that sufficiently high concentrations of counterions suppress both isotropic-isotropic and isotropic-nematic coexistence. A key prediction of theory is the result that the concentration of counterions in the dense (or coacervate) phase is slightly higher than in the dilute (or supernatant) phase. Also, comparison of the phase separation boundaries between comparable rodlike and flexible polyelectrolyte mixtures reveals that the isotropic-isotropic macrophase separation regime is broader in the case of weakly charged flexible polyelectrolytes. Furthermore, in this work, we have limited ourselves to the phase separation regime in the mixtures containing weakly charged polyelectrolytes close to the critical point. We have found that the critical points for the isotropic-isotropic and isotropic-nematic phase transitions exist at very low number density of rods ($n/\Omega \sim 1/L^2 d$) and weak electrostatic interaction strengths (i.e., $l_B/l \ll 1$). At this point, we remark on the range of validity of the theory to describe the phase boundaries and some of the future directions. There are three key issues, which limits the validity of the theory. In future, we’ll extend the theory to higher electrostatic interaction strengths by addressing the issues mentioned below. First issue is the use of the random phase approximation (RPA) to compute the contribution to the free energy coming from the electrostatic correlations. A typical way to judge the validity of the RPA is to compare the mean field contribution with the correlation term in the free energy. Due to the fact that the mean field contribution to the free energy coming from the electrostatics is zero and correlation terms beyond the RPA are not available, the range of validity of RPA can’t be inferred directly. However, general consensus[@borue_complexation; @muthu_double; @joanny_leibler; @kaji_kanaya] is that the RPA for flexible polymers is valid for concentrations above the overlap concentration. In this work, the coacervate phase has concentration above the overlap concentration and the supernatant phase is very dilute. Indeed, the supernatant phase is not well described by the RPA. On the other hand, the RPA is suitable for a very good quantitative description of the phase boundary describing the coacervate phase. Remarkably, the RPA predicts a supernatant phase with almost zero density in qualitative agreement with the experiments[@dubin_review; @pe_complex_reviews; @voorn_review; @veis_complexation; @kabanov_work1; @kabanov_work2; @dautzenberg_work1; @pogodina_work; @dautzenberg_work2]. Furthermore, the RPA describing the coacervate phase boundary in the case of flexible polyelectrolyte mixtures has been compared extensively with the experiments[@leibler_borukhov] and simulations[@jay_yuri_2]. Indeed, agreement between the theory, simulations and experiments is remarkable. In order to go beyond the RPA, Hartree approximation[@glenn_book] can be used, which can provide a more quantiative description for the supernatant phase also. However, we leave it for future work. Second issue arises due to the use of the Onsager second virial approach to describe the steric effects in the case of long rods. It is well-known[@edwardsbook; @straley_review] that the approach only works in the limit of very long aspect ratio of the rods. In fact, the limit of validity of the Onsager corresponds to $L/d \gg 10$. Furthermore, the second virial approach is strictly valid for low number density of the rods. For dense systems, higher order terms needs to be considered. For very long rods, the isotropic-nematic and isotropic-isotropic phase transitions occur at low enough number densities of the rods (which is of the order of overlap concentration). This is the reason the second virial approach is able to correctly predict these phase transitions at low electrostatic interaction strengths. However, an increase in the electrostatic interaction strength causes the density of the coacervate phase to increase and the virial approach breaks down. This deficiency of the theory can be removed by considering the effect of higher order terms[@parsons_paper; @lee_higher], which we’ll consider in future. Third issue is the ignorance of charge renormalization while describing phase separation. In this work, the phase separation regime corresponding to very low electrostatic interaction strengths is described assuming that there is no counterion adsorption on the polyelectrolytes. In this regime, the issue of charge renormalization during phase separation can be safely ignored. However, with the increase in the electrostatic interaction strength, counterions can adsorb on the polyelectrolytes and modulate their charge. For this part of the phase diagram, the degree of ionization for each polyelectrolyte in each phase needs to be computed, while equating the chemical potential and pressure of each component in the two phases[@levin_charge_regularization; @muthu_charge_regularization]. In other words, the possibility of different degree of ionizations in daughter phases needs to be explored. In this work, we have limited the theoretical study to $l_B/l < 1$, where issue of counterion adsorption can be safely ignored. A related issue is the consideration of ionic clusters (such as dipoles, quadrapoles etc.) formed as a result of strong electrostatic attraction between different oppositely charged components. It has been shown[@fisher_levin; @fisher] in the literature that the issue of ion-pairing has to be taken into account in the case of hard sphere model for simple electrolytes (also known as restricted primitive model (RPM)) to correctly match the simulation results in the critical regime. This is a manifestation of the fact that the critical point for the RPM, as predicted by the Debye-Hückel theory, corresponds[@fisher_levin] to $q^2 l_B^\star/a = 16$, where $q$ is the charge per sphere, $l_B^\star$ is the Bjerrum length at the critical point and $a$ is the diameter of the hard spheres. For symmetrical electrolytes, $a$ is the same for both kinds of charged spheres. Also, for monovalent electrolytes, $q=1$ and the critical point exists at very low temperature such that $l_B^\star/a = 16$. At such low temperatures, indeed one has to extend the Debye-Hückel theory (which is a RPA like) by including atleast dipolar interactions in the regime near the critical point. However, in contrast to the symmetric electrolytic mixtures, the phase separation regime close to the critical point in the case of mixtures containing long flexible or rodlike polyelectrolytes can be well described within RPA without any consideration of ion-ion, ion-polyelectrolyte (which is the same as the counterion adsorption) or polyelectrolyte-polyelectrolyte pairs. This is an outcome of the fact that now the phase separation takes place at relatively higher temperatures and low densities. In this regime, the mixtures of oppositely charged polyelectrolytes behave as weakly correlated liquids[@rubinstein_pairing]. However, if we increase the electrostatic interaction strength and go far from the critical regime, there is a competition for pairing between different oppositely charged species. A rough estimate for the minimum driving force for pairing between the ions is given by Bjerrum’s theory[@ebeling_work; @levin_review] of ion-pairs. According to the theory, life time of the paired state in the case of two oppositely charged ions (while undergoing thermal motion) significantly increases when $q^2 l_B/r > 2$, $r$ being the distance between the ions. In this regime, the electrostatic attraction takes over the thermal energy of ions and ion-ion pair (or dipole) needs to be considered as a new species. This relation can be cast in terms of number density of ions ($= n_c/\Omega$) by using $r \sim (n_c/\Omega)^{-1/3}$ so that $q^2l_B/l > 1/(n_c/\Omega)^{1/3}$. Issue of ion pairs/clusters formed as a result of binding, which involves charged monomers is more suble compared to the issue of pairing in small ions. However, we can estimate the regime, where one has to explicitly consider the binding between the polyelectrolytes and oppositely charged counterions or polyelectrolytes. Carrying out a single chain analysis, it has been shown that the binding of counterions on flexible and rodlike polyelectrolytes becomes important[@manning; @counterion_adsorption_muthu] roughly around $l_B/l > 1$. Similarly, an analysis of the system containing two oppositely charged flexible polyelectrolytes[@zhaoyang_complex] reveals that the complexation takes place only for strong electrostatic interaction strengths (i.e., $\sigma^2 l_B > 1$). For rodlike polyelectrolytes, the electrostatic interaction strength required for complexation needs to be stronger compared to the flexible polyelectrolytes due to weaker (logarithmic) electrostatic potential for rodlike polymers. These single chain analyses provide a clear picture about the dilute solution regime. In the regime above the overlap concentration for polymers, situation is more complicated and one has to consider the multi-chain effects. However, one can carry out a mean-field analysis[@monica_complexation_2; @semenov_rubinstein] to estimate the fraction of charged monomers involved in binding (say, $\Gamma$). It can be readily shown[@monica_complexation_2] that the fraction is given by the relation $\Gamma/(1-\Gamma)^2 = (\sigma/l )\left[c_p/(2 Ld)\right]\exp(\|E/k_B T\|)$, $E/k_BT \sim l_B$ being the energy gain per pair. In this work, we have considered weakly charged polyelectrolyte solutions at very low electrostatic interaction strengths so that the fraction of charged monomers involved in pairing is close to zero. At present we are not aware of experimental data sets sufficient for a comprehensive test of the theoretical predictions made here. With this aim, we welcome interactions with experimental groups to define appropriate systems and experimental protocols. ACKNOWLEDGEMENT {#acknowledgement .unnumbered} =============== \[acknowledgement\] We are grateful to Prof. Philip A. Pincus and Dr. Yongseok Jho for useful discussions on the phase behavior of polyelectrolytes. Financial support was provided by the UCSB-MIT-Caltech Institute for Collaborative Biotechnologies and the Materials Research Laboratory (MRL) at UCSB.. This work made use of the MRL Computing Facilities supported by the MRSEC Program of the National Science Foundation under Award No. DMR05-20415. APPENDIX A : Partition function for mixtures of rodlike polyelectrolytes {#app:A .unnumbered} ========================================================================= Here, we present the details of our derivation of the free energy for the mixtures of rodlike polyelectrolytes in the presence of counterions as described in Section II. Different parameters representing the number of rods and counterions, length and diameter of the rods, and charge along the rods have already been described in Section II. In terms of these parameters, the partition function can be written as $$\begin{aligned} Z &=& \frac{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)} d\mathbf{r}_{j}d\mathbf{u}_{j} \right]}{\prod_{k=+,-}n_k!\prod_{j' = c+,c-}n_{j'}!} \prod_{j' = c+,c-} \prod_{j'' = 1}^{n_{j'}} \int d\mathbf{r}_{j''}\exp \left [- \frac{H_{int}}{k_BT} \right ]\end{aligned}$$ where $$\begin{aligned} \frac{H_{int}}{k_BT} &=& \frac{H_{w}}{k_BT} + \frac{H_{e}}{k_BT}, \\ \frac{H_{w}}{k_BT} &=& \frac{1}{2}\sum_{j=1}^{n_+ + n_-}\sum_{k=1}^{n_+ + n_-} W(\mathbf{r}_j,\mathbf{u}_{j},\mathbf{r}_k,\mathbf{u}_{k}), \\ \frac{H_{e}}{k_BT} &=& \frac{l_B}{2}\int d\mathbf{r} \int d\mathbf{r}' \frac{\hat{\rho}_e(\mathbf{r})\hat{\rho}_e(\mathbf{r}')}{\|\mathbf{r} - \mathbf{r}'\|}.\end{aligned}$$ Here, $p_k(\mathbf{u}_{j},\nh)$ for $k=+,-$ is the probability of finding a rod of type $k$ oriented along unit vector $\mathbf{u}_j$ when the director is taken to be along the unit vector $\nh$. Subscript $p_k(\mathbf{u},\nh)$ under the integral symbol means that the integration need to be carried out under the constraint that the orientational distribution function for the positive and negative rods are $p_+(\mathbf{u},\nh)$ and $p_-(\mathbf{u},\nh)$, respectively. Also, hamiltonian $H_{int}$ is divided into contributions coming from the short range excluded volume interactions ($H_w$) and the long range electrostatic interactions ($H_e$). Furthermore, in the expression for $H_w, W(\mathbf{r}_j,\mathbf{u}_{j},\mathbf{r}_k,\mathbf{u}_{k})$ characterizes the excluded volume interactions between two rods whose centers are located at $\mathbf{r}_j$ and $\mathbf{r}_k$ with their axes oriented along $\mathbf{u}_j$ and $\mathbf{u}_k$, respectively. Also, $l_B$ is Bjerrum length and $\hat{\rho}_e$ in the expression for $H_e$ is the local charge density defined as $$\begin{aligned} \hat{\rho}_e(\mathbf{r}) &=& Z_{+}\sigma_+ \hat{\rho}_+(\mathbf{r}) + Z_{-}\sigma_- \hat{\rho}_-(\mathbf{r}) + \sum_{j' = c+,c-} Z_{j'} \hat{\rho}_{j'}(\mathbf{r}) \\ \hat{\rho}_{k}(\mathbf{r}) &=& N_k \sum_{j=1}^{n_{k}}\frac{1}{L_k}\int_{-L_k/2}^{L_k/2} ds_{j} \, \delta (\mathbf{r}_{j} + s_{j} \mathbf{u}_{j} - \mathbf{r}), \quad \mbox{for} \quad k = +,-,\\ \hat{\rho}_{j'}(\mathbf{r}) &=& \sum_{j''=1}^{n_{j'}} \delta (\mathbf{r}-\mathbf{r}_{j''}), \quad \mbox{for} \quad j' = c+,c-,\end{aligned}$$ where $Z_{\gamma}$ is the valency (with sign) of the charged species of type $\gamma$ and $N_k$ is the number of monomers for the rod of type $k$ so that $L_k = N_k l_k$, $l_k$ being the monomer length. Also, $s_j$ in the expression for $\hat{\rho}_k(\mathbf{r})$ is the contour variable used to locate any monomer along the $j^{th}$ rod of type $k$. In order to proceed further, we rewrite $Z$ as $$\begin{aligned} Z& = & Z_o \left < \exp \left [- \frac{H_{int}}{k_BT} \right ]\right >_{p_+,p_-} \label{eq:parti}\end{aligned}$$ where $Z_o$ is the partition function for a non-interacting system with the same orientational distributions (i.e., $p_+,p_-$) and also the normalization factor in the expression for $Z$, given by $$\begin{aligned} Z_o &=& \exp\left[- \frac{F_o}{k_BT}\right] = \frac{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u},\nh)}d\mathbf{r}_{j}d\mathbf{u}_{j} \right]}{\prod_{k=+,-}n_k!\prod_{j' = c+,c-}n_{j'}!}\int \prod_{j' = c+,c-} \prod_{j'' = 1}^{n_{j'}} d\mathbf{r}_{j''} \nonumber \\ &=& \prod_{\gamma = +,-,c+,c-}\left[\frac{\Omega^{n_\gamma}}{n_\gamma!}\right]\prod_{k=+,-}\prod_{j=1}^{n_k} \int_{p_k(\mathbf{u},\nh)}d\mathbf{u}_{j} . \end{aligned}$$ Using Stirling’s approximation for factorials $ n! \simeq n (\ln n - 1)$ and considering different ways of distributing $n_k$ rods for a given orientational probability distribution function $p_k(\mathbf{u}_{j},\nh)$ along the surface of a unit sphere[@edwardsbook] $$\begin{aligned} \frac{F_o}{k_BT} & = & \sum_{j=+,-}n_j \int d\mathbf{u}_{j} p_j(\mathbf{u}_{j},\nh) \ln \left[4\pi p_j(\mathbf{u}_{j},\nh)\right] + \sum_{\gamma = +,-,c+,c-} n_{\gamma} \left[\ln \frac{n_{\gamma}}{\Omega} - 1\right].\end{aligned}$$ Also, $$\begin{aligned} \left<\exp \left [- \frac{H_{int}}{k_BT} \right ]\right>_{p_+,p_-} &=& \frac{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)} d\mathbf{r}_{j}d\mathbf{u}_{j} \right]\prod_{j' = c+,c-} \prod_{j'' = 1}^{n_{j'}} \int d\mathbf{r}_{j''}\exp \left [- \frac{H_{int}}{k_BT} \right ] }{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[ \int_{p_k(\mathbf{u}_{j},\nh)} d\mathbf{r}_{j}d\mathbf{u}_{j}\right]\prod_{j' = c+,c-} \prod_{j'' = 1}^{n_{j'}} \int d\mathbf{r}_{j''}}. \nonumber \\ &&\end{aligned}$$ Using the Hubbard-Stratonovich transformation[@glenn_book] for the electrostatic part i.e., $H_e$ and using Fourier transform defined for any arbitrary function $f(\mathbf{r})$ by $$\begin{aligned} f(\mathbf{r}) &=& \int \frac{d^{3}\mathbf{q}}{(2\pi)^3}f_q e^{i \mathbf{q}\cdot \mathbf{r}},\end{aligned}$$ the partition function can be written as $$\begin{aligned} \frac{Z}{Z_o} &=& \frac{ \int \prod_{q}D[\psi_q] \exp\left[-\frac{1}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q \frac{q^2}{4\pi l_B}\psi_{-q} + \sum_{j' = c+,c-}n_{j'}\ln Q_{j'}\left\{\psi_q\right\} \right ] }{\int \prod_{q}D[\psi_q]\exp\left[-\frac{1}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q \frac{q^2}{4\pi l_B}\psi_{-q}\right ] } \nonumber \\ && \frac{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)}d\mathbf{r}_{j}d\mathbf{u}_{j} \right] \exp \left [- \frac{H_{w}}{k_BT} + i \sum_{k=+,-}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q Z_k \sigma_k \rho_{k,-q}\right] }{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)} d\mathbf{r}_{j}d\mathbf{u}_{j}\right] }\nonumber \\ && \label{eq:parti_all}\end{aligned}$$ where $$\begin{aligned} \rho_{k,q} &=& \int\, d\mathbf{r} \hat{\rho}_k(\mathbf{r}) e^{i\mathbf{q}\cdot\mathbf{r}} = \sum_{i=1}^{n_{k}}N_k e^{i\mathbf{q}.\mathbf{r}_i} \frac{\sin\left[\left(\mathbf{q}\cdot\mathbf{u}_i\right)L_k/2\right]}{\left[\left(\mathbf{q}\cdot\mathbf{u}_i\right)L_k/2\right]}, \quad \mbox{for} \quad k = +,- \label{eq:fourier_den}\end{aligned}$$ and $$\begin{aligned} Q_{j'}\left\{\psi_q \right\} & = & \frac{1}{\Omega}\int d\mathbf{r} \exp \left[i \int \frac{d^{3}q}{(2\pi)^3}\psi_q Z_{j'}e^{i\mathbf{q}\cdot\mathbf{r}} \right] \label{eq:partition_smallions}\end{aligned}$$ So far the partition function is exact. However, evaluation of the exact partition function is a tedious task. Useful insights can be obtained by invoking the following approximations. From Eq. \[eq:partition\_smallions\] $$\begin{aligned} \sum_{j' = c+,c-}\ln Q_{j'}^{n_{j'}}\left\{\psi_q\right\} &\simeq& \sum_{j' = c+,c-}\ln \left[1 - \frac{Z_{j'}^2 n_{j'}}{2\Omega}\int \frac{d^{3}q}{(2\pi)^3}\psi_q \psi_{-q}\right ] \nonumber \\ &\simeq & - \sum_{j' = c+,c-}\frac{Z_{j'}^2 n_{j'}}{2\Omega}\int \frac{d^{3}q}{(2\pi)^3}\psi_q \psi_{-q}. \label{eq:log_exp}\end{aligned}$$ Note here that the linear terms arising from the expansion of the exponential in $Q_{j'}$ vanish in the sum due to the presence of oppositely charged counterions, i.e. $Z_{c+} = - Z_{c-}$. Furthermore, approximating the logarithm by the first term in the expansion in Eq. \[eq:log\_exp\] is strictly valid for dilute concentrations of the counterions so that $n_{j'}/\Omega$ is small. Using this approximation, Eq. \[eq:parti\_all\] becomes $$\begin{aligned} \frac{Z}{Z_o} &=& \frac{ \int \prod_{q}D[\psi_q] \exp\left[-\frac{1}{8\pi l_B}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q \left(q^2 + \kappa^2\right)\psi_{-q} \right ] }{\int \prod_{q}D[\psi_q]\exp\left[-\frac{1}{8\pi l_B}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q q^2\psi_{-q}\right ] } \nonumber \\ && \frac{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)}d\mathbf{r}_{j}d\mathbf{u}_{j} \right] \exp \left [- \frac{H_{w}}{k_BT} + i \sum_{k=+,-}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q Z_k \sigma_k \rho_{k,-q}\right] }{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)} d\mathbf{r}_{j}d\mathbf{u}_{j}\right] }\nonumber \\ && \label{eq:parti_all2}\end{aligned}$$ where we have defined $\kappa^2 = 4\pi l_B \sum_{j'} Z_{j'}^2 n_{j'}/\Omega $ so that $\kappa^{-1}$ is the Debye length. Now the integrals over the spatial and orientational degrees of freedom under the constraint of the given distribution functions can be carried out using the approximation described in Ref. [@edwardsbook]. To be explicit, $$\begin{aligned} I_{p}\left\{\psi_q\right\} &=& \frac{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)}d\mathbf{r}_{j}d\mathbf{u}_{j} \right] \exp \left [- \frac{H_{w}}{k_BT} + i \sum_{k=+,-}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q Z_k \sigma_k \rho_{k,-q}\right] }{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)} d\mathbf{r}_{j}d\mathbf{u}_{j}\right] }\nonumber \\ &=& \frac{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)}d\mathbf{r}_{j}d\mathbf{u}_{j} \right] \left[1 - \left\{1 - \exp \left (- \frac{H_{w}}{k_BT}\right) \right\}\right] \exp\left[i \sum_k\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q Z_k \sigma_k \rho_{k,-q}\right] }{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)} d\mathbf{r}_{j}d\mathbf{u}_{j}\right] } \nonumber \\ &=& 1 - \frac{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)}d\mathbf{r}_{j}d\mathbf{u}_{j} \right] \left\{1 - \exp \left (- \frac{H_{w}}{k_BT}\right) \right\}}{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)} d\mathbf{r}_{j}d\mathbf{u}_{j}\right]} \nonumber \\ && - \frac{1}{2}\frac{\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)}d\mathbf{r}_{j}d\mathbf{u}_{j} \right]\left[\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \psi_q \sum_k Z_k \sigma_k \rho_{k,-q}\right]^2 } {\prod_{k=+,-}\prod_{j=1}^{n_k}\left[\int_{p_k(\mathbf{u}_{j},\nh)} d\mathbf{r}_{j}d\mathbf{u}_{j}\right]} + \cdots \label{eq:orientation_int}\end{aligned}$$ where we have ignored a cross-term between the excluded volume interactions and the electrostatics part to keep the calculations analytically tractable. Now, assuming that the excluded volume interactions between the rods occur independently of each other, second term in the series in Eq. \[eq:orientation\_int\] can be evaluated.[@edwardsbook] Third term in the series is also straigthforward to evaluate after plugging the expression for $\rho_{k,q}$ given in Eq. \[eq:fourier\_den\]. Now, exponentiating the series $$\begin{aligned} \frac{Z}{Z_o} &=& \exp\left[-\frac{1}{2\Omega} \sum_{j,k=+,-}' n_j n_k \int d\mathbf{u} \int d\mathbf{u}' p_j(\mathbf{u},\nh) \left [2L_j L_k d \|\mathbf{u}\times\mathbf{u}'\| \right]p_k(\mathbf{u}',\nh) \right]\nonumber \\ && \frac{ \int \prod_{q}D[\psi_q] \exp\left[-\frac{1}{8\pi l_B}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q \left\{q^2 + \kappa^2 + \sum_{k=+,-}\kappa_k^2 t_k(qL_k)\right\}\psi_{-q}\right ] }{\int \prod_{q}D[\psi_q]\exp\left[-\frac{1}{8\pi l_B}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q q^2 \psi_{-q}\right ] } \nonumber \\ &&\end{aligned}$$ where we have defined $\kappa_k^2 = 4\pi l_B Z_{k}^2 \sigma_{k}^2 L_k^2 n_k/\Omega$ and $$\begin{aligned} t_k(q L_k) &=& \int d\mathbf{u} p_{k}(\mathbf{u},\nh)\left[\frac{\sin\left[\left(\mathbf{q}.\mathbf{u}\right)L_k/2\right]}{\left[\left(\mathbf{q}.\mathbf{u}\right)L_k/2\right]}\right]^2, \quad \mbox{for} \quad k = +,-.\end{aligned}$$ Also, the primed superscript means that the $j=k$ terms are omitted from the double sum. By carrying out the Gaussian functional integrals over $\psi_q$ and subtracting out the free energy in the low density limit, Eq. \[eq:free\_rods\] is obtained. APPENDIX B : Limiting cases for the electrostatic part of the free energy of the isotropic phase {#app:B .unnumbered} ================================================================================================ Here, we provide a derivation of the approximate form of the electrostatic contribution to the free energy of the isotropic phase in the limiting cases of $L_k \rightarrow \infty$ (cf. Eqs. \[eq:fe\_limit1\] and \[eq:fe\_limit2\]). In particular, we focus on the logarithmic corrections to the free energy of the mixture of charged rods as already described using scaling analysis in section \[sec:scaling\_section\]. The electrostatic component of the free energy is given by $$\begin{aligned} \frac{F_e\left\{L_k \rightarrow \infty \right\}}{k_BT} &=& \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{\kappa^2 }{q^2} + \frac{\pi \kappa_p^2}{q^3}\right] - \frac{\kappa^2 }{q^2} - \frac{\pi \kappa_p^2}{q^3} \right \} \end{aligned}$$ For weakly charged rods so that $\kappa_p^2 \ll 1$, $$\begin{aligned} \frac{F_e\left\{L_k \rightarrow \infty \right\}}{k_BT} &=& \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{\kappa^2 }{q^2} \right] - \frac{\kappa^2 }{q^2} \right \}- \frac{\Omega}{2}\int\frac{d^{3}\mathbf{q}}{(2\pi)^3} \frac{\pi \kappa_p^2 \kappa^2}{q^3 \left(q^2 + \kappa^2\right)} \nonumber \\ && - \frac{\Omega}{4}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \frac{\pi^2 \kappa_p^4}{q^2 \left(q^2 + \kappa^2\right)^2} + \cdots \nonumber \\ &\simeq& -\frac{\Omega \kappa^3}{12 \pi} - \frac{\Omega \kappa_p^2}{4 \pi}\ln \left[\frac{\kappa}{q}\right]_{q\rightarrow 0} - \frac{\Omega \pi \kappa_p^4}{32 \kappa^3} + \cdots, \end{aligned}$$ which is Eq. \[eq:fe\_limit1\]. On the other hand, for strongly charged rods in weakly screened solutions $\kappa_p^2 \gg 1$ and $\kappa \rightarrow 0$. In this limit, $$\begin{aligned} \frac{F_e\left\{L_k \rightarrow \infty \right\}}{k_BT} &\simeq& \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left \{\ln \left[1 + \frac{\kappa^2 }{q^2} \right] - \frac{\kappa^2 }{q^2}\right\} \nonumber \\ && + \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{\pi \kappa_p^2}{q^3 }\left\{ 1 - \frac{\kappa^2}{q^2} + \frac{\kappa^4}{q^4}\right\}\right] - \frac{\pi \kappa_p^2}{q^3 } \right \}\nonumber \\ &=& -\frac{\Omega \kappa^3}{12 \pi} + \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{\pi \kappa_p^2}{q^3 }\right] - \frac{\pi \kappa_p^2}{q^3 } \right \} + \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \ln \left[ 1 - \frac{\frac{\kappa^2}{q^2} - \frac{\kappa^4}{q^4}}{1 + \frac{\pi \kappa_p^2}{q^3 }}\right] \nonumber \\ &=& -\frac{\Omega \kappa^3}{12 \pi} + \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \left\{\ln \left[1 + \frac{\pi \kappa_p^2}{q^3 }\right] - \frac{\pi \kappa_p^2}{q^3 } \right \} - \frac{\Omega}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3} \frac{\frac{\kappa^2}{q^2} - \frac{\kappa^4}{q^4}}{1 + \frac{\pi \kappa_p^2}{q^3 }} + \cdots \nonumber \\ &=& -\frac{\Omega \kappa^3}{12 \pi} - \frac{\Omega \kappa_p^2}{6 \pi} \ln \left[\frac{\sqrt{\pi}\kappa_p}{q}\right]_{q\rightarrow 0} + \frac{\Omega \kappa^2 \kappa_p^{2/3}}{6\sqrt{3}\pi^{2/3}} - \frac{\Omega \kappa^4}{6\sqrt{3}\pi^{4/3}\kappa_p^{2/3}}- \frac{\Omega \kappa^2}{4 \pi^2} \left[q\right]_{q\rightarrow \infty} + \cdots ,\nonumber \\ && \end{aligned}$$ which is Eq. \[eq:fe\_limit2\]. APPENDIX C : Partition function for mixtures of flexible polyelectrolytes {#app:C .unnumbered} ========================================================================== Here, we present the derivation of the free energy for mixtures of oppositely charged flexible polyelectrolytes in the presence of their counterions. The flexible polyelectrolytes are represented by continuous curves so that $\mathbf{R}_{jk}(s_j)$ is the position vector of the $s^{th}$ monomer along the $j^{th}$ chain of type $k$. Furthermore, the contour lengths of chains of type $k$ is taken to be $L_k = N_k l_k$, $l_k$ being the Kuhn’s segment length. Accounting for the conformational degrees of freedom of the flexible chains by the path integral representation, the partition function for this system can be written as $$\begin{aligned} Z &=& \frac{\prod_{k=+,-}\prod_{j=1}^{n_k} \int D\left[\mathbf{R}_{jk}\right]\exp\left[-\frac{H_0}{k_B T}\right]}{{\prod_{k=+,-}n_k!\prod_{j' = c+,c-}n_{j'}!}}\prod_{j' = c+,c-}\prod_{j'' = 1}^{n_{j'}}\int d\mathbf{r}_{j''} \exp \left [- \frac{H_{int}}{k_BT} \right ]. \end{aligned}$$ where $$\begin{aligned} \frac{H_{0}}{k_BT} &=& \sum_{k=+,-}\sum_{j=1}^{n_k}\frac{3}{2l_k^2}\int_{0}^{N_k}ds_j \left(\frac{\partial \mathbf{R}_{jk}(s_j)}{\partial s_j}\right)^2 \\ \frac{H_{int}}{k_BT} &=& \frac{H_{w}}{k_BT} + \frac{H_{e}}{k_BT} \\ \frac{H_{w}}{k_BT} &=& \frac{w}{2}\int d\mathbf{r} \left[\hat{\rho}_+(\mathbf{r}) + \hat{\rho}_-(\mathbf{r})\right]^2 \\ \frac{H_{e}}{k_BT} &=& \frac{l_B}{2}\int d\mathbf{r} \int d\mathbf{r}' \frac{\hat{\rho}_e(\mathbf{r})\hat{\rho}_e(\mathbf{r}')}{\|\mathbf{r} - \mathbf{r}'\|}.\end{aligned}$$ Note here that in contrast to the rodlike polymers, the Hamiltonian for the flexible polyelectrolytes has an additional contribution coming from the chain connectivity ($H_0$) in addition to the contributions coming from short range excluded volume ($H_w$) and long range electrostatic interactions ($H_e$). Furthermore, in the spirit of polymer field theories[@glenn_book], we have replaced $H_w$ by the delta functional form and defined a excluded volume parameter $w$ to take care of the short range interactions. Also, the microscopic densities in the above equations are defined as $$\begin{aligned} \hat{\rho}_e(\mathbf{r}) &=& Z_{+}\sigma_+ \hat{\rho}_+(\mathbf{r}) + Z_{-}\sigma_- \hat{\rho}_-(\mathbf{r}) + \sum_{j'=c+,c-} Z_{j'} \hat{\rho}_{j'}(\mathbf{r}) \\ \hat{\rho}_{k}(\mathbf{r}) &=& \sum_{j=1}^{n_{k}}\int_{0}^{N_k} ds_{j} \, \delta \left[\mathbf{r} - \mathbf{R}_{jk}(s_j)\right], \quad \mbox{for} \quad k = +,-,\\ \hat{\rho}_{j'}(\mathbf{r}) &=& \sum_{j''=1}^{n_{j'}} \delta (\mathbf{r}-\mathbf{r}_{j''}), \quad \mbox{for} \quad j' = c+,c-.\end{aligned}$$ To proceed further, we rewrite the partition function as $$\begin{aligned} Z& = & Z_o \left< \exp \left [- \frac{H_{int}}{k_BT} \right ]\right>_{R_+,R_-} \label{eq:parti_flex}\end{aligned}$$ where $Z_o$ is the partition function of a mixture of non-interacting chains and counterions. Explicitly, it is given by $$\begin{aligned} Z_o &=& \frac{\prod_{k=+,-}\prod_{j=1}^{n_k} \int D\left[\mathbf{R}_{jk}\right]\exp\left[-\frac{H_0}{k_BT}\right]}{\prod_{k=+,-}n_k!\prod_{j'=c+,c-}n_{j'}!} \prod_{j' = c+,c-}\prod_{j'' =1}^{n_{j'}}\int d\mathbf{r}_{j''}. \end{aligned}$$ Using the Stirling’s approximation $n! \simeq n \ln n - n$, we get $$\begin{aligned} \frac{F_o}{k_BT} & = & - \ln Z_o = -\sum_{k=+,-} n_k \ln Q_k^0 + \sum_{\gamma = +,-,c+,c-} n_{\gamma} \left[\ln \frac{n_{\gamma}}{\Omega} - 1\right],\end{aligned}$$ where $F_o$ is the Helmholtz free energy of the mixtures of non-interacting chains and counterions. Also, $Q_k^0$ is the partition function of a single Gaussian chain of type $k$, given by $$\begin{aligned} Q_k^0 &=& \frac{1}{\Omega}\int D\left[\mathbf{R}_{k}\right]\exp\left[-\frac{3}{2l_k^2}\int_{0}^{N_k}ds \left(\frac{\partial \mathbf{R}_{k}(s)}{\partial s}\right)^2\right].\end{aligned}$$ In writing the above equation, we have dropped the index representing the chain number in the path integral. Furthermore, $$\begin{aligned} \left<\exp \left [- \frac{H_{int}}{k_BT} \right ]\right>_{R_+,R_-} &=& \frac{ \prod_{k}\prod_{j=1}^{n_k} \int D\left[\mathbf{R}_{jk}\right]\exp\left[-\frac{H_0}{k_BT}\right]\prod_{j'}\prod_{j''=1}^{n_{j'}}\int d\mathbf{r}_{j''} \exp \left [- \frac{H_{int}}{k_BT} \right ] }{ \prod_{k}\prod_{j=1}^{n_k}\int D\left[\mathbf{R}_{jk}\right]\exp\left[-\frac{H_0}{k_BT}\right]\prod_{j'}\prod_{j''=1}^{n_{j'}}\int d\mathbf{r}_{j''}}, \nonumber \\ &&\end{aligned}$$ where $k=+,-$ and $j' = c+,c-$. Using Hubbard-Stratonovich transformation[@glenn_book] for the excluded volume and electrostatic parts in the Hamiltonian, and using three dimensional Fourier transforms as defined in Appendix A, the partition function can be written as $$\begin{aligned} \frac{Z}{Z_o'} &=& \frac{ \int \prod_{q}\left[D[\phi_q] D[\psi_q]\right]\exp\left[-\frac{1}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\left[\phi_q \frac{1}{w}\phi_{-q} + \psi_q \frac{q^2}{4\pi l_B}\psi_{-q} \right ]\right]}{\int \prod_{q}\left[D[\phi_q] D[\psi_q]\right]\exp\left[-\frac{1}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\left(\phi_q \frac{1}{w}\phi_{-q} + \psi_q \frac{q^2}{4\pi l_B}\psi_{-q}\right)\right ] }\times \nonumber \\ && \exp\left[ \sum_{k=+,-}n_k \ln Q_k\left\{\phi_q,\psi_q \right\} + \sum_{j'=c+,c-}n_{j'}\ln Q_{j'}\left\{\psi_q\right\}\right]\end{aligned}$$ where $$\begin{aligned} Z_o' &=& Z_o\exp\left[-\frac{w}{2 \Omega}\left( n_+ N_+ + n_- N_-\right)^2 \right ]\end{aligned}$$ and $Q_{j'}$ is the partition function for a single small ion of type $j'$, given by Eq. \[eq:partition\_smallions\]. Also, $Q_k$ is the partition function for a single chain of type $k$, given by $$\begin{aligned} Q_{k}\left\{\phi_q,\psi_q \right\} & = & \frac{\int D\left[\mathbf{R}_k\right]\exp\left[-\frac{3}{2l_k^2}\int_{0}^{N_k}ds \left(\frac{\partial \mathbf{R}_k(s)}{\partial s}\right)^2 + i \int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\left( \phi_q + Z_k \sigma_k \psi_q \right) \rho_{k,-q} \right]}{\int D\left[\mathbf{R}_k\right]\exp\left[-\frac{3}{2l_k^2}\int_{0}^{N_k}ds \left(\frac{\partial \mathbf{R}_k(s)}{\partial s}\right)^2\right]}.\nonumber \\ &&\end{aligned}$$ Here, $\rho_{k,q}$ is the Fourier component of the microscopic density of a single chain of type $k$, given by $$\begin{aligned} \rho_{k,q} &=& \int_{0}^{N_k} ds e^{i\mathbf{q}\cdot\mathbf{R}_k(s)}, \quad \mbox{for} \quad k = +,-.\end{aligned}$$ Expanding in powers of $\rho_{k,q}$ and using translational invariance $$\begin{aligned} Q_{k} & \simeq & 1 - \frac{1}{2\Omega} \int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\left( \phi_q + Z_k \sigma_k \psi_q \right) \left<\rho_{k,q} \rho_{k,-q}\right> \left( \phi_{-q} + Z_k \sigma_k \psi_{-q} \right) \nonumber \\ &=& 1 - \frac{1}{2\Omega} \int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\left( \phi_q + Z_k \sigma_k \psi_q \right) N_k^2g\left(q^2N_kl_k^2/6 \right)\left( \phi_{-q} + Z_k \sigma_k \psi_{-q} \right) \nonumber \\ & \simeq & \exp \left[- \frac{1}{2\Omega} \int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\left( \phi_q + Z_k \sigma_k \psi_q \right) N_k^2g\left(q^2N_kl_k^2/6 \right)\left( \phi_{-q} + Z_k \sigma_k \psi_{-q} \right) \right]\end{aligned}$$ where $g(x)$ is Debye function, given by $$\begin{aligned} g(x) &=& \frac{2}{x^2}\left[e^{-x} - 1 + x\right]\end{aligned}$$ Consider the special case of symmetric mixtures so that $Z_+ \sigma_+ n_+ N_+^2 g\left(q^2N_+ l_+^2/6 \right) = -Z_- \sigma_- n_- N_-^2 g\left(q^2N_-l_-^2/6 \right)$. For this particular case, the cross terms containing $\phi \psi$ vanishes. Using the approximation for the partition function of small ions as given by Eq. \[eq:log\_exp\], the partition function for the symmetric mixtures of oppositely charged flexible polyelectrolytes becomes $$\begin{aligned} \frac{Z}{Z_o'} &=& \frac{ \int \prod_{q}D[\phi_q] \exp\left[-\frac{1}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\phi_q \left\{\frac{1}{w} + \frac{1}{\Omega}\sum_{k=+,-}N_k^2 n_k g\left(q^2N_k l_k^2/6\right)\right\}\phi_{-q}\right ] }{\int \prod_{q}D[\phi_q]\exp\left[-\frac{1}{2}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\phi_q \frac{1}{w} \phi_{-q}\right ] }\times \nonumber \\ && \frac{ \int \prod_{q}D[\psi_q] \exp\left[-\frac{1}{8\pi l_B}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q \left\{q^2 + \kappa^2 + \sum_{k=+,-}\kappa_k^2 g\left(q^2N_k l_k^2/6\right)\right\}\psi_{-q}\right ] }{\int \prod_{q}D[\psi_q]\exp\left[-\frac{1}{8\pi l_B}\int \frac{d^{3}\mathbf{q}}{(2\pi)^3}\psi_q q^2 \psi_{-q}\right ] } \nonumber \\ &&\end{aligned}$$ where $\kappa^2 = 4\pi l_B \sum_{j'=c+,c-} Z_{j'}^2 n_{j'}/\Omega $ and $\kappa_k^2 = 4\pi l_B Z_{k}^2 \sigma_{k}^2 N_k^2 n_k/\Omega$ are the same as defined in Appendix A for rodlike polyelectrolytes. 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The symbols “I” and “N” denote regions of two-phase coexistence involving isotropic and nematic phases, respectively. $2n/\Omega, \; L$ and $d$ are the number density of positive and negative polyelectrolytes, the length, and the diameter of the charged rods, respectively. The y-axis is the ratio of the Bjerrum length $l_B$ to the monomer length $l$ and is a measure of electrostatic strength that is inversely proportional to temperature.[]{data-label="fig:phase_cartoon"}](phase_dia.eps){width="3in" height="3in"} $\begin{array}{c@{\hspace{.001in}}c@{\hspace{.001in}}} \includegraphics[width=3.5in,height=3.5in]{stability_nocount.eps}& \\ \textbf{(a)} & \\ & \\ \includegraphics[width=3.5in,height=3.5in]{stability_count.eps} & \\ \textbf{(b)} & \end{array}$ $\begin{array}{c@{\hspace{.001in}}c@{\hspace{.001in}}} \includegraphics[width=3.5in,height=3.5in]{compare_order_lbp7_N1000_dia1by50_nc.eps}& \\ \textbf{(a)} & \\ & \\ \includegraphics[width=3.5in,height=3.5in]{compare_order_lbp7_N1000_dia1by50.eps} & \\ \textbf{(b)} & \end{array}$ $\begin{array}{c@{\hspace{.001in}}c@{\hspace{.001in}}} \includegraphics[width=3.5in,height=3.5in]{poly_alpp1_N1000_dia1by50_count_effect.eps}& \\ \textbf{(a)} & \\ & \\ \includegraphics[width=3.5in,height=3.5in]{ions_alpp1_N1000_dia1by50_count_effect.eps} & \\ \textbf{(b)} & \end{array}$ ![Comparison of the envelopes for isotropic-isotropic phase separation in symmetric mixtures of the oppositely charged flexible and rodlike polyelectrolytes. Dashed curves correspond to the spinodals and solid curves represent the binodals. The curves correspond to $N = 1000,\; \tilde d = 1/50,\; \sigma = 0.1$. “I” and “I-I” denote one-phase isotropic and two-phase isotropic-isotropic coexistence regimes, respectively.[]{data-label="fig:rods_flex_bp1"}](compare_flex_rods_alpp1_N1000_dia1by50.eps){width="15cm"} $\begin{array}{c@{\hspace{.001in}}c@{\hspace{.001in}}} \includegraphics[width=3.5in,height=3.5in]{phase_diagram_nocount.eps}& \\ \textbf{(a)} & \\ & \\ \includegraphics[width=3.5in,height=3.5in]{coexist_order_nc_alpha_effect.eps} & \\ \textbf{(b)} & \end{array}$ $\begin{array}{c@{\hspace{.001in}}c@{\hspace{.001in}}} \includegraphics[width=3.5in,height=3.5in]{phase_diagram_count.eps}& \\ \textbf{(a)} & \\ & \\ \includegraphics[width=3.5in,height=3.5in]{coexist_order_count_effect.eps} & \\ \textbf{(b)} & \end{array}$ ![Counterion densities in the two coexisting phases shown in Fig. \[fig:phase\_diagram\_rods\_count\]. The counterions are uniformly distributed between the two phases for these set of parameters.[]{data-label="fig:counterions_coexisting"}](coexist_counterions_in.eps){width="15cm"} ![Effect of the total number density of rodlike polyelectrolytes on the phase diagram in symmetric mixtures in the presence of counterions and at fixed linear charge density. These plots correspond to $N = 1000, \; d/l = 1/50$ and $\sigma = 0.1$. Dashed lines in Fig. (a) correspond to the boundary beyond which the isotropic phase becomes unstable and the completely ordered nematic phase becomes one of the coexisting phases.[]{data-label="fig:den_effect_phase"}](count_effect_rods_full.eps){width="15cm"} ![Counterion densities in the coexisting phases corresponding to I-I and I-N phase transitions as shown in Fig. \[fig:den\_effect\_phase\]. In these figures, the right branch of each coexisting curve corresponds to the counterion density in the phase dense in rodlike polyelectrolytes (coacervate). It is shown that for strong enough electrostatics, the counterions also phase segregate and the counterion density is higher in the dense phase (see the results for $2n_t L^2 d/\Omega = 0.1$). These plots correspond to $N = 1000, \; d/l = 1/50$ and $\sigma = 0.1$. []{data-label="fig:den_effect_phase2"}](count_effect_count_full.eps){width="15cm"} ![For table of contents use only](rodlike_TOC.eps){height="3.6cm" width="4.6cm"}
--- author: - Xuedong Hu title: 'Spin-Based Quantum Dot Quantum Computing' --- Introduction ============ During the past decade, the study of quantum computing and quantum information processing has generated wide spread interest among physicists from areas ranging from atomic physics, optics, to various branches of condensed matter physics [@Reviews; @nielsen]. The key thrust behind the rush toward a working quantum computer (QC) was a quantum algorithm designed by a computer scientist, Peter Shor from AT&T, that can factor large numbers exponentially faster than any available classical algorithms [@Shor]. This exponential speedup is due to the intrinsic parallelism in the superposition principle and unitary evolution of quantum mechanics, thus it requires a computer that is made up of quantum mechanical parts (qubits), whose evolution is governed by quantum mechanics. Since the invention of Shor’s factoring algorithm, it has also been proved that error correction can be done to a quantum system [@error], so that a practical QC does not have to be forever perfect to be useful. After these two important developments, the field of quantum computation has seen an explosive growth. Many physical systems have been proposed as candidates for qubits in a QC. Prominent examples include trapped ions [@Zoller; @iontrap], photons or atoms in cavities [@Sleator; @cavityQED], nuclear spins in a liquid NMR system [@NMR], electron spin in semiconductor quantum dots [@LD; @Imam], donor electron or nuclear spins in semiconductors [@Kane1; @Privman; @Vrijen], structures consisting of superconducting Josephson junctions [@Schon], and many more. The experimental demonstration of quantum coherence and maneuverability of these physical systems can be characterized as very fruitful in some cases, such as trapped ions [@phasegate] and liquid state NMR [@factor15], to preliminary in many other cases, such as most solid state schemes. Although experimental progress in many solid state schemes has been slow to come, they are still often considered promising in the long term because of their perceived scalability. After all, the present computer technology is based on semiconductor integrated circuits with ever smaller feature size. However, it is still to be demonstrated whether and how the available semiconductor technology can help scale up an architecture that is quantum mechanically coherent. Here we first present a brief overview of the current theoretical and experimental progresses in the study of quantum dot-based quantum computing schemes. We then focus on the spin-based varieties, which are generally regarded as the most scalable because of the relatively long coherence times of electron and nuclear spins. General Features of the Quantum Dot Quantum Computing Schemes ============================================================= Classification of the QC Schemes -------------------------------- Many proposals have been made to use quantum dot (QD) and various electron and nuclear degrees of freedom to process quantum information. Crudely these proposals can be classified as charge-based and spin-based (both electron and nuclear spins). We comment on a few of the charged-based proposals below and then discuss in more detail the particular spin-based schemes on which we focus in this lecture. One of the earliest charge-based proposal is to use the lowest two orbital energy levels of a single electron QD [@Barenco]. When an external electric field is applied to the QD, the ground and first excited states would acquire opposite electric dipole moments (the so-called quantum-confined Stark effect [@Miller]). With the interdot tunneling completely suppressed, the inter-qubit coupling is then dominated by dipolar interaction. It creates energy shifts in the energy levels of one QD depending on the state of its neighboring dot, thus provides the physical basis for conditional two-qubit operations [@Barenco; @Brum]. Resonant optical pulses can then be used to implement the conditional excitations. Several charge-based proposals have since been suggested with more concrete architectures and more detailed physical descriptions. Examples include those using pillars of vertically stacked QDs [@Sanders], and chains of horizontal double dot [@Tanamoto; @Fed]. Similar dipole-interaction-based proposals were also put forward in other physical systems such as trapped neutral atoms [@Lukin]. Alternatively, there have also been suggestions using cavity modes (instead of dipole interaction) to couple electronic orbital states [@Sherwin]. Indeed, the prominent charge-based qubit is the Cooper pair box proposal in superconductor, where the charged and uncharged state of a small superconducting grain form the basis of a qubit [@Schon; @Bouchiat; @Nakamura; @Vion; @Yamamoto]. The semiconductor analogue of the coherent charge oscillation experiment [@Nakamura] has just recently been done [@Hayashi], although with much shorter coherent time compared to the superconducting counterpart. The great experimental successes in Cooper pair boxes have also prompted searches for other systems that share some common features with Cooper pair boxes. Examples include such exotic systems like quantum Hall bilayers [@YangS; @SPS]. All the charge-based schemes mentioned so far use singly charged semiconductor QDs. The associated strong Coulomb interaction provide a convenient means for fast qubit manipulation, but can also lead to fast decoherence. One way to alleviate this problem is to use neutral excitations such as excitons as qubits, where there is the added benefit that excitons can be precisely controlled optically. Indeed, uncharged QDs have been proposed as possible candidates for quantum information processing [@Zanardi; @Biolatti; @PCChen; @Troiani; @Brown], and many experiments have been done to demonstrate exciton coherence and control in a single QD [@Heberle; @Bonadeo; @Bayer; @Cole; @GChen; @Li]. Here single excitons are optically excited in individual QDs and can be coherently manipulated optically. The presence and absence of an exciton in a QD provide the two states of a qubit. Again, entanglement between different qubits is based on Coulomb renormalization of the energy levels. The exciton-based QC proposals clearly illustrate the dichotomy faced by all QC architectures: excitons are neutral, therefore are more insulated from their environment and decohere more slowly than the single charge based schemes. However, the charge neutrality also strongly reduces the interaction between spatially separated excitons, thus rendering it more difficult to perform entangling operations. In the following we will focus on spin-based QD QC architectures. A fermionic spin, more specifically a spin-1/2, being a quantum two-level system in a finite magnetic field, is a natural qubit with its spin-up and spin-down states. For example, the most successful experimental demonstration of quantum control and entanglement is in a trapped ion system using two hyperfine split nuclear spin levels for qubit [@Zoller]. It was discussed in the mid-1990s that Ising interaction between electron spins in solids can produce the desired entanglement for a QC [@DPD]. Here, the specific common thread among the schemes we will concentrate on is that in all of them direct electron exchange coupling plays a crucial role. There are certainly proposals where electron spin interaction takes other forms (such as cavity photon mediated [@Imam], free electron mediated [@Privman; @Mozy], optical RKKY interaction [@Pier], and dipole coupling [@Calarco]), and there have been a tremendous amount of research done on the optical characterization of electron and nuclear spins [@optor; @Kikkawa; @Gammon; @Kato]. Nevertheless, we are going to be mostly focused on the electrical control of spins and their interaction. GaAs Quantum Dot QC Architecture -------------------------------- One of the earliest proposed solid state QC schemes uses the spin of a single electron trapped in a GaAs QD as its qubit (see [@DPD1; @LD; @DPD2; @BLD; @HD1] and references therein, and Fig. \[fig:QDQC\]). Local magnetic fields are used to manipulate single spins, in the sense that it creates a local Zeeman splitting, which can then be accessed by a resonant RF pulse. Inter-dot exchange interaction, which is a purely spin interaction in the form $J {\bf S}_1 \cdot {\bf S}_2$ but of electrostatic origin, is used to couple neighboring spins and introduce two-qubit entanglement. A single trapped electron in a GaAs QD ground orbital state means very low spin-orbit coupling as the electrons occupy states at the bottom of the GaAs conduction band and have essentially S type states [@HD1]. Thus the electron spin coherence time should be even longer than in the bulk, where electron spin decoherence is already very slow. ![A schematic of a QD QC.[]{data-label="fig:QDQC"}](QDQC.eps){height="4cm"} Some of the critical issues regarding GaAs QD QC are trapping a single electron in a gated QD, producing a local electron Zeeman splitting that is different from all its neighbors (so that resonant single spin rotations can be performed), creating and controlling a finite exchange coupling between electrons in neighboring QDs, and last but not least, measuring the single electron spins with high fidelity. Si Quantum Dot QC Architecture ------------------------------ The GaAs QD QC architecture we discussed above can be relatively easily extended to a Si/SiGe material system [@Friesen]. Here the electrons are confined in the pure (but strained) silicon region (instead of SiGe alloys), while the confinement is produced by the SiGe alloys along the growth direction and surface gates in the in-plane directions (Fig. \[fig:QDQC\]). Compared to GaAs, electron spin decoherence due to hyperfine coupling (as we will discuss in Section 3) in such a system can be suppressed by using purified $^{28}$Si as the host material. On the other hand, the more complicated band structure of silicon and the less well-controlled interface may pose problems to the coherent manipulation of a quantum device, which needs to be further studied. Si Donor Nuclear Spin QC Architecture ------------------------------------- The QDs underlying the previous two QC schemes are essentially artificial atoms, where electron confinement is provided by the barrier materials and the external electrostatic potential from the metallic gates on the surface of the devices. A naturally occurring alternative is the weakly bound donor states. Take an example of a monovalent donor, where one extra proton is present at the donor nuclear site while an extra donor electron is loosely bound to this proton. Now the donor location becomes the tag for the bound electron spin or the resident nuclear spin, and identical copies of such donors can be easily made in a semiconductor. One of the most intriguing and influential QC schemes is the donor nuclear spin based Si QC [@Kane1], as is shown in Fig. \[fig:SiQC\](a). Here spin-1/2 $^{31}$P donor nuclei are qubits, while donor electrons together with external gates provide single-qubit (using external magnetic field) and two-qubit operations (using hyperfine and electron exchange interactions). Specifically, the single donor nuclear spin splitting is given by [@Kane1] $$\hbar \omega_A = 2 g_n \mu_n B +2A + \frac{2A^2}{\mu_B B} \,,$$ where $g_n$ is the nuclear spin g-factor (1.13 for $^{31}$P [@Kane1]), $\mu_n$ is the nuclear magneton, $A$ is the strength of the hyperfine coupling between the $^{31}$P nucleus and the donor electron spin, and $B$ is the applied magnetic field. It’s clear that by changing $A$ one can effectively change the nuclear spin splitting, thus allow resonant manipulations of individual nuclear spins. If the donor electrons of two nearby donors are allowed to overlap, the interaction part of the spin Hamiltonian for the two electrons and the two nuclei include electron-nuclear hyperfine coupling and electron-electron exchange coupling [@Kane1]: $$\begin{aligned} H & = & H_{\rm Zeeman} + H_{\rm int} \nonumber \\ & = & H_{\rm Zeeman} + A_1 {\bf S}_1 \cdot {\bf I}_1 + {\bf A}_2 {\bf S}_2 \cdot {\bf I}_2 + J {\bf S}_1 \cdot {\bf S}_2 \,,\end{aligned}$$ where ${\bf S}_1$ and ${\bf S}_2$ represent the two electron spins, ${\bf I}_1$ and ${\bf I}_2$ represent the two nuclear spins, $A_1$ and $A_2$ are the hyperfine coupling strength at the two donor sites, and $J$ is the exchange coupling strength between the two donor electrons, which is determined by the overlap of the donor electron wavefunctions. The lowest order perturbation calculation (assuming $A_1= A_2= A$ and $J$ is much smaller than the electron Zeeman splitting) results in an effective exchange coupling between the two nuclei and the coupling strength is $$J_{nn} = \frac{4A^2 J}{\mu_B B (\mu_B B - 2J)} \,.$$ Now the two donor electrons are essentially shuttles between different nuclear spin qubits and are controlled by external gate voltages. The final measurement is done by first transferring nuclear spin information into electron spins using hyperfine interaction, then converting electron spin information into charge states such as charge locations [@Kane2]. A significant advantage of silicon is that its most abundant isotope $^{28}$Si is spinless, thus providing a “quiet” environment for the donor nuclear spin qubits. In addition, Si also has smaller intrinsic spin-orbit coupling than other popular semiconductors such as GaAs. In general, nuclear spins have very long coherence times because they do not strongly couple with their environment, and are thus good candidates for qubits. However, this isolation from the environment also brings with it the baggage that individual nuclear spins are difficult to control and measure. This is why donor electrons play a crucial role in the Si QC scheme. On top of the good material properties Si possesses, there is another potential advantage of a QC based on Si: the prospect of using the vast resources available from the Si-based semiconductor chip industry. In Bruce Kane’s own words, it is always advantageous if one can have the eight hundred pound gorilla on his/her side. ![Schematics of (a) a Si donor nuclear spin QC and (b) a Si donor electron spin QC.[]{data-label="fig:SiQC"}](SiQC.eps){height="5cm"} Si Donor Electron Spin QC Architecture -------------------------------------- A direct Si donor electron analogue to the GaAs QD QC scheme has also been proposed [@Vrijen]. In this scheme the $^{31}$P donor electron spins are employed as qubits (Fig. \[fig:SiQC\](b)). The phosphorus donors are located in a Si layer sandwiched in between SiGe alloy layers. By moving a donor electron into alloy regions of a different g-factor, its Zeeman splitting can be tuned significantly, which then allows selective single-qubit operations using resonant RF pulses. Similarly, different alloy regions also present different electron effective masses, which affect the size of the donor electron wavefunction sensitively. Such property can then be used to tune the exchange coupling between two bound donor electrons, and two-qubit operations are again provided by the direct exchange interaction between neighboring donor electrons. This all-electron proposal has much faster gate operations compared to the nuclear-electron hybrid scheme mentioned above. However, the alloy environment has to be more thoroughly studied for such a QC scheme to be considered more practical. Electron Spin Coherence in Semiconductors ========================================= To use spins as qubit for a QC, they should possess very long coherence time. Here “long” is in the sense that within a characteristic spin coherence time a large number of operations can be performed. The criteria for the large number is determined by the requirements of successfully performing quantum error correction, which lead to numbers in the range of $10^4$ to $10^6$ [@Reviews]. For example, if a gate operation on an electron spin can be performed in 1 ns, then its coherence time needs to be longer than 10 $\mu$s. Spin coherence (whether it is electron spin or nuclear spin) is regularly described in terms of a longitudinal (or spin-lattice) relaxation time $T_1$ and a spin-spin relaxation time $T_2$ [@HSD_Rev]. These descriptions originate from the magnetic resonance studies of these spin species going back to the 1940s [@Slichter; @Abragam]. $T_1$ generally describes processes that involve energy transfer between a spin and its environment, while $T_2$ describes everything that disrupts the quantum coherence of a spin, thus is generally much shorter than $T_1$. Notice that spin-flip processes cause both population relaxation and dephasing, contributing to both rates $1/T_{1}$ and $1/T_{2}$. However, in a real physical system the longitudinal and transverse directions are often affected differently by the environment. Indeed, there exist pure dephasing processes that affect only $T_{2}$ but not $T_{1}$. One example is the colliding molecules (which can be described as a pseudospin system, with spin up and down referring to the two relevant internal levels of the molecules) in an optically active gaseous medium, where molecules constantly undergo collisions with each other, some of them inelastic, but most of them elastic. During an inelastic collision, electrons in the molecules undergo transitions that correspond roughly to spin relaxation (or complete loss of the system as an electron gets excited out of the two optically active levels). During an elastic collision, the single molecule energy spectrum changes due to the presence of the other molecule nearby. This shift in energy levels (particularly the two active levels) is dependent on the details of the collision, and is thus a random variable, which we refer to as $\delta \omega(t)$. Including this frequency shift, the differential equation for the off-diagonal density matrix element $\rho_{\uparrow \downarrow}$ (representing the coherence between the up and down levels) becomes [@MS] $${\dot \rho}_{\uparrow \downarrow}(t)=-i[\omega + \delta \omega(t)] \rho_{\uparrow \downarrow}. \label{fluctrho}$$ Note that in the first order approximation the population in the up level, $\rho_{\uparrow \uparrow}$, is not affected by this level splitting fluctuation since its equation of motion is independent of the energy difference $\omega$ [@MS] (in other words, the fluctuation in the level splitting does not lead to population relaxation). $\delta\omega(t)$ is a random variable that averages to zero, $\langle \delta \omega (t)\rangle=0$. Within the Markovian approximation, the random fluctuation in energy level splitting of the two level system causes a pure dephasing effect: $$\langle \rho_{\uparrow \downarrow}(t)\rangle = \langle \rho_{\uparrow \downarrow}(0)\rangle e^{-i\omega t} e^{-\gamma_{ph}t}.$$ This pure dephasing only contributes to $T_2$ of a spin or pseudospin system, but not to $T_1$. Another well-known example of pure dephasing is the dipolar spin-spin interaction between nuclear spins in a solid, which produces effective local magnetic field fluctuations for each nuclear spin and hence contributes essentially only to $T_2$ (the corresponding effect on $T_1$ is extremely small). What is important for dephasing is that some change in the state of the environment must occur due to its interaction with the system—dephasing does not require an inelastic scattering process in the system, although all inelastic scatterings necessarily produce dephasing. In fact, as mentioned before, $T_2$ in the context of electron spin resonance (ESR) and nuclear magnetic resonance (NMR) is often called the spin-spin relaxation time because the most important intrinsic effect contributing to $T_2$ is the dipolar interaction among various spins in the system, which, while transferring energy among the spins themselves, does not lead to overall energy relaxation from the total spin system (and does not change the total magnetic moment). By contrast, spin-lattice interactions lead to energy relaxation (via spin-flip processes) from the spin system to the lattice, and thus contribute to $T_1^{-1}$, the spin-lattice relaxation rate. In short, $T_2$ sets the time scale for the spin system to achieve equilibrium within itself whereas $T_1$ sets the time scale for the global thermodynamic equilibrium between the spin system and the lattice. For the purpose of quantum computing, it is obvious that $T_2$ is the directly relevant time scale, because we need to keep all the spins completely quantum coherent. As we mentioned before, it is imperative that $T_2$ for an electron spin in a single QD is a factor of $10^{4}$ or so greater than the typical gating time in a QD QC [@nielsen] in order for the quantum computing process to be fault tolerant. For $B=1$ T, the Zeeman splitting in a GaAs QD is about $0.03$ meV, which yields $100$ ps for the precession time of one spin, which can be used as the one qubit gate (the two qubit gate time is shorter, $\hbar/J\sim 50$ ps for $J\sim 0.1$ meV). Therefore for quantum error correction to be performed reliably, $T_{2}$ for the trapped electron spin needs to be on the $\mu$s time scale, which may very well be the case at low enough temperatures in a single QD. We note that the existing experimental estimates of free electron spin relaxation time $T_2$ in GaAs (for $T = 1$–4 K) is around 10–100 ns [@ESR], which is obviously a lower bound. Spin Decoherence Channels in Semiconductors ------------------------------------------- In doped semiconductors there are three major spin relaxation mechanisms: Elliot-Yafet (EY), Dyakonov-Perel’ (DP), and Bir-Aronov-Pikus (BAP) mechanisms, for conduction electrons [@optor; @Jaro] at not-too-low temperatures. The origin of the EY mechanism is spin-orbit coupling. Spin-orbit coupling does not lead to spin relaxation by itself, but it mixes the electron orbital and spin degrees of freedom. When combined with another scattering mechanism, such as phonon emission/absorption and impurity scattering, electron spin-flip can occur (spin-flip now means the dominant spin component in the Bloch state changes). In the DP mechanism, the splitting of spin up and down conduction bands due to lack of inversion symmetry (as in III-V semiconductors such as GaAs, which has the zinc-blende lattice structure) acts as an effective momentum dependent magnetic field ${\mathbf B}({\mathbf k})$. An electron with momentum ${\bf k}$ and spin ${\bf S}$ precesses in this effective field ${\bf B}({\bf k})$ and loses its spin memory. As this electron is scattered into a different ${\bf k}$ state, its spin will start to precess around the new effective field. This constant change of effective magnetic field actually reduces the electron spin relaxation, so that the spin relaxation time is inversely proportional to the momentum relaxation time in this mechanism (cf motional narrowing). Lastly, the BAP mechanism is given by the exchange interaction between electrons and holes. Electronic spins move in an effective field produced by the hole spins, and relaxation takes place when hole spins change in a rate much faster than the electron precession frequency. For the purpose of QD quantum computing, electrons are individually confined in the QDs or around donor nuclei. This quantization of the electron orbital motion should significantly reduce the cross-section of the spin relaxation channels discussed above. However, if the confined electrons are close to a heterogeneous interface, the associated electric field would increase the strength of spin-orbit coupling for such electrons (for example, in GaAs, whose lattice lacks inversion symmetry). Overall, spin-orbit coupling is relatively weak in the regime for quantum computing and is less important than at higher temperatures and electron densities. Furthermore, without scattering off phonons or impurities, spin-orbit coupling can actually be useful rather than harmful to quantum computing [@Wu; @Wu2]. At this limit, other sources of electron spin decoherence, such as electron-nuclear spin coupling, can be more significant, as we will discuss in the next section. Phonon-assisted spin flip rates due to spin-orbit coupling in a single electron GaAs QD has been calculated [@Khaetskii]. As discussed above, due to wave function localization, the spin orbit relaxation mechanisms for a free electron (EY and DP) are strongly suppressed in a QD, giving a long spin flip time: $T_{1}\approx 1$ ms for $B=1$ T and $T=0$ K. It was further noticed that spin relaxation is dominated by the EY mechanism, which yields $T_{1}\propto B^{-5}$ for transitions between Zeeman sublevels in a one electron QD. These calculations are consistent with recent transport measurements of spin relaxation in both vertical and horizontal QDs [@Fuji1; @Fuji2; @Hanson]. Pulses of current were injected into a QD coupled to leads in the Coulomb Blockade regime, where the decay rate from excited states can be measured by analyzing the currents generated by the pulses. The results indicate that, for $T=150$ mK and $B=0-2$ T, spin relaxation times ($T_{1}$) are longer than at least a few $\mu$s in a many-electron QD (less than 50 electrons), and are longer than 50 $\mu$s in single electron dots. This is encouraging from the perspective of the spin-based solid state QC architectures where spin relaxation times of $\mu$s or longer are most likely necessary for large scale QC operation. It is important to keep in mind that $T_2$ is a more directly relevant decoherence time for quantum computing. Therefore experimental determination of electron and nuclear spin $T_2$ in semiconductor nanostructures are crucial, but are still to be performed. ESR combined with transport techniques in principle could be used to probe $T_{2}$ in a QD in the Coulomb Blockade regime, just like transport techniques were used to detect ESR in two dimensional electron systems [@ESR]. For example, it was proposed [@Engel] that by applying an AC pump field to a single electron QD subjected to a magnetic field, the stationary current through this QD will exhibit a peak as a function of the pump frequency, whose width will yield a lower bound on $T_{2}$. Spectral Diffusion for Electron Spins ------------------------------------- Spin-orbit coupling does not lead to pure dephasing effect, so that one should expect to have $T_1 = 2 T_2$ in materials where spin-orbit coupling in combination with phonon/impurity scattering dominates the spin relaxation, such as high quality GaAs at higher electron density. When the strength of spin-orbit coupling or the cross-section of scattering is reduced, other possible spin decoherence channels have to be considered. In GaAs, all nuclear isotopes ($^{69}$Ga, $^{71}$Ga, and $^{75}$As) have spin $3/2$, and there is finite hyperfine interaction between conduction electrons and all these isotopes [@Paget]. The hyperfine interaction is essentially an on-site dipole-dipole interaction between electron and nuclear spins and is $\propto {\bf S} \cdot {\bf I}$, which includes terms that lead to simultaneous spin flip-flop of both electron and nuclear spins. At a finite magnetic field, the Zeeman splittings of electron and nuclear spins are different by three orders of magnitude, thus energy conservation requires another process to be involved in the transition, therefore reducing the cross-section of such type of processes [@Erlingsson]. However, in the hyperfine interaction there is also a term that is proportional to $S_z I_z$, where the nuclear spins basically produce an effective magnetic field for the electron (assuming one electron trapped in a QD in our situation). If the nuclear spins are all frozen in their respective states, they would simply produce a random but fixed field, which would result in the so-called inhomogeneous broadening for the electron and the effect can be accounted for by calibration and spin echo techniques [@Slichter; @Abragam]. If the nuclear spins are dynamically coupled (through dipolar coupling, for example), though, the trapped electron would be in a magnetic field that fluctuates both spatially and temporally. This fluctuating field makes the electron Zeeman splitting a random variable that undergoes the so-called spectral diffusion, which results in pure dephasing for the electron spin [@Roger_SD]. The calculated results for GaAs QDs are in the order of tens of $\mu$s. Theoretically, it can be envisioned that if all the nuclear spins are polarized, the corresponding electron spin spectral diffusion and dephasing can be suppressed [@BLD]. However, 100% polarized nuclear spins would create a significant effective magnetic field, which can have its own negative side effect. Furthermore, the creation of this high degree of polarization is also nontrivial. In other words, ingenious approaches need to be devised to deal with the nuclear spins in a GaAs QD, and we may not have seen the optimal approach just yet. Spectral diffusion can occur in donor electrons in Si as well, when the host Si material is not purified: Naturally occurring Si contains more than 95% of $^{28}$Si and $^{30}$Si, which have no nuclear spin, and nearly 5% $^{29}$Si, which has nuclear spin $1/2$. For a confined donor electron (at a phosphorus site, for example), which has a Bohr radius about 30 nm, there can be a lot of nuclear spin-$1/2$s (more than $10^4$ of them within a sphere of radius of 30 nm) with finite hyperfine coupling to the electron, and thus can produce spectral diffusion in this electron as discussed above [@Roger_SD]. Fortunately, in Si there is a way to reduce the effect from the nuclear spins of $^{29}$Si: isotopic purification. If a complete purification can be achieved, the nuclear spin induced electron spin spectral diffusion can be completely suppressed [@Roger_SD]. Spin Manipulations and Exchange =============================== Spin Hamiltonian in a GaAs Double Quantum Dot: Coulomb Interaction and Pauli Principle -------------------------------------------------------------------------------------- One of the key issues in the spin-based QD QC involving exchange coupling in a coupled QD is to accurately describe the electron interaction in terms of a spin Hamiltonian such as a Heisenberg exchange Hamiltonian with appropriate correction terms. Ideally, for small QDs at low temperatures, the electron orbital degrees of freedom are frozen, so that the only states two electrons can possibly occupy in a double dot are the ground spin singlet and triplet states, whose splitting is the exchange splitting $J$. However, it is important to clarify whether the ground state manifold is well separated from the excited states, and whether the exchange splitting $J$ in the ground state manifold is sufficiently large to support a practical QC. The Hamiltonian for two electrons in an electrostatic confinement produced by surface gates and growth direction barriers can be written as $$H=\sum_{i=1,2} \left[ \frac{1}{2m^} \left( {\bf p} + \frac{e}{c}{\bf A}({\bf r}_i) \right)^2 + V({\bf r}_i) \right] + \frac{e^2}{\epsilon r_{12}} + \sum_{i=1,2} g^ \mu_B {\bf B}({\bf r}_i) \cdot {\bf S}_i \,. \label{eq:Hamiltonian}$$ This is an effective mass Hamiltonian where the underlying Bloch function at $\Gamma$ point (the bottom of the GaAs conduction band) is already factored out. Notice that none of the spin-dependent terms except Zeeman coupling are included in this Hamiltonian. In essence, electrostatic interaction is much stronger than direct magnetic interactions such as magnetic dipole interaction. Neglected interactions include electron spin-orbit interaction (which will be discussed later on in this subsection), electron-electron magnetic dipole interaction, electron-nuclear spin contact hyperfine interaction (discussed in the previous section on electron spin spectral diffusion), electron-nuclear spin dipole interaction, and other higher order interactions. In fact, in arriving at the effective mass Hamiltonian (\[eq:Hamiltonian\]), small spin-mixing terms inversely proportional to the conduction band gap are also neglected [@Bastard]. Here we have separated the two-electron-interaction related terms of the Hamiltonian from those that involves interactions between either or both of the electrons and the surrounding environments that include the crystal lattice (in terms of phonons), the nuclear spins of the ions, and other electrons present in the system. The effects of these interactions are categorized as decoherence, in the sense that as soon as electron spin coherence passes into these channels, the chance of a revival of the coherence is vanishingly small. The theoretical calculation of electron spin exchange in a double dot was first done using the Heitler-London approach, in which the electron orbital states are limited to the ground states in the two single QDs that form the double dot [@BLD], and double occupied single dot states are excluded. The effects of the lowest two double-occupied states (where both electrons are in the same dot) are included in the so-called Hund-Mulliken calculation [@BLD]. Both calculations indicated that the exchange constant can be sizable in a GaAs QD system. More accurate calculations of the exchange coupling and the overall spectrum of the two QD-confined electrons can be performed with larger basis of single dot states. For example, we performed a molecular orbital calculation to further clarify the properties of the exchange splitting in a GaAs horizontal double QD by including the excited P orbital states of the two QDs [@HD1]. The inclusion of the anisotropic P orbitals provides more flexibility to electron wavefunction deformation and bonding, thus leads to more faithful description of the exchange splitting of the two-electron ground states. After all, molecular bonding is strongly affected by electron distribution in space, while the system we considered is essentially an effective two-dimensional hydrogen molecule. The system we considered is formed from two-dimensional electron gas by surface gate depletion (the so-called horizontal or lateral QDs). The growth direction confinement (due to AlGaAs alloys) is so strong that the excitation energy scale along that direction is much higher than the horizontal direction and excitation along that direction is neglected (thus the name horizontal or lateral QD). One important advantage of the horizontally coupled QDs is that the inter-dot coupling can be easily tuned with surface gate potential adjustments. Our numerical results showed that the inclusion of the P orbitals indeed affect the exchange coupling quite significantly, generally causing an increase about 20% compared to the Hund-Mulliken calculation [@HD1] (see Figs. \[fig:spec\_2e\] and \[fig:J\_2e\]). The size of the QDs we considered is quite small (in the order of 40 to 50 nm in diameter), somewhat smaller than the state of the art experimental value of about 100 nm. The increase in size invariably leads to smaller on-site Coulomb repulsion, smaller single particle excitation energy, and generally smaller exchange coupling. ![Two-electron energy spectrum of a GaAs horizontal double QD [@HD1] as a function of applied perpendicular magnetic field. The ground singlet-triplet manifold is well separated from the excited states. Here the inter-dot distance is 30 nm, and the ground electron wavefunction radius is about 10 nm.[]{data-label="fig:spec_2e"}](spec_2e.eps){height="6cm"} ![Exchange splitting between the ground singlet and unpolarized triplet states of a GaAs horizontal double QD [@HD1].[]{data-label="fig:J_2e"}](J_2e.eps){height="6cm"} The spectroscopic results for double QD [@BLD; @HD1] show that exchange coupling should be sufficiently strong to implement quantum computing operations, where a basic controlled-NOT operation can be built from single qubit spin rotations and two so-called square-root-of-swap gates [@LD]: $$U_{CNOT} = e^{i\frac{\pi}{4} \sigma_{2y}} \ e^{i\frac{\pi}{4} \sigma_{1z}} \ e^{-i\frac{\pi}{4} \sigma_{2z}} \ U_{sw}^{\frac{1}{2}} \ e^{i\frac{\pi}{2} \sigma_{1z}} \ U_{sw}^{\frac{1}{2}} \ e^{-i\frac{\pi}{4} \sigma_{2y}} \,,$$ where $\sigma$ are Pauli matrices, and $U_{sw} = \exp (i\pi \sigma_1 \cdot \sigma_2)$. For example, an exchange splitting of 0.1 meV corresponds to a swap gate with duration as short as 100 ps. Compared to the electron spin decoherence time that might be as long as milliseconds, such gating time is sufficiently short for quantum error correction codes to work properly. By including only the lowest spin singlet and triplet states, we have implicitly assumed that the higher energy two-electron states can be neglected. Such an assumption is based on the electron states being manipulated adiabatically. On the other hand, the gate operations are limited in duration by the electron spin decoherence time. It is thus imperative to determine what the adiabatic requirement is in the present architecture. Such calculation was carried out within the Hund-Mulliken model by assuming a particular temporal shape of the exchange splitting $J$ in a double QD [@Schliemann]. We performed such a calculation with calculated two-electron energy spectra and wavefunctions [@HD_gate]. Instead of assuming a particular temporal profile for the exchange coupling $J$, we assumed a temporal variation in the inter-dot barrier height, which is a quantity that can be directly tuned by external voltages applied to the surface gates. Our results show that errors due to non-adiabaticity decreases rapidly as the gate operations become longer, and the requirement for adiabatic operation is not overbearing on the spin-based GaAs QD QC [@HD_gate]. The calculations on the energy spectra and adiabatic manipulation thus justify the use of Heisenberg spin exchange Hamiltonian to describe the low energy dynamics in a two-electron double QD (whose effective mass Hamiltonian is given by Eq. \[eq:Hamiltonian\]) and the related two-qubit operations in a spin-based QD QC. In these calculations spin-orbit coupling has been neglected because of their small magnitude for the conduction electrons in bulk GaAs (near the bottom of the conduction band the electron wavefunction is mostly formed from the atomic S orbitals). However, the horizontal QDs are made from two-dimensional electron gas confined in heterostructures or quantum wells, where the sharp surfaces and asymmetry lead to strong Rashba type spin-orbit coupling (even for symmetric quantum wells, this spin-orbit coupling does not vanish because of the lack of inversion symmetry in GaAs), which in turn leads to finite anisotropic exchange in the spin interaction [@Kavokin] in the form of ${\bf h} \cdot ({\bf S}_1 \times {\bf S}_2)$ plus higher order corrections. Here ${\bf h}$ is a vector determined by spin-orbit interaction. The inclusion of these terms does not take away the capability of QDs to perform quantum logic operations (indeed, there has been a proposal to use the anisotropic exchange coupling to perform quantum logic operations [@Wu; @Wu2]). Nevertheless, they do add more complexity to the aesthetically simple and beautiful isotropic Heisenberg exchange coupling. Fortunately, it has been proved that by carefully choosing the temporal profile of the inter-dot coupling (basically maintaining time reversal symmetry), it is possible to largely eliminate the effects of the anisotropic exchange due to the spin-orbit interaction [@Bone]. Even when the two-electron interaction in a double dot can be characterized by the Heisenberg spin exchange Hamiltonian, inhomogeneity in the single spin environment can still cause problems in the two-qubit quantum logic operations. For example, we showed that inhomogeneous Zeeman coupling leads to incomplete swap operations [@HSD]. This means that swap cannot be accomplished by a single pulse of exchange gate anymore. Instead, several pulses (at least 3) have to be used for large inhomogeneity [@HD3], while smaller inhomogeneities such as those due to trapped charges nearby can be and have to be corrected [@HSD]. Interestingly, inhomogeneous Zeeman coupling can also be utilized for the purpose of qubit encoding [@Levy], which leads to all-exchange logical operations (eliminating the need for local magnetic field and/or g-factor engineering) [@DPD3] that originate from the concept of decoherence free subspace [@DFS]. To relax the requirement of spin-based QD quantum computation, multi-electron QDs have also been studied as candidates for qubits [@HD2]. For instance, we performed a configuration interaction calculation for a double dot with six electrons (three per dot) to explore whether the low-energy dynamics is still entirely dominated by spin dynamics. Our results showed that the ground state complex is well separated from the higher energy excited states (Fig. \[fig:spec\_6e\]) at relatively high magnetic fields, and the splitting between the lowest-energy singlet and triplet states can be sizable as well (Fig. \[fig:J\_6e\]). However, our results also showed that orbital level degeneracy can lead to the participation of multiple states in the low energy dynamics at zero or low magnetic fields, therefore causing serious complexity and difficulty in spin exchange. To solve this problem, external means such magnetic field or quantum dot deformation has to be applied to lift the orbital degeneracy so that the electron cloud in each QD can again be described by an effective spin-1/2 entity. Another theoretical study has also clearly demonstrated a variety of difficulties in the control of gate operations when one attempts to use multi-electron QDs as qubits [@Mucc]. ![Six-electron energy spectrum of a GaAs horizontal double QD [@HD2].[]{data-label="fig:spec_6e"}](spec_6e.eps){height="6cm"} ![Energy splitting of the two lowest energy states of a six-electron GaAs horizontal double dot [@HD2].[]{data-label="fig:J_6e"}](J_6e.eps){height="6cm"} In using a tunable exchange interaction for quantum gates, one needs to control the magnitude and duration of the interaction precisely. Since exchange interaction depends sensitively on the wavefunction overlap, its precise control is of critical importance. One suggestion on how a good control can be achieved is to control the way the wavefunction overlap is tuned by altering the geometry of the QD designs and is called a pseudo-digital approach [@Friesen3]. As the experimental study of QD QC pushes toward two and more qubits and qubit manipulations, such considerations are directly relevant to the designs of next generation architectures and ultimately the scale-up of any of the QD-based QC schemes. Implications of Si Conduction Band Structure to Electron Exchange ----------------------------------------------------------------- As we mentioned before, Si possesses a variety of nice material properties (small spin-orbit coupling, spinless isotopes, etc.) for the purpose of quantum computing, so that it is clearly one of the favored host materials for a solid-state QC. However, Si does have one complexity that GaAs, the other popular semiconductor material, does not have: The Silicon conduction band has six minima close to the X points of the silicon First Brillouin Zone [@Kohn], so that donor electron wavefunctions have to be expanded on the basis of the six Bloch functions at these points. It was pointed out in the context of donor magnetic phase transition that the presence of degenerate conduction valleys leads to valley interferences and a shift to smaller values for average electron exchange coupling [@Andres]. The potential problem such interference effects may cause to a donor based Si QC was also mentioned in the original proposal [@Kane1]. To quantitatively address this concern over donor exchange coupling in Si, which is a crucial link for two-qubit operations in the Si QC architecture, we have performed a series of Heitler-London type calculations of the donor electron exchange coupling. Such a calculation is based upon the single donor wavefunctions, which can be expressed as [@Kohn; @Andres; @KHD1; @KHD2] $$\psi ({\bf r}) = \sum_{\mu = 1}^6 \alpha_\mu F_{\mu} ({\bf r}) \phi_\mu(\bf r) = \sum_{\mu = 1}^6 \alpha_\mu F_{\mu} ({\bf r}) u_\mu({\bf r}) e^{i {\bf k}_{\mu} \cdot {\bf r}}\,, \label{eq:sim}$$ where $F_{\mu}$ are the so-called envelope functions while $\phi_\mu(\bf r)$ are the Bloch functions at the bottoms of the Si conduction band [@KHD1]. $\|{\bf k}_\mu\| \approx 0.85 \cdot 2\pi/a$ is the location of the conduction band minima and is very close to the X points. The presence of these plane wave phase factors leads to a significantly more complicated expression (compared to, for example, GaAs) for the electron exchange splitting [@KHD2]: $$J({\bf R}) = \sum_{\mu, \nu} \|\alpha_\mu\|^2 \ \|\alpha_\nu\|^2 {\cal J}_{\mu \nu} ({\bf R}) \cos ({\bf k}_{\mu}-{\bf k}_{\nu})\cdot {\bf R}\,. \label{eq:Si_exch}$$ Here ${\cal J}$ represents integrals over the envelope functions and is thus a slowly varying function of donor positions. The key fact here is that the ${\bf R}$-dependence of $J({\bf R})$ is strongly oscillatory because of the sinusoidal factors $\cos ({\bf k}_{\mu}-{\bf k}_{\nu})\cdot {\bf R}$. Our numerical results showed that the inter-valley interference indeed leads to strong atomic scale oscillations in the inter-donor electron exchange (see Figs. 7 and 8), which potentially presents a significant difficulty in the control of two-qubit operations [@KHD1]. Uniaxial strain can be used to break the Si lattice symmetry and partially lift the degeneracy between the valleys, so that as few as two valleys make up the bottom of the conduction band. Then the sum over $\mu$ in Eq. (\[eq:Si\_exch\]) is much simplified, but the sinusoidal factor will still remain, so that care still has to be taken in controlling the donor exchange (Figs. 9 and 10) [@KHD2]. These results have been corroborated by another calculation that also considered higher order corrections in Coulomb interaction energy [@Wellard]. More recently we have attempted to relax the Heitler-London approximation to minimize the two-electron energy [@KCHD]. However, the results showed that in the cases of donors, which have very low single particle potential energy near the donor nuclei, the two-electron contribution to total energy is completely dominated by the single particle contributions, thus the Heitler-London results based on single donor states are quite robust [@KCHD]. ![Donor electron exchange splitting in relaxed bulk Si. The two $^{31}$P donors are aligned along the \[100\] direction [@KHD1].[]{data-label="fig:J_Si_100"}](J_Si_100.eps){height="7cm"} ![Donor electron exchange splitting in bulk Si. The two $^{31}$P donors are almost aligned along the \[100\] direction, separated by about 18 lattice constants, and with one of the donors displaced into its nearest neighbor lattice sites [@KHD1].[]{data-label="fig:J_Si_NN"}](J_Si_NN.eps){height="6cm"} ![Donor electron exchange splitting in Si uniaxially strained along \[001\] direction ($\chi=-20$). The two donors are approximately aligned along the \[100\] direction, with one of them displaced along the \[010\] direction [@KHD2].[]{data-label="fig:J_SG_010"}](J_SG_010.eps){height="6cm"} ![Donor electron exchange splitting in both relaxed bulk Si and Si strained along the \[001\] direction. The two donors are both in the (001), or $xy$, plane. We consider a situation where one of the donors is located in any of the possible lattice positions between two concentric circles of radii 90 Å and 180 Å with the other donor positioned at the center of the circles. The data points correspond to the exchange calculated at all relative positions considered. The solid line is $J({\bf R})$ for ${\bf R}$ along the \[100\] direction for $\chi = -20$ [@KHD2].[]{data-label="fig:J_SG_inplane"}](J_SG_inplane.eps){height="6cm"} As we mentioned before, there have also been proposals using electron spins in Si or SiGe host materials for quantum computing. The problem with oscillatory exchange also plagues the donor-electron-based scheme. In the case of QDs in Si, the problem becomes more subtle. In such a system there does not exist a strong scattering center such as a donor, so that the different valleys in Si do not couple significantly, so that the validity of Eq. (\[eq:sim\]), with its strongly pinned phase for each valley, becomes questionable. The valley degeneracy is lifted by the quantum well for the QDs, but the energy splitting is quite small [@Boykin]. More theoretical and experimental studies are needed to further clarify this situation. One way to avoid the potential problems associated with the control of inter-donor exchange coupling is to move a donor electron around between ionized nulei [@Lari; @Skinner]. Here nuclear spins are still the qubits, while a [single electron]{} is used as a shuttle to physically move among nuclei to enable effective two-qubit coupling and operations. Since on-site hyperfine coupling has been shown to be quite robust against Si bandstructure complexities [@Smit; @Martins], this approach thus maintains the long coherence time advantage of a donor nuclear spin in Si while completely removes the requirement of inter-donor exchange coupling. However, a potential disadvantge of such a charge-coupled device may be associated with the charge movement and the corresponding spin decoherence. Further analysis is still needed to clarify the physical picture here. Single Spin Detection Schemes ----------------------------- Single spin detection is crucial for any spin-based QC architecture to work properly, both in terms of quantum error correction, and in terms of reading out the final results of a calculation. However, since magnetic force is weak (compared to electrical force), and a single Bohr magneton is a very small magnetic moment, direct measurement within a short duration (such as using the most sensitive magnetometer available at present, a SQUID magnetometer) is almost impossible. However, several techniques are being actively studied and have produced some promising progresses. One approach to single spin detection is to first convert the electron spin states into electron charge states, for example through spin blockade effects [@SB_Weinmann; @Ciorga1; @SB; @Hutt]. It is well established now that single electron transistors (SET) and quantum point contacts (QPC) are extremely sensitive charge detectors [@SCT]. Thus, if one can establish a correlation between electron charge location and spin states, a means to determine spin states can be established by observing the current or conductance of the SET/QPC charge detector. One shortcoming of the conventional DC-biased SETs and QPCs is that they are relatively slow detectors, with bandwidth in the order of 10 kHz. However, an alternative approach to DC-SET has been proposed [@RFSET]. Instead of measuring current directly, here pulsed radio-frequency field is sent into a circuit containing an SET and the circuit response is measured. The relevant physical quantity in such a measurement is conductance, which can be determined without collecting a large number of electrons. The bandwidth of these so-called Radio Frequency SETs (RF-SET) has been shown to reach above 10 MHz, so that they are very promising candidates for quantum detectors. Now, if spin states can be efficiently converted to charge locations, single spin detection also becomes possible. One of the early examples of such conversion was suggested in connection with the donor in Si architecture, where the on-site exchange splitting between a singlet and a triplet state on a double valent donor, together with an SET charge detector, is used for spin detection [@Kane2]. A similar scheme can also be constructed for artificial QD-based structures. Quantum measurement of hyperfine split nuclear spin levels in trapped ion systems has been achieved using the so-called quantum jump technique, in which an ancilla electron orbital level and light absorption/emission between this level and the qubit levels are used to read out the quantum state of the qubit with basically 100% efficiency. Such ancilla levels certainly also exist for semiconductor QDs, so that similar quantum jump processes have been suggested for single electron spin detection [@Calarco]. Concrete examples of such a scheme have been described and simulated for an electron spin in a QD [@Friesen2; @Pazy]. In the original proposal for QD QC in GaAs, the so-called spin valve effect is suggested for spin measurement [@LD], where electron tunneling into another QD depends on its spin orientation, in analogy to giant magnetic resistance [@Prinz]. An alternative is to prepare a supercooled paramagnetic dot, so that when an electron tunnel into this QD, a large ferromagnetic domain could quickly nucleate along the incoming spin direction. This more macroscopic magnetization can then be detected in a more traditional way [@LD]. Another approach toward spin measurement is a direct magnetic force detection using the so-called magnetic resonance force microscopy (MRFM) [@Sidles]. In MRFM a small magnet at the tip of a cantilever creates a strongly inhomogeneous magnetic field near the surface of a sample containing paramagnetic electron spins. When a RF field with a certain frequency is applied, only those spins that are Zeeman split by the right amount can get into resonance with the external RF field, so that these spins can apply controlled forces on the cantilever. By measuring the motion of the cantilever, one can infer information of those electron spins that are on resonance (thus the name MRFM). Single spin detection is not needed for the purpose of characterizing a single or double QD/donor system. For example, we proposed a scheme to use resonant micro-Raman scattering to measure inter-donor exchange coupling [@KHDD]. The polarization- and temperature-dependence of Raman scattering provides abundant information of a two-donor system, while resonant photons enhances the cross-section of such scattering process so that single pairs of donors can be observable. Similarly, transport and/or optical techniques can be used to measure the ground exchange splitting in a double QD as well [@QD-rev]. Approaches to generate and detect electron spin entanglement in quantum dots ---------------------------------------------------------------------------- Two basic ingredients of spin-based QD QC are single spin manipulation and particularly single spin detection. These are very hard tasks and are attracting plenty of attention from experimentalists and theorists alike. In the meantime, traditional ensemble-averaged experiments can also be used to demonstrate quantum coherent properties of electron spins. For example, electron spin entanglement has never been experimentally demonstrated in condensed matter systems in terms of measured correlations, because of the usual presence of strong interaction and the associated difficulty in isolating the target electrons. However, with the help of QDs and transport through them, it is possible to generate and detect electron spin entanglement. From the perspective of QD QC, a reliable source of spin entangled electrons is very important to tasks such as error correction. Many approaches for creating/detecting entanglement in solid state systems have been theoretically proposed in the literature. For example, Cooper pairs in a superconductor are (usually) in a spin singlet (and entangled) state, thus it is quite natural to consider extracting them with control to make up a source of pairs of spin-entangled electrons [@DFHZ; @Lesovik; @Cht; @Recher1; @Recher2; @Bena]. The key here is to separate the two electrons into different drain electrodes using, for example, Coulomb blockade in quantum dots or wires. There are many other physical systems within solid state that have electron spin entangled states as part of their eigenstate spectrum and have been suggested as sources of entangled electrons. Examples include quantum dot and quantum wire electron singlet states in semiconductors [@Ionicioiu; @WOliver; @Saraga; @Entangle; @XX]. Here we briefly discuss our idea of using a double dot to create spin entangled electron pairs [@Entangle]. As discussed in the previous sections, at zero or low magnetic field, the ground state of a two-electron double dot is a spin singlet state, where the spins of the two electrons are entangled ($(\|\uparrow \downarrow \rangle - \|\downarrow \uparrow \rangle)/\sqrt{2}$). However, sequential tunneling through a double dot is generally dominated by the lowest order processes, so that the current through a double dot is generally made up of mostly single electrons tunneling out of the entangled state. To extract pairs of spin-entangled electrons, a relatively straightforward approach is to introduce time-dependent (specifically, periodic) tunnel barriers, so that during part of one cycle the electrons form molecular states (such as the spin singlet state) in the double dot, while during the rest of the cycle the electrons are emptied out into the drain electrodes. Such time-dependent manipulation of tunnel barriers is quite well-understood for single quantum dots, and in principle feasible for a double dot. We estimated various parameters for such a double dot turnstile and found that they are quite reasonable and fall within the capability range of currently available technology. A possible method to observe solid state electronic entanglement within ensemble-averaged approach is to use two-electron interference in transport. One example is to use a beam splitter [@BLS; @Entangle], into which pairs of entangled electrons are injected. Although this detection scheme is not a true Bell-type measurement, it is nonetheless an important first step as it deals with correlations between electrons that have been extracted from their entangler and separated. Current Experimental Status =========================== Single electron trapping in horizontal QDs ------------------------------------------ Since the invention of gated semiconductor QDs, there has been a continuous trend to fabricate structures that can hold smaller and smaller number of electrons. Single electron QD through electrical transport was achieved in vertically contacted InGaAs QDs several years ago [@vertQD]. However, since confinement in a gated dot is given by an electrostatic field produced by metallic gates that are 50 to 100 nm away, and the feature width of a metallic gate is generally above 20 nm, horizontal QDs made from depleted two-dimensional electron gas are generally large in size but energetically shallow. In these QDs it is difficult to simultaneously increase depletion (to reduce number of trapped electrons) and maintain a finite coupling to the source and drain leads (so that transport characterization can be done). A solution to this problem was found several years ago, when a new design with source and drain coupling controlled by a single gate was studied [@Ciorga]. The design has since been used to create single electron QDs and two-electron double QDs successfully [@Andy; @Elzerman1]. Furthermore, recent experiments have shown that electrons trapped in such a QD have significant (and consistent with expectation) Zeeman splitting, and that spin-flip time is extremely long in these confined states [@Hanson]. Single spin detection --------------------- Although many theoretical proposals have been published on how to achieve single spin detection, experimental demonstration remains a challenging and hotly pursued goal, particularly for solid state spin-based QC architectures, though impressive progress have been made along a variety of lines of research. Recently, QPCs have been used to successfully measure single QD properties such as QD charging and excited state spectroscopy [@Elzerman1; @Elzerman2]. The simpler structure of a QPC makes it an enticing alternative to an SET detector, and may be easier to integrate into larger structures. In analogy to the quantum jump technique in trapped ion QC schemes [@iontrap], optical transitions have been used to observe single spin states of nitrogen-vacancy defect centers in diamond [@Jelezko]. These observations clearly demonstrated that although solid state systems general have much more complexity compared to atomic systems, many of the techniques and concepts can be transferred with truly fruitful consequences. It has recently been reported that as few as two electron spins can be reliably detected with a particular realization of the MRFM [@Mamin]. Spin diffusion suppression, which is intimately related to the interaction between the cantilever and the spin being observed, has also been characterized in the inhomogeneous magnetic field produced by the small magnet at the end of the cantilever of an MRFM [@Budakian]. Electron-nuclear spin interaction in QDs ---------------------------------------- As we mentioned previously, electron-nuclear spin hyperfine coupling can lead to electron spin spectral diffusion and pure dephasing. Thus it is an important environmental issue for electron spins, especially for QD host materials that have a lot of nuclear spins, such as GaAs and InGaAs (the preferred materials for gated horizontal and vertical QDs, respectively). There have been quite extensive experimental studies of electron-nuclear spin coupling in bulk semiconductors and more recently semiconductor heterostructures [@optor; @Gammon-science; @Kawa; @SB-NS]. Dynamical nuclear spin polarization has recently been demonstrated in a spin blocked semiconductor double QD made of Ga$_{0.95}$In$_{0.05}$As [@QD-rev; @SB-NS; @SB]. Coulomb interaction and Pauli principle means that in a double QD two-electron singlet and triplet states are split by the exchange interaction, so that proper voltage offset and bias between the double dot leads to significant suppression in the tunnel current due to occupied triplet state (thus the so-called spin blockade regime, which has been suggested for single spin detection, a crucial component of quantum information processing) [@SB]. One way to lift this blockade is to apply an appropriate magnetic field, so that singlet-triplet spin-flip transition can be facilitated by the electron-nuclear spin hyperfine coupling as one of the polarized triplet state is energetically degenerate with the singlet state [@SB-NS]. This selective transition can then dynamically polarize the nuclear spins in the double QD. Indeed, strong experimental evidences have been observed that nuclear spins are indeed being polarized [@SB-NS], though many physical issues still have to be sorted out before a more thorough understanding of the coupled electron-nuclear spin system can be achieved. Fabrication of donor arrays in Si --------------------------------- Currently there are active experimental research efforts in attempting to fabricate well-controlled $^{31}$P donor arrays in a Si crystal, and plenty of experimental progress have been made [@Aussi_review]. Two experimental approaches are adopted, attacking the problem from opposite directions [@IQC1; @Aussi_review]. One uses ion implantation by bombarding $^{31}$P ions into crystalline silicon, thus it is also called a top-down approach. The other uses MBE to grow the system layer by layer and STM to identify donor locations, and is called the bottom-up approach. In the top-down approach, annealing is needed after the bombardment to make the $^{31}$P donors substitutional so that they become shallow donors. If they remain interstitial, they would behave as deep centers [@Altarelli], which have different electronic structures and thus are not useful for the purpose of quantum computing within the Kane proposal. Single electron transistors are used to monitor the presence of donors since they are very sensitive to net charges [@Aussi_review]. At present the precise positioning of the donors and annealing of the Si host lattice are being actively studied [@Schenkel; @Yang]. In the bottom-up approach, a clean Si surface is first hydrogenated. An STM is then used to pick off hydrogen atoms at desired locations, after which the surface is doused with PH$_3$ gas so that phosphorus atoms would tend to attach to the surface at the vacancies. This way an ordered array of donors can be fabricated with a high degree of regularity. Using this approach, it has been shown that precise positioning of phosphorus donors into a linear array can be achieved on a Si \[001\] surface [@OBrien]. More recently, the incorporation of the donors into the bulk of Si has also been demonstrated [@Schofield]. Athough much more experimental efforts have to be invested, and most probably more sophisticated technologies in donor positioning and manipulation have to be invented for this QC architecture to be fabricated with precision, recent experimental progresses are nonetheless quite impressive and promising. Summary ======= We have presented a brief review of the theoretical and experimental progresses related to spin-based QD QC architectures. We introduced the most prominent proposals of QD QC, outlined the basics on how these proposals might work, explored potential problems with the different material systems, and finally discussed where the present experimental studies stand. We thank US ARO, ARDA, and LPS for continued financial support to our research. The results presented here are the products of collaborations with S. Das Sarma, B. Koiller, D. Drew, R. de Sousa, R.B. Capaz, and fruitful discussions with J. Fabian, I. Zǔtić, A. Kaminski, M. Friesen, R.A. Webb, Y.Z. Chen, B.E. Kane, D. Romero, S. Tarucha, K. Ono, R. Hanson, and R. Budakian.
--- abstract: 'Let $M_i$ and $N_i$ be path-connected locally uniquely geodesic metric spaces that are not points and $f:\prod_{i=1}^m M_i\rightarrow \prod_{i=1}^n N_i$ be an isometry where $\prod_{i=1}^n N_i$ and $\prod_{i=1}^m M_i$ are given the sup metric. Then $m=n$ and after reindexing $M_i$ is isometric to $N_i$ for all $i$. Moreover $f$ is a composition of an isometry that reindexes the factor spaces and an isometry that is a product of isometries $f_i:M_i\rightarrow N_i$.' author: - William Malone bibliography: - 'prodRM8.bib' title: 'Isometries of Products of Path-Connected Locally Uniquely Geodesic Metric Spaces with the Sup Metric are Reducible.' --- Introduction ============ For the duration a product $\prod_{i=1}^m M_i$ of metric spaces is always considered to be endowed with the sup metric. In these metric spaces there are two obvious types of isometries $f:\prod_{i=1}^m M_i\rightarrow \prod_{i=1}^m M_i$. The first is a product $f=f_1\times\dots\times f_m$ where $f_j:M_j\rightarrow M_j$ is an isometry. The second is a reindexing of the form $$g(\alpha_1,\dots,\alpha_m)=(\alpha_{\pi{^{-1}}(1)},\dots,\alpha_{\pi{^{-1}}(m)})$$ for some permutation $\pi$ of $\{1,\dots,m\}$. This leads to the following definition. An isometry $f:\prod_{i=1}^m M_i\rightarrow \prod_{i=1}^m N_i$ is reducible if there is a permutation $\pi$ of $\{1,2,\dots,m\}$ and isometries $f_i:M_i\rightarrow N_{\pi(i)}$ such that $$f(\alpha_1,\dots,\alpha_m)=(f_{\pi{^{-1}}(1)}(\alpha_{\pi{^{-1}}(1)}),\dots,f_{\pi{^{-1}}(m)}(\alpha_{\pi{^{-1}}(m)})).$$ The main result states that in certain cases all isometries are reducible. Suppose $M_i$ and $N_i$ are path-connected locally uniquely geodesic metric spaces that are not points with $f:\prod_{i=1}^m M_i\rightarrow \prod_{i=1}^n N_i$ an isometry. Then $m=n$ and $f$ is reducible.\[main\] The heart of the proof lies in the fact that particular isometric embeddings of a graph in the product space $\prod_{i=1}^m M_i$ are invariant under any isometry. This will quickly lead to the number of factors in the product being an isometry invariant as well as forcing $f$ to take the aspect of a product map. A proof for the case where $m=2$ occurs in the appendix of [@FarbMasur08], and a discussion of the background on geodesic metric spaces can be found in [@BriHae99]. Acknowledgements ---------------- I would like to thank Mladen Bestvina, Yael Algom-Kfir, and Erika Meucci for their many helpful comments and discussions. I would also like to thank Benson Farb for mentioning this problem during a talk at the University of Utah. Proof of Main Theorem ===================== In this section we define a useful family of subsets of $\prod_{i=1}^m M_i$. Utilizing the Main Lemma, a proof of the Main Theorem is then provided. A $k$-slice $\Omega$ is a subset of $\prod_{i=1}^m M_i$ of the form $$\alpha_1\times \dots \times \alpha_{k-1}\times \Omega_k\times \alpha_{k+1}\times \dots\times\alpha_m$$ where $\alpha_i\in M_i$ and $\Omega_k\subset M_k$ with $\|\Omega_k\|\geq 2$. The $\alpha_i$ are called the fixed coordinates of $\Omega$.\[slice\] Suppose that $M_i$ and $N_i$ are path-connected locally uniquely geodesic metric spaces and $f:\prod_{i=1}^m M_i\rightarrow \prod_{i=1}^n N_i$ an isometry. If $\Omega\subset \prod_{i=1}^m M_i$ is a $k$-slice then $f(\Omega)$ is a $j$-slice. Moreover $\Omega$ and $\Omega'$ are both $k$-slices if and only if $f(\Omega)$ and $f(\Omega')$ are both $j$-slices. In order to prove the Main Lemma and hence the Main Theorem the fact that $m=n$ for any isometry $f:\prod_{i=1}^m M_i\rightarrow \prod_{i=1}^n N_i$ is essential. However the proof of this fact is more technical and will be postponed. (of Main Theorem) Assuming $m=n$, let $\Phi\subset\prod_{i=1}^m M_i$ be a $k$-slice for $k\in\{1,\dots,m\}$. By the Main Lemma the image $f(\Phi)$ is a $j$-slice for some $j$. The second part of the Main Lemma gives more information namely that the image of any $k$-slice $\Omega\subset\prod_{i=1}^m M_i$ will be a $j$-slice. To complete the proof it suffices to show that if two points, $$a=(\alpha_1,\alpha_2,\dots,\alpha_{k-1},\gamma,\alpha_{k+1},\dots,\alpha_m)$$ and $$b=(\beta_1,\beta_2,\dots,\beta_{k-1},\gamma,\beta_{k+1},\dots,\beta_m)$$ such that $\alpha_i\neq \beta_i$ for all $i$, have the same $k^{th}$ coordinate then $f(b)$ and $f(a)$ have the same $j^{th}$ coordinate. This will show the existence of isometries $f_i:M_k\rightarrow N_j$ and complete the proof of reducibility after post composing by the reindexing isometry $\pi:\prod_{i=1}^mN_i\rightarrow \prod_{i=1}^m N_i$ where $\pi(N_j)=N_k$. Consider the points $$\begin{array}{l} a_0=a=(\alpha_1,\alpha_2,\dots,\alpha_{k-1},\gamma,\alpha_{k+1},\dots,\alpha_m)\\ a_1=(\beta_1,\alpha_2,\dots,\alpha_{k-1},\gamma,\alpha_{k+1},\dots,\alpha_m)\\ a_2=(\beta_1,\beta_2,\alpha_3,\dots,\alpha_{k-1},\gamma,\alpha_{k+1},\dots,\alpha_m)\\ \vdots\\ a_{k-1}=(\beta_1,\beta_2,\dots,\beta_{k-1},\gamma,\alpha_{k+1},\dots,\alpha_m)\\ a_{k}=(\beta_1,\beta_2,\dots,\beta_{k-1},\gamma,\beta_{k+1},\alpha_{k+2},\dots,\alpha_m)\\ \vdots\\ a_{m-2}=(\beta_1,\beta_2,\dots,\beta_{k-1},\gamma,\beta_{k+1},\dots,\beta_{m-1},\alpha_m)\\ a_{m-1}=b=(\beta_1,\beta_2,\dots,\beta_{k-1},\gamma,\beta_{k+1},\dots,\beta_m).\\ \end{array}$$ Notice that $\{a_i,a_{i+1}\}$ is an $i+1$-slice for $i\in\{0,1,\dots,k-2\}$ and an $i+2$ slice for $i\in \{k-1,\dots,m-2\}$. Since no pair $\{a_i,a_{i+1}\}$ is a $k$-slice, the pair of points $\{f(a_i),f(a_{i+1})\}$ is never a $j$-slice by the Main Lemma. Thus the $j^{th}$ coordinate of $f(a_i)$ and $f(a_{i+1})$ are identical for all $i$. The Graph $\mathcal Q^m_r$ ========================== In this section we define a metric graph, give examples of particularly nice isometric embeddings of this graph in $\prod_{i=1}^m M_i$, show that $m=n$ for any isometry $f:\prod_{i=1}^m M_i\rightarrow \prod_{i=1}^n N_i$, and prove the Main Lemma. Observe that if $\alpha_i,\beta_i\in M_i$ are connected by a unique geodesic of length $r$ for $i=1,\dots,m$ then $a=(\alpha_1,\dots,\alpha_m)$ and $b=(\beta_1,\dots,\beta_m)$ are connected by a unique geodesic in $\prod M_i$. This fact that in certain directions within the product pairs of points are joined by unique geodesics is the main motivation for the following graph. A quadrilateral graph of dimension $m$ with length $r$ denoted $\mathcal Q^m_r\subset \prod_{i=1}^m {\mathbb R}$ is the metric graph whose vertices are of the form $\pm e_i^r=(0,0,\dots,0,\pm 2r,0,\dots,0)$ (non-zero in $i^{th}$ coordinate) or $(\pm r,\pm r,\dots,\pm r)$. Connect two vertices by an edge of length $r$ if their distance is $r$ in $\prod_{i=1}^m {\mathbb R}$. The first two examples of this family of graphs (pictured in Figure 1) are a quadrilateral with subdivided sides and the one skeleton of Kepler’s Rhombic dodecahedron [@BallCox87]. ![$\mathcal Q^3_1\subset \prod_{i=1}^3 {\mathbb R}$ (left) and $\mathcal Q^2_1\subset \prod_{i=1}^2 {\mathbb R}$ (right) ](graph23) \[graph\] Let $M_i$ be a uniquely geodesic metric space. An example of an isometric embedding $\iota:\mathcal Q^m_r\rightarrow \prod_{i=1}^m M_i$ is given by the following construction. Pick a geodesic segment $\lambda_i\subset M_i$ of length $4r$. Let $\lambda_i(0)=\alpha_i$, $\lambda_i(r)=\theta_i$ $\lambda_i(2r)=\beta_i$, $\lambda_i(3r)=\varphi_i$ and $\lambda_i(4r)=\omega_i$. Then let $$\begin{array}{ll}& \iota(e^r_k)=\{(\beta_1,\beta_2,\dots,\beta_{k-1},\omega_k,\beta_{k+1},\dots,\beta_m\\|k\in \{1,2,\dots,m\}\}\\ &\iota(-e^r_k)=\{(\beta_1,\beta_2,\dots,\beta_{k-1},\alpha_k,\beta_{k+1},\dots,\beta_m\\|k\in \{1,2,\dots,m\}\}\\ and\,\,as \,\,a \,\,set& \iota(\pm r, \pm r,\dots,\pm r)=\{(\pi_1,\dots,\pi_m)\|\pi_i\in\{\theta_i,\varphi_i\}\} \end{array}$$ which defines the isometry on the set of vertices. One can check that appropriate vertices are connected by unique geodesics of length $r$. \[graph\] This construction shows that isometric embeddings $\iota:\mathcal Q^m_r\rightarrow \prod_{i=1}^m M_i$ are easy to construct when the $M_i$ are uniquely geodesic metric spaces. If the spaces $M_i$ are only path-connected locally uniquely geodesic metric spaces then in general an isometric embedding $\iota:\mathcal Q^m_r\rightarrow \prod_{i=1}^m M_i$ will not exist for large $r$. However for every $a\in \prod M_i$ there exists a metric ball $B_{\epsilon}(a)$ such that $B_{\epsilon}(a)$ is a product of uniquely geodesic metric spaces. Thus for $4r<\epsilon$ an isometrically embedded $\mathcal Q^m_r$ can be constructed in the same way. Let $\iota:\mathcal Q^k_r\rightarrow \prod_{i=1}^m M_i$ be an isometric embedding. If the image of every edge is a uniquely geodesic segment in $\prod M_i$ then call $\iota:\mathcal Q^k_r\rightarrow \prod_{i=1}^m M_i$ admissible. If $\iota:\mathcal Q^k_r\rightarrow \prod_{i=1}^m M_i$ is admissible and $f:\prod_{i=1}^m M_i\rightarrow \prod_{i=1}^n N_i$ is an isometry then $f\circ \iota:\mathcal Q^k_r \rightarrow \prod_{i=1}^n N_i$ is admissible. This is clear from definitions. Let $\iota:\mathcal Q^k_r\rightarrow \prod_{i=1}^m M_i$ be admissible. The map $\iota$ is standard if for all $j$ there exists an $l$ such that $\{\iota(e^r_j),\iota(-e^r_j)\}$ is an $l-$slice. The proof of the Main Lemma requires a result establishing the fact that every admissible isometric embedding $\iota:\mathcal Q^m_r\rightarrow \prod_{i=1}^m M_i$ is standard. The following discussion works toward establishing this fact. First some results concerning the behavior of points in the image of an admissible $\iota:\mathcal Q^m_r\rightarrow \prod_{i=1}^m M_i$ are required. For any vertex $x_j\in Q^m_r$ of the form $\{\pm e^r_1,\dots,\pm e^r_m\}$ and any pair of points $\{e_i^r,-e_i^r\}$ with $x_j\not \in \{e_i^r,-e_i^r\}$ there exists a geodesic segment $\Lambda\subset \mathcal Q^m_r$ with endpoints $\{e_i^r,-e_i^r\}$ such that $x_j\in \Lambda$. \[prop1\] Constructing such a piecewise uniquely geodesic path in $\mathcal Q^m_r$ is straightforward. Note that there are precisely $2^{2m-4}$ paths connecting pairs of endpoints $\{e_i^r,-e_i^r\}$ so $\Lambda$ is unique only when $m=2$. Assume that $M_i$ is a uniquely geodesic metric space. Let $p$ and $q$ be points in $\prod_{i=1}^m M_i$ and $\Lambda$ a geodesic with endpoints $\{p,q\}$. Let $d_{M_i}(p_i,q_i)$ be the distance between the $i^{th}$ coordinates of $p$ and $q$. Note that $d_{M_i}(p_i,q_i)=d(p,q)$ if and only if the $i^{th}$ coordinate path of $\Lambda$ (denoted $\lambda_i$) is a geodesic of length $d(p,q)$. Moreover for any point $z\in \Lambda$ the $i^{th}$ coordinate of $z$ is uniquely determined by $d(p,z)$ (since $\lambda_i$ is the only geodesic connecting its endpoints in $M_i$). \[prop2\] Property \[prop2\] follows directly from the definition of a geodesic and will allow us to characterize the possible admissible $\iota:\mathcal Q^m_r\rightarrow \prod_{i=1}^m M_i$ when $M_i$ is uniquely geodesic. Assume that $M_i$ is a uniquely geodesic metric space and let $\iota:\mathcal Q^k_r\rightarrow \prod_{i=1}^m M_i$ be admissible. Let $$\begin{array}{c}\iota(e_j^r)=(\alpha_1,\dots,\alpha_m)\\ \iota(-e_j^r)=(\beta_1,\dots,\beta_m)\\ \iota(e_t^r)=(\gamma_1,\dots,\gamma_m)\\ \iota(-e_t^r)=(\kappa_1,\dots,\kappa_m)\\ \end{array}.$$ If $d(\iota(e_t^r),\iota(-e_t^r))=d_{M_l}(\gamma_l,\kappa_l)=4r$ then $\alpha_l=\beta_l$. \[disjoint\] By property \[prop1\] there exists geodesics $\Lambda,\Lambda'\subset \iota(\mathcal Q^k_r)$ such that the endpoints of $\Lambda$ and $\Lambda '$ are $\iota(\pm e_t^r)$ with $\iota(e_j^r)\in \Lambda$ and $\iota(-e_j^r)\subset \Lambda '$. By property \[prop2\] and the fact that $d(\iota(e_t^r), \iota(e_j^r))=d(\iota(e_t^r), \iota(-e_j^r))=2r$ we see that $\alpha_l=\beta_l $. For an admissible $\iota:\mathcal Q^k_r\rightarrow\prod_{i=1}^m M_i$ associate a number $$q_j=\|\{l\in\{1,\dots,m\} \| d(\iota(e_j^r),\iota(-e_j^r))=d_{M_l}(\alpha_l,\beta_l)\}\|$$ to each pair $\iota(\pm e_j^r)$. Clearly $q_j\geq 1$ for all $j$. If follows immediately from Lemma \[disjoint\] that two pairs $\{\iota(e_j^r),\iota(-e_j^r)\}$ and $\{\iota(e_t^r),\iota(-e_t^r)\}$ cannot attain the distance $4r$ in the same coordinate. These two observations imply $q_j=1$ for all $j$ when $k=m$. Let $M_i$ and $N_i$ be path-connected locally uniquely geodesic metric spaces and $f:\prod_{i=1}^m M_i \rightarrow \prod_{i=1}^n N_i$ be an isometry. Then $m=n$.\[m=n\] (Of lemma \[m=n\]) By invariance of an admissible $\iota(\mathcal Q^k_r)$ under isometry it follows that the largest $k$ such that there exists an admissible $\iota:\mathcal Q^k_r\rightarrow \prod_{i=1}^m M_i$ is an isometry invariant. Let this number be denoted by $L(\prod_{i=1}^m M_i)$. There exists a ball $B(a,4r)$ around any point $a\in \prod_{i=1}^m M_i$ such that $B(a,4r)$ is a product of uniquely geodesic metric spaces. An admissible $\mathcal Q^m_r$ can be constructed in $B(a,4r)$ by Example 3.2 which shows $L(\prod_{i=1}^m M_i)\geq m$. Since the number $q_j\geq1$ for all pairs $\{\iota(e_j^r),\iota(-e_j^r)\}$ in any admissible $\iota:\mathcal Q^k_r\rightarrow \prod_{i=1}^m M_i$ and $\sum_{j=1}^m q_j\leq m$ it is obvious that for $k>m$ no such $\iota$ can exist and thus $L(\prod_{i=1}^m M_i)\leq m$. Let $M_i$ be a uniquely geodesic metric space and $\iota:\mathcal Q^m_r \rightarrow \prod_{i=1}^m M_i$ admissible. Then $\iota:\mathcal Q^m_r\rightarrow \prod_{i=1}^m M_i$ is standard.\[standard\] (Of Lemma \[standard\]) Using the above fact that $q_j=1$ for all $j$ and Lemma \[disjoint\] the collection of pairs $$\left(\cup_{i=1}^{j-1} \{\iota(e_i^r),\iota(-e_i^r)\}\right)\cup\left(\cup_{i=j+1}^n\{\iota(e_i^r),\iota(-e_i^r)\}\right)$$ forces the distance between coordinates of $\{\iota(e_j^r),\iota(-e_j^r)\}$ to be either $4r$ or $0$, thus making $\{\iota(e_j^r),\iota(-e_j^r)\}$ a $l$-slice for some $l$. We now restate and prove the Main Lemma. Suppose that $M_i$ and $N_i$ are path-connected locally uniquely geodesic metric spaces and $f:\prod_{i=1}^m M_i\rightarrow \prod_{i=1}^n N_i$ an isometry. If $\Omega\subset \prod_{i=1}^m M_i$ is a $k$-slice then $f(\Omega)$ is a $j$-slice. Moreover $\Omega$ and $\Omega'$ are both $k$-slices if and only if $f(\Omega)$ and $f(\Omega')$ are both $j$-slices. By lemma \[m=n\] $m=n$. Three things must be shown. 1. (Part A) Given three distinct points $a,b,c$ such that $\{a,b\}$ is a $j$-slice, $\{b,c\}$ is a $k$-slice, and $\{a,c\}$ is an $i$-slice then $i=j=k$. (Part B) Given 4 distinct points $a,b,c,d$ such that $\{a,b\}$, $\{b,c\}$, $\{c,d\}$, and $\{a,d\}$ are $i,j,k,$ and $l$-slices respectively then either $i=j=k=l$ or $i=k$ and $j=l$. 2. If $\Omega$ is a $k$-slice then $f(\Omega)$ is a $j$-slice. 3. If $\Omega$ and $\Omega'$ are $k$-slices then $f(\Omega)$ and $f(\Omega')$ are $j$-slices. (Proof of Step 1 Part A) By definition of a $k$-slice if $i=j$ then $i=j=k$. Suppose $i\neq j$ then $a$ and $c$ differ in the $i^{th}$ and $j^{th}$ coordinate. But $\{a,c\}$ is a $k$-slice which is a contradiction. (Proof of Step 1 Part B) Observe that since $\{a,b\}$ is a $i$-slice and $\{b,c\}$ is a $j$-slice the points $a$ and $c$ differ in the $i^{th}$ and $j^{th}$ coordinates. Similarly they differ in the $l^{th}$ and $k^{th}$ coordinate. Thus $\{i,j\}=\{k,l\}$. If $i=l$ then $i=j=k=l$ otherwise $i=k$ and $j=l$. (Proof of Step 2) Any $k$-slice $\Omega$ is contained in a $k-$slice of the form $$\Phi=\alpha_1\times\dots\alpha_{k-1} \times M_k\times \alpha_{k+1}\dots \times\alpha_m$$ for $\alpha_i\in M_i$ so it suffices to show the result for $\Phi$. Let $\eta\in M_k$ be an arbitrary point. Let $\delta_s$ and $\bar{\epsilon_{\eta}}$ be the uniquely geodesic constants for $\alpha_s\in M_s$ and $\eta\in M_k$ respectively. Define $$W_{\eta}=\left(\prod_{s=1}^{k-1} B(\alpha_s,\epsilon_{\eta})\right)\times B(\eta_t,\epsilon_{\eta})\times \left(\prod_{s=k+1}^{m} B(\alpha_s,\epsilon_{\eta})\right)$$ where $\epsilon_{\eta}=min\{\bar{\epsilon_{\eta}},\delta_1,\dots,\delta_m\}$. Let $\Psi=W_{\eta}\cap \Phi$. For any three points $x_1,x_2,x_3\in \Psi$ construct three isometrically embedded graphs $\iota_{i}: \mathcal Q^m_r\rightarrow W_{\eta}$ with $r=\frac{1}{4}d(x_i,x_{i+1})$, $\iota_i(e_1^r)=x_i$, and $\iota_i(-e_1^r)=x_{i+1}$ for $i\in \{1,2,3\}$ via Example 3.2 (note that $x_4=x_1$). Since $W_{\eta}$ is a product of uniquely geodesic metric spaces we can apply lemma \[standard\] to $f\circ \iota_i$ to see that $\{f(x_1),f(x_2),f(x_3)\}$ satisfy the hypothesis of Step 1 part A. Thus $\{f(x_1),f(x_2),f(x_3)\}$ is a $j$-slice and since $x_i$ were arbitrary $f(\Psi)$ is a $j$-slice. If $\Psi'$ is a defined about a point $\eta'\in M_k$ sufficiently close to $\eta$ then $f(\Psi)\cap f(\Psi')$ consists of more than two points and the set $f(\Psi)\cup f(\Psi')$ is a $j$-slice. This implies $f(\Phi)$ is a $j-slice$ since $M_k$ has a single path component. (Proof of Step 3) Note that $$\Omega\subset \alpha_1\times\dots\times\alpha_{k-1} \times M_k\times\alpha_{k+1}\times\dots\times\alpha_m=\Phi$$ and $$\Omega'\subset \beta_1\times\dots\times\beta_{k-1} \times M_k\times\beta_{k+1}\times\dots\times\beta_m=\Phi'$$ for some $M_k$ and hence it suffices to show the result for $\Phi,\Phi'$. We can also assume that $\alpha_i=\beta_i$ for all $i\neq 1$ since we can change one coordinate at a time to interpolate between $\Phi$ and $\Phi'$. Now there clearly exist distinct points $w_1,w_2\in \Phi$ and $w_1',w_2'\in \Phi'$ such that $\{w_1,w_1'\}$ and $\{w_2,w_2'\}$ are $1$-slices. By step 2 $f(w_1),f(w_1'),f(w_2),f(w_2')$ satisfy the hypothesis of Step 1 part B. If $\{f(w_1),f(w_1'),f(w_2),f(w_2')\}$ was a $j$-slice then by step 2 after applying the inverse isometry $\{w_1,w_2,w_1,'w_2'\}$ would be a $k$-slice or a $1$-slice which is a contradiction. William Malone: Dept. of Mathematics, University of Utah 155 S 1400 E Room 233, Salt Lake City, UT 84112-0090 Email: malone@math.utah.edu
--- abstract: 'The possibility of meson condensation in stars and in heavy-ion collisions has been discussed in the past. Here, we study whether rho meson condensation ($\rho^{-}$) can occur in very dense matter and determine the effect of strong magnetic fields on this condensation. We find that rho meson condensates can appear in the core of the neutron star assuming a rho mass which is reduced due to in-medium effects. We find that the magnetic field has a non-negligible effect in triggering condensation.' author: - Ritam Mallick - Stefan Schramm - Veronica Dexheimer - Abhijit Bhattacharyya title: 'On the possibility of rho-meson condensation in neutron stars' --- The study of strongly interacting matter at high densities and/or temperatures is a central topic of modern nuclear physics. From the experimental side the main approaches to learn about matter under these conditions are the study of relativistic heavy-ion collisions (HIC) and observations of neutron stars, where the former approach aims at studying high-temperature and the latter one high-density and low-temperature matter. In addition, the future experiments at GSI/ FAIR with beam energies $\leq$ 30 GeV per nucleon and the beam energy scan at RHIC aim to study the properties of matter not only at high temperature but also at comparatively high densities. Increasing the density of the nuclear matter it is expected that the spontaneously broken chiral symmetry, characterized by a large quark condensate, is at least partially restored. However, a clear observable signature of this effect is still not well established. From the theory side, an early conjecture suggested by Brown and Rho [@brown], argued that hadrons (except for the pseudo-Goldstone bosons like pions) experience a mass reduction in nuclear matter, which is proportional to the in-medium quark condensate. More recent calculations, using QCD sum rules [@hatsuda], quark-meson models [@saito] and hadronic models of vacuum polarization [@jean] also suggest such mass reduction. However, a full understanding of the restoration of chiral symmetry in dense matter remains an open and much-discussed topic. More direct evidence of a potential mass reduction of vector mesons, the enhancement in the production of low invariant-mass dileptons in heavy-ion collisions has been investigated extensively, following such observations for an invariant mass around $400-500$ MeV, as measured by the CERES experiment at the CERN-SPS [@agakichiev]. Various model studies ranging from simple thermal models to more detailed transport calculations, predicted enhanced dilepton production [@cassing; @srivastava; @koch; @bratkovskaya]. One of the conjectures which explains such enhanced dilepton production in HIC, suggested the reduction of the in-medium masses of the vector mesons. The theoretical results were consistent with experimental data. However, later dynamical studies hinted that the earlier studies overestimated the effect [@cassing1]. The enhancement in dilepton production due to vector meson mass reduction also differed from experiment to experiment (consistent at SPS energies but not with DLS results [@bratkovskaya1]). Overall the situation with respect to dilepton enhancement is still not unambiguously resolved. More recent experiments of $\gamma - A$ [@trnka] reactions provided s clearer signature of an in-medium $\omega$ mass reduction. In the experiment by the TAPS collaboration [@trnka] the modification of the $\omega $ in nuclei was measured in photo-production experiments, where its mass was found to be $m_{\omega}^* = 722 \pm 4$ MeV at $0.6\, n_0$. Similar numbers were also found in $12$ GeV photon-nuclear reaction by Naruki et al. [@naruki]. The $\omega$ meson mass and quark-condensate were studied in more recent NJL model calculations [@huguet], where they approximately recovered the B-R scaling law using constraints from the TAPS experiment. However, for the $\rho$ meson the picture is still not clear. At the other extreme QCD exhibits exciting physics in a strong background magnetic field, since quarks are electrically charged and can be strongly affected by the field [@chernodub2]. It changes the chiral symmetry breaking by increasing the quark-condensate [@schramm1; @shovkovy]. However, the strength of the magnetic field which typically can influence QCD effects is about $eB \sim m_{\pi}^2 \sim 3\times 10^{18}$ G, where $e$ is the charge of the electron, $B$ is the magnetic field and $m_{\pi}$ denotes the mass of pion. Quarks and antiquarks can form condensates in the presence of strong magnetic fields. For the quark-antiquark bound state with light quarks, i.e. for the $\rho$ meson, strong magnetic fields can enhance the formation of a condensate. As will be discussed further below, the key idea lies in the fact that a sufficiently strong magnetic field can trigger an instability in the vacuum leading to condensates in specific quark channels, especially with respect to the charged vector mesons generating non-zero $\rho$ condensates [@schramm1; @chernodub2]. Huge magnetic fields are generated at the periphery of heavy-ion collisions, where the large nuclear charges and momenta magnify the magnetic field [@skokov]. At LHC energies of $3.5$ TeV, the strength of the magnetic field could be as high as $10^{20}-10^{21}$ G [@schramm]. However, such a field only exists for a very short time and is restricted to a small volume. On the other hand, the extreme conditions discussed above can occur naturally in neutron stars (NS), where the nuclear density is several times the nuclear saturation density ($n_0$) and the magnetic field can be as high as $10^{18}-10^{19}$ G in their cores [@ferrer; @dexheimer]. The high density might reduce the $\rho$ meson mass due to in-medium effects and with the aid of strong magnetic field, a condensate could subsequently form. The possibility of pion and/or kaon condensation in neutron stars has been investigated in numerous studies in the past. Since it was first suggested by Migdal [@migdal1; @migdal2], pion condensation and its impact on NS physics became the focus of intense discussion. There were calculations, both supporting and challenging a condensed state in the dense cores of compact stars. Similarly, for kaon ($K^{-}$) condensation [@kaplan], it was argued that as the electron chemical potential is an increasing function of density, whereas the effective (anti-)kaon mass decreases with density, condensation sets in at some density. Depending on the kaon-nucleon sigma term and the specific model, kaon condensation might occur already at about $3-4$ times saturation density, when the kaon energy becomes equal to the electron chemical potential, and above this density it is favourable for neutrons to decay to protons and kaons rather than protons and electrons, giving rise to a kaon condensate. Such densities can easily be achieved in the cores of neutron stars and, therefore, condensation can take place. However, non-linear terms and the occurrence of hyperons tend to shift the onset of kaon condensation to higher densities [@Mishra:2009bp], beyond values reached in neutron stars. One general problem for all condensate studies is their effect on the maximum possible neutron star masses. Since the discovery of pulsars PSR J1614-2230 and PSR J0348+0432 [@demorest; @antonidis] with masses approximately $2 M_{\odot}$ puts severe constraints on the stiffness of the equation of state (EoS) of the core region of stellar matter. Therefore, a potential onset of a condensate is important since (zero-momentum) condensates soften the equation of state thereby reducing maximum star masses. In this Letter, we study the possibility of rho meson ($\rho^{-}$) condensation occurring inside a NS and look into the conditions which might make rho condensation possible. There are various aspects that increase the possibility of condensation in a neutron star environment. Firstly, in the core of the star extreme densities exist, which could strongly amplify a potential density-dependence of the meson mass. Secondly, as the neutron star is in a charge-neutral state, condensation of negatively charged particles set in when their mass drops below the lepton Fermi energy, in striking contrast to the situation in a heavy-ion collision, where there is no such effect. Finally, there is a possibility of strong extended magnetic fields in at least certain classes of neutron stars like magnetars, which can generate a substantial effect for a charged spin-1 particle. We use a standard relativistic mean field approach to study the possibility of rho-meson condensation. Within this approach the nucleons interact through meson exchange represented by their mean field values. The scalar meson ($\sigma$) provides the attractive force, whereas the vector mesons ($\omega,\rho$) generate repulsion. In order to investigate whether the rho meson will condense in a neutron star we assume a simple relation for the medium-dependent $\rho$ mass. Certainly, in more extensive calculations a fuller study of density- (or field-) dependent masses of all involved hadrons should be performed. Here, in order to determine whether such a condensation could take place in principle, for simplicity we adopt a linear dependence of the rho mass on the scalar field $$m_{\rho}^=m_{\rho}-g \sigma$$ where $g$ is a dimensionless constant and $m_{\rho}=776$ MeV is the vacuum mass of the rho meson. This term effectively introduces a coupling between the vector and scalar fields. In the presence of a background magnetic field of strength $B$, the energy level $E_{n,S_x}$ of a particle of mass $m$, charge $e$, and spin $s$ in the Landau level $n$ is given by [@skalozub; @ambjorn; @schramm; @chernodub1; @chernodub2] $$E_{n,S_x} =\sqrt{p^2+m^2+(2n-g_B S_x+1)eB}$$ where $S_x$ is the spin projection onto the magnetic field axis, and $g_B$ is gyromagnetic ratio of the particle which is taken to be $2$ for the $\rho$ meson [@samsonov; @bhagwat; @hedditch]. Therefore, the energy squared of the lowest state for a charged rho meson corresponding to $p=0,n=0$ and $S_x=1$ is $${m_{\rho^{-}}^2}^=m_{\rho^{-}}^2-eB.$$ The upper equations point to the fact that in principle, if the in-medium effect or magnetic field is high enough, the effective mass squared of the $\rho$ meson can be negative and condensation can take place. However, for neutron stars the situation is less restrictive as mentioned above, since the mass of the $\rho^{-}$ only has to fall below the electron chemical potential, so that electrons can be replaced by negative rho mesons for generating a charge-neutral system. In this case, we obtain a neutron star with rho meson condensation. To have a better idea about if and how the rho meson condensation takes place in a neutron star, we have to adopt a specific relativistic mean field description including electrons to ensure charge neutrality of the matter. We choose the GM3 model for our calculation as its equation of state (EoS) has frequently been used in the literature to describe nuclear matter in NSs [@glendenning]. This EoS can generate $2$ solar mass NSs, which is an important criterion. We solve the mean field equations at finite density including the additional term Eq. (1). As this term does not affect isospin symmetric matter the isospin-independent quantities of saturated matter for the GM3 parameters do not change. However, the GM3 value of the asymmetry energy ($a_{sym} = 32.5$ MeV) changes. Therefore, while varying the field-dependence of the meson mass with the coupling $g$ from Eq. (1), we accordingly adjust the $g_{N\rho}$ coupling of the nucleon to the rho meson in order to maintain the original value of $a_{sym}$. 0.2in The results for stellar matter calculations are shown in Fig 1. We plot the variation of the electron chemical potential and the effective mass of the rho meson as a function of normalized baryon density. The dashed lines represent the rho meson effective mass and the solid lines depict the electron chemical potential. In Fig. 1a the black lines shows the respective values without any in-medium effect or magnetic field ($g=0$, $B=0$). In this case, we find that there is no crossing point between the lines, which means the rho meson does not condense in this environment for any reasonable density value. The inclusion of the magnetic field ($B\ne 0$) shifts the curves but is still not able to generate a crossing (red lines), which points to the fact that the magnetic field alone cannot give rise to condensation in a compact star. In order to investigate the maximum size of the effect with some reasonable assumptions, we considered a magnetic field of a strength of $7.1\times10^{18}$ G ($5\times 10^5$ MeV$^2$). In order to produce rho condensation solely due to a strong magnetic field, much higher values for the magnetic field is required ($3-4\times10^{19}$ G). However, studies suggest that such high magnetic fields are difficult, if not impossible, to realize in a NS; however, they may locally appear for a short time during heavy ion collisions. For the crossing to take place, the rho mass modification due to in-medium effect must be included. Taking this effect into account we find that the blue lines cross each other at about $7.6$ times nuclear saturation density. For such variation, we have assumed a vector-scalar coupling $g = -4$. We have checked the parameter setting with nuclear matter constraints, and all are well within the accepted values. Recent measurements of pulsar masses suggest that the central energy density of neutron stars might lie around $5-6$ times nuclear saturation values [@demorest]. In order for condensation to occur at such densities one must consider the effects of density-dependence of the mass and the effect of magnetic fields simultaneously. In Fig 1b. we plot curves for the case in which both (in-medium and magnetic) effects are taken into account. With the above mentioned values of $B$ and $g$ the condensate sets in at approximately $6.9$ times saturation density. However, if we now increase the value of $g$, i.e. assuming a larger rho mass modification due to in-medium effects, the condensate sets in at smaller densities. For $g=-6$, the condensate appears around $4.4$ times nuclear saturation density. Therefore, for a neutron star with a central density of $6$ times nuclear density, a rho meson condensate appears in the core of the star and extends up to the radius where the nuclear density is $4.4$ times nuclear density. 0.2in To study how rho condensation impacts the properties of a neutron star, we solve the Tolman-Oppenheimer-Volkov equation. In Fig. 2, we plot the sequence of stars obtained using the GM3 EoS. The black line represents the results for a NS where there is no $\rho^{-}$ condensate. The maximum mass achieved is about $2.05 M_{\odot}$. The red and blue curves show the sequences where rho meson condensation takes place. The solid lines in the curve represent stars which do not contain rho condensate, whereas the dotted points of the curve indicate stars that have some amount of $\rho^{-}$ condensate in their cores. For these curves the magnetic field is kept constant at $B=7.1\times10^{15}$ G. We find that as we increase $g$, the condensate appears much earlier i.e. more and more stars of the sequence have a condensate region in their cores. At this point we have determined the onset of condensation but so far have not performed a full calculation of the star structure including a condensate. However, assuming a drastic softening of the equation of state, the biggest possible shift of the maximum mass is given by the star mass when condensation starts to set in, which, for this specific model, yields a shift downward by about $0.2 M_{\odot}$ for a coupling of $g=-6$ and including magnetic fields as can be inferred from Fig. 2. To summarize, we studied the possibility of rho meson condensation within a standard relativistic mean field approach. While not rigorously proving the existence of the condensate, we have shown that such a state can be achieved, especially in a stellar environment, using quite reasonable assumptions. We conclude that the appearance of the condensate requires a modification of rho meson mass in the dense medium. In addition, magnetic fields have a non-negligible effect in reducing the threshold for condensation further. For that the magnetic field has to be quite large, at least in the high-density core region of the star; however, the exact upper limit of potential magnetic fields in the interior of a neutron star is still an open discussion. Note that although we had to use a specific model for obtaining numerical results all the arguments are quite generic and hold as well for other models. 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--- abstract: \| In the early 1990s, a family of combinatorial CW-complexes named permutoassociahedra was introduced by Kapranov, and it was realised by Reiner and Ziegler as a family of convex polytopes. The polytopes in this family are “hybrids” of permutohedra and associahedra. Since permutohedra and associahedra are simple, it is natural to search for a family of simple permutoassociahedra, which is still adequate for a topological proof of Mac Lane’s coherence. This paper presents such a family. [ ([ 2000]{}): 52B11, 18D10, 52B12, 51M20]{} [: associativity, commutativity, coherence, simple polytopes, geometric realisation]{} address: - - - author: - 'Djordje Baralić, Jelena Ivanović and Zoran Petrić' title: A simple permutoassociahedron --- [The paper is dedicated to Jim Stasheff whose work is the cornerstone of this beautiful mathematics.]{} Introduction ============ A geometry of commutativity and associativity interaction is demonstrated by polytopes called permutoassociahedra. This family of polytopes arises as a “hybrid” of two more familiar families—permutohedra and associahedra. Kapranov, [@K93], defined a combinatorial CW-complex $KP_n$ (denoted by $\mathbf{KPA}_{n-1}$ in [@RZ94]), whose vertices correspond to all possible complete bracketings of permuted products of $n$ letters. He showed that for every $n$, this CW-complex is an $(n-1)$-ball, and found a three dimensional polytope that realises $KP_4$. Reiner and Ziegler, [@RZ94], realised all these CW-complexes as convex polytopes. However, for every $n\geq 4$, the polytope $KP_n$ is not simple. Our goal is to define a new family of polytopes $\mathbf{PA}_{n-1}$, having the same vertices as $KP_n$, which are all simple, and which may still serve as a topological proof of Mac Lane’s coherence. The combinatorics of the family $\mathbf{PA}_n$ is introduced in [@P14]. This family is called permutohedron-based associahedra in that paper, and for the sake of simplicity, we call it here simple permutoassociahedra. From [@P14 Proposition 10.1], it follows that there exist simple polytopes that realise these combinatorially defined objects. Our polytopes $\mathbf{PA}_n$ belong to a family that generalises polytopes called nestohedra or hypergraph polytopes. The work of Fulton and MacPherson, [@FMP94], De Concini and Procesi, [@DeCP95], Stasheff and Shnider, [@S97], Gaiffi, [@Ga03], [@Ga04], Feichtner and Kozlov, [@FK04], Feichtner and Sturmfels, [@FS05], Carr and Devadoss, [@CD06], Postnikov, Reiner and Williams, [@PRW08], Postnikov, [@P09], Došen and the third author, [@DP10], and of many others established the family of nestohedra as an important tool joining algebra, combinatorics, geometry, logic, topology and some other fields of mathematics. We pay attention in particular to 0, 1 and 2-faces of polytopes $\mathbf{PA}_n$. These faces suggest a choice of generators and corresponding equations relevant for building a symmetric monoidal category freely generated by a set of objects. All this is important for Mac Lane’s coherence. However, the main result of the paper is an explicit realisation of the family $\mathbf{PA}_n$ given by systems of inequalities representing halfspaces in $\mathbf{R}^{n+1}$. Our realisation has its roots in the realisation of associahedra and cyclohedra given in [@S97]. Throughout the text, the subset relation is denoted by $\subseteq$, while the proper subset relation is denoted by $\subset$. Varying the combinatorics by generators ======================================= In this section we try to formalise the main differences between the approach of [@K93], leading to the family of permutoassociahedra $KP_n$, and our approach, leading to a family of simple permutoassociahedra. Roughly speaking, the main difference is the choice of arrows that generate symmetry in a symmetric monoidal category. The reader interested mainly in combinatorics of the new family of polytopes, may skip this section and just take a look at the commutative diagrams given below. The notion of symmetric monoidal categories (not under this name) is given, and a coherence result is proven by Mac Lane in [@ML63]. (For the definition of symmetric monoidal categories we refer to [@ML71 XI.1]). The original proof of coherence relies on a strictification of the monoidal structure of such a category followed by the standard presentation of symmetric groups by generators and relations (for details see [@DP04 §3.3, Chapter 5]). Since a coherence result of a class of categories deals with canonical arrows, it is natural to formulate it in terms of a category in this class freely generated by a set of objects. This idea is suggested by Voreadou, [@V77], and it is fairly used in [@DP04]. There is no explicit referring to this approach in [@K93]—however, one may reconstruct the instant one step proof of Mac Lane’s coherence, suggested by the author, through the symmetric monoidal category freely generated by an infinite set of objects. Also, this provides us a possibility to compare the two families of permutoassociahedra. Given the set $\omega=\{0,1,2,\ldots\}$ of generating objects, what is a symmetric monoidal category $\mathbf{SM}$ with the strict unit $I$, freely generated by $\omega$? For the set of objects of this category, there is no doubt—this set contains $I$, and all the other objects are built out of elements of $\omega$ with the help of binary operation denoted here by $\cdot$, which is the object part of the required bifunctor. For example, $(2\cdot((4\cdot 2)\cdot 0))$ is an object of this category and we omit, as usual, the outermost parentheses denoting this object by $2\cdot((4\cdot 2)\cdot 0)$. The arrows of $\mathbf{SM}$ are equivalence classes of terms. One can choose here different languages to build these terms, and hence to have different equivalence relations, generated by commutative diagrams, that produce the arrows of $\mathbf{SM}$. Our reconstruction of the approach of [@K93] is that the terms are built inductively in the following manner. (${\mbox{\bf 1}}$.1) For an object $A$, the identity ${\mbox{\bf 1}}_A:A{\rightarrow}A$ is an ${\mbox{\bf 1}}$-term; (${\mbox{\bf 1}}$.2) for $f$ and $g$, ${\mbox{\bf 1}}$-terms, $(f\cdot g)$ is an ${\mbox{\bf 1}}$-term. ($\alpha$.1) For objects $A$, $B$ and $C$ distinct from $I$, $\alpha_{A,B,C}:A\cdot(B\cdot C){\rightarrow}(A\cdot B)\cdot C$ and $\alpha^{-1}_{A,B,C}:(A\cdot B)\cdot C{\rightarrow}A\cdot (B\cdot C)$ are $\alpha$-terms; ($\alpha$.2) for $f$ an $\alpha$-term and $g$ an ${\mbox{\bf 1}}$-term, $(f\cdot g)$ and $(g\cdot f)$ are $\alpha$-terms. ($\tau$.1) For $p,q\in\omega$, $\tau_{p,q}:p\cdot q{\rightarrow}q\cdot p$ is a $\tau$-term; ($\tau$.2) for $f$ a $\tau$-term and $g$ an ${\mbox{\bf 1}}$-term, $(f\cdot g)$ and $(g\cdot f)$ are $\tau$-terms. ($t$.1) Every ${\mbox{\bf 1}}$-term, $\alpha$-term and $\tau$-term is a term; ($t$.2) if $f:A{\rightarrow}B$ and $g:B{\rightarrow}C$ are terms, then $g\circ f:A{\rightarrow}C$ is a term. For example, $$\alpha^{-1}_{2\cdot 4,1,0}\circ(\alpha_{2,4,1}\cdot {\mbox{\bf 1}}_0)\circ(({\mbox{\bf 1}}_2\cdot \tau_{1,4})\cdot {\mbox{\bf 1}}_0)\circ (\alpha^{-1}_{2,1,4}\cdot{\mbox{\bf 1}}_0):((2\cdot 1)\cdot 4)\cdot 0{\rightarrow}(2\cdot 4)\cdot(1\cdot 0)$$ is a term. The identities are neutrals for composition. Moreover, ${\mbox{\bf 1}}_A\cdot{\mbox{\bf 1}}_B={\mbox{\bf 1}}_{A\cdot B}$ and the following diagrams commute for $f$, $g$ and $f\cdot(g\cdot h)$ being $\alpha$-terms or $\tau$-terms. (120,50) (0,40)[(0,0)[$A\cdot B$]{}]{} (0,10)[(0,0)[$A\cdot B'$]{}]{} (120,40)[(0,0)[$A'\cdot B$]{}]{} (120,10)[(0,0)[$A'\cdot B'$]{}]{} (0,30)[(0,-1)[10]{}]{} (120,30)[(0,-1)[10]{}]{} (25,40)[(1,0)[70]{}]{} (25,10)[(1,0)[70]{}]{} (-15,25)[(0,0)[${\mbox{\bf 1}}\cdot g$]{}]{} (135,25)[(0,0)[${\mbox{\bf 1}}\cdot g$]{}]{} (60,46)[(0,0)[$f\cdot{\mbox{\bf 1}}$]{}]{} (60,4)[(0,0)[$f\cdot{\mbox{\bf 1}}$]{}]{} (200,25)[(0,0)[(1)]{}]{} (120,50) (0,40)[(0,0)[$A\cdot(B\cdot C)$]{}]{} (0,10)[(0,0)[$A\cdot(B\cdot C)$]{}]{} (120,40)[(0,0)[$(A\cdot B)\cdot C$]{}]{} (120,10)[(0,0)[$(A\cdot B)\cdot C$]{}]{} (0,30)[(0,-1)[10]{}]{} (120,30)[(0,-1)[10]{}]{} (30,40)[(1,0)[60]{}]{} (30,10)[(1,0)[60]{}]{} (90,35)[(-3,-1)[60]{}]{} (-10,25)[(0,0)[${\mbox{\bf 1}}$]{}]{} (130,25)[(0,0)[${\mbox{\bf 1}}$]{}]{} (60,45)[(0,0)[$\alpha$]{}]{} (60,5)[(0,0)[$\alpha$]{}]{} (55,29)[(0,0)[$\alpha^{-1}$]{}]{} (200,25)[(0,0)[(2)]{}]{} (120,110) (60,100)[(0,0)[$A\cdot(B\cdot(C\cdot D))$]{}]{} (0,70)[(0,0)[$A\cdot((B\cdot C)\cdot D)$]{}]{} (0,40)[(0,0)[$(A\cdot(B\cdot C))\cdot D$]{}]{} (60,10)[(0,0)[$((A\cdot B)\cdot C)\cdot D$]{}]{} (120,55)[(0,0)[$(A\cdot B)\cdot (C\cdot D)$]{}]{} (50,90)[(-3,-1)[30]{}]{} (0,60)[(0,-1)[10]{}]{} (20,30)[(3,-1)[30]{}]{} (70,90)[(2,-1)[40]{}]{} (110,40)[(-2,-1)[40]{}]{} (20,88)[(0,0)[${\mbox{\bf 1}}\cdot \alpha$]{}]{} (-10,55)[(0,0)[$\alpha$]{}]{} (20,22)[(0,0)[$\alpha\cdot{\mbox{\bf 1}}$]{}]{} (97,82)[(0,0)[$\alpha$]{}]{} (97,28)[(0,0)[$\alpha$]{}]{} (200,55)[(0,0)[(3)]{}]{} (120,50) (0,40)[(0,0)[$A\cdot(B\cdot C)$]{}]{} (0,10)[(0,0)[$A'\cdot(B'\cdot C')$]{}]{} (120,40)[(0,0)[$(A\cdot B)\cdot C$]{}]{} (120,10)[(0,0)[$(A'\cdot B')\cdot C'$]{}]{} (0,30)[(0,-1)[10]{}]{} (120,30)[(0,-1)[10]{}]{} (30,40)[(1,0)[60]{}]{} (30,10)[(1,0)[60]{}]{} (-25,25)[(0,0)[$f\cdot(g\cdot h)$]{}]{} (145,25)[(0,0)[$(f\cdot g)\cdot h$]{}]{} (60,45)[(0,0)[$\alpha$]{}]{} (60,5)[(0,0)[$\alpha$]{}]{} (200,25)[(0,0)[(4)]{}]{} (120,55) (10,40)[(0,0)[$p\cdot q$]{}]{} (10,10)[(0,0)[$p\cdot q$]{}]{} (110,40)[(0,0)[$q\cdot p$]{}]{} (10,30)[(0,-1)[10]{}]{} (30,40)[(1,0)[60]{}]{} (90,35)[(-3,-1)[60]{}]{} (0,25)[(0,0)[${\mbox{\bf 1}}$]{}]{} (60,45)[(0,0)[$\tau$]{}]{} (60,18)[(0,0)[$\tau$]{}]{} (200,25)[(0,0)[(5)]{}]{} (120,145) (25,130)[(0,0)[$p\cdot (q\cdot r)$]{}]{} (-10,106)[(0,0)[$(p\cdot q)\cdot r$]{}]{} (-30,82)[(0,0)[$(q\cdot p)\cdot r$]{}]{} (-30,58)[(0,0)[$q\cdot(p\cdot r)$]{}]{} (-10,34)[(0,0)[$q\cdot(r\cdot p)$]{}]{} (25,10)[(0,0)[$(q\cdot r)\cdot p$]{}]{} (95,130)[(0,0)[$p\cdot (r\cdot q)$]{}]{} (130,106)[(0,0)[$(p\cdot r)\cdot q$]{}]{} (150,82)[(0,0)[$(r\cdot p)\cdot q$]{}]{} (150,58)[(0,0)[$r\cdot(p\cdot q)$]{}]{} (130,34)[(0,0)[$r\cdot(q\cdot p)$]{}]{} (95,10)[(0,0)[$(r\cdot q)\cdot p$]{}]{} (-2,28)[(2,-1)[22]{}]{} (-26,53)[(1,-1)[13]{}]{} (-30,76)[(0,-1)[12]{}]{} (-13,100)[(-1,-1)[13]{}]{} (20,122)[(-2,-1)[22]{}]{} (122,28)[(-2,-1)[22]{}]{} (146,52)[(-1,-1)[13]{}]{} (150,76)[(0,-1)[12]{}]{} (133,100)[(1,-1)[13]{}]{} (100,122)[(2,-1)[22]{}]{} (50,130)[(1,0)[20]{}]{} (50,10)[(1,0)[20]{}]{} (60,115)[(0,0)]{} (60,5)[(0,0)]{} (-2,20)[(0,0)[$\alpha$]{}]{} (-32,45)[(0,0)[${\mbox{\bf 1}}\cdot \tau$]{}]{} (-40,70)[(0,0)[$\alpha^{-1}$]{}]{} (-32,95)[(0,0)[$\tau\cdot{\mbox{\bf 1}}$]{}]{} (-2,120)[(0,0)[$\alpha$]{}]{} (122,20)[(0,0)[$\alpha$]{}]{} (152,45)[(0,0)[${\mbox{\bf 1}}\cdot \tau$]{}]{} (162,70)[(0,0)[$\alpha^{-1}$]{}]{} (152,95)[(0,0)[$\tau\cdot{\mbox{\bf 1}}$]{}]{} (122,120)[(0,0)[$\alpha$]{}]{} (60,137)[(0,0)[${\mbox{\bf 1}}\cdot \tau$]{}]{} (60,3)[(0,0)[$\tau\cdot{\mbox{\bf 1}}$]{}]{} (200,70)[(0,0)[(6)]{}]{} All these equations are assumed to hold in contexts. For example, the equation (4) in the context $\cdot D$ reads (120,50) (0,40)[(0,0)[$(A\cdot(B\cdot C))\cdot D$]{}]{} (-5,10)[(0,0)[$(A'\cdot(B'\cdot C'))\cdot D$]{}]{} (120,40)[(0,0)[$((A\cdot B)\cdot C)\cdot D$]{}]{} (125,10)[(0,0)[$((A'\cdot B')\cdot C')\cdot D$]{}]{} (0,30)[(0,-1)[10]{}]{} (120,30)[(0,-1)[10]{}]{} (40,40)[(1,0)[40]{}]{} (40,10)[(1,0)[40]{}]{} (-35,25)[(0,0)[$(f\cdot(g\cdot h))\cdot{\mbox{\bf 1}}$]{}]{} (155,25)[(0,0)[$((f\cdot g)\cdot h)\cdot{\mbox{\bf 1}}$]{}]{} (60,45)[(0,0)[$\alpha\cdot{\mbox{\bf 1}}$]{}]{} (60,5)[(0,0)[$\alpha\cdot{\mbox{\bf 1}}$]{}]{} Out of the category $\mathbf{SM}$, one can build a (pseudo)graph $\mathcal{G}$ (in the terminology of [@H69]) whose vertices are the objects of $\mathbf{SM}$, while $A$ and $B$ are joined by an edge in $\mathcal{G}$, when there is an $\alpha$-term, or a $\tau$-term $f:A{\rightarrow}B$. Since every such term has an inverse, the edges are undirected. There are no multiple edges. However, this graph contains loops since, for example, there is an edge joining $p\cdot p$ with itself, witnessed by the term $\tau_{p,p}$. Every connected component of $\mathcal{G}$ contains the objects with the same number of occurrences of elements of $\omega$. Hence, the object $I$, as well as every element of $\omega$, makes a singleton connected component of $\mathbf{SM}$. For the coherence question, we are interested in diversified connected components, i.e. the connected components containing the objects of $\mathbf{SM}$ without repeating the elements of $\omega$. In particular, for every $n\in\omega$, we are interested in the connected component $\mathcal{G}_n$ of $\mathcal{G}$ containing the object of the form $$0\cdot(1\cdot(\ldots\cdot n)\ldots).$$ It is straightforward to see that $\mathcal{G}_n$ does not contain loops. In order to establish the symmetric monoidal coherence (cf.[@DP04 §3.3,Chapter 5]), it suffices to show that for every $n\in\omega$, every diagram involving the objects of $\mathcal{G}_n$ commutes in $\mathbf{SM}$. A topological proof of this fact consists in finding an $n$-dimensional CW-ball, whose 0-cells are the vertices of $\mathcal{G}_n$, 1-cells are the edges of $\mathcal{G}_n$, and 2-cells correspond to quadrilaterals (1) and (4), pentagons (3) and dodecagons (6) given above. This all is done by Kapranov in [@K93]. Our approach is a bit different. We construct a category $\mathbf{SM}$\* with the same objects as $\mathbf{SM}$, for which we show that is, up to renaming of arrows, the same as $\mathbf{SM}$. To fix the notation we define the left, right and middle contexts in the following way. The symbol $\square$ is a left and a right context. If $L$ is a left and $R$ is a right context, while $A$ is an object of $\mathbf{SM}$\, then $(A\cdot L)$ is a left context, $(R\cdot A)$ is a right context, and $(R\cdot L)$ is a middle context. If $L$, $R$ and $M$ are left, right and middle contexts, respectively, and $a,b\in\omega$, then $La$ denotes the object of $\mathbf{SM}$\ obtained by replacing the occurrence of $\square$ in $L$ by $a$, and we define the objects $bR$ and $bMa$ in the same way. It is obvious that for every object $A$ of $\mathbf{SM}$\* different from $I$, there exists a unique left context $L$ and a unique $a\in\omega$, namely the rightmost occurrence of an element of $\omega$ in $A$, such that $A$ is $La$. The same holds for right contexts and if $A$ contains at least one $\cdot$, the same holds for middle contexts. Our definition of terms of $\mathbf{SM}$\* is as for $\mathbf{SM}$ save that the clauses ($\tau$.1) and ($\tau$.2) are replaced by the clause ($\sigma$) For objects $La$ and $bR$, $\sigma_{La,bR}:La\cdot bR{\rightarrow}Lb\cdot aR$ is a $\sigma$-term, and in ($t$.1), “$\tau$-term” is replaced by “$\sigma$-term”. For example, $$\sigma_{2\cdot 1,4\cdot 0}\circ \alpha^{-1}_{2\cdot 1,4,0}:((2\cdot 1)\cdot 4)\cdot 0{\rightarrow}(2\cdot 4)\cdot(1\cdot 0)$$ is a term. To define the arrows of $\mathbf{SM}$\, we use the equivalence relation on terms, which is generated by the commutative diagrams (1), (2), (3) and (4) (save that $f$, $g$ and $f\cdot(g\cdot h)$ are $\alpha$-terms) and the following commutative diagrams involving $\sigma$’s. (120,50) (0,40)[(0,0)[$La\cdot bR$]{}]{} (0,10)[(0,0)[$La\cdot bR$]{}]{} (120,40)[(0,0)[$Lb\cdot aR$]{}]{} (0,32)[(0,-1)[14]{}]{} (30,40)[(1,0)[60]{}]{} (90,35)[(-3,-1)[60]{}]{} (-8,25)[(0,0)[${\mbox{\bf 1}}$]{}]{} (60,45)[(0,0)[$\sigma$]{}]{} (60,18)[(0,0)[$\sigma$]{}]{} (200,25)[(0,0)[(7)]{}]{} For the diagram (8), it is assumed that $f:La{\rightarrow}L'a$ and $g:bR{\rightarrow}bR'$ are $\alpha$-terms and that $f^a_b:Lb{\rightarrow}L'b$ and $g^b_a:aR{\rightarrow}aR'$ are corresponding $\alpha$-terms obtained by substituting $b$ for the distinguished occurrence of $a$ in $f$, and vice versa for $g$. (200,60)(0,-5) (0,40)[(0,0)[$La\cdot bR$]{}]{} (0,10)[(0,0)[$Lb\cdot aR$]{}]{} (70,40)[(0,0)[$L'a\cdot bR$]{}]{} (70,10)[(0,0)[$L'b\cdot aR$]{}]{} (0,32)[(0,-1)[14]{}]{} (70,32)[(0,-1)[14]{}]{} (20,40)[(1,0)[30]{}]{} (20,10)[(1,0)[30]{}]{} (-7,25)[(0,0)[$\sigma$]{}]{} (77,25)[(0,0)[$\sigma$]{}]{} (35,47)[(0,0)[$f\cdot {\mbox{\bf 1}}$]{}]{} (35,3)[(0,0)[$f^a_b\cdot {\mbox{\bf 1}}$]{}]{} (240,25)[(0,0)[(8)]{}]{} (130,40)[(0,0)[$La\cdot bR$]{}]{} (130,10)[(0,0)[$Lb\cdot aR$]{}]{} (200,40)[(0,0)[$La\cdot bR'$]{}]{} (200,10)[(0,0)[$Lb\cdot aR'$]{}]{} (130,30)[(0,-1)[10]{}]{} (200,30)[(0,-1)[10]{}]{} (150,40)[(1,0)[30]{}]{} (150,10)[(1,0)[30]{}]{} (123,25)[(0,0)[$\sigma$]{}]{} (207,25)[(0,0)[$\sigma$]{}]{} (165,47)[(0,0)[${\mbox{\bf 1}}\cdot g$]{}]{} (165,3)[(0,0)[${\mbox{\bf 1}}\cdot g^b_a$]{}]{} (120,95) (10,82)[(0,0)[$La\cdot (bMc\cdot dR)$]{}]{} (-20,58)[(0,0)[$(La\cdot bMc)\cdot dR$]{}]{} (-20,34)[(0,0)[$(La\cdot bMd)\cdot cR$]{}]{} (10,10)[(0,0)[$La\cdot (bMd\cdot cR)$]{}]{} (110,82)[(0,0)[$Lb\cdot (aMc\cdot dR)$]{}]{} (140,58)[(0,0)[$(Lb\cdot aMc)\cdot dR$]{}]{} (140,34)[(0,0)[$(Lb\cdot aMd)\cdot cR$]{}]{} (110,10)[(0,0)[$Lb\cdot (aMd\cdot cR)$]{}]{} (-13,27)[(2,-1)[22]{}]{} (-20,52)[(0,-1)[12]{}]{} (10,75)[(-2,-1)[22]{}]{} (133,27)[(-2,-1)[22]{}]{} (140,52)[(0,-1)[12]{}]{} (110,75)[(2,-1)[22]{}]{} (50,82)[(1,0)[20]{}]{} (50,10)[(1,0)[20]{}]{} (-20,21)[(0,0)[$\alpha^{-1}$]{}]{} (-30,45)[(0,0)[$\sigma$]{}]{} (-15,71)[(0,0)[$\alpha$]{}]{} (135,21)[(0,0)[$\alpha^{-1}$]{}]{} (150,45)[(0,0)[$\sigma$]{}]{} (135,71)[(0,0)[$\alpha$]{}]{} (60,89)[(0,0)[$\sigma$]{}]{} (60,3)[(0,0)[$\sigma$]{}]{} (200,50)[(0,0)[(9)]{}]{} (120,145) (20,130)[(0,0)[$La\cdot (b\cdot cR)$]{}]{} (-10,106)[(0,0)[$(La\cdot b)\cdot cR$]{}]{} (-30,82)[(0,0)[$(La\cdot c)\cdot bR$]{}]{} (-30,58)[(0,0)[$La\cdot (c\cdot bR)$]{}]{} (-10,34)[(0,0)[$Lc\cdot (a\cdot bR)$]{}]{} (20,10)[(0,0)[$(Lc\cdot a)\cdot bR$]{}]{} (100,130)[(0,0)[$Lb\cdot (a\cdot cR)$]{}]{} (130,106)[(0,0)[$(Lb\cdot a)\cdot cR$]{}]{} (150,82)[(0,0)[$(Lb\cdot c)\cdot aR$]{}]{} (150,58)[(0,0)[$Lb\cdot (c\cdot aR)$]{}]{} (130,34)[(0,0)[$Lc\cdot (b\cdot aR)$]{}]{} (100,10)[(0,0)[$(Lc\cdot b)\cdot aR$]{}]{} (-2,28)[(2,-1)[22]{}]{} (-26,53)[(1,-1)[13]{}]{} (-30,76)[(0,-1)[12]{}]{} (-13,100)[(-1,-1)[13]{}]{} (20,122)[(-2,-1)[22]{}]{} (122,28)[(-2,-1)[22]{}]{} (146,52)[(-1,-1)[13]{}]{} (150,76)[(0,-1)[12]{}]{} (133,100)[(1,-1)[13]{}]{} (100,122)[(2,-1)[22]{}]{} (50,130)[(1,0)[20]{}]{} (50,10)[(1,0)[20]{}]{} (-2,20)[(0,0)[$\alpha$]{}]{} (-32,45)[(0,0)[$\sigma$]{}]{} (-40,70)[(0,0)[$\alpha^{-1}$]{}]{} (-32,95)[(0,0)[$\sigma$]{}]{} (-2,120)[(0,0)[$\alpha$]{}]{} (122,20)[(0,0)[$\alpha$]{}]{} (152,45)[(0,0)[$\sigma$]{}]{} (162,70)[(0,0)[$\alpha^{-1}$]{}]{} (152,95)[(0,0)[$\sigma$]{}]{} (122,120)[(0,0)[$\alpha$]{}]{} (60,137)[(0,0)[$\sigma$]{}]{} (60,3)[(0,0)[$\sigma$]{}]{} (200,70)[(0,0)[(10)]{}]{} The equations (1), (2), (3) and (4) are assumed to hold in contexts, while the contexts make no sense for the equations (7), (8), (9) and (10). It is more or less straightforward to define the $\sigma$-terms by $\tau$-terms and $\alpha$-terms, and to show by induction that the equations (7), (8), (9) and (10) hold in $\mathbf{SM}$ for defined $\sigma$’s. The equations (5), (4) with $f\cdot(g\cdot h)$ being a $\tau$-term, (1) with $f$, $g$ being $\tau$-terms and (6), respectively, are essential for this proof. However, if one is not interested in a new topological proof of Mac Lane’s coherence, then this coherence result instantly delivers the commutativity of the above diagrams in $\mathbf{SM}$. For the other direction, the $\tau$-terms are easily defined in terms of $\sigma$-terms and $\alpha$-terms. Our family of polytopes (see Section \[realisation\]) will guarantee that all the equations of $\mathbf{SM}$ hold in $\mathbf{SM}$\ for defined $\tau$-terms. Hence, $\mathbf{SM}$ and $\mathbf{SM}$\* are, up to renaming of arrows, the same. Let $\mathcal{G}^$ be a (pseudo)graph with the same vertices* as $\mathcal{G}$, while $A$ and $B$ are joined by an edge in $\mathcal{G}^$, when there is an $\alpha$-term, or a $\sigma$-term $f:A{\rightarrow}B$. As in the case of $\mathcal{G}$, we denote by $\mathcal{G}^_n$ the connected component of $\mathcal{G}^$ containing the object $$0\cdot(1\cdot(\ldots\cdot n)\ldots).$$ It is evident that $\mathcal{G}^_n$ is $n$-regular, i.e. it has exactly $n$ edges incident to every vertex. A graph $G$ is a graph of a polytope $P$ when the vertices and edges of $P$ are in one-to-one correspondence with the vertices and edges of $G$, and this correspondence respects the incidence relation. By a result of Blind and Mani-Levitska, [@BM87] (see also Kalai’s elegant proof, [@K88]), one can conclude that if there is an $n$-dimensional simple polytope whose graph is $\mathcal{G}^_n$, then all such polytopes are combinatorially equivalent. Our goal is to geometrically realise such a polytope for every $n$, and to show that its 2-faces correspond to quadrilaterals (1), (4) and (8), pentagons (3), octagons (9) and dodecagons (10). A two-fold nested set construction {#two-fold} ================================== A family of simplicial complexes that are closely related to the face lattices of a family of polytopes, which we call simple permutoassociahedra, is introduced in this section. Our construction of this family of simplicial complexes is an example of the two-fold construction of complexes of nested sets for simplicial complexes, which is thoroughly investigated in [@P14]. Since we are here interested in a single family of complexes, there is no reason to repeat the theory of nested set complexes in its full generality and we present only the facts necessary for our construction. We refer to [@P14] for a detailed treatment of the theory. An abstract simplicial complex* on a set $T$ is a non-empty collection $K$ of finite subsets of $T$ closed under taking inclusions, i.e. if $\sigma \in K$ and $\tau\subseteq \sigma$, then $\tau \in K$. The elements of $K$ are called simplices. It is assumed that $\{v\}$ is a simplex of $K$ for every element $v\in T$. An abstract simplicial complex $K$ is finite when the number of its simplices is finite, otherwise it is infinite. In this contribution we consider only finite abstract simplicial complexes, and we call them shortly simplicial complexes. A simplex $\tau\in K$ is a face of a simplex $\sigma \in K$ if $\tau\subseteq \sigma$. A simplex $\sigma \in K$ is called maximal when it is not a face of some other simplex from $K$. If $V_1,\ldots, V_m$ are the maximal simplices of $K$, than $K$ has a presentation as $$K=\mathcal{P}(V_1)\cup\ldots\cup \mathcal{P}(V_m).$$ Observe that the maximal simplices of a simplicial complex $K$ are incomparable (with respect to $\subseteq$) finite sets. \[d1\] A collection $\mathcal{B}$ of non-empty subsets of a finite set $V$ containing all singletons $\{v\}, v\in V$ and satisfying that for any two sets $S_1, S_2\in \mathcal{B}$ such that $S_1\cap S_2 \neq\emptyset$, their union $S_1\cup S_2$ also belongs to $\mathcal{B}$, is called a building set of $\mathcal{P}(V)$. Let $K$ be a simplicial complex and let $V_1,\ldots,V_m$ be the maximal simplices of $K$. A collection $\mathcal{B}$ of some simplices of $K$ is called a building set of $K$ when for every $i=1,\ldots,m$, the collection $$\mathcal{B}^{V_i}=\mathcal{B} \cap \mathcal{P}(V_i)$$ is a building set of $\mathcal{P}(V_i)$. The notion of a building set of a finite-meet semilattice introduced in [@FK04 Definition 2.2] reduces to this notion when the semilattice is a simplicial complex (cf.[@P14 Section 3]). For a family of sets $N$, we say that $\{X_1,\ldots,X_m\}\subseteq N$ is an $N$-antichain, when $m\geq 2$ and $X_1,\ldots,X_m$ are mutually incomparable with respect to $\subseteq$. \[d2\] Let $\mathcal{B}$ be a building set of a simplicial complex $K$. We say that $N\subseteq \mathcal{B}$ is a nested set with respect to $\mathcal{B}$, when the union of every $N$-antichain is an element of $K-\mathcal{B}$. This definition is in accordance with [@FK04 Definition 2.7] (cf. [@P14 Section 3]). Now we proceed to the main construction of the paper. We start with the set $X=\{0,\ldots,n\}$, $n\geq 1$, i.e. the ordinal $n+1$. (The case when $n=0$ is trivial.) Let $C_0$ be the simplicial complex $\mathcal{P}(X)-\{X\}$, i.e. the family of subsets of $X$ with at most $n$ elements, and let $\mathcal{B}_0=C_0-\{\emptyset\}$. In the literature, $C_0$ is also known as the boundary complex $\partial \Delta^n$ of the abstract $n$-simplex $\Delta^n$, and obviously $\mathcal{B}_0$ is a building set of the simplicial complex $C_0$ according to Definition \[d1\]. A set $N\subseteq \mathcal{B}_0$ such that the union of every $N$-antichain belongs to $C_0-\mathcal{B}_0$ is called 0-nested. According to Definition \[d2\], every 0-nested set is a nested set with respect to $\mathcal{B}_0$. Since $C_0-\mathcal{B}_0=\{\emptyset\}$, every two members of a 0-nested set are comparable. It is evident that a subset of a 0-nested set is a 0-nested set—thus the family of all 0-nested sets makes a simplicial complex, which we denote by $C_1$. The maximal 0-nested sets are of the form $$\bigl\{ \{i_n,\ldots,i_1\},\ldots,\{i_n,i_{n-1}\},\{i_n\} \bigr\},$$ where $i_1,\ldots,i_n$ are mutually distinct elements of $X$. Such a 0-nested set corresponds to the permutation $$i_0 i_1\ldots i_n$$ of $X$, where $\{i_0\}=X-\{i_1,\ldots,i_n\}$. For example, when $X=\{0,1,2,3\}$, $$\bigl\{ \{1,0,3\},\{1,0\},\{1\} \bigr\}$$ is a maximal 0-nested set that corresponds to the permutation $2301$ (see the picture at the end of Section \[realisation\]). We say that a polytope $P$ (geometrically) realises a simplicial complex $K$, when the semilattice obtained by removing the bottom (the empty set) from the face lattice of $P$ is isomorphic to $(K,\supseteq)$. The $n$-dimensional permutohedron (cf. [@M66 Definition 4.1]) realises the simplicial complex $C_1$. To prove this, one has to rely on [@P14 Proposition 3.4] and well known facts about nested set complexes related to permutohedra (cf. [@FS05], [@CD06], [@P09] and [@DP10]). In this one-to-one correspondence, the vertices of the permutohedron correspond to the maximal 0-nested sets. Hence, there are $(n+1)!$ maximal 0-nested sets. Also, the facets of the permutohedron correspond to the singleton nested sets of the form $\{A\}$, where $A\in \mathcal{B}_0$. It is more intuitive to denote the facets of this permutohedron by ordered partitions of $X$ with two blocks. With singleton nested sets we just economise a little bit—our nested set $\{A\}$ corresponds to the ordered partition whose first block is $X-A$ and the second is $A$. For a maximal 0-nested set $$\bigl\{ \{i_n,\ldots,i_1\},\ldots,\{i_n,i_{n-1}\},\{i_n\} \bigr\},$$ consider the following path graph with $n$ vertices and $n-1$ edges: $$\{i_n,\ldots,i_1\}-\ldots-\{i_n,i_{n-1}\}-\{i_n\}.$$ We say that a set of vertices of this graph is connected, when this is the set of vertices of a connected subgraph of this graph. Let $\mathcal{B}_1\subseteq C_1$ be the family of all sets of the form $$\bigl\{ \{i_{k+l},\ldots,i_k,\ldots,i_1\},\ldots,\{i_{k+l},\ldots,i_k,i_{k-1}\},\{i_{k+l},\ldots,i_k\} \bigr\},$$ where $1\leq k\leq k+l\leq n$ and $i_1,\ldots,i_{k+l}$ are mutually distinct elements of $X$, i.e. $\mathcal{B}_1$ is the set of all non-empty connected sets of vertices of the path graphs that correspond to all maximal 0-nested sets. According to Definition \[d1\], we have that $\mathcal{B}_1$ is a building set of the simplicial complex $C_1$. Every member of $C_1-\mathcal{B}_1$ is of the form $\{A_1,\ldots,A_r\}$, where $$2\leq r\leq n-1 \;\&\; X\supset A_1\supset\ldots\supset A_r \;\&\; \exists i \:\|A_i-A_{i+1}\| \geq 2.$$ \[e1\] For $X=\{0,1,2\}$, we have that $\mathcal{B}_1$ is $$\Bigl\{ \bigl\{\{0\}\bigr\},\;\bigl\{\{1\}\bigr\},\;\bigl\{\{2\}\bigr\},\; \bigl\{\{0,1\}\bigr\},\; \bigl\{\{0,2\}\bigr\},\; \bigl\{\{1,2\}\bigr\} ,$$ $$\bigl\{\{0,1\},\{0\}\bigr\},\; \bigl\{\{0,2\},\{0\}\bigr\},\; \bigl\{\{0,1\},\{1\}\bigr\},$$ $$\bigl\{\{1,2\},\{1\}\bigr\},\; \bigl\{\{0,2\},\{2\}\bigr\},\; \bigl\{\{1,2\},\{2\}\bigr\} \Bigr\}.$$ A set $N\subseteq \mathcal{B}_1$ is 1-nested when the union of every $N$-antichain belongs to $C_1-\mathcal{B}_1$. According to Definition \[d2\], every 1-nested set is a nested set with respect to $\mathcal{B}_1$. Again, it is evident that the family of all 1-nested sets makes a simplicial complex, which we denote by $C$. As a corollary of the following proposition, we have that $C$ is a flag complex. \[p2\] A set $N\subseteq \mathcal{B}_1$ is 1-nested when the union of every two incomparable elements of $N$ belongs to $C_1-\mathcal{B}_1$. Let $\mathcal{A}$ be an $N$-antichain. By our assumption, every two different members of $\mathcal{A}$ are of the form $$\{A_{i1},\ldots,A_{ir}\}\quad{\rm and}\quad \{A_{j1},\ldots,A_{jq}\},$$ with $A_{i1}\supset\ldots\supset A_{ir}\supset A_{j1}\supset\ldots\supset A_{jq}$, and $$\|A_{ik}-A_{i(k+1)}\|=1=\|A_{jk}-A_{j(k+1)}\|,\quad \|A_{ir}- A_{j1}\|\geq 2.$$ Hence $\mathcal{A}$ can be linearly ordered in a natural way, and we may conclude that $\bigcup\mathcal{A}$ is a subset of a maximal 0-nested set, which means that it is in $C_1$. The “gaps” arose by provisos $\|A_{ir}- A_{j1}\|\geq 2$ guarantee that $\bigcup\mathcal{A}$ is not in $\mathcal{B}_1$. \[e8\] For $X=\{0,1,2\}$, we have the following 12 maximal 1-nested sets: $$\begin{array}{lll} \Bigl\{\bigl\{\{1,2\},\{2\}\bigr\},\bigl\{\{1,2\}\bigr\}\Bigr\}, & \Bigl\{\bigl\{\{1,2\},\{2\}\bigr\},\bigl\{\{2\}\bigr\}\Bigr\}, & \Bigl\{\bigl\{\{0,2\},\{2\}\bigr\},\bigl\{\{2\}\bigr\}\Bigr\}, \\[1ex] \Bigl\{\bigl\{\{0,2\},\{2\}\bigr\},\bigl\{\{0,2\}\bigr\}\Bigr\}, & \Bigl\{\bigl\{\{0,2\},\{0\}\bigr\},\bigl\{\{0,2\}\bigr\}\Bigr\}, & \Bigl\{\bigl\{\{0,2\},\{0\}\bigr\},\bigl\{\{0\}\bigr\}\Bigr\}, \\[1ex] \Bigl\{\bigl\{\{0,1\},\{0\}\bigr\},\bigl\{\{0\}\bigr\}\Bigr\}, & \Bigl\{\bigl\{\{0,1\},\{0\}\bigr\},\bigl\{\{0,1\}\bigr\}\Bigr\}, & \Bigl\{\bigl\{\{0,1\},\{1\}\bigr\},\bigl\{\{0,1\}\bigr\}\Bigr\}, \\[1ex] \Bigl\{\bigl\{\{0,1\},\{1\}\bigr\},\bigl\{\{1\}\bigr\}\Bigr\}, & \Bigl\{\bigl\{\{1,2\},\{1\}\bigr\},\bigl\{\{1\}\bigr\}\Bigr\}, & \Bigl\{\bigl\{\{1,2\},\{1\}\bigr\},\bigl\{\{1,2\}\bigr\}\Bigr\}, \end{array}$$ and $C$ is the union of power sets of these sets. \[e2\] If $X=\{0,1,2,3\}$ and $M=\{\{1,0,3\},\{1,0\},\{1\}\}$, then $$N=\Bigl\{M,\bigl\{\{1\}\bigr\},\bigl\{\{1,0,3\}\bigr\}\Bigr\}, \quad \Bigl\{M,\bigl\{\{1,0,3\},\{1,0\}\bigr\},\bigl\{\{1,0,3\}\bigr\}\Bigr\},$$ $$\Bigl\{M,\bigl\{\{1,0\},\{1\}\bigr\},\bigl\{\{1\}\bigr\}\Bigr\}, \quad\quad \Bigl\{M,\bigl\{\{1,0,3\},\{1,0\}\bigr\},\bigl\{\{1,0\}\bigr\}\Bigr\},$$ $${\rm \it and}\quad\Bigl\{M,\bigl\{\{1,0\},\{1\}\bigr\},\bigl\{\{1,0\}\bigr\}\Bigr\},$$ are 5 out of 120 maximal 1-nested sets. In Example \[e2\], every maximal 1-nested set, except $N$, is such that all of its members are comparable (it contains no antichains), while $N$ contains the $N$-antichain $\{\{\{1\}\},\{\{1,0,3\}\}\}$ whose union is $\{\{1\},\{1,0,3\}\}\not\in \mathcal{B}_1$. If $N$ is a 1-nested set, then there is a maximal 0-nested set $M$ such that every member of $N$ is a subset of $M$, i.e.$N\subseteq \mathcal{B}_1^M=\mathcal{B}_1\cap \mathcal{P}(M)$. In this case, we say that $N$ is derived from $M$. In Example \[e2\], the five displayed 1-nested sets are derived from $M$. If $N$ is a maximal 1-nested set, then there is unique such $M$, namely the maximal 0-nested set contained in $N$. This also holds for every 1-nested set that contains a maximal 0-nested set. Otherwise, if $N$ does not contain a maximal 0-nested set, then there is not a unique such $M$. \[r1\] Every maximal 0 and 1-nested set is of cardinality $n$. \[r5\] Every element of $\mathcal{B}_1$ is contained in a maximal 1-nested set. For a maximal 0-nested set $M$, let $C^M$ be the link of $M$ in $C$, i.e. $C^M$ is the set of subsets not containing $M$ of the elements of $C$ that contain $M$. This simplicial complex is related to the $(n-1)$-dimensional associahedron (Stasheff polytope, see [@S63] and [@S97]) in the same way as the simplicial complex $C_1$ has been related to the $n$-dimensional permutohedron. The vertices of the $(n-1)$-dimensional associahedron correspond to all the $$c_n=\frac{1}{n+1}\binom{2n}{n}$$ different ways of bracketing a string of $n+1$ letters. The number $c_n$ is the $n$th Catalan number. Two vertices are adjacent if they correspond to a single application of the associative law. In general, $k$-faces of the associahedron are in bijection with the set of correct bracketings of a string of $n+1$ letters with $n-k$ pairs of brackets. Two vertices lie in the same $k$-face if and only if the corresponding complete bracketings could be reduced, by removing $k$ pairs of brackets, to the same bracketing of the string of $n+1$ letters with $n-k$ pairs of brackets. Figure \[s:asn\] depicts the $(n-1)$-dimensional associahedra for $n=2,3$ and $4$. ![$K_3$, $K_4$ and $K_5$[]{data-label="s:asn"}](asociedri.pdf){width="80.00000%"} The $(n-1)$-dimensional associahedron realises $C^M$. This is again proved by relying on well known facts about nested set complexes related to associahedra (cf. [@FS05], [@CD06], [@P09] and [@DP10]). Hence, there are $c_n$ maximal 1-nested sets derived from a maximal 0-nested set. From this, one concludes that there are all in all $$\frac{(2n)!}{n!}$$ maximal 1-nested sets. A correspondence between the maximal 1-nested sets and complete bracketings of permuted products of the elements of $\{0,\ldots,n\}$ is given in Section \[faces\]. The main question related to the simplicial complex $C$ is the following. \[q1\] Is there an $n$-dimensional realisation of $C$? The positive answer to this question is given in Section \[realisation\]. But first, in Section \[faces\], we show that if such a polytope exists, then its graph is $\mathcal{G}^_n$ and the 2-faces of this polytope correspond to the diagrams (1), (4), (8), (3), (9) and (10). The 0, 1 and 2-faces of $C$ {#faces} =========================== In order to establish the correspondence announced above, we investigate the elements of the simplicial complex $C$ that should correspond to the vertices, edges and two dimensional faces of a polytope, which provides the positive answer to Question \[q1\]. Respectively, we call these elements of $C$ the 0-faces, 1-faces and 2-faces. 0-faces ------- The 0-faces* of $C$ are the maximal 1-nested sets. Every 0-face corresponds to a unique complete bracketing of a permuted product of the elements of $\{0,\ldots,n\}$ in the following way. Let $V$ be a 0-face derived from $$M=\bigl\{\{i_n,\ldots,i_1\},\ldots,\{i_n,i_{n-1}\},\{i_n\}\bigr\}.$$ We start with $n+1$ place holders separated by $n$ dots, which are labelled by the elements of $M$ in the descending (with respect to $\subseteq$) order: $$\underline{\hspace{1em}} \stackrel{\{i_n,\ldots,i_1\}}{\cdot} \underline{\hspace{1em}} \ldots \underline{\hspace{1em}} \stackrel{\{i_n,i_{n-1}\}}{\cdot} \underline{\hspace{1em}} \stackrel{\{i_n\}}{\cdot} \underline{\hspace{1em}}.$$ Each label denotes the elements of $X$ to the right of the labelled dot. Hence we have the product: $$i_0 \stackrel{\{i_n,\ldots,i_1\}}{\cdot} i_1 \stackrel{\{i_n,\ldots,i_2\}}{\cdot} \ldots \stackrel{\{i_n,i_{n-1}\}}{\cdot} i_{n-1} \stackrel{\{i_n\}}{\cdot} i_n,$$ where $\{i_0\}=X-\{i_1,\ldots,i_n\}$. Eventually, for every element $A$ of $V$, one has to insert the pair of brackets that include the labels contained in $A$. \[e3\] For $X=\{0,1,2,3\}$, $M=\{\{1,0,3\},\{1,0\},\{1\}\}$, and $V=\{M,\{\{1\}\},$\ $\{\{1,0,3\}\}\}$, we have that $V$ corresponds to the following complete bracketing $$((2 \stackrel{\{1,0,3\}}{\cdot} 3 )\stackrel{\{1,0\}}{\cdot} (0 \stackrel{\{1\}}{\cdot} 1)).$$ In the picture given at the end of Section \[realisation\], the 0-face $V$ corresponds to the vertex incident to the pentagon covering the vertex labelled by 2301, dodecagon covering the label $\{1\}$ and dodecagon covering the label $\{0,1,3\}$. Also, for every complete bracketing $P$ of a product $i_0\cdot i_1\cdot\ldots\cdot i_n$, one can find a unique 0-face of $C$ that corresponds to $P$. Hence, the 0-faces of $C$ are in one-to-one correspondence with the vertices of $\mathcal{G}^_n$. \[ziegler\] Since the face lattice of a simple polytope is completely determined by the bipartite graph of incidence relation between the vertex set and the set of facets, we have the following alternative way to define the lattice structure, which may replace our simplicial complex $C$. This is suggested to us by G.M.Ziegler. As the vertex set take all the complete bracketings of permuted products of $X=\{0,\ldots,n\}$, and as the set of facets take all the ordered partitions of $X$ into $k$ blocks, $k\geq 2$, where only the first and the last block are allowed to contain more than one element. A vertex $V$, whose underlying permutation is $i_0 i_1\ldots i_n$, and a facet $F$ are incident when there is a pair of corresponding brackets in $V$ with $i_j$ being the leftmost and $i_k$ being the rightmost element in the scope of these brackets, such that $F$ is the ordered partition of $X$ whose first block is $\{i_0,\ldots,i_j\}$ followed by $k-j-1$ singleton blocks $\{i_{j+1}\},\ldots,\{i_{k-1}\}$, and the last block is $\{i_k,\ldots,i_n\}$. If we assign to an element of $\mathcal{B}_1$ of the form $$\bigl\{ \{i_{k+l},\ldots,i_k,\ldots,i_1\},\ldots, \{i_{k+l},\ldots,i_k,i_{k-1}\},\{i_{k+l},\ldots,i_k\} \bigr\},$$ the ordered partition of $X$ whose first block is $X-\{i_{k+l},\ldots,i_k,\ldots,i_1\}$, the second block is $\{i_1\}$, and so on up to the penultimate block, which is $\{i_{k-1}\}$, while the last block is $\{i_{k+l},\ldots,i_k\}$, then the above analysis of 0-faces explains the incidence relation given in this remark. 1-faces ------- The 1-faces* of $C$ are obtained from the 0-faces by removing a single element. There are two main types of 1-faces of $C$; a 1-face of the first type contains a maximal 0-nested set, and a 1-face of the second type is obtained from a 0-face by removing its maximal 0-nested set. If $E$ is a 1-face of the first type, and $$M=\{\{i_n,\ldots,i_1\},\ldots,\{i_n,i_{n-1}\},\{i_n\}\}$$ is the maximal 0-nested set contained in $E$, then a unique bracketing of the product $i_0\cdot i_1\cdot\ldots\cdot i_n$, with one pair of brackets omitted, is assigned to $E$ in the same way as a complete bracketing has been assigned to a 0-face. There are exactly two 0-faces that include a 1-face $E$ of the first type. They correspond to two possible insertions of pairs of brackets in the bracketing corresponding to $E$. Hence, every 1-face of the first type corresponds to an $\alpha$-term. \[e4\] For $X=\{0,1,2,3\}$, $M=\{\{1,0,3\},\{1,0\},\{1\}\}$, and $E=\{M,\{\{1\}\}\}$, we have that $E$ corresponds to the following bracketing $$(2 \stackrel{\{1,0,3\}}{\cdot} 3 \stackrel{\{1,0\}}{\cdot} (0 \stackrel{\{1\}}{\cdot} 1)).$$ The 0-face $V$ from Example \[e3\] and the 0-face $V'=\{M,\{\{1,0\},\{1\}\},\{\{1\}\} \}$, which corresponds to the complete bracketing $$(2 \cdot (3 \cdot (0 \cdot 1))),$$ are the 0-faces that include $E$. In the picture given at the end of Section \[realisation\], the 1-face $E$ corresponds to the edge incident to the pentagon covering the vertex labelled by 2301 and dodecagon covering the label $\{1\}$. As before, for every bracketing $P$ of a product $i_0\cdot i_1\cdot\ldots\cdot i_n$, with one pair of brackets omitted, one can find a unique 1-face of the first type that corresponds to $P$. In order to describe the 1-faces of the second type, we define some notions, whose more general form is introduced in [@DP10]. A construction of a path graph $$v_1-v_2-\ldots-v_n$$ $n\geq 1$, is a subset of $\mathcal{P}(\{v_1,\ldots,v_n\})$ inductively defined as follows: \(1) if $n=1$, then $\{\{v_1\}\}$ is the construction of this graph with a single vertex $v_1$; \(2) if $n>1$, $1\leq j\leq n$ and $L$, $R$ are constructions of the path graphs $$v_1-\ldots-v_{j-1}\quad{\rm and}\quad v_{j+1}-\ldots-v_n$$ (if $j=1$, then $L=\emptyset$, and if $j=n$, then $R=\emptyset$), then $$\bigl\{\{v_1,\ldots,v_n\}\bigr\}\cup L\cup R$$ is a construction of the path graph $$v_1-v_2-\ldots-v_n.$$ \[r6\] If $K$ is a construction of $v_1-\ldots-v_n$, then ${\{v_1,\ldots,v_n\}\in K}$. \[r3\] Every construction of $v_1-\ldots-v_n$ has exactly $n$ elements. The following proposition stems from [@DP10 Proposition 6.11]. \[p1\] $V$ is a 0-face derived from $$\bigl\{\{i_n,\ldots,i_1\},\ldots,\{i_n,i_{n-1}\},\{i_n\}\bigr\}$$ iff $V$ is a construction of the path graph $$\{i_n,\ldots,i_1\}-\ldots-\{i_n,i_{n-1}\}-\{i_n\}.$$ From this proposition and the relation between the simplicial complex $C^M$ and the $(n-1)$-dimensional associahedron mentioned in Section \[two-fold\], one can extract the following recurrence relation for the Catalan numbers $$c_0=1\quad{\rm and}\quad c_{n+1}=\sum_{i=0}^n c_i c_{n-i}.$$ For $K\in C$ and $A\in K$, we say that $a\in A$ is $A$-superficial with respect to $K$, when for every $B\in K$ such that $B\subset A$, we have that $a\not\in B$. The number of $A$-superficial elements with respect to $K$ is denoted by $\nu_K(A)$. From the definition of construction and Proposition \[p1\] we have the following. \[r2\] If $K$ is a 0-face, then for every $A\in K$ we have that $\nu_K(A)=1$. Let $E$ be a 1-face of the second type, i.e. $E$ is obtained from a 0-face $V$ by removing its maximal 0-nested set $M$. By Remark \[r2\], $\nu_V(M)=1$, and hence, $\bigcup E$ is of the form $$\bigl\{\{i_n,\ldots,i_1\},\ldots,\{i_n,\ldots,i_{j-1}\},\{i_n,\ldots,i_{j+1}\}, \ldots,\{i_n\}\bigr\},$$ which in the limit cases reads $$\bigl\{\{i_n,\ldots,i_2\},\ldots,\{i_n\}\bigr\} \quad {\rm or} \quad \bigl\{\{i_n,\ldots,i_1\},\ldots,\{i_n,i_{n-1}\}\bigr\}.$$ By repeating the procedure we used for 0-faces and 1-faces of the first type, starting with $\bigcup E$, the following product (with two place holders vacant) is obtained $$i_0 \stackrel{\{i_n,\ldots,i_1\}}{\cdot} i_1 {\cdot}\ldots {\cdot} i_{j-2} \stackrel{\{i_n,\ldots,i_{j-1}\}}{\cdot} \underline{\hspace{1em}} \cdot \underline{\hspace{1em}} \stackrel{\{i_n,\ldots,i_{j+1}\}}{\cdot} i_{j+1} {\cdot} \ldots \stackrel{\{i_n\}}{\cdot} i_n,$$ and a complete bracketing of this product is reconstructed in the same way as for 0-faces. (Formally, the outermost brackets are omitted, but we take them for granted.) Note that this product must contain the following brackets: $$(i_0 \stackrel{\{i_n,\ldots,i_1\}}{\cdot} i_1 {\cdot} \ldots {\cdot} i_{j-2} \stackrel{\{i_n,\ldots,i_{j-1}\}}{\cdot} \underline{\hspace{1em}}\,) \cdot (\,\underline{\hspace{1em}} \stackrel{\{i_n,\ldots,i_{j+1}\}}{\cdot} i_{j+1} {\cdot} \ldots \stackrel{\{i_n\}}{\cdot} i_n).$$ There are exactly two 0-faces that include a 1-face $E$ of the second type. They correspond to two possible permutations of $i_{j-1},i_j$ in the vacant positions of the bracketing corresponding to $E$. Hence, every 1-face of the second type corresponds to a $\sigma$-term. \[e5\] For $X=\{0,1,2,3\}$, and $E=\{\{\{1\}\},\{\{1,0,3\}\}\}$, we have that $E$ corresponds to the following bracketing $$(2 \stackrel{\{1,0,3\}}{\cdot} \underline{\hspace{1em}}\,) \cdot (\,\underline{\hspace{1em}} \stackrel{\{1\}}{\cdot} 1).$$ The 0-face $V$ from Example \[e3\] and the 0-face $$V''=\{\{\{1,3,0\},\{1,3\},\{1\}\},\{\{1\}\},\{\{1,3,0\}\} \},$$ which corresponds to the complete bracketing $$((2 \cdot 0) \cdot (3 \cdot 1)),$$ are the 0-faces that include $E$. In the picture given at the end of Section \[realisation\], the 1-face $E$ corresponds to the edge incident to the dodecagon covering the label $\{1\}$ and dodecagon covering the label $\{0,1,3\}$. As before, for every complete bracketing $P$ of a product with two vacant positions, one being the rightmost in the left factor of $P$, and the other being the leftmost in the right factor of $P$, one can find a unique 1-face of the second type that corresponds to $P$. This, together with the analogous fact concerning the 1-faces of the first type, entails that the 1-faces of $C$ are in one-to-one correspondence with the edges of $\mathcal{G}^_n$. 2-faces ------- The 2-faces* of $C$ are obtained from the 1-faces by removing a single element. There are two main types of 2-faces of $C$; a 2-face of the first type contains a maximal 0-nested set, and a 2-face of the second type does not contain a maximal 0-nested set. If $F$ is a 2-face of the first type, and $$M=\bigl\{\{i_n,\ldots,i_1\},\ldots,\{i_n,i_{n-1}\},\{i_n\}\bigr\}$$ is the maximal 0-nested set contained in $F$, then a unique bracketing of the product $i_0\cdot i_1\cdot\ldots\cdot i_n$, with two pairs of brackets omitted, is assigned to $F$ in the same way as a complete bracketing has been assigned to a 0-face. There are three possible situations: (1.1) there is $A\in F$ such that $\nu_F(A)=3$; (1.2) there are incomparable $A,B\in F$ such that $\nu_F(A)=\nu_F(B)=2$; (1.3) there are $A, B\in F$ such that $B\subset A$ and $\nu_F(A)=\nu_F(B)=2$. By a straightforward analysis, in the case (1.1), one concludes that $F$ is included in exactly five 1-faces and five 0-faces and it corresponds to Mac Lane’s pentagon (3). In the cases (1.2) and (1.3), $F$ is included in exactly four 1-faces (corresponding to $\alpha$-terms) and four 0-faces. In the case (1.2), $F$ corresponds to a functoriality quadrilateral (1), while in the case (1.3) it corresponds to a naturality quadrilateral (4). \[e6\] Let $X=\{0,1,2,3,4,5\}$ and let $M$ be the maximal 0-nested set $$\bigl\{\{0,1,2,3,4\},\{0,1,2,3\},\{0,1,2\}, \{0,1\},\{0\}\bigr\}.$$ The 2-face $F_1=\{M,\{\{0,1,2,3\},\{0,1,2\},\{0,1\},\{0\}\},\{\{0\}\}\}$, corresponds to the bracketing $(5\cdot(4\cdot 3\cdot 2\cdot(1\cdot 0)))$. (120,110) (60,100)[(0,0)[$5\cdot(4\cdot(3\cdot (2\cdot(1\cdot 0))))$]{}]{} (-10,70)[(0,0)[$5\cdot(4\cdot((3\cdot 2)\cdot(1\cdot 0)))$]{}]{} (-10,40)[(0,0)[$5\cdot((4\cdot(3\cdot 2))\cdot(1\cdot 0))$]{}]{} (60,10)[(0,0)[$5\cdot(((4\cdot 3)\cdot 2)\cdot(1\cdot 0))$]{}]{} (130,55)[(0,0)[$5\cdot((4\cdot 3)\cdot (2\cdot(1\cdot 0)))$]{}]{} (50,90)[(-3,-1)[30]{}]{} (0,60)[(0,-1)[10]{}]{} (20,30)[(3,-1)[30]{}]{} (70,90)[(2,-1)[40]{}]{} (110,40)[(-2,-1)[40]{}]{} (-5,88)[(0,0)[$5\cdot(4\cdot(3\cdot 2\cdot(1\cdot 0)))$]{}]{} (-40,55)[(0,0)[$5\cdot(4\cdot(3\cdot 2)\cdot(1\cdot 0))$]{}]{} (-5,22)[(0,0)[$5\cdot((4\cdot 3\cdot 2)\cdot(1\cdot 0))$]{}]{} (130,82)[(0,0)[$5\cdot(4\cdot 3\cdot (2\cdot(1\cdot 0)))$]{}]{} (130,27)[(0,0)[$5\cdot((4\cdot 3)\cdot 2\cdot(1\cdot 0))$]{}]{} The 2-face $F_2=\{M,\{\{0,1,2,3,4\},\{0,1,2,3\}\},\{\{0,1\},\{0\}\}\}$, corresponds to the bracketing $((5\cdot 4\cdot 3)\cdot(2\cdot 1\cdot 0))$ and it is easy to make the corresponding functoriality quadrilateral. The 2-face $F_3=\{M,\{\{0,1,2,3\},\{0,1,2\},\{0,1\},\{0\}\}, \{\{0,1\},\{0\}\}\}$ corresponds to the bracketing $(5\cdot (4\cdot 3\cdot(2\cdot 1\cdot 0)))$ and it is easy to make the corresponding naturality quadrilateral. If $F$ is a 2-face of the second type, then there are two possibilities $$(2.1)\;\; \|\bigcup F\|=n-1\quad {\rm or}\quad (2.2)\;\; \|\bigcup F\|=n-2.$$ In the case (2.1), a quadrilateral of a form of the diagrams (8) corresponds to $F$. In the case (2.2), $\bigcup F$ is of the form: $$\bigl\{\{i_n,\ldots,i_1\},\ldots,\{i_n,i_{n-1}\},\{i_n\}\bigr\}- \bigl\{\{i_n,\ldots,i_j\},\{i_n,\ldots,i_k\}\bigr\},$$ for $1\leq j<k\leq n$, and we distinguish the following situations: (2.2.1) if $k-j>1$, then an octagon of a form of the diagram (9) corresponds to $F$; (2.2.2) if $k-j=1$, then a dodecagon of a form of the diagram (10) corresponds to $F$. \[e7\] For $X=\{0,1,2,3,4,5\}$, the 2-face $$F_4=\Bigl\{\bigl\{\{0,1,2,3,4\}\bigr\},\; \bigl\{\{0,1,2\},\{0,1\},\{0\}\bigr\},\; \bigl\{\{0\}\bigr\}\Bigr\}$$ describes the bracketing $(5\cdot\underline{\hspace{1em}}\,)\cdot(\,\underline{\hspace{1em}}\cdot 2\cdot(1\cdot 0))$. (120,80)(0,-10) (-25,50)[(0,0)[$(5\cdot 4)\cdot(3\cdot(2\cdot(1\cdot 0)))$]{}]{} (-25,10)[(0,0)[$(5\cdot 3)\cdot(4\cdot(2\cdot(1\cdot 0)))$]{}]{} (145,50)[(0,0)[$(5\cdot 4)\cdot((3\cdot 2)\cdot(1\cdot 0))$]{}]{} (145,10)[(0,0)[$(5\cdot 3)\cdot((4\cdot 2)\cdot(1\cdot 0))$]{}]{} (-20,40)[(0,-1)[20]{}]{} (140,40)[(0,-1)[20]{}]{} (30,50)[(1,0)[60]{}]{} (30,10)[(1,0)[60]{}]{} (-65,30)[(0,0)[$(5\cdot \underline{\hspace{.5em}}\,)\cdot(\,\underline{\hspace{.5em}}\cdot(2\cdot(1\cdot 0)))$]{}]{} (185,30)[(0,0)[$(5\cdot \underline{\hspace{.5em}}\,)\cdot((\,\underline{\hspace{.5em}}\cdot 2)\cdot(1\cdot 0))$]{}]{} (60,60)[(0,0)[$(5\cdot 4)\cdot(3\cdot 2\cdot (1\cdot 0))$]{}]{} (60,0)[(0,0)[$(5\cdot 3)\cdot(4\cdot 2\cdot (1\cdot 0))$]{}]{} The 2-face $F_5=\{\{\{0,1,2,3,4\}\},\{\{0,1,2\}\},\{\{0\}\}\}$, describes the bracketing $$(5\cdot \underline{\hspace{1em}}\,) \cdot (\,\underline{\hspace{1em}} \cdot \underline{\hspace{1em}}\,)\cdot (\,\underline{\hspace{1em}}\cdot 0),$$ where the first two vacant positions are reserved for 3 and 4, while the last two are reserved for 1 and 2. This face corresponds to an octagon of a form of the diagram (9). The 2-face $F_6=\{\{\{0,1,2,3,4\}\},\{\{0,1\},\{0\}\},\{\{0\}\}\}$ describes the bracketing $$(5\cdot \underline{\hspace{1em}}\,)\cdot \underline{\hspace{1em}}\cdot (\,\underline{\hspace{1em}}\cdot (1\cdot 0))$$ and it corresponds to a dodecagon of a form of the diagram (10). Simple permutoassociahedra {#realisation} ========================== In this section we present a polytope $\mathbf{PA}_n$, the $n$-dimensional simple permutoassociahedron, and give the positive answer to Question \[q1\]. We start with the following notation. For $1\leq k\leq k+l\leq n$, let $$\kappa(k,l)=\frac{3^{k+l+1}-3^{l+1}}{2}+\frac{3^k-3k}{3^n-n-1}.$$ For an element $$\beta=\bigl\{\{i_{k+l},\ldots,i_k,\ldots,i_1\},\ldots,\{i_{k+l},\ldots,i_k,i_{k-1}\},\{i_{k+l},\ldots,i_k\} \bigr\},$$ of $\mathcal{B}_1$, let $\beta^=$ be the equation (hyperplane in $\mathbf{R}^{n+1}$) $$x_{i_1}+2x_{i_2}+\ldots+k(x_{i_k}+\ldots+x_{i_{k+l}})=\kappa(k,l),$$ and let the halfspace $\beta^{\geq}$ and the open halfspace $\beta^>$ be defined as $\beta^=$, save that “$=$” is replaced by “$\geq$” and “$>$”, respectively. \[r4\] We have that $\kappa(k,l)=3^{l+k}+\ldots+3^{l+1}+\varepsilon(k)$, for $$\varepsilon(k)=\frac{3^k-3k}{3^n-n-1}<1.$$ For $\pi$ being the hyperplane $x_0+\ldots +x_n=3^{n+1}$ in $\mathbf{R}^{n+1}$, let $$\mathbf{PA}_n=(\bigcap \{\beta^{\geq}\mid \beta\in \mathcal{B}_1\})\cap \pi.$$ The rest of this section is devoted to a proof of the following result. \[t1\] $\mathbf{PA}_n$ is a simple $n$-dimensional polytope that realises $C$. For every $0\leq i\leq n$, we have that $\{\{i\}\}\in \mathcal{B}_1$. Hence, $\mathbf{PA}_n$ is a subset of the $n$-simplex $$x_0+\ldots+x_n=3^{n+1},\quad\quad x_i\geq 3,\quad 0\leq i\leq n,$$ and therefore a polytope, and not just a polyhedron. For $\mathcal{B}_1^M$ being defined as $\mathcal{B}_1\cap \mathcal{P}(M)$, we have the following. \[l4\] If $\beta,\gamma\in \mathcal{B}_1$ are such that $\beta^{=}\cap \gamma^{=}\cap \mathbf{PA}_n\neq\emptyset$, then there is a maximal 0-nested set $M$ such that $\beta,\gamma\in\mathcal{B}_1^M$. Suppose that - there is no maximal 0-nested set $M$ such that $\beta,\gamma\in\mathcal{B}_1^M$. Let $\beta^{=}$ be the hyperplane $$y_1+2y_2+\ldots+k(y_k+\ldots+y_{k+l})=\kappa(k,l),$$ and let $\gamma^{=}$ be the hyperplane $$z_1+2z_2+\ldots+m(z_m+\ldots+z_{m+p})=\kappa(m,p),$$ where $Y=\{y_1,\ldots,y_{k+l}\}$ and $Z=\{z_1,\ldots,z_{m+p}\}$ are subsets of the set of variables $\{x_0,\ldots,x_n\}$. Let $\Sigma$ be the following sum: $$y_1+2y_2+\ldots+k(y_k+\ldots+y_{k+l})+ z_1+2z_2+\ldots+m(z_m+\ldots+z_{m+p}).$$ We show that $\Sigma>\kappa(k,l)+\kappa(m,p)$, for every point $a(x_0,x_1,\ldots,x_n)\in \mathbf{PA}_n$, which entails that $\mathbf{PA}_n$ and the hyperplane $\Sigma=\kappa(k,l)+\kappa(m,p)$ are disjoint. Since $\beta^{=}\cap \gamma^{=}$ is included in this hyperplane, we conclude that $a\not\in \beta^{=}\cap \gamma^{=}$. In order to prove this, we use the following fact several times: if $i_1,\ldots,i_m$ are mutually distinct elements of $\{0,\ldots,n\}$, then $\{\{i_1,\ldots,i_m\}\}\in \mathcal{B}_1$, and for $a(x_0,x_1,\ldots,x_n)\in \mathbf{PA}_n$, we have that $x_{i_1}+\ldots+x_{i_m}\geq 3^m$. \(1) If $Y$ and $Z$ are incomparable and $Y\cup Z=\{u_1,\ldots,u_q\}$, then we have that $q>k+l,m+p$ and $$\Sigma\geq u_1+\ldots+ u_q\geq 3^q\geq \frac{3^{k+l+1} + 3^{m+p+1}}{2}>\kappa(k,l)+\kappa(m,p).$$ \(2) If $Z\subset Y$, then, by $(\ast)$, we conclude that either (2.1) $z_1=y_i$ for some $i\leq k-2$ and $ \{z_2,\ldots,z_{m+p}\}\subseteq\{y_k,\ldots,y_{k+l}\}$, or (2.2) $z_1=y_{k-1}$ and $\{z_2,\ldots,z_{m+p}\}\subset\{y_k,\ldots,y_{k+l}\}$, or (2.3) $\{z_2,\ldots,z_{m+p}\}\not\subseteq\{y_k,\ldots,y_{k+l}\}$. In the case (2.1), we have that $k+l-i\geq p+m$, and $\Sigma$ = $\geq (y_1+\ldots+y_{l+k})+\ldots+ (y_{i-1}+\ldots+y_{l+k})+ 2(y_i+\ldots+y_{l+k})$\ $\geq 3^{l+k}+\ldots+3^{l+k-i+2}+2\cdot 3^{l+k-i+1}$\ $> 3^{l+k}+\ldots+3^{l+k-i+2}+ 3^{l+k-i+1}+ 2(3^{l+k-i}+\ldots+1)$\ $\geq 3^{l+k}+\ldots+3^{l+1}+1+ 3^{p+m}+ \ldots+3^{p+1}+1$, since $k+l-i\geq p+m$\ $> \kappa(k,l)+\kappa(m,p)$. In the case (2.2), we have that $l+1\geq p+m$, and we repeat the above calculation with $i$ replaced by $k-1$. In the case (2.3), let $i$ be the smallest element of $\{1,\ldots,k-1\}$ such that $y_i=z_j$ for some $j\in\{2,\ldots,m+p\}$. We have that $k\!+\!l\!-\!i\!+\!1\geq p\!+\!m$ and $\Sigma$ = $\geq (y_1+\ldots+y_{l+k})+\ldots+ (y_{i-1}+\ldots+y_{l+k})+ 3(y_i+\ldots+y_{l+k})$\ $\geq 3^{l+k}+\ldots+3^{l+k-i+2}+3\cdot 3^{l+k-i+1}$\ $> 3^{l+k}+\ldots+3^{l+k-i+2}+ 2(3^{l+k-i+1}+\ldots+1)$\ $\geq 3^{l+k}+\ldots+3^{l+1}+1+ 3^{p+m}+ \ldots+3^{p+1}+1$, since $k\!+\!l\!-\!i\!+\!1\geq p\!+\!m$\ $> \kappa(k,l)+\kappa(m,p)$. \(3) If $Z=Y$, then, by $(\ast)$, there is $i\in\{1,\ldots,k-1\}$ such that for some $r\geq 1$, $\gamma^{=}$ is of the form $$y_1+\ldots+(i-1)y_{i-1}+(i+r)y_i+\ldots=\kappa(m,p),$$ where the coefficient of every $y_j$, $j>i$, hidden in “$\ldots$”, is greater or equal to $i$. Hence, $\Sigma$ = $\geq 2(y_1+\ldots+y_{l+k})+\ldots+ 2(y_{i-1}+\ldots+y_{l+k})+ 3(y_i+\ldots+y_{l+k})$\ $\geq 2(3^{l+k}+\ldots+3^{l+k-i+1})+ 3^{l+k-i+1}$\ $> 2(3^{l+k}+\ldots+1)$, since $3^{l+k-i+1}-1=2(3^{k+l-i}+\ldots+1)$\ $> \kappa(k,l)+\kappa(m,p)$. The following lemma is the basis of the inductive proof of [@DP10 Lemma 9.1]. \[l3\] Let $Y$ and $Z$ be two incomparable subsets of $\{0,\ldots,n\}$. If for every $i\in Y\cup Z$ we have that $x_i\geq 0$, and $\sum_{i\in Y}x_i\leq 3^{\|Y\|}$, $\sum_{i\in Z}x_i\leq 3^{\|Z\|}$, then $$\sum_{i\in Y\cup Z}x_i< 3^{\|Y\cup Z\|}.$$ In the sequel, we rely on the following affine transformation that normalises the equations of hyperplanes localised at $\mathcal{B}_1^M$ for $$M=\bigl\{\{n,\ldots,1\},\ldots,\{n,n-1\},\{n\}\bigr\}.$$ Let $p$ be the orthogonal projection from the hyperplane $\pi$ to the hyperplane ${\pi^0: x_0=0}$, and let $\mathbf{PA}_n^0$ be the $p$-image of $\mathbf{PA}_n$. Since $p$ is an affine bijection, the polytopes $\mathbf{PA}_n$ and $\mathbf{PA}_n^0$ are combinatorially equivalent and $p$ maps a face of $\mathbf{PA}_n$ to the corresponding face of $\mathbf{PA}_n^0$. Consider the $n\times n$ matrices $L$ and $L^{-1}$ $$\frac{1}{3^n\!-\!n\!-\!1}\left( \begin{array}{ccccc} 1 & -1 & 0 & \ldots & 0 \\ 0 & 1 & -1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \end{array}\right), \quad (3^n\!-\!n\!-\!1)\left( \begin{array}{ccccc} 1 & 1 & 1 & \ldots & 1 \\ 0 & 1 & 1 & \ldots & 1 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \end{array}\right)$$ and the vector $$\mathbf{w}=\left(\begin{array}{c} 3^n-3^{n-1} \\ 3^{n-1}-3^{n-2} \\ \vdots \\ 6 \\ 3 \end{array} \right)$$ Let $h:\mathbf{R}^n{\rightarrow}\pi^0$ be the affine bijection mapping $(x_1',\ldots,x_n')\in \mathbf{R}^n$ into the point $(x_0,\ldots,x_n)\in\pi^0$, such that $x_0=0$ and $$\left(\begin{array}{c} x_1 \\ \vdots \\ x_n \end{array} \right)= \mathbf{w} + L \left(\begin{array}{c} x_1'-3 \\ \vdots \\ x_n'-3 \end{array} \right).$$ Let $u=h^{-1}\circ p$ and let $\mathbf{PA}_n'$ be the polytope obtained as the $u$-image of $\mathbf{PA}_n$. The polytopes $\mathbf{PA}_n$ and $\mathbf{PA}_n'$ are again combinatorially equivalent. What are the hyperplanes of $\mathbf{R}^n$ that correspond by $u$ to $\beta^=\cap\pi$, where $\beta\in\mathcal{B}_1^M$? Every such $\beta$ is of the form $$\bigl\{\{n,n-1,\ldots,n-l,\ldots,n-l-k+1\},\ldots,\{n,n-1,\ldots,n-l\}\bigr\},$$ and let us define the $u$-correspondent of $\beta$ to be the set $Y=\{n-l-k+1,\ldots,n-l\}$. The hyperplane $\beta^{=}$ in $\mathbf{R}^{n+1}$ is of the form $$x_{n-l-k+1}+2x_{n-l-k+2}+\ldots+(k-1)x_{n-l-1}+k(x_{n-l}+\ldots+x_n)=\kappa(k,l),$$ and let us denote the left-hand side of this equation by $\Xi$. For $Y$ being the $u$-correspondent of $\beta$, we define the hyperplane $Y^{=}$ in $\mathbf{R}^n$ by the equation $$\sum_{i\in Y} x'_i=3^{\|Y\|},\quad{\rm i.e.}\quad x_{n-l-k+1}'+\ldots+x_{n-l}'=3^k.$$ We show that $\beta^{=}\cap\pi^0$ is the $h$-image of $Y^{=}$. By the definition of $h$, we have that $$\begin{array}{rl} x_{n-l-k+1}\!\!\!\! & =2\cdot 3^{l+k-1}+\frac{1}{3^n-n-1}(x_{n-l-k+1}'-x_{n-l-k+2}') \\ \vdots & \quad\quad\vdots \\ x_{n-l}\!\!\!\! & =2\cdot 3^{l}+\frac{1}{3^n-n-1}(x_{n-l}'-x_{n-l+1}') \\ \vdots & \quad\quad\vdots \\ x_{n}\!\!\!\! & =3+\frac{1}{3^n-n-1}(x_{n}'-3),\; {\rm and} \end{array}$$ $$\begin{array}{rl} \Xi\!\!\!\! & =2\cdot 3^{l+k-1}+ 2\cdot 2 \cdot 3^{l+k-2}+\ldots+(k-1) 2\cdot 3^{l+1}+k(2\cdot 3^l+\ldots +2\cdot 3+3) \\[1ex] & \;\;\;+ \frac{1}{3^n-n-1}(x_{n-l-k+1}'+\ldots+x_{n-l}'-3 k) \\[2ex] & =3^{l+k}+3^{l+k-1}+\ldots+3^{l+1}+ \frac{1}{3^n-n-1}(x_{n-l-k+1}'+\ldots+x_{n-l}'-3 k). \end{array}$$ Hence, $$\begin{array}{rcl} \Xi=\kappa(k,l) & \Leftrightarrow & \frac{1}{3^n-n-1}(x_{n-l-k+1}'+\ldots+x_{n-l}'-3k)= \frac{3^k-3k}{3^n-n-1} \\[1ex] & \Leftrightarrow & x_{n-l-k+1}'+\ldots+x_{n-l}'= 3^k, \end{array}$$ which means that $\beta^{=}\cap\pi^0$ is the $h$-image of $Y^{=}$, i.e. the hyperplane $Y^=$ of $\mathbf{R}^n$ corresponds by $u$ to $\beta^=\cap\pi$. The same holds when the superscript “$=$” is replaced by “$\geq$” in which case the equation $Y^=$ is replaced by the appropriate inequality. Hence the $u$-correspondence between the members of $\mathcal{B}_1^M$ and the members of the set $$\mathcal{B}=\bigl\{\{n-l,\ldots,n-l-k+1\}\mid 1\leq k\leq k+l\leq n\bigr\},$$ which is a bijection, is lifted to a bijection between a set of hyperplanes (halfspaces) tied to $\mathbf{PA}_n$ and a set of hyperplanes (halfspaces) tied to $\mathbf{PA}_n'$. Note that $\mathcal{B}$ is the set of all non-empty connected sets of vertices of the path graph $1-2-\ldots-n$ and it is a building set of $\mathcal{P}(\{1,\ldots,n\})$. It is easy to check that the following implications and its converses hold. If two members of $\mathcal{B}_1^M$ are incomparable, then the $u$-correspondents are incomparable. If the union of two members of $\mathcal{B}_1^M$ is in $\mathcal{B}_1^M$, then the union of the $u$-correspondents is in $\mathcal{B}$. If $N\subseteq \mathcal{B}_1^M$ is 1-nested, then the set of $u$-correspondents of members of $N$ is a nested set with respect to $\mathcal{B}$. If $N\subseteq \mathcal{B}_1^M$ is a construction of the path graph $$\{n,\ldots,1\}-\ldots-\{n,n-1\}-\{n\},$$ then the set of $u$-correspondents of members of $N$ is a construction of the path graph $1-2-\ldots-n$. \[l5\] If $\beta,\gamma\in \mathcal{B}_1$ are such that $\beta^{=}\cap \gamma^{=}\cap \mathbf{PA}_n\neq\emptyset$, and $\beta$, $\gamma$ are incomparable, then $\beta\cup\gamma\in C_1-\mathcal{B}_1$. From Lemma \[l4\], we conclude that there is a maximal 0-nested set $M$ such that $\beta,\gamma\in \mathcal{B}_1^M$. By symmetry, we may assume that $$M=\bigl\{\{n,\ldots,1\},\ldots,\{n,n-1\},\{n\}\bigr\}.$$ (In that case $x_0$ occurs neither in $\beta^{=}$ nor in $\gamma^{=}$, since $\beta,\gamma\in \mathcal{B}_1^M$.) Let $Y,Z\in\mathcal{B}$ be the $u$-correspondents of $\beta$ and $\gamma$, respectively. Since $\beta$ and $\gamma$ are incomparable, $Y$ and $Z$ are incomparable too. Let $a\in \beta^{=}\cap \gamma^{=}\cap \mathbf{PA}_n$ and let $a'=u(a)$. Then the coordinates $(x_1',\ldots,x_n')$ of $a'$ satisfy the equations $Y^=$ and $Z^=$, and hence, the inequalities $Y^\leq$ and $Z^\leq$. This means that $Y$, $Z$ and the coordinates of $a'$, satisfy the conditions of Lemma \[l3\] (the coordinates of $a'$ are positive since $x_i'\geq 3$ holds for every $1\leq i\leq n$). Hence, we have that $Y\cup Z\not\in \mathcal{B}$, otherwise, the equation $(Y\cup Z)^{\geq}$, i.e. $\sum_{i\in Y\cup Z} x_i'\geq 3^{\|Y\cup Z\|}$ would hold, which contradicts Lemma \[l3\]. From this we conclude that $\beta\cup\gamma\not\in \mathcal{B}_1$, and since both $\beta$ and $\gamma$ are in $\mathcal{B}_1^M$, we have that $\beta\cup\gamma\in C_1-\mathcal{B}_1$. As a consequence of Lemma \[l5\] and Proposition \[p2\] we have the following. \[c2\] If $\beta_1,\ldots,\beta_k\in \mathcal{B}_1$ are such that $\beta_1^{=}\cap\ldots\cap \beta_k^{=}\cap \mathbf{PA}_n\neq\emptyset$, then $\{\beta_1,\ldots,\beta_k\}$ is 1-nested. \[p3\] For every vertex $v$ of $\mathbf{PA}_n$, there is a 0-face $V$ of $C$ such that $$\{v\}=(\bigcap \{\beta^{=}\mid \beta\in V\})\cap \pi,$$ and for every $\gamma\in \mathcal{B}_1-V$, $v\not\in\gamma^{=}$. Let $v$ be a vertex of $\mathbf{PA}_n\subseteq \mathbf{R}^{n+1}$. Since the dimension of this space is $n+1$, one concludes that there are $\beta_1,\ldots,\beta_n\in \mathcal{B}_1$ such that $$\{v\}=\beta_1^{=}\cap\ldots\cap \beta_n^{=} \cap \pi.$$ From Corollary \[c2\], it follows that $V=\{\beta_1,\ldots,\beta_n\}$ is a 1-nested set, and since it has $n$ elements, by Remark \[r1\], it is a 0-face of $C$. It remains to prove that for every $\gamma\in \mathcal{B}_1 -V$, we have that $v\not\in\gamma^{=}$. This is straightforward, since by maximality of $V$, we have that $V\cup \{\gamma\}$ is not 1-nested, and by Corollary \[c2\], the vertex $v$ cannot be in $\gamma^{=}$. \[c1\] $\mathbf{PA}_n$ is a simple $n$-dimensional polytope. This is obvious after Proposition \[p3\], or we may apply [@DP10 Proposition 9.7], since every 0-face has exactly $n$ elements. \[p4\] For every 0-face $V$ of $C$, there is a vertex $v$ of $\mathbf{PA}_n$ such that $$\{v\}=(\bigcap \{\beta^{=}\mid \beta\in V\})\cap \pi.$$ We show that for every 0-face $V$ of $C$, $(\bigcap \{\beta^{=}\mid \beta\in V\})\cap \pi$ is a singleton $\{v\}$ such that $v\in \gamma^>$, for every $\gamma\in \mathcal{B}_1-V$, from which the proposition follows. Again, by symmetry, we may assume that $V=\{\beta_1,\ldots,\beta_n\}\subseteq \mathcal{B}_1^M$, for $$M=\bigl\{\{n,\ldots,1\},\ldots,\{n,n-1\},\{n\}\bigr\}.$$ Let $X_i$, for every $i\in\{1,\ldots,n\}$, be the $u$-correspondent of $\beta_i\in V$, for $u$ being the affine transformation introduced after Lemma \[l3\]. Since $V$ is a construction of $$\{n,\ldots,1\}-\ldots-\{n,n-1\}-\{n\},$$ we have that $N=\{X_1,\ldots,X_n\}$ is a construction of the path graph $1-2-\ldots-n$. By induction on $n\geq 1$, we show that $X_1^{=}\cap\ldots\cap X_n^{=}$ is a singleton. If $n=1$, then $X_1^{=}$ is $x_1'=3$, and we are done. If $n>1$, let $L$ and $R$ be the constructions of the path graphs $$1-\ldots-(j-1)\quad {\rm and}\quad (j+1)-\ldots-n,$$ such that $N=\{\{1,\ldots,n\}\}\cup L\cup R$. By the induction hypothesis, we have that $\bigcap\{\beta_i^{=}\mid \beta_i\in L\}$ and $\bigcap\{\beta_i^{=}\mid \beta_i\in R\}$ are singletons in the corresponding $(j-1)$-dimensional and $(n-j)$-dimensional spaces. Hence, the coordinates $x_1',\ldots,x_{j-1}',x_{j+1}',\ldots,x_n'$ of a point belonging to the above intersection are determined, and it remains to determine the coordinate $x_j'$. It has a unique value evaluated from the equation $x_1'+\ldots+x_{j-1}'+x_j'+x_{j+1}'+\ldots+ x_n'=3^n$ and the values of $x_1',\ldots,x_{j-1}',x_{j+1}',\ldots,x_n'$. From this we conclude that $$(\bigcap \{\beta^{=}\mid \beta\in V\})\cap \pi$$ is a singleton $\{v\}$. It remains to show that $v\in \gamma^>$, for every $\gamma\in \mathcal{B}_1-V$. Note that in the above calculation of coordinates of $u(v)$, for the $j$th coordinate we have $$x_j'=3^n-(3^{j-1}+3^{n-j})>3^{n-1},$$ since $x_1'+\ldots+x_{j-1}'=3^{j-1}$ and $x_{j+1}'+\ldots+x_n'=3^{n-j}$. Note also that $j\in\{1,\ldots,n\}$ was the $\{1,\ldots,n\}$-superficial element with respect to $N$. In the same way, we may conclude that if $m$ is the $X_i$-superficial with respect to $N$, then the $m$th coordinate of $u(v)$ satisfies $x_m'>3^{\|X_i\|-1}$. For $\gamma\in \mathcal{B}_1^M -V$, let $Y=\{n-l,\ldots,n-l-k+1\}$ be its $u$-correspondent in $\mathcal{B}$. Since $Y\in \mathcal{B}-N$, it is easy to prove, by induction on $n$, that there exists $X_i\in N$ such that $Y\subset X_i$ and $Y$ contains the $X_i$-superficial element with respect to $N$ (cf.[@DP10 Lemma 6.14]). Hence, there is $m\in\{n-l,\ldots,n-l-k+1\}$, which is the $X_i$-superficial, and $k<\|X_i\|$. From the preceding paragraph, we conclude that the $m$th coordinate of $u(v)$ satisfies $x_m'>3^{\|X_i\|-1}$, and since all the other coordinates are positive, we have that $u(v)\in Y^>$, i.e.$$x_{n-l-k+1}'+\ldots+x_{n-l}'>3^{\|X_i\|-1}\geq 3^k,$$ hence, $v\in \gamma^>$. For every other $\gamma\in \mathcal{B}_1 -V$, i.e. when $\gamma\in \mathcal{B}_1-\mathcal{B}_1^M$, consider the permutohedron $P_n$ obtained as the intersection of the hyperplane $\pi$ and the halfspaces of the form $$x_{i_1}+\ldots+x_{i_k}\geq 3^k,$$ where $i_1,\ldots,i_k$ are mutually distinct elements of $\{0,\ldots,n\}$. This permutohedron realises the simplicial complex $C_1$. The intersection of $P_n$ and the hyperplane $$x_1+2x_2+\ldots+nx_n=\kappa(n,0)$$ is an $(n-1)$-simplex. It is the convex hull of the set of points $\{a_1,\ldots,a_n\}\subseteq \mathbf{R}^{n+1}$ (note that $\varepsilon(n)=\frac{3^n-3n}{3^n-n-1}$): $$\begin{array}{l} a_1(3^{n+1}-3^n-\varepsilon(n), 3^{n}-3^{n-1}+\varepsilon(n), 3^{n-1}-3^{n-2},\ldots,3), \\[1ex] a_2(3^{n+1}-3^n, 3^{n}-3^{n-1}-\varepsilon(n), 3^{n-1}-3^{n-2}+\varepsilon(n),\ldots,3), \\[1ex] \quad\vdots \\[1ex] a_n(3^{n+1}-3^n,\ldots, 3^3-3^2, 3^2-3-\varepsilon(n), 3+\varepsilon(n)). \end{array}$$ (The coordinates of $a_i$ are obtained as the solution of the system $$\begin{array}{c r l} & x_1+2x_2+\ldots+n x_n & =\kappa(n,0) \\[1ex] & x_0+x_1+\;\;x_2+\ldots+\;\;x_n & =3^{n+1} \\[1ex] (1) & x_1+\;\; x_2+\ldots +\;\; x_n & =3^n \\[1ex] (2) & x_2+\ldots + \;\;x_n & =3^{n-1} \\[1ex] \vdots & & \\[1ex] (n) & x_n & =3, \end{array}$$ with the equation $(i)$ omitted.) Since $\gamma\in \mathcal{B}_1-\mathcal{B}_1^M$, the hyperplane $\gamma^{=}$ is of the form $$x_{i_1}+2 x_{i_2}+\ldots+k(x_{i_k}+\ldots+x_{i_{k+l}})=\kappa(k,l),$$ where $i_k<i_{k+1}<\ldots<i_{k+l}$ and - $i_1,i_2,\ldots, i_{k+l}$ are not consecutive members of the sequence $1,2,\ldots,n$. Denote the left-hand side of the above equation by $\Xi$. We show that for every $i\in\{1,\ldots,n\}$, the coordinates $(x_0,\ldots,x_n)$ of $a_i$ satisfy $\gamma^>$. From this, since $v$ is in the convex hull of $\{a_1,\ldots, a_n\}$, one obtains that $v\in\gamma^>$. \(1) If for some $0\leq m\leq n-l-k$, $x_m$ occurs in $\Xi$, then $$\begin{array}{rl} \Xi\!\!\!\! & > x_m > 3^{n+1-m} - 3^{n-m}-1= 2\cdot 3^{n-m} -1= 2(3^{n-m}-1)+1 \\[2ex] & > \frac{3}{2}(3^{n-m}-1)+1= \frac{3^{n-m+1}-3}{2}+1 \geq \frac{3^{l+k+1}-3}{2}+1 \\[2ex] & > \kappa(k,l). \end{array}$$ \(2) If $i_1,\ldots,i_{l+k}\in \{n-l-k+1,\ldots,n\}$, then $\Xi$ is of the form $$a_1x_{n-l-k+1}+ a_2 x_{n-l-k+2}+\ldots +a_{l+k}x_n,$$ where $a_m\geq 0$, for every $0\leq m\leq l+k$. Since $(\ast\ast)$ holds, there is $j\in\{1,\ldots,k-1\}$ such that $a_j>j$ and for every $m\in\{1,\ldots,j-1\}$, we have that $a_m=m$. Therefore, $$\begin{array}{rl} \Xi\!\!\!\! & \geq x_{n-l-k+1}+\ldots+(j-1) x_{n-l-k+j-1} + j(x_{n-l-k+j}+\ldots+ x_n)+x_{n-l-k+j} \\[2ex] & = (x_{n-l-k+1}+\ldots+ x_n) +\ldots + (x_{n-l-k+j}+\ldots+x_n)+x_{n-l-k+j}. \end{array}$$ For $i$ such that $n-l-k+1\leq i\leq n-l-k+j$, the coordinates $(x_0,\ldots,x_n)$ of the point $a_i$ satisfy the inequality $$(x_{n-l-k+1}+\ldots+ x_n) +\ldots + (x_{n-l-k+j}+\ldots+x_n)> 3^{l+k}+\ldots+3^{l+k-j+1}.$$ For the other values of $i$, the above inequality converts into equality. Also the $(n-l-k+j)$th coordinate $x_{n-l-k+j}$ of $a_i$ is greater than $3^{l+k-j+1}-3^{l+k-j}-1$. Hence, $$\begin{array}{rl} \Xi\!\!\!\! & > 3^{l+k-j+1}{\displaystyle \frac{3^j-1}{2}} + 3^{l+k-j+1} -3^{l+k-j}-1 \\[2ex] & = {\displaystyle \frac{3^{l+k+1}-3^{l+k-j+1}+ 2\cdot 3^{l+k-j+1} -2\cdot 3^{l+k-j}}{2}}-1 \\[2ex] & = {\displaystyle \frac{3^{l+k+1}+ 3^{l+k-j}}{2}}-1 \\[2ex] & \geq {\displaystyle \frac{3^{l+k+1}+ 3^{l+1}}{2}}-1,\quad {\rm since\;} k\geq j+1 \\[2ex] & > {\displaystyle \frac{3^{l+k+1}- 3^{l+1}}{2}}+1 > \kappa(k,l). \end{array}$$ From Propositions \[p3\], \[p4\] and Corollary \[c1\], we conclude that the face lattice of $\mathbf{PA}_n$, with bottom removed, is given by $$\Bigl(\bigcup\bigl\{\mathcal{P}(\{\beta^{=}\mid \beta\in V\})\mid V\;\mbox{\rm is a 0-face of }C\bigr\},\supseteq \Bigr),$$ from which the theorem follows. ![$\mathbf{PA}_1$, $\mathbf{PA}_2$ and $\mathbf{PA}_3$[]{data-label="s:pasn"}](PA123.pdf){width="100.00000%"} Figure \[s:pasn\] illustrates the $n$-dimensional simple permutoassociahedra for $n=1,2$ and $3$. We conclude the paper with a picture that indicates the connection between the 3-dimensional simple permutoassociahedron and the 3-dimensional permutohedron. The facets of the permutohedron are labelled by the elements of $$\mathcal{B}_0=\mathcal{P}(\{0,1,2,3\})-\{\{0,1,2,3\}, \emptyset\}.$$ As it has been already noted, it is more intuitive to transform every such $A\in\mathcal{B}_0$ into the ordered partition of $\{0,1,2,3\}$, with $\{0,1,2,3\}-A$ as the first block and $A$ as the second block. (300,330)(0,-15) (0,0) (300,0) (0,300) (300,300) (40,40) (260,40) (40,260) (260,260) (120,60) (180,60) (100,100) (200,100) (60,120) (240,120) (130,130) (170,130) (130,170) (170,170) (60,180) (240,180) (100,200) (200,200) (120,240) (180,240) (150,30)[(0,0)[$\{3\}$]{}]{} (150,95)[(0,0)[$\{1,2,3\}$]{}]{} (150,150)[(0,0)[$\{1,2\}$]{}]{} (150,205)[(0,0)[$\{0,1,2\}$]{}]{} (150,270)[(0,0)[$\{0\}$]{}]{} (150,-7)[(0,0)[$\{0,3\}$]{}]{} (150,307)[(0,0)[$\{0,3\}$]{}]{} (-13,150)[(0,0)[$\{0,3\}$]{}]{} (313,150)[(0,0)[$\{0,3\}$]{}]{} (75,75)[(0,0)[$\{2,3\}$]{}]{} (225,75)[(0,0)[$\{1,3\}$]{}]{} (75,225)[(0,0)[$\{0,2\}$]{}]{} (225,225)[(0,0)[$\{0,1\}$]{}]{} (30,150)[(0,0)[$\{0,2,3\}$]{}]{} (95,150)[(0,0)[$\{2\}$]{}]{} (205,150)[(0,0)[$\{1\}$]{}]{} (270,150)[(0,0)[$\{0,1,3\}$]{}]{} (0,-5)[(0,0)[$1203$]{}]{} (300,-5)[(0,0)[$2103$]{}]{} (0,305)[(0,0)[$1230$]{}]{} (300,305)[(0,0)[$2130$]{}]{} (27,40)[(0,0)[$1023$]{}]{} (273,40)[(0,0)[$2013$]{}]{} (120,55)[(0,0)[$0123$]{}]{} (180,55)[(0,0)[$0213$]{}]{} (112,100)[(0,0)[$0132$]{}]{} (188,100)[(0,0)[$0231$]{}]{} (70,122)[(0,0)[$1032$]{}]{} (230,122)[(0,0)[$2031$]{}]{} (120,132)[(0,0)[$0312$]{}]{} (180,132)[(0,0)[$0321$]{}]{} (120,168)[(0,0)[$3012$]{}]{} (180,168)[(0,0)[$3021$]{}]{} (70,178)[(0,0)[$1302$]{}]{} (230,178)[(0,0)[$2301$]{}]{} (112,200)[(0,0)[$3102$]{}]{} (188,200)[(0,0)[$3201$]{}]{} (120,245)[(0,0)[$3120$]{}]{} (180,245)[(0,0)[$3210$]{}]{} (27,260)[(0,0)[$1320$]{}]{} (273,260)[(0,0)[$2310$]{}]{} (0,0)[(1,0)[300]{}]{} (0,0)[(0,1)[300]{}]{} (300,0)[(0,1)[300]{}]{} (0,300)[(1,0)[300]{}]{} (0,0)[(1,1)[40]{}]{} (300,0)[(-1,1)[40]{}]{} (0,300)[(1,-1)[40]{}]{} (300,300)[(-1,-1)[40]{}]{} (40,40)[(4,1)[80]{}]{} (40,40)[(1,4)[20]{}]{} (260,40)[(-4,1)[80]{}]{} (260,40)[(-1,4)[20]{}]{} (40,260)[(4,-1)[80]{}]{} (40,260)[(1,-4)[20]{}]{} (260,260)[(-4,-1)[80]{}]{} (260,260)[(-1,-4)[20]{}]{} (100,100)[(-2,1)[40]{}]{} (100,100)[(1,-2)[20]{}]{} (200,100)[(2,1)[40]{}]{} (200,100)[(-1,-2)[20]{}]{} (100,200)[(-2,-1)[40]{}]{} (100,200)[(1,2)[20]{}]{} (200,200)[(2,-1)[40]{}]{} (200,200)[(-1,2)[20]{}]{} (120,60)[(1,0)[60]{}]{} (120,240)[(1,0)[60]{}]{} (60,120)[(0,1)[60]{}]{} (240,120)[(0,1)[60]{}]{} (100,100)[(1,1)[30]{}]{} (200,100)[(-1,1)[30]{}]{} (100,200)[(1,-1)[30]{}]{} (200,200)[(-1,-1)[30]{}]{} (130,130)[(1,0)[40]{}]{} (130,130)[(0,1)[40]{}]{} (170,170)[(-1,0)[40]{}]{} (170,170)[(0,-1)[40]{}]{} (105,50) (100,60) (145,60) (110,70) (120,75) (100,60) (100,60) (110,70) (145,60) (145,60) (100,90) (90,100) (110,95) (110,110) (95,110) (100,90) (100,90) (110,110) (110,110) (90,100) (23.5,23.5) (65,40) (60,50) (50,60) (40,65) (65,40) (65,40) (40,65) (40,65) (50,60) (50,105) (60,100) (60,145) (70,110) (75,120) (60,100) (60,100) (70,110) (60,145) (60,145) (120,120) (145,125) (125,145) (145,135) (135,145) (120,120) (120,120) (145,135) (145,135) (125,145) (-5,20) (5,20) (20,-5) (20,5) (15,15) (-5,20) (-5,20) (15,15) (15,15) (20,-5) (65,40) (60,50) (40,65) (50,60) (70,110) (75,120) (100,90) (110,95) (180,120) (155,125) (165,145) (155,135) (175,145) (180,120) (180,120) (155,135) (155,135) (165,145) (180,180) (155,165) (165,155) (155,175) (175,155) (180,180) (180,180) (155,175) (155,165) (165,155) (120,180) (145,165) (125,155) (145,175) (135,155) (120,180) (120,180) (145,175) (135,155) (125,155) (100,210) (90,200) (110,205) (110,190) (95,190) (110,205) (90,200) (110,190) (110,190) (95,190) (200,210) (210,200) (190,205) (190,190) (205,190) (200,210) (200,210) (190,190) (190,190) (205,190) (210,100) (200,90) (190,95) (190,110) (205,110) (200,90) (200,90) (190,110) (190,110) (205,110) (155,60) (200,60) (195,50) (180,75) (190,70) (200,60) (200,60) (190,70) (155,60) (155,60) (145,240) (100,240) (105,250) (110,230) (120,225) (100,240) (100,240) (110,230) (145,240) (145,240) (155,240) (200,240) (195,250) (190,230) (180,225) (200,240) (200,240) (190,230) (155,240) (155,240) (225,120) (250,105) (240,100) (240,145) (230,110) (240,100) (240,100) (230,110) (240,145) (240,145) (50,195) (60,200) (60,155) (70,190) (75,180) (60,200) (60,200) (70,190) (60,155) (60,155) (250,195) (240,200) (240,155) (230,190) (225,180) (240,200) (240,200) (230,190) (240,155) (240,155) (276.5,23.5) (235,40) (240,50) (250,60) (260,65) (235,40) (235,40) (260,65) (260,65) (250,60) (23.5,276.5) (65,260) (60,250) (50,240) (40,235) (65,260) (65,260) (40,235) (40,235) (50,240) (276.5,276.5) (235,260) (240,250) (250,240) (260,235) (235,260) (235,260) (260,235) (260,235) (250,240) (305,20) (295,20) (280,-5) (280,5) (285,15) (305,20) (305,20) (285,15) (285,15) (280,-5) (305,280) (295,280) (280,305) (280,295) (285,285) (305,280) (305,280) (285,285) (285,285) (280,305) (-5,280) (5,280) (20,305) (20,295) (15,285) (-5,280) (-5,280) (15,285) (15,285) (20,305) (260,65) (250,60) (260,235) (250,240) (40,235) (50,240) (235,40) (240,50) (235,260) (240,250) (65,260) (60,250) (230,110) (225,120) (230,190) (225,180) (70,190) (75,180) (200,90) (190,95) (200,210) (190,205) (100,210) (110,205) (125,155) (135,155) (145,165) (145,175) (165,155) (175,155) (145,125) (145,135) (-5,20) (5,20) (295,20) (305,20) (20,-5) (20,5) (20,295) (20,305) A vertex of the permutohedron is labelled by the permutation corresponding to the maximal 0-nested set, i.e. to the set of all the facets incident to this vertex. The facets of the permutoassociahedron correspond to the elements of $\mathcal{B}_1$. For example, the pentagon that covers the vertex labelled by 2301 corresponds to the set $\{\{0,1,3\},\{0,1\},\{1\}\}$. We are grateful to Pierre-Louis Curien for a suggestion how to improve the notation in our paper. We are grateful to G" unter M. Ziegler for some useful comments, including Remark \[ziegler\], and for some corrections concerning a previous version of this paper. The first author is grateful to the Vietnam Institute for Advanced Studies in Mathematics in Hanoi for the hospitality, excellent working conditions and generous financial support while finishing this research. The first, the second and the third author were partially supported by the Grants $174020$, III$44006$ and $174026$ of the Ministry for Education and Science of the Republic of Serbia, respectively. [99]{} and [P. Mani-Levitska]{}, [On puzzles and polytope isomorphisms]{}, *Aequationes mathematicae, vol. 34 (1987) pp.287-297 and [S.L. Devadoss]{}, [Coxeter complexes and graph-associahedra]{}, Topology and its Applications, vol. 153 (2006) pp. 2155-2168 and [C. Procesi]{}, [Wonderful models of subspace arrangements]{}, Selecta Mathematica* (N.S.), vol.1 (1995) pp. 459-494 and [Z. Petri' c]{}, *Proof-Theoretical Coherence, KCL Publications, London, 2004 (revised version available at: <http://www.mi.sanu.ac.yu/~kosta/publications.htm>) ——–, [Hypergraph polytopes]{}, Topology and its Applications, vol. 158 (2011) pp. 1405-1444 and [D.N. Kozlov]{}, [Incidence combinatorics of resolutions]{}, Selecta Mathematica* (N.S.), vol. 10 (2004) pp.37-60 and [B. Sturmfels]{}, [Matroid polytopes, nested sets and Bergman fans]{}, *Portugaliae Mathematica* (N.S.), vol. 62 (2005) pp. 437-468 and [R. 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--- abstract: 'We introduce the notion of an asymptotically Poincaré family of surfaces in an end of a quasi-Fuchsian manifold. We show that any such family gives a foliation of an end by asymptotically parallel convex surfaces, and that the asymptotic behavior of the first and second fundamental forms determines the projective structure at infinity. As an application, we establish a conjecture of Labourie from [@labourie1992] regarding constant Gaussian curvature surfaces. We also derive consequences for constant mean curvature surfaces.' address: 'Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL' author: - Keaton Quinn date: 'July 31, 2019 (v1: November 21, 2018)' title: 'Asymptotically Poincaré surfaces in quasi-Fuchsian manifolds' --- Introduction ============ Epstein in [@epstein1984] describes a map that takes a domain $\Omega$ in ${\mathbb{C}\mathrm{P}}^1$ and a conformal metric $\sigma$ on $\Omega$ and gives an Epstein surface $\mathrm{Ep}_\sigma : \Omega \to {\mathbb{H}}^3$ in hyperbolic space. A variant of this construction associates to any conformal metric $\sigma$ on the Riemann surface at infinity $X$ of an end of a quasi-Fuchsian manifold $M$, a map $X \to M$, which we also call an Epstein surface. We use this construction to study surfaces in quasi-Fuchsian manifolds. The hyperbolic metric $h$ of $X$ gives a distinguished Epstein surface that we call the Poincaré surface of $X$. Scalar multiples of $h$ give a family of Epstein surfaces, which we call , that foliate the end. Similarly, a family of conformal metrics that is $C^\infty$-asymptotic to the family of scalar multiples of $h$ gives rise to a family of Epstein surfaces that we call an asymptotically Poincaré family. These definitions are made precise in Section \[asym-def-section\] (see Definition \[asym-def\]). Our main result concerns the first and second fundamental forms, $I_\epsilon$ and $I\!I_\epsilon$, of an asymptotically Poincaré family $S_\epsilon$, for $\epsilon \in (0,1)$. These both give families of Riemannian metrics on $X$, and hence, points in Teichmüller space $[I_\epsilon]$ and $[I\!I_\epsilon]$. We prove the following regarding their asymptotic behavior. \[big-thm-intro\] In a quasi-Fuchsian manifold, let $S_\epsilon$, for $\epsilon \in (0,1)$, be an asymptotically Poincaré family of surfaces with surface at infinity $X$. If $I_\epsilon$ and $I\!I_\epsilon$ are the corresponding first and second fundamental forms then $$[I_\epsilon] \to [h] \quad \text{ and } \quad [I\!I_\epsilon] \to [h] \quad \text{ as } \epsilon \to 0$$ and the tangent vector to $[I\!I_\epsilon]$ at $[h]$ vanishes while the tangent vector to $[I_\epsilon]$ is determined by the holomorphic quadratic differential $\phi$ that measures the difference between the induced projective structure and the Fuchsian projective structure on $X$. That is, $$\dot{[I_\epsilon]} = c \, \mathrm{Re}(\phi) \quad \text{and } \quad \dot{[I\!I_\epsilon]} = 0$$ for a positive constant $c$. Following [@schlenker2017], we call $\phi$ the holomorphic quadratic differential at infinity of $M$. Also, $[I_\epsilon]$ refers to the conformal class of the Riemannian metric $I_\epsilon$ and $\dot{[I_\epsilon]}$ denotes $\left. \frac{\partial}{\partial \epsilon} [I_\epsilon] \right\|_{\epsilon = 0}$. However, a comment is needed regarding $[I\!I_\epsilon]$. Due to the convenient choice of normal vector as described in Section \[asym-def-section\], the second fundamental form is actually negative definite. Its negative, therefore, defines a point in Teichmüller space and so we will use $[I\!I_\epsilon]$ to denote the conformal class of $-I\!I_\epsilon$. As an application of our results we answer a conjecture posed by Labourie. In [@labourie1991] he proves that an end of a quasi-Fuchsian manifold admits a unique foliation by $k$-surfaces: surfaces $S_k$, for $k$ in $(-1,0)$, such that the Gaussian curvature of $S_k$ is identically $k$. In [@labourie1992], Labourie notes how $k$-surfaces may be interpreted as a path in Teichmüller space and he asks what the tangent vectors to the paths $[I_k]$ and $[I\!I_k]$ are at $k=0$. He conjectures that they are related to the holomorphic quadratic differential at infinity. We show in Section \[k-surfaces-section\] that these $S_k$ form an asymptotically Poincaré family of surfaces. Theorem \[big-thm-intro\] then proves his conjecture and gives the relationship explicitly. \[k-surfaces-intro\] Let $I_k$ and $I\!I_k$ be the first and second fundamental forms of the $k$-surface $S_k$. Let $\phi$ be the holomorphic quadratic differential at infinity of $M$. Then, as $k \to 0$, the tangent vectors to $[I_k]$ and $[I\!I_k]$ in the Teichmüller space of $X$ are given by $$\dot{[I_k]}= - \mathrm{Re}(\phi) \quad \text{and } \quad \dot{[I\!I_k]} = 0.$$ We consider another application of Theorem \[big-thm-intro\] in Section \[mean-curvature-section\]. The work of Mazzeo and Pacard in [@mazzeo-pacard2011] show that the ends of a quasi-Fuchsian manifold admits a unique foliation by surfaces of constant mean curvature. We prove that this family of surfaces forms an asymptotically Poincaré family of surfaces by constructing, for each negative $k$ near zero, an Epstein surface whose mean curvature is identically $-\sqrt{1+k}$. Since asymptotically Poincaré surfaces foliate an end of $M$, by the uniqueness result of [@mazzeo-pacard2011] these surfaces are those shown to exist by Mazzeo and Pacard. Therefore, Theorem \[big-thm-intro\] describes the behavior of this family in Teichmüller space. \[cmc-intro\] Let $I_k$ and $I\!I_k$ be the first and second fundamental forms of the Epstein surface with constant mean curvature $-\sqrt{1+k}$. Let $\phi$ be the holomorphic quadratic differential at infinity of $M$. Then, as $k \to 0$, the tangent vectors to $[I_k]$ and $[I\!I_k]$ in Teichmüller space are given by $$\dot{[I_k]}= - \mathrm{Re}(\phi) \quad \text{and } \quad \dot{[I\!I_k]} = 0.$$ Acknowledgments {#acknowledgments .unnumbered} --------------- I would like to thank my Ph.D. thesis advisor David Dumas for suggesting this problem and for the many helpful conversations that followed. The author was partially supported in Summer 2018 by a research assistantship under NSF DMS-1246844, RTG: Algebraic and Arithmetic Geometry, at the University of Illinois at Chicago. Preliminaries ============= Quasi-Fuchsian Manifolds ------------------------ For a discrete subgroup $\Gamma$ of $\mathrm{SL}(2,{\mathbb{C}})$, let $\Lambda(\Gamma)$ denote its limit set and $\Omega(\Gamma) = {\mathbb{C}\mathrm{P}}^1 - \Lambda(\Gamma)$ the domain of discontinuity. If the limit set $\Lambda(\Gamma)$ is a Jordan curve then the domain of discontinuity is separated into two domains $\Omega(\Gamma) = \Omega_+ \cup \Omega_-$. When $\Omega_+$ and $\Omega_-$ are each $\Gamma$-invariant, the subgroup $\Gamma$ is called quasi-Fuchsian and the manifold $M = {\mathbb{H}}^3/\Gamma$ is a quasi-Fuchsian manifold. The convex hull of $\Lambda(\Gamma)$ in hyperbolic space is $\Gamma$-invariant and its quotient in $M$ is called the convex core. The complement of the convex core in $M$ consists of two ends diffeomorphic to $\Omega_{\pm}/\Gamma \times (0,\infty)$, and the surfaces $\Omega_{\pm}/\Gamma$ are called the surfaces at infinity for their respective ends. We will be focusing on one end of a quasi-Fuchsian manifold with smooth surface at infinity $\Omega/\Gamma$. Projective Structures and the Schwarzian derivative --------------------------------------------------- A complex projective structure on a surface $S$ is an atlas of charts to ${\mathbb{C}\mathrm{P}}^1$ whose transition functions are restrictions of Möbius transformations. The surface at infinity $\Omega/\Gamma$ of an end of $M$ is both a Riemann surface $X$ and a complex projective surface; we denote the latter by $X_M$ to emphasize its dependence on the manifold $M$. The Riemann surface $X$ also has its standard Fuchsian projective structure $X_F$ coming from the uniformization theorem. The difference between these two projective structures is the quadratic differential $\phi = X_M - X_F$, which is holomorphic with respect to $X$ (see e.g., [@dumas2009]). We call this the holomorphic quadratic differential at infinity of $M$. Here is a more concrete description of $\phi$. Recall that if $f$ is a locally injective holomorphic function on a domain in ${\mathbb{C}}$ or ${\mathbb{C}\mathrm{P}}^1$ then the Schwarzian derivative $\mathcal{S}(f)$ of $f$ is a holomorphic quadratic differential. Then $\tilde{\phi} = \mathcal{S}(f)$, where $f: \Omega \to {\mathbb{H}}$ is a Riemann map, is a $\Gamma$-invariant tensor that induces the holomorphic quadratic differential at infinity $\phi$. Epstein Surfaces ================ Conformal Metrics ----------------- A Riemann surface structure on a surface $S$ distinguishes a conformal class of metrics on $S$, called conformal metrics. These are metrics $\sigma$ that, in a complex coordinate chart $z$, can be written as $\sigma = e^{2\eta} \|dz\|^2$, for some real valued function $\eta$. This $\eta$ is called the log density function of $\sigma$ and we say the metric $\sigma$ is of class $C^k$ if $\eta$ is a $C^k$ function. For a $C^2$ conformal metric $\sigma = e^{2\eta}\|dz\|^2$, the Gaussian curvature is the function $K(\sigma) = -4 e^{-2\eta}\eta_{z\bar{z}}$. For a compact Riemann surface with genus larger than 1, this conformal class contains a unique hyperbolic metric which we denote by $h$. If we call $X$ the Riemann surface structure on $S$ then we will denote the set of all smooth conformal metrics on $S$ by $\mathrm{Conf}^\infty(X)$. The Visual Metric ----------------- A natural trivialization of the unit tangent bundle of hyperbolic space is $U {\mathbb{H}}^3 \cong {\mathbb{H}}^3 \times {\mathbb{C}\mathrm{P}}^1$ given by sending a tangent vector $v$ at a point $p$ to the ideal endpoint of the geodesic through $p$ in the direction $v$. For each $p$ we may use this trivialization to push forward the induced metric on $U_p{\mathbb{H}}^3$ to ${\mathbb{C}\mathrm{P}}^1$, obtaining $V_p$, a conformal metric on ${\mathbb{C}\mathrm{P}}^1$ called the visual metric from $p$. As an example, the visual metric from the origin in the ball model $V_0$ is just the spherical metric on $S^2$, which is identified with ${\mathbb{C}\mathrm{P}}^1$ in this model. In general, if $M$ is a Möbius transformation taking $0$ to the point $p$, then $V_p = M_V_0$. The Epstein Map --------------- In [@epstein1984] Epstein gives a way of constructing surfaces in hyperbolic space from domains in the Riemann sphere. These Epstein surfaces will be our main tool for studying asymptotically Poincaré surfaces. Our exposition of this Epstein construction follows [@dumas2017] (see also [@anderson1998]). Let $\Omega$ be a domain in ${\mathbb{C}\mathrm{P}}^1$ and $\sigma$ a $C^k$ conformal metric on $\Omega$, then there exists a unique $C^{k-1}$ map $\mathrm{Ep}_\sigma : \Omega \to {\mathbb{H}}^3$, called the Epstein map of $\Omega$ for the metric $\sigma$, such that for all $z \in \Omega$, $$V_{\mathrm{Ep}_\sigma(z)}(z) = \sigma(z).$$ Moreover, the image of a point $z$ depends only on the 1-jet of $\sigma$ at $z$. Epstein’s original construction gave a formula for $\mathrm{Ep}$ in the ball model of hyperbolic space. Dumas gives an $\text{SL}(2,{\mathbb{C}})$-frame field description of the map as follows. Choose an affine chart $z$ on ${\mathbb{C}\mathrm{P}}^1$ that distinguishes a point $0 \in \Omega$ and $\infty \notin \Omega$. Then, on the geodesic in ${\mathbb{H}}^3$ with ideal endpoints $0$ and $\infty$, there exists a unique point $p$ such that the visual metric from $p$ at $0$ is the Euclidean metric of this affine chart, $V_p(0) = \|dz\|^2$. The Epstein map is an $\mathrm{SL}(2,{\mathbb{C}})$-frame orbit of this point. \[dumas-def\] On a domain $\Omega$ in ${\mathbb{C}\mathrm{P}}^1$ write $\sigma = e^{2\eta}\|dz\|^2$. Define the $\mathrm{SL}(2,{\mathbb{C}})$-frame field $\widetilde{\mathrm{Ep}}_\sigma: \Omega \to \mathrm{SL}(2,{\mathbb{C}})$ by $$\widetilde{\mathrm{Ep}}_\sigma(z) = \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \eta_z & 1 \end{pmatrix} \begin{pmatrix} e^{-\eta/2} & 0 \\ 0 & e^{\eta/2} \end{pmatrix},$$ then the Epstein map is given by $$\mathrm{Ep}_\sigma(z) = \widetilde{\mathrm{Ep}}_\sigma(z) \cdot p.$$ Even though we call the image an Epstein surface, the Epstein map need not be an immersion. Indeed, if $\sigma$ is itself a visual metric then the Epstein map for $\sigma$ is constant. However, the lift of $\mathrm{Ep}_\sigma$ from $\Omega$ to the unit tangent bundle of hyperbolic space given by $\widehat{\mathrm{Ep}}_\sigma(z) = (\text{Ep}_\sigma(z), z)$ is an immersion (recall the trivialization $U{\mathbb{H}}^3 \cong {\mathbb{H}}^3 \times {\mathbb{C}\mathrm{P}}^1$). This lift can be thought of as providing a unit “normal” vector field for the Epstein surface even when the Epstein map is not an immersion. Indeed, this lift agrees with a unit normal vector field when the surface is immersed and so we will simply refer to it as the normal field from now on. Because the Epstein map is unique, it is natural with respect to the action of $\mathrm{SL}(2,{\mathbb{C}})$ in the following sense. Suppose $M$ is a Möbius transformation, then the following diagram commutes: $$\begin{tikzpicture}[scale=0.7] \node (1) at (0,2) {$(\Omega,\sigma)$}; \node (2) at (3,2) {$(M(\Omega),M_\sigma)$}; \node (3) at (0,0) {${\mathbb{H}}^3$}; \node (4) at (3,0) {${\mathbb{H}}^3$}; \draw[->] (1) to node [above] {$M$} (2); \draw[->] (1) to node [left] {$\mathrm{Ep}$} (3); \draw[->] (3) to node [above] {$M$} (4); \draw[->] (2) to node [right] {$\mathrm{Ep}$} (4); \end{tikzpicture}$$ That is, $\mathrm{Ep}_{M_\sigma}(M(z)) = M( \mathrm{Ep}_{\sigma}(z))$. See [@anderson1998]. This allows us to define Epstein maps on certain quotients. Suppose in general that $\Gamma$ is a subgroup of $\mathrm{SL}(2,{\mathbb{C}})$ acting freely and properly discontinuously on ${\mathbb{H}}^3 \cup {\mathbb{C}\mathrm{P}}^1$ leaving a domain $\Omega$ invariant. Then $\Omega/\Gamma$ inherits a Riemann surface structure. Call this structure $X$ and let $\sigma$ be a conformal metric on $X$. Lift this to $\tilde{\sigma}$ on $\Omega$, which is $\Gamma$-invariant. Then $\mathrm{Ep}_{\tilde{\sigma}}: \Omega \to {\mathbb{H}}^3$ is $\Gamma$-equivariant and therefore descends to a map $\mathrm{Ep}_\sigma : X \to {\mathbb{H}}^3/ \Gamma$. In particular, when $\Gamma$ is a quasi-Fuchsian group and $\Omega$ a component of the domain of discontinuity, each conformal metric $\sigma$ on the surface at infinity $X$ gives rise to a map from $X$ into the quasi-Fuchsian manifold $M$. Parallel Surfaces ----------------- Let $g^t : U {\mathbb{H}}^3 \to {\mathbb{H}}^3$ denote the time-$t$ geodesic flow projected down to ${\mathbb{H}}^3$. Thus for a unit tangent vector $v$ on ${\mathbb{H}}^3$ we have $g^t(v) = \exp_p(tv)$. Using the lift of an Epstein surface to $U{\mathbb{H}}^3$ described above, each Epstein surface gives rise to a family of surfaces by applying the geodesic flow (and projecting to ${\mathbb{H}}^3$). That is, we have the flowed surfaces $g^t \circ \widehat{\mathrm{Ep}}_\sigma(\Omega)$. In fact, these surfaces are themselves Epstein surfaces corresponding to scalar multiples of $\sigma$: \[epstein-flow\] Let $\Omega$ be a domain in ${\mathbb{C}\mathrm{P}}^1$ and $\sigma$ a conformal metric on $\Omega$. Then $$g^t \circ \widehat{\mathrm{Ep}}_\sigma = \mathrm{Ep}_{e^{2t} \sigma}.$$ That is, flowing the Epstein surface for $\sigma$ for time $t$ corresponds to taking the Epstein surface for the metric $e^{2t}\sigma$. Schwarzian Derivatives of Conformal Metrics ------------------------------------------- In general, the Schwarzian derivative of one conformal metric with respect to another on a Riemann surface is the $(2,0)$ part of the Schwarzian tensor defined by Osgood and Stowe in [@osgood-stowe1992]. For two conformal metrics $\sigma_1$ and $\sigma_2$ and a coordinate chart $z$, write $\sigma_i = e^{2\eta_i} \|dz\|^2$, then the Schwarzian derivative of $\sigma_2$ with respect to $\sigma_1$ is the quadratic differential $$B(\sigma_1,\sigma_2) = \left( (\eta_2)_{zz} - (\eta_2)_z^2 - (\eta_1)_{zz} + (\eta_1)_z^2 \right) dz^2.$$ As opposed to the Schwarzian derivative of a function, this need not be holomorphic. The two are related, though. For $f$ locally injective and holomorphic, we have $$\mathcal{S}(f) = 2B(\|dz\|^2,f^\|dz\|^2).$$ We also have naturality $f^B(\sigma_1,\sigma_2) = B(f^\sigma_1,f^\sigma_2)$ (again, for $f$ holomorphic) and a cocycle property $B(\sigma_1,\sigma_3) = B(\sigma_1,\sigma_2) + B(\sigma_2,\sigma_3)$. When $\sigma$ is a conformal metric on a domain in ${\mathbb{C}}$ with $B(\|dz\|^2,\sigma) = 0$ it is called a Möbius flat metric. We will denote any such metric by $g_{{\mathbb{C}\mathrm{P}}^1}$. The cocycle property gives us that the Schwarzian derivative of $\sigma = e^{2\eta}\|dz\|^2$ relative to any Möbius flat metric is $$B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma) = (\eta_{zz} - \eta_{z}^2 )dz^2.$$ Geometry of Epstein Surfaces ---------------------------- The first fundamental form of the Epstein surface for the metric $\sigma$ is given by $I(\sigma) = \mathrm{Ep}_\sigma^(g_{{\mathbb{H}}^3})$ for $g_{{\mathbb{H}}^3}$ the metric of ${\mathbb{H}}^3$. It is given by $$I(\sigma) = \frac{4}{\sigma}\|B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)\|^2 + \frac{1}{4}(1-K(\sigma))^2\sigma + 2(1-K(\sigma))\text{Re}(B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)).$$ The second fundamental form (relative to the normal lift $\widehat{\mathrm{Ep}}_\sigma$) is $$I\!I(\sigma) = \frac{4}{\sigma}\|B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)\|^2 - \frac{1}{4} (1 - K(\sigma)^2)\sigma - 2 K(\sigma) \text{Re}(B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma))$$ (see [@dumas2017 Eqns. 3.2-3.3]). Here $K(\sigma)$ is the Gaussian curvature of $\sigma$ and $B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)$ the Schwarzian derivative of $\sigma$ with respect to a Möbius flat metric. With $I(\sigma)$ and $I\!I(\sigma)$ we can compute the Gaussian curvature by $K(I(\sigma)) = -1 + \det(I(\sigma)^{-1}I\!I(\sigma))$ and the mean curvature by $H(\mathrm{Ep}_\sigma) = \frac{1}{2}\mathrm{tr}(I(\sigma)^{-1}I\!I(\sigma))$. We obtain $$K(I(\sigma)) = \frac{4K(\sigma)}{(1-K(\sigma))^2 - \frac{16}{\sigma^2} \|B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)\|^2}$$ and $$H(\mathrm{Ep}_\sigma) = \frac{K(\sigma)^2 - 1 - \frac{16}{\sigma^2}\|B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)\|^2}{(K(\sigma) - 1)^2 - \frac{16}{\sigma^2}\|B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)\|^2}.$$ In the quasi-Fuchsian setting, if $\sigma$ is a $\Gamma$-invariant conformal metric on $\Omega$ then each term in the above equations is also $\Gamma$-invariant. This is maybe less clear for the quadratic differential $B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)$ since the Möbius flat metric $g_{{\mathbb{C}\mathrm{P}}^1}$ is not itself $\Gamma$-invariant. However, we see that for $\gamma \in \Gamma$ we have $\gamma^B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma) = B(\gamma^g_{{\mathbb{C}\mathrm{P}}^1},\gamma^\sigma) = B(\gamma^ g_{{\mathbb{C}\mathrm{P}}^1},\sigma)$, by naturality of Schwarzian derivatives of conformal metrics. The metric $\gamma^* g_{{\mathbb{C}\mathrm{P}}^1}$ is still a Möbius flat metric, and so $B(\gamma^* g_{{\mathbb{C}\mathrm{P}}^1},\sigma) = B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)$, implying $B(g_{{\mathbb{C}\mathrm{P}}^1}, \sigma)$ is $\Gamma$-invariant. Therefore, $B(g_{{\mathbb{C}\mathrm{P}}^1},\sigma)$ induces a quadratic differential on $X$, which we will denote by $B(\sigma)$. In summary of the above, we have the following Gaussian and mean curvatures of the Epstein surfaces in $M$. \[curvature-epstein\] The Gaussian curvature for the Epstein surface $\mathrm{Ep}_\sigma : X \to M$ is given by $$K(I(\sigma)) = \frac{4K(\sigma)}{(1-K(\sigma))^2 - \frac{16}{\sigma^2} \|B(\sigma)\|^2},$$ and the mean curvature by $$\pushQED{\qed} H(\mathrm{Ep}_\sigma) = \frac{K(\sigma)^2 - 1 - \frac{16}{\sigma^2}\|B(\sigma)\|^2}{(K(\sigma) - 1)^2 - \frac{16}{\sigma^2}\|B(\sigma)\|^2}. \qedhere \popQED$$ These are now equations on the compact Riemann surface $X$. Asymptotically Poincaré Families of Surfaces {#asym-def-section} ============================================ Previously, we have discussed Epstein surfaces for domains and for quotients. Generalizing this, we say an embedded closed surface $i : S \to M$ of genus greater than one is an Epstein surface of $X$ if there exists a $\Gamma$-invariant conformal metric $\tilde{\sigma}$ on $\Omega$ and a diffeomorphism $\varphi : X \to S$ such that the diagram $$\begin{tikzpicture}[scale=0.7] \node (1) at (0,2) {$\Omega$}; \node (2) at (3,2) {${\mathbb{H}}^3$}; \node (3) at (0,0) {$S$}; \node (4) at (3,0) {$M$}; \draw[->] (1) to node [above] {$\mathrm{Ep}_{\tilde{\sigma}}$} (2); \draw[->] (1) to node [left] {$\varphi \circ \pi_X$} (3); \draw[->] (3) to node [above] {$i$} (4); \draw[->] (2) to node [right] {$\pi_M$} (4); \end{tikzpicture}$$ commutes. Here the $\pi_X$ and $\pi_M$ are the respective quotient maps $\Omega \to X$ and ${\mathbb{H}}^3 \to M$. When $S$ is an Epstein surface of $X$, the conformal metric $\tilde{\sigma}$ induces a conformal metric $\sigma$ on $X$ that we call the conformal metric at infinity. We have $\mathrm{Ep}_{\sigma} = \varphi \circ i$. The Epstein surface $S$ will then refer to the embedded image of $\mathrm{Ep}_{\sigma}: X \to M$. \[asym-def\] Let $S_\epsilon$ for $\epsilon$ in $(0,1)$ be a family of Epstein surfaces with conformal metrics at infinity $\sigma(\epsilon)$. We call this family asymptotically Poincaré if 1. there exists a scaling function $f:[0,1) \to [0,\infty)$ so that the path $$f\sigma:(0,1) \to \mathrm{Met}^\infty(X)$$ is differentiable and converges to the hyperbolic metric on $X$ as $\epsilon \to 0$, that is, $f(\epsilon)\sigma(\epsilon) \to h$ as $\epsilon \to 0$, 2. the function $f$ is smooth and has simple zero at 0, and 3. the continuous extension $\gamma:[0,1) \to \mathrm{Met}^\infty(X)$ of $f \sigma$ is differentiable. As will be shown, as $\epsilon$ tends towards zero, the family of surfaces is leaving the end of the manifold. The following lemma shows that the surfaces in an asymptotically Poincaré family are asymptotically parallel. \[asym-parallel-lemma\] Suppose $S_\epsilon$ is an asymptotically Poincaré family of surfaces. Let $t > 0$. If $g^t$ is the geodesic flow operator defined above, then $$d_M \left(g^t (\widehat{\mathrm{Ep}}_{\sigma(\epsilon)}(z)), \mathrm{Ep}_{\sigma(e^{-2t} \epsilon)}(z) \right) \to 0 \quad \text{ as } \epsilon \to 0$$ uniformly in $z$. That is, the distance between the surface $S_\epsilon$ flowed for time $t$ and the surface $S_{e^{-2t}\epsilon}$ tends towards zero as $\epsilon$ does. We work with the universal covers. Lift $\sigma(\epsilon)$ to $\tilde{\sigma}(\epsilon)$ and $\gamma(\epsilon) = f(\epsilon)\sigma(\epsilon)$ to $\tilde{\gamma}(\epsilon)$ on $\Omega$. Write $\tilde{\sigma}(\epsilon) = e^{2\eta(\epsilon)}\|dz\|^2$ and $\tilde{\gamma}(\epsilon) = e^{2\lambda(\epsilon)}\|dz\|^2$, then $\eta(\epsilon) = \lambda(\epsilon) - (1/2)\ln(f(\epsilon))$. Recall from Lemma \[epstein-flow\] that $g^t \circ \widehat{\mathrm{Ep}}_{\tilde{\sigma}(\epsilon)} = \mathrm{Ep}_{e^{2t}\tilde{\sigma}(\epsilon)}$. For ease of notation, let $c = e^{-2t}$. Then a straightforward calculation using Proposition \[dumas-def\] gives the distance between $\mathrm{Ep}_{c^{-1} \tilde{\sigma}(\epsilon)}$ and $\mathrm{Ep}_{\tilde{\sigma}(c\epsilon)}$ to be $$2 \mathrm{arctanh} \left(\sqrt{ \frac {(1 - \sqrt{\frac{c f(\epsilon)}{f(c \epsilon)}}e^{\lambda(c\epsilon) - \lambda(\epsilon)})^2 + 4 c f(\epsilon)e^{-2\lambda(\epsilon)}\|\lambda_z(c\epsilon) - \lambda_z(\epsilon)\|^2} {(1 + \sqrt{\frac{c f(\epsilon)}{f(c \epsilon)}}e^{\lambda(c\epsilon) - \lambda(\epsilon)})^2 + 4 c f(\epsilon)e^{-2\lambda(\epsilon)}\|\lambda_z(c\epsilon) - \lambda_z(\epsilon)\|^2} } \right).$$ Since $\tilde{\gamma}(\epsilon)$ converges in $\mathrm{Met}^\infty(X)$, the function $\lambda$ has a $C^2$ limit as $\epsilon \to 0$. Therefore, the argument of $\mathrm{arctanh}$ converges to zero uniformly in $z$. Thus the same holds in the quotient $M$ as well. We have required that an asymptotically Poincaré family consist of embedded surfaces. The next proposition gives a useful condition for a family of conformal metrics to give rise to an asymptotically Poincaré family of surfaces. \[asym-family-prop\] Let $\sigma: (0,1) \to \mathrm{Conf}^\infty(X)$ be a family of conformal metrics on $X$. Suppose there exists a smooth function $f:[0,1) \to [0,\infty)$ with simple zero at $0$, such that $f\sigma \to h$ as $\epsilon \to 0$ and such that the extension $\gamma: [0,1) \to \mathrm{Conf}^\infty(X)$ is differentiable. Then there exists an $\epsilon_0 >0$ so that for $\epsilon < \epsilon_0$, the Epstein map $\mathrm{Ep}_{\sigma(\epsilon)}$ is an embedding. Hence, the Epstein surfaces $\mathrm{Ep}_{\sigma(\epsilon)}:X \to M$, for $\epsilon < \epsilon_0$, form an asymptotically Poincaré family. Let $\tilde{\sigma}:(0,1) \to \mathrm{Conf}^\infty(\Omega)$ be the lift of the family $\sigma$. Define the Epstein family map $\mathrm{Ep}_{\tilde{\sigma}}: \Omega \times (0,1) \to {\mathbb{H}}^3$ by $\mathrm{Ep}_{\tilde{\sigma}}(z,\epsilon) = \mathrm{Ep}_{\tilde{\sigma}(\epsilon)}(z)$. It follows from the $\mathrm{SL}(2,{\mathbb{C}})$-frame definition of the Epstein map that in the upper half space model ${\mathbb{H}}^3 \cong {\mathbb{C}}\times {\mathbb{R}}^+$, the family map is given by $$\mathrm{Ep}_{\tilde{\sigma}}(z,\epsilon) = (z,0) + \frac{2}{e^{2\eta} + 4 \|\eta_z\|^2}\left(2 \eta_{\bar{z}}, e^\eta \right).$$ Writing $\tilde{\sigma}(\epsilon) = e^{2 \eta(z,\epsilon)}\|dz\|^2$ and $\tilde{\gamma}(\epsilon) = e^{2 \lambda(z,\epsilon)}\|dz\|^2$, the condition $\tilde{\gamma}(\epsilon) = f(\epsilon)\tilde{\sigma}(\epsilon)$ becomes $\eta(z,\epsilon) = \lambda(z,\epsilon) - (1/2) \ln(f(\epsilon))$. Note that $\lambda(z,\epsilon) \to \rho(z)$ as $\epsilon \to 0$, uniformly in $z$, where $\rho$ is the log density of the Poincaré metric of $\Omega$. Hence we can rewrite $\mathrm{Ep}_{\tilde{\sigma}}$ as $$\mathrm{Ep}_{\tilde{\sigma}}(z,\epsilon) = (z,0) + \frac{2}{e^{2\lambda} + 4 f(\epsilon) \|\lambda_z\|^2} \left( 2 f(\epsilon) \lambda_{\bar{z}}, \sqrt{f(\epsilon)}e^{\lambda} \right)$$ and see that $$\lim_{\epsilon \to 0} \mathrm{Ep}_{\tilde{\sigma}} (z,\epsilon) = (z,0).$$ So we may extend $\mathrm{Ep}_{\tilde{\sigma}}$ to a map $\Omega \times [0,1) \to {\mathbb{H}}^3 \sqcup \Omega$, which is the identity on the boundary $\Omega \times \{0\} \to \Omega$. While this map is not differentiable at $\epsilon = 0$ (due to the $\sqrt{f(\epsilon)}$), the map $F: \Omega \times [0,1) \to {\mathbb{H}}^3 \sqcup \Omega$ given by $F(z,\epsilon) = \mathrm{Ep}_{\tilde{\sigma}}(z,\epsilon^2)$ satisfies $F(z,0) = (z,0)$, is differentiable at $\epsilon = 0$, and has derivative $$d F_{(z,0)} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \sqrt{f'(0)} e^{-\rho(z)} \end{pmatrix}.$$ Since this is invertible, $F$ is a local $C^1$-diffeomorphism at the boundary of $\Omega \times [0,1) \to {\mathbb{H}}^3 \sqcup \Omega$. Define $\bar{M} = ( {\mathbb{H}}^3 \sqcup \Omega ) /\Gamma = M \sqcup X$. Then $\bar{M}$ is a smooth manifold with compact boundary $X$. By $\Gamma$-equivariance of $\mathrm{Ep}_{\tilde{\sigma}}$, $F$ descends to a map $X \times[0,1) \to \bar{M}$ that is the identity on the boundary $\partial(X \times [0,1)) \to \partial{\bar{M}} = X$ and that is a local diffeomorphism there. Therefore, the restriction of $F$ to $X \times [0, \delta)$, for some small enough $\delta$, is a diffeomorphism onto a collar neighborhood of $\partial \bar{M} = X$. Unraveling, we get that $\mathrm{Ep}_{\sigma}$ is a diffeomorphism from a collar neighborhood $X \times (0,\sqrt{\delta})$ to a neighborhood of infinity of $M$. In particular, each Epstein map $\mathrm{Ep}_\sigma(\cdot, \epsilon) = \mathrm{Ep}_{\sigma(\epsilon)}$, for $\epsilon < \sqrt{\delta}$, is an immersion and injective with compact domain $X$. Hence each Epstein surface, for $\epsilon < \sqrt{\delta}$, is embedded. To complete the proof take $\epsilon_0 = \sqrt{\delta}$. In the preceeding proof, we have the map $F: X \times [0,\delta) \to \bar{M}$ is a diffeomorphism onto its image. Hence we have the following result. \[foliation\] If $S_\epsilon$ is an asymptotically Poincaré family of surfaces, then there exists an $\epsilon_0 > 0$ such that for $\epsilon < \epsilon_0$ the surfaces $S_\epsilon$ form a foliation of the end of $M$ whose surface at infinity is $X$. We now turn to our main result, but first note that the co-orientation on an Epstein surface we are using is that induced by the lift $\widehat{\mathrm{Ep}}_\sigma$, which points towards the surface at infinity $X$. This implies that $I\!I(\sigma)$ is negative definite (for small enough $\epsilon$), and so $-I\!I(\sigma)$ is a smooth Riemannian metric. In our setting it is natural to work with the Riemannian model of Teichmüller space. That is, we use $$\mathcal{T}(X) = \mathrm{Met}^\infty(X)/ \mathrm{Diff}_0^\infty(X) \rtimes P^\infty(X),$$ where $P^\infty(X)$ is the set of smooth positive functions on $X$ and $\mathrm{Diff}_0^\infty(X)$ is the group of smooth diffeomorphisms isotopic to the identity. See [@tromba1992] for details. The smooth topology, however, is difficult to work with directly. So, we first work in the Sobolev setting $\mathrm{Met}^s(X)$ and $H^s(X)$ of Sobolev tensors and functions of a fixed regularity $s > 3$. Since $K$ and $B$ are smooth functions of $\sigma$ and its derivatives we have that they both extend to functions on Sobolev classes of metrics. Hence if $\sigma \in \mathrm{Conf}^s(X)$ then $I(\sigma)$ and $-I\!I(\sigma)$ belongs to $\mathrm{Met}^{s-2}(X)$. We will obtain results with these extensions and then argue our results are independent of the chosen $s$. \[thm-in-sobolev\] Suppose $S_\epsilon$ is an asymptotically Poincaré family of surfaces. Let $\gamma : [0,1) \to \mathrm{Met}^s(X)$ be the extension of $f\sigma$ thought of as taking values in a class of Sobolev metrics for a fixed $s > 3$. Then the first and second fundamental forms $I \circ \sigma: (0,1) \to \mathrm{Met}^{s-2}(X)$ and $I\!I \circ \sigma: (0,1) \to \mathrm{Met}^{s-2}(X)$ satisfy $$I_\epsilon = 4 f'(0) \epsilon I(\sigma(\epsilon)) \to h \quad \text{ and } \quad I\!I_\epsilon = - 4 f'(0) \epsilon I\!I(\sigma(\epsilon)) \to h \quad \text{ as } \epsilon \to 0.$$ Moreover, $I_\epsilon$ and $I\!I_\epsilon$ are differentiable at $\epsilon = 0$ and their tangent vectors are given by $$\dot{I_\epsilon} = \dot{\gamma} + 2 f'(0) h + 4 f'(0) \mathrm{Re}(\phi) \quad \text{ and } \quad \dot{I\!I_\epsilon} = \dot{\gamma}$$ We have that $\gamma:[0,1) \to \mathrm{Met}^\infty(X)$ is continuous and differentiable at $\epsilon = 0$. Therefore, as $\mathrm{Met}^\infty(X) = \cap_{s > 3} \mathrm{Met}^s(X)$ we also have that $\gamma$ is continuous to $\mathrm{Met}^s$ and differentiable at $\epsilon = 0$. Then, we have $I(\sigma(\epsilon)) = I(\frac{1}{f(\epsilon)} \gamma(\epsilon))$ is equal to $$4 f(\epsilon) \frac{\|B(\gamma(\epsilon))\|^2}{\gamma(\epsilon)} + \frac{1}{4 f(\epsilon)}(1 - f(\epsilon) K(\gamma(\epsilon)))^2 \gamma(\epsilon) + 2(1 - f(\epsilon)K(\gamma(\epsilon)))\mathrm{Re}(B(\gamma(\epsilon))),$$ which is a smooth tensor independent of $s$. Since $f:[0,1) \to {\mathbb{R}}$ and $\gamma:[0,1) \to \mathrm{Met}^s(X)$ are differentiable at 0 and since $K: \mathrm{Met}^s(X) \to H^{s-2}(X)$ and $B: \mathrm{Conf}^s(X) \to \Gamma^{s-2}(\Sigma^{2}(X))$ are differentiable at the hyperbolic metric $h$, we can write $$\begin{aligned} f(\epsilon) = \epsilon f'(0) + O(\epsilon^2), \quad K(\gamma(\epsilon)) = -1 + O(\epsilon), \quad B(\gamma(\epsilon)) = \frac{1}{2} \phi + O(\epsilon).\end{aligned}$$ Substitution and some simplification gives $$I(\sigma(\epsilon)) = \frac{1}{4\epsilon f'(0)} h + \frac{1}{4 f'(0)} \dot{\gamma} + \frac{1}{2} h + \mathrm{Re}(\phi) + O(\epsilon).$$ Consequently, $$I_\epsilon = h + \epsilon(\dot{\gamma} + 2 f'(0)h + 4 f'(0) \mathrm{Re}(\phi)) + O(\epsilon^2).$$ The same reasoning will give $I\!I_\epsilon = h + \epsilon \dot{\gamma} + O(\epsilon^2)$. The result then follows. The space $\mathrm{Met}^s(X)$ is an open subset of the vector space of symmetric 2-tensors $\Gamma^s(\Sigma^2(X))$. So, we may canonically identify each tangent space to $\mathrm{Met}^s(X)$ with this space. For each $g \in \mathrm{Met}^s(X)$, there is an $L^2$ orthogonal decomposition of the tangent space at $g$ as $$T_g \mathrm{Met}^s(X) = \{\dot{g} \ \| \ \mathrm{tr}_g (\dot{g}) = 0 = \delta_g(\dot{g}) \} \oplus \{\mathcal{L}_X g + fg \ \|\ f \in H^s \text{ and } X \in \Gamma(TX) \},$$ where $\delta_g(\dot{g})$ is the divergence of the 1-1 tensor $g^{-1}\dot{g}$. The second summand is tangent to the $\mathrm{Diff}_0^s(X) \rtimes P^s(X)$ orbit of $g$. The first summand is referred to as transverse traceless tensors and consists of elements that are smooth, trace-free, and divergence-free (see [@fischer-marsden1975] for more details). Since the transverse traceless tensors are orthogonal to the group orbit they may be identified with the tangent space $T_{[g]} \left( \mathrm{Met}^s(X)/ \mathrm{Diff}_0^s(X) \rtimes P^s(X) \right)$ to the quotient at $[g]$. Moreover transverse traceless tensors are a set of smooth tensors that are $L^2$ orthogonal to the $\mathrm{Diff}_0^s(X) \rtimes P^s(X)$ orbit for any $s$. Consequently, they may be identified with the tangent space to the quotient $\mathrm{Met}^\infty(X)/(\mathrm{Diff}_0^\infty(X) \rtimes P^\infty(X))$. That is, they may be naturally identified with the tangent space to Teichmüller space at $[g]$: $$T_{[g]} \mathcal{T}(X) = \{\dot{g} \ \| \ \mathrm{tr}_g (\dot{g}) = 0 = \delta_g(\dot{g}) \}.$$ Under this identification, the action of the derivative of the projection $\pi: \mathrm{Met}^\infty(X) \to \mathcal{T}(X)$ at $g$ is given by orthogonal projection onto the transverse traceless tensors $T_g \mathrm{Met}^\infty(X) \to \{\dot{g} \ \| \ \mathrm{tr}_g (\dot{g}) = 0 = \delta_g(\dot{g}) \}.$ We make one more comment before we prove our main Theorem. Let $X$ be a Riemann surface and $h$ the corresponding hyperbolic metric. If $\phi$ is a holomorphic quadratic differential, then $\mathrm{Re}(\phi)$ is a smooth $h$-transverse-traceless tensor. We now prove our main result, Theorem \[big-thm-intro\] from the Introduction. Recall that $[I\!I]$ denotes the point in Teichmüller space corresponding to the conformal class of $-I\!I$. \[main-result\] Let $S_\epsilon$ for $\epsilon \in (0,1)$ be an asymptotically Poincaré family of surfaces with metrics at infinity $\sigma(\epsilon)$. If $h$ is the hyperbolic metric of $X$ and $\phi$ the holomorphic quadratic differential at infinity, then in Teichmüller space $\mathcal{T}(X)$ we have $$[I(\sigma(\epsilon))] \to [h] \quad \text{ and } \quad [I\!I(\sigma(\epsilon))] \to [h] \quad \text{ as } \epsilon \to 0.$$ Moreover, the tangent vectors in $T_{[h]} \mathcal{T}(X)$ are given by $$\dot{[I(\sigma(\epsilon))]} = 4 f'(0) \mathrm{Re}(\phi) \quad \text{ and } \quad \dot{[I\!I(\sigma(\epsilon))]} = 0.$$ Using the notation from Proposition \[thm-in-sobolev\], for each $s > 3$ we have that $I_\epsilon$ and $I\!I_\epsilon$, which are paths through smooth tensors, converge in $\mathrm{Met}^s(X)$ to the hyperbolic metric $h$. Since $\mathrm{Met}^\infty(X) = \cap \mathrm{Met}^s(X)$ we know that $I_\epsilon$ and $I\!I_\epsilon$ converge to $h$ in $\mathrm{Met}^\infty(X)$. Moreover, since the projection $\mathrm{Met}^\infty(X) \to \mathcal{T}(X)$ is continuous, and since $[I(\sigma(\epsilon))] = [I_\epsilon]$ and $[I\!I(\sigma(\epsilon))] = [I\!I_\epsilon]$, we have that $$[I(\sigma(\epsilon))] \to [h] \quad \text{ and } \quad [I\!I(\sigma(\epsilon))] \to [h] \quad \text{ as } \epsilon \to 0.$$ Since both paths converge to $[h]$ at $\epsilon = 0$ we can extend them to continuous paths $[0,1) \to \mathcal{T}(S)$. Since $[I(\sigma(\epsilon))]$ and $[I\!I(\sigma(\epsilon))]$ agree with $[I_\epsilon]$ and $[I\!I_\epsilon]$, respectively, we also have they are differentiable at $\epsilon = 0$. From Proposition \[thm-in-sobolev\] we know that $$\dot{I_\epsilon} = \dot{\gamma} + 2 f'(0) h + 4 f'(0) \mathrm{Re}(\phi) \quad \text{ and } \quad \dot{I\!I_\epsilon} = \dot{\gamma}.$$ Since $\phi$ is holomorphic, $4 f'(0) \mathrm{Re}(\phi)$ is trace-free and divergence-free. On the other hand, $2 f'(0) h$ is pure trace and so belongs to the $\mathrm{Diff}_0^s(X) \rtimes P^s(X)$ orbit of $h$. Furthermore, for any $s$ it belongs to the group orbit of $h$. Hence, the derivative of the projection at $h$ removes this term. Similarly, $\gamma$ is a path in the group orbit of $h$ and so $\dot{\gamma}$ projects to 0. We then have $$\begin{aligned} \dot{[I(\sigma(\epsilon))]} = \dot{[I_\epsilon]} = d \pi_h (\dot{\gamma} + 2 f'(0) h + 4 f'(0) \mathrm{Re}(\phi)) = 4 f'(0) \mathrm{Re}(\phi)\end{aligned}$$ and $$\dot{[I\!I(\sigma(\epsilon))]} = \dot{I\!I_\epsilon} = d \pi_h (\dot{\gamma}) = 0,$$ as claimed. $k$-Surfaces in Quasi-Fuchsian Manifolds {#k-surfaces-section} ======================================== The $k$-Surface Equation ------------------------ We now prove that $k$-surfaces form an asymptotically Poincaré family of surfaces. To do this we derive an equation for a conformal metric that implies its Epstein surface is a $k$-surface and we show this equation has a unique smooth solution for each $k$ near zero. In the proof of existence of solutions we will see that these conformal metrics satisfy the hypothesis of Proposition \[asym-family-prop\]. This also gives an alternative proof to Labourie’s theorem on the existence of $k$-surfaces, in this case for quasi-Fuchsian manifolds. Finding a metric $\sigma$ whose Epstein surface has constant Gaussian curvature $k$ is finding a metric $\sigma$ that solves $K(I(\sigma)) = k$, which from Lemma \[curvature-epstein\] is solving $$\frac{4K(\sigma)}{(1-K(\sigma))^2 - \frac{16}{\sigma^2} \|B(\sigma)\|^2} = k.$$ For now we will focus on solving $$\label{k-surface-equation} 4K(\sigma) = k \left((1-K(\sigma))^2 - \frac{16}{\sigma^2} \|B(\sigma)\|^2 \right),$$ as we will see that for small enough $k$, $K(I(\sigma)) = k$. We are interested in obtaining solutions to (\[k-surface-equation\]) for $k$ near zero. This is hampered by the fact that there are no solutions to (\[k-surface-equation\]) when $k = 0$. Indeed we would be asking for $K(\sigma)=0$, which is impossible on a surface with genus bigger than 1. In an attempt to obtain better asymptotics we consider the case when $\Gamma$ is Fuchsian. Working in the universal covers, $\mathrm{Ep}_h : \Omega \to {\mathbb{H}}^3$ gives a totally geodesic copy of ${\mathbb{H}}^2$ in ${\mathbb{H}}^3$ with constant curvature $-1$. For $-1<k<0$ the $k$-surfaces are given by equidistant copies of this ${\mathbb{H}}^2$. The conformal metric whose Epstein surface is the $k$-surface is then $c(k)h$ for some function of $k$ satisfying $K(I(c(k)h)) = k$. Since $B(g_{{\mathbb{C}\mathrm{P}}^1},c(k)h) = 0$ we get the defining equation of $c$ as $4K(c(k)h) = k(1 - K(c(k)h))^2$. More explicitly $c$ is given by $$\label{c-def} c(k) = \frac{1+\sqrt{1+k}}{1-\sqrt{1+k}}.$$ Suppose we have solution metrics $\sigma_k$ on $X$ that give Epstein surfaces with constant Gaussian curvature $k$. As we just saw, in the Fuchsian case we have $\sigma_k = c(k) h$. In the general quasi-Fuchsian case, define $f(k) = c(k)^{-1}$ and let $\tau_k = f(k)\sigma_k$, so that in the Fuchsian case the metric $\tau_k$ is the hyperbolic metric for all $k$. Since $\sigma_k$ solves (\[k-surface-equation\]), by substitution and some simplification we get that $\tau = \tau_k$ solves the equation $$\label{scaled-equation} (2+k)(1+K(\tau))^2 + 2\sqrt{1+k}\left(1-K(\tau)^2\right) + 16\left(2\sqrt{1+k} - 2 - k \right)\frac{\|B(\tau)\|^2}{\tau^2} = 0.$$ In the limiting case $k=0$ we see that the hyperbolic metric $h$ on $X$ solves (\[scaled-equation\]). Solutions actually exists in a neighborhood of $(k,\tau) = (0,h)$ as we will show. To apply PDE theory we define the function $F : U \times \mathrm{Conf}^\infty(X) \to C^\infty(X)$ given by $$\begin{gathered} F(k,\tau) = (2+k)(1+K(\tau))^2 \\ + 2\sqrt{1+k}\left(1-K(\tau)^2 \right) +16\left(2\sqrt{1+k} - 2 - k \right)\frac{\|B(\tau)\|^2}{\tau^2}.\end{gathered}$$ Here, $U$ is an open interval around $0$ small enough not to contain $-1$ so that $F$ is smooth on its domain. Points $(k,\tau)$ where $F(k,\tau) = 0$ are solutions to the scaled equation (\[scaled-equation\]); we have already found $F(0,h) = 0$. Solutions to the $k$-Surface Equation ------------------------------------- We will use the Implicit Function Theorem in the Banach space setting to obtain solutions and so we first work with $\mathrm{Conf}^s(X) = \{ p h \ \|\ p \in H^s(X,{\mathbb{R}}^+)\}$, where $H^s(X,{\mathbb{R}}^+)$ is the Sobolev space of functions on $X$ taking positive values. With the norm $\\|\tau\\|_s := \\| \tau/h\\|_s$, the set $\mathrm{Conf}^s(X)$ is naturally identified with an open subset of the Banach space $H^s(X) = H^{s}(X,{\mathbb{R}})$. Fix $s>3$ from now on. Extend $F$ to a function $U \times \mathrm{Conf}^s(X) \to H^{s-2}(X)$ so that $F$ is then defined on an open subset of a Banach space. \[weak-solutions\] There is a neighborhood $V$ of $0$ such that for each $k \in V$, there exists a unique $\tau \in \mathrm{Conf}^s(X)$ such that $F(k,\tau) = 0$. Note that since the constituent parts of $F$ are smooth on $U$, $F$ is smooth. Furthermore, $D_2F_{(0,h)} : H^s(X) \to H^{s-2}(X)$ is an isomorphism, where $D_2F$ is the partial derivative of $F$ with respect to its second argument. Indeed, a direct computation of the derivative of $F$ at $(0,h)$ yields $$DF_{(0,h)}(\dot{k},\dot{\tau}) = 4 \, D K_h(\dot{\tau}).$$ Note that $D_1F_{(0,h)} = 0$ and that $D_2F_{(0,h)} = 4 \, DK_h$. The differential of the curvature function $D K_h$ is given by a formula of Lichnerowicz: $$4\ DK_h(\dot{\tau}) = -2(\Delta_h - Id)\frac{\dot{\tau}}{h},$$ which is an isomorphism $H^s(X) \to H^{s-2}(X)$ (see [@tromba1992 Page 33]). Consequently, by the Banach Implicit Function Theorem (see [@gilbarg-trudinger2001 Theorem 17.6]) there exists a neighborhood $V$ of $0$ and a curve $\gamma : V \to \mathrm{Conf}^s(X)$ with $\gamma(0) = h$ and $F(k, \gamma(k)) = 0$. Moreover, these are the only solutions to $F(k,\tau) = 0$ in $V$, and $\gamma$ is smooth since $F$ is. Regularity of Solutions ----------------------- Theorem \[weak-solutions\] furnishes weak solutions $\gamma(k)$ that vary smoothly as a map $V \to \mathrm{Conf}^s(X)$. The Sobolev Embedding Theorem immediately strengthens the regularity of the individual $\gamma(k)$ to at least $C^{2,\alpha}$, for each $k$ (see [@aubin1982]), and so we have strong solutions $F(k,\gamma(k)) = 0$. A nonlinear equation $A(u) = 0$ is said to be elliptic at $u$ if the derivative of $A$ at $u$, $DA_u$, is an elliptic linear operator. When $A$ is smooth, solutions $u$ where $A$ is elliptic are also smooth ([@gilbarg-trudinger2001 Lemma 17.16]). In our case, since $D_2F_{(0,h)} = -2(\Delta_h - Id)$ is an elliptic operator we have $F(0, \tau) = 0$ is elliptic at $\tau = h$. Moreover, ellipticity is an open condition in ${\mathbb{R}}\times C^2(X)$, so there exists an open interval $(-\delta,\delta)$ such that the linearization $D_2F(k,\gamma(k))$ is elliptic for all $k \in (-\delta,\delta)$. This has two consequences. First, it shows the solutions given by Theorem \[weak-solutions\] are individually smooth conformal metrics. We conclude the following theorem. \[k-surfaces-existence\] There exists an $\delta > 0$ such that for all $-\delta < k < 0$ there exists a unique smooth metric $\sigma_k$ whose Epstein surface is a $k$-surface. We have a curve $\gamma$ defined on an interval $(-\delta, \delta)$ such that the smooth metric $\gamma(k)$ satisfies $F(k,\gamma(k)) = 0$. This means that $\gamma(k)$ solves the scaled equation (\[scaled-equation\]) and so, for $k \in (-\delta,0)$, the metric $\sigma(k) = c(k) \gamma(k)$ (where $c(k)$ is defined by (\[c-def\])) satisfies the $k$-surface equation (\[k-surface-equation\]). Finally, we must show that $(1-K(\sigma))^2 - \frac{16}{\sigma^2} \|B(\sigma)\|^2$ in (\[k-surface-equation\]) is everywhere nonzero so that (\[k-surface-equation\]) is equivalent to the desired Guassian curvature condition. But this follows from Nehari’s Theorem (see [@lehto-1987 Theorem 1.3], or the original paper [@nehari-1949]), which gives an a priori bound on $\|B(h)\|^2/h^2$, implying that $\|B(\sigma_k)\|^2/\sigma_k^2 \to 0$ as $k \to 0$. Second, ellipticity also implies the family $\gamma(k)$ varies smoothly in $k$ in the $C^\infty$ topology. Indeed, it implies the Banach Implicit Function Theorem may be applied to $D_2F(k,\gamma(k)): H^t(X) \to H^{t-2}(X)$ for any $t > s$ and any $k \in (-\delta,\delta)$ to get that $\gamma$ is also a smooth path into $\mathrm{Conf}^t(X)$. We have the following corollary. \[k-surfaces-parallel\] The family of $k$-surfaces forms an asymptotically Poincaré family. \[k-surfaces-cor\] We show that $\tilde{\sigma}(\epsilon) = \sigma_{-\epsilon}$ satisfies the hypotheses of Proposition \[asym-family-prop\]. Since $\mathrm{Conf}^\infty(X) = \cap \mathrm{Conf}^s(X)$ and since $\gamma$ is smooth on $(-\delta,\delta)$ into $\mathrm{Conf}^s(X)$ for all $s > 3$, we have the $\gamma$ is smooth into $\mathrm{Conf}^\infty(X)$. Now, define $\tilde{f}(\epsilon) = f(- \epsilon)$ and $\tilde{\gamma}(\epsilon) = \gamma(-\epsilon)$ for $\epsilon \in (0,\delta)$. Then the Corollary follows from Proposition \[asym-family-prop\] since $\tilde{f}$ is smooth on $[0,\delta)$ with derivative $\tilde{f}'(0) = 1/4$, since $\tilde{\gamma}$ is differentiable, and since $\tilde{f}(\epsilon) \tilde{\sigma}(\epsilon) = \tilde{\gamma}(\epsilon) = \gamma(k) \to h$ as $\epsilon \to 0$. A Conjecture of Labourie ------------------------ A geometrically finite end of a hyperbolic 3-manifold admits a foliation by $k$-surfaces. Labourie first proved their existence in [@labourie1991]. In [@labourie1992] Labourie discusses how the conformal classes $[I_k]$ and $[I\!I_k]$ behave as paths in the Teichmüller space of $X$. When $k \to -1$, $[I\!I_k]$ approaches the point at infinity of $\mathcal{T}(X)$ corresponding to the measured geodesic lamination on the convex core of the quasi-Fuchsian manifold. When $k \to 0$, both $[I_k]$ and $[I\!I_k]$ converge to the same point: the complex structure $X$ on the surface at infinity. He conjectures that the tangent vectors to these paths are related to the holomorphic quadratic differential at infinity $\phi$. Our Theorem \[k-surfaces-existence\] is another proof of the existence of these $k$-surfaces, at least for $k$ close to zero. By the uniqueness result of Theorem 1.10 in [@labourie1992], given an end of $M$ and $k \in (-1,0)$, there exists a unique immersed incompressible $k$-surface. Since Epstein surfaces are incompressible, the $k$-surfaces produced by our Theorem \[k-surfaces-existence\] are the $k$-surfaces Labourie obtains. Our approach here is more concrete than the methods used by Labourie. In return for sacrificing the generality of Labourie’s pseudoholomorphic methods, a proof of Labourie’s conjecture follows easily. Corollary \[k-surfaces-parallel\] gives that, for $k$ near zero, these $k$-surfaces form an asymptotically Poincaré family, and so Theorem \[main-result\] resolves Labourie’s conjecture. Let $I_k$ and $I\!I_k$ be the first and second fundamental forms of the $k$-surface. Let $\phi$ be the holomorphic quadratic differential at infinity of $M$. Then, as $k \to 0$, the tangent vectors to $[I_k]$ and $[I\!I_k]$ in Teichmüller space are given by $$\dot{[I_k]} = - \mathrm{Re}(\phi) \quad \text{and } \quad \dot{[I\!I_k]} = 0.$$ We take $\epsilon = -k$. The derivative $f'(0) = -1/4$ can be computed directly. Theorem \[main-result\] now gives the theorem. Constant Mean Curvature Surfaces in Quasi-Fuchsian Manifolds {#mean-curvature-section} ============================================================ We now prove that there exists an asymptotically Poincaré family of surfaces $S_k$ for $k$ near zero, such that the mean curvature of the surface $S_k$ is $-\sqrt{1+k}$. This is done similarly to the previous section: we derive an equation for a conformal metric that implies its Epstein surface has mean curvature $-\sqrt{1+k}$ and we show this equation has a unique smooth solution for each $k$ near zero. The proof of existence of solutions will show that these conformal metrics satisfy the hypothesis of Proposition \[asym-family-prop\]. Recall from Lemma \[curvature-epstein\] that the mean curvature of $\mathrm{Ep}_\sigma: X \to M$ is given by $$H(\mathrm{Ep}_\sigma) = \frac{K(\sigma)^2 - 1 - \frac{16}{\sigma^2}\|B(\sigma)\|^2}{(K(\sigma) - 1)^2 - \frac{16}{\sigma^2}\|B(\sigma)\|^2}.$$ To find a metric whose Epstein surface has constant mean curvature $-\sqrt{1+k}$ we must solve the equation $H(\mathrm{Ep}_\sigma) = -\sqrt{1+k}$, which simplifies to $$\label{mean-curvature-equation} 1-K(\sigma)^2 - \sqrt{1+k}(1-K(\sigma))^2 + (1 + \sqrt{1+k})\frac{16}{\sigma^2}\|B(\sigma)\|^2 = 0.$$ As in the $k$-surfaces case it suffices to solve (\[mean-curvature-equation\]) since $(K(\sigma) - 1)^2 - \frac{16}{\sigma^2}\|B(\sigma)\|^2$ will eventually be nonzero. Furthermore, we scale the equation by assuming $\sigma_k$ solves (\[mean-curvature-equation\]) and defining $\tau_k = f(k) \sigma_k$ for $f(k) = \frac{1-\sqrt{1+k}}{1+\sqrt{1+k}}$. If $\sigma_k$ solves (\[mean-curvature-equation\]) then $\tau = \tau_k$ solves $G(k,\tau) = 0$ for $G: U \times \mathrm{Conf}^\infty(X) \to C^\infty(X)$ defined by $$G(k,\tau) = 1+\sqrt{1+k} + 2\sqrt{1+k}K(\tau) + (-1 + \sqrt{1+k})(K(\tau)^2 - \frac{16}{\tau^2}\|B(\tau)\|^2).$$ Here $U$ is a small enough open set around zero not containing $-1$ so that $G$ is smooth on its domain. There exists a neighborhood $W$ of 0 so that for each $k \in W$, there exists a unique $\tau \in \mathrm{Conf}^\infty(X)$ so that $G(k,\tau) = 0$. Extend $G$ to a map $U \times \mathrm{Conf}^s(X) \to H^{s-2}(X)$ for $s > 3$. When $k = 0$ we have the hyperbolic metric $h$ as solution $G(0,h) = 0$. The map $G$ is smooth. The derivative of $G$ at $(0,h)$ is given by $$dG_{(0,h))}(\dot{k},\dot{\tau}) = -4 \frac{\|\phi\|^2}{h^2} \dot{k} + 2 D K_h(\dot{\tau}).$$ Notice that $D_2 G_{(0,h)} = D K_h$, which—as in the proof of Theorem \[weak-solutions\]—is an isomorphism $H^s(X) \to H^{s-2}(X)$. Hence, by the Banach Implicit Function Theorem there exists an open set $W$ and a smooth curve $\gamma: W \to \mathrm{Conf}^{s}(X)$ such that $G(k,\gamma(k)) = k$. Moreover, these are the only solutions to $G(k,\tau) = 0$ in $W$. The same regularity arguments apply here as in the $k$-surface case and imply the existence of an $\delta > 0$ so that when $k \in (-\delta,\delta)$, each metric $\gamma(k)$ is smooth and the family $\gamma$ varies smoothly in $k$ in the $C^\infty$ topology. This implies that the mean curvature surfaces form an asymptotically Poincaré family of surfaces. There exists a $\delta > 0$ so that for $k \in (-\delta, 0)$ the family of surfaces $S_k$ where $S_k$ has constant mean curvature $-\sqrt{1+k}$ forms an asymptotically Poincaré family of surfaces. Since $\gamma(k)$ solves $G(k,\gamma(k)) = 0$ for $k \in (-\delta,\delta)$, the metric $\gamma(k)$ solves the scaled mean curvature equation. So, defining $\sigma_k = \frac{1 + \sqrt{1+k}}{1 - \sqrt{1+k}} \ \gamma(k)$ for $k \in (-\delta,0)$, we see that $\sigma_k$ solves the mean curvature equation (\[mean-curvature-equation\]), which implies that $\mathrm{Ep}_{\sigma_k}$ has constant mean curvature $-\sqrt{1+k}$. The same arguments as in Corollary \[k-surfaces-cor\] show that these metrics satisfy the hypothesis of Proposition \[asym-family-prop\]. So, the surfaces $S_k = \mathrm{Ep}_{\sigma_{k}}(X)$ form an asymptotically Poincaré family of surfaces. Consequently, we obtain Theorem \[cmc-intro\]. [[Sch]{}17]{} C. Gregory Anderson, Projective structures on [R]{}iemann surfaces and developing maps to $\mathbb{H}^3$ and $\mathbb{C}\mathrm{P}^n$, Ph.D. thesis, University of California, Berkeley, 1998. Thierry Aubin, Nonlinear analysis on manifolds. [M]{}onge-[A]{}mp[è]{}re equations, Grundlehren der Mathematischen Wissenschaften, vol. 252, Springer-Verlag, New York, 1982. 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--- abstract: 'A quantum system may be purified, i.e., projected into a pure state, faster if one applies feedback operations during the measurement process. However existing results suggest that such an enhancement is only possible when the measurement efficiency exceeds 0.5, which is difficult to achieve experimentally. We address the task of finding the global optimal feedback control for purifying a single qubit in the presence of measurement inefficiency. We use the Bloch vector length, a more physical and practical quantity than purity, to assess the quality of the state, and employ a backward iteration algorithm to find the globally optimal strategy. Our results show that a speedup is available for quantum efficiencies well below 0.5, which opens the possibility of experimental implementation in existing systems.' author: - 'Yuxiao Jiang$^{1,3} $' - 'Xiyue Wang$^{1,2}$' - 'Leigh Martin$^{1,2}$' - 'K. Birgitta Whaley$^{1,3}$' title: Optimality of feedback control for qubit purification under inefficient measurement --- [^1] [^2] Introduction {#sec:introduction} ============ To implement quantum information processing, it is essential to have full knowledge of the quantum system at hand. When the state of a system is perfectly known, it is referred to as a pure state. However, under the influence of environmental noise or imperfect operations on the quantum state, a pure state will experience relaxation and decoherence, which in general can result in transformation to an unknown, or mixed state. While quantum error correction can be used to protect a computation or other specific operations on the qubit from decoherence, pure state initialization, i.e., generation of pure states from arbitrary mixed states is a far more basic task that underpins this, (e.g., in the production of clean ancillas), as well as countless other elementary tasks. A simple way to implement state purification is to perform a measurement. Although the elementary model of a measurement is a sudden interaction with a measurement device that gives rise to instantaneous wave function collapse, realistic measurements always take a finite amount of time and do not generally provide complete collapse. In the limit of a continuous measurement, information about the system state flows continuously from system to observer at a finite rate. Jacobs showed in 2003 that under continuous monitoring, when the measurement process is slow relative to the timescale of available unitary operations, feedback can increase the purification rate of qubits over what would be possible using measurement alone [@Jacobs2003]. The task of finding the optimal control protocol for qubit state purification under continuous measurements has since been studied in a number of works [@Jacobs2003; @Wiseman2006reconsidering; @Wiseman2008; @Combes:2011wt]. In discussions of such optimizations, it is important to understand the differences between locally and globally optimal control strategies. A locally optimal strategy refers to a control strategy that optimizes the figure of merit at an instant in time. In contrast, a globally optimal strategy provides the control protocol that optimizes the figure of merit over the finite time period between the allotted start and finish of the control task. The resulting instantaneous control protocol at any given time within this period will generally not be equal to the corresponding locally optimal control protocol. In Ref. [@Wiseman2006reconsidering], Wiseman and Ralph emphasized the distinction between two goals in qubit purification: i) the max purity goal, i.e., maximizing the purity at final time, and ii) the min time goal, i.e., minimizing the time required to achieve a given purity. Subsequent work established the globally optimal strategies for these two goals in the limit of perfect measurement efficiency, $\eta =1$ [@Wiseman2008]. Li et al. first discussed the global (over a finite time period) and local (instantaneous) optimality of qubit purification in the presence of finite measurement inefficiency, $\eta < 1$ [@Li2013]. Using the verification theorem, they proved that the optimal strategy for the max purity goal with efficiency $\eta$ up to $1/2$ is to measure the state without applying feedback. This result is significant because routinely achieved quantum efficiencies are around $0.4$, and efficiencies well above $0.5$ would be required to demonstrate a significant purification enhancement. Many other imperfections like finite coherence time can be ameliorated by increasing the measurement rate, but efficiency is limited by challenging hardware constraints. The choice of figure of merit for such a control problem strongly affects the optimal protocol, due to the stochastic nature of the system’s evolution. Up to now, most work has used the state purity metric, defined as $\text{Tr[}\rho^{2}\text{]}$, to measure the quality of the state during the purification process. In practice however, as recognized in Ref. [@Wiseman2008], the length of the Bloch vector $r$ is arguably more relevant to many applications, primarily because it is linearly related to the fidelity between the instantaneous state and a target pure state, whereas the purity is nonlinear in this fidelity and not directly observable. However an analytic solution for the globally optimal protocol under the Bloch vector length metric does not appear to be available except for the case of unit efficiency [@Wiseman2008], which has impeded further investigation to the realistic situation of finite measurement inefficiency. In this paper, we use Bloch vector length instead of purity as a metric to quantify state purification, and determine both the local and global optimality of control strategies for state purification in the presence of inefficient measurements. We develop a general method to numerically search for the globally optimal strategy and apply it to find a lookup table for the optimal control protocol. Our main result is that unlike the purity metric, the optimization according to the Bloch vector metric $r$ benefits from feedback control for any measurement efficiency $\eta\in [0,1]$. To simplify the search process, we further make use of the stochastic version of Pontryagin’s maximum principle (Appendix A) to restrict the control landscapes. The robustness and error estimates for our methods are discussed in Appendix B. In the main text we take $\eta=0.3$ as an example to demonstrate the performance of the optimal protocol. Results for other values of $\eta$, both below and above 1/2, are provided in Appendix C. Continuous Measurements and Feedback on a Qubit {#sec:Setup} =============================================== We consider a qubit system subjected to weak, continuous measurement of operator X with measurement efficiency $\eta$ and strength $k$. The evolution of the system is described by stochastic master equation (SME): $$\label{eq:SMEgeneral} \begin{aligned} d\rho &= 2k\mathcal{D}[X]\rho dt + \sqrt{2\eta k}\mathcal{H}[X]\rho dW. \end{aligned}$$ where $\mathcal{D}[A]\rho=A\rho A^{\dagger}-\frac{1}{2}(A^{\dagger}A\rho+\rho A^{\dagger}A)$ and $\mathcal{H}[X]=A\rho+\rho A^{\dagger}-\text{Tr}[(A+A^{\dagger})\rho]\rho$. $dW$ is a zero-mean, Gaussian random variable with variance $dt$. We assume the system evolves without the influence of relaxation and decoherence. The information that flows from system to observer is denoted by $dR=\sqrt{4k}\langle X\rangle dt+dW/\sqrt{\eta}$. The assumption that unitary operations may be implemented quickly relative to the measurement timescale implies that any feedback control can effectively be implemented instantaneously and with no delay. This means after each measurement time step, feedback applies a time-dependent unitary operation $U_{t}$ on the state such that $\rho(t+dt)=U_{t}\rho(t) U_{t}^{\dagger}$. We now apply Eq. to a qubit using the Bloch sphere representation, where any single-qubit density operator can be identified with a Bloch vector $\bm{r}=(x,y,z)$ in $\mathbb{R}^{3}$ using the relation $\rho=(\mathbb{I}+\bm{r}\cdot\bm{\sigma})/2$. Measurement perturbs the qubit state so that in the Bloch sphere representation, the dynamical evolution will appear as a random trajectory in $\mathbb{R}^{3}$. Without loss of generality, we consider here the purification effect of a weak measurement $X=\sigma_{z}/2$, starting from an initial state with zero $y$ component. We take further advantage of the symmetry of the Bloch sphere by assuming the unitary feedback operators to be rotations about the $y$ axis, so that the trajectory can be restricted to the $xz$ plane at all times. Under this simplification, each feedback protocol can be described with a single parameter $\theta(t)$, where at each time step the feedback control operator puts the state vector at some angle $\theta$ from the $+z$ axis. We can then parameterize each protocol by its function $u(r,t)$, where $$\label{eq:u_def} u(r,t) = \text{cos}\theta(t) = \frac{z(t)}{r(t)},$$ with $r$ the length of Bloch vector, which is our figure of merit. The actual unitary control operation is then of the form $U_{t}=\text{exp}\{-i\alpha(t)\sigma_{y}\}$, where $\alpha(t)$ is determined by a combination of the instantaneous state and $\theta$. Note that the control value $u(r,t) = 1$ corresponds to no-feedback, and all other values including $u(r,t)=0$ correspond to active feedback. Substituting the measurement operator $X=\frac{\sigma_{z}}{2}$ into Eq. , we can isolate the $x$ and $z$ components of $d\rho$. Using $r=\sqrt{x^{2}+z^{2}}$ and Itô’s rule, we arrive at an equation of motion for $r$ [@Li2013] $$\label{eq:SMEr} \begin{aligned} dr = k(r-\frac{\eta}{r})(u^{2}-1) dt + \sqrt{2k\eta}(1-r^{2})u dW, \end{aligned}$$ where we have suppressed the arguments of the variables $r(t)$ and $u(r,t)$. For notational convenience we shall continue to use the abbreviation whenever display of the arguments is not essential to the discussion. Under this setup, our optimal purification problem can be stated quite generally as minimizing some cost function $\mathcal{C}[u]$ given an initial state $r_0$, where $\mathcal{C}$ is a functional of the control function $u(r,t)$, $\mathbf{E}$ denotes the expectation value, and $T$ is the allowed duration of the purification protocol. For the ideal situation with perfect measurement efficiency, $\eta = 1$, measurement in a fixed (diagonal) basis without feedback has been identified as the globally optimal protocol for the “min time” goal [@Wiseman2006reconsidering]. Feedback that keeps the qubit unbiased with respect to the measurement basis i.e., “always on" feedback, $u(r,t)=0$ for all $t$) is the globally optimal protocol for both the “max purity” and “max Bloch vector length” goals when $\eta = 1$ [@Wiseman2008; @Combes:2011wt]. Less is known about the situation with inefficient measurements, $\eta < 1$. The no-feedback protocol (i.e., $u=1$) for the “min time” goal has been explicitly shown to be globally optimal for all values of $\eta$, with numerical evidence also showing that this global optimality is maintained in the presence of decoherence [@Li2013]. For the “max purity” goal with $\eta < 0.5$, measurement without feedback is known to be globally optimal (following an initial rotation to the measurement axis) [@Li2013]. However for $1/2 < \eta < 1$, the globally optimal protocol for the “max purity” goal is unknown. Furthermore, the optimal strategy for any non-unit value of $\eta$ is unknown for “max Bloch vector length,” arguably one of the most physically relevant goals. As mentioned earlier, one of the motivations of this work is to overcome the impracticality of using state purity as a metric for purification, by using instead the Bloch vector length $r$ to measure the quality of the state. The cost function is then $\mathcal{C}[u]=\textbf{E[}1-r(T)\textbf{]}$, which is both measurable and linearly related to the state fidelity when the orientation of the Bloch vector is known. One may start by solving for the locally optimal protocol, i.e., the protocol that yields the greatest expected Bloch vector length increase $dr$ during a time interval $dt$. This can be done analytically; it amounts to maximizing the $dt$ term in Eq. , since the $dW$ contribution averages to zero. We thereby arrive at the locally optimal protocol $$\label{eq:localOpt} u_{lo,r}(r,t) = \begin{cases} 0 & \text{if $r\leq r^{}$} \\ 1 & \text{if $r> r^{}$}, \end{cases}$$ where $r^{} = \sqrt{\eta}$ (see also Ref. [@Combes:2011wt]). The same form was derived from a previous purification analysis of Eq. with the purity metric when the measurement inefficiency is greater than 0.5, i.e., $1/2<\eta\leq 1$, where $r^{}_{p}=\sqrt{2-1/\eta}$ [@Li2013]. However, with the purity metric, the locally (and global) optimal strategy was found to be $u_{lo,p}(r,t) = \pm 1$ when $\eta\leq 1/2$. In contrast to the local optimality problem, the global optimality problem appears to be analytically intractable. The difficulty arises in finding a general solution of Eq. for $r(T)$, given an arbitrary $u(r,t)$ and arbitrary initial conditions. Furthermore, the size of the search space is formidable, since $u(r,t)$ could be any function $f:[0,T]\times[0,1]\rightarrow[-1,1]$. However, the verification theorem [@Shabani2008] provides a way to numerically generate the globally optimal protocol while only exploring a small fraction of the full parameter space. In addition, Pontryagin’s maximum principle [@yong1999stochastic] provides a useful necessary condition for the optimal control, namely that the control Hamiltonian must take an extreme value over all permissible controls. As shown in Appendix A, the stochastic form of the maximum principle can be used to show that for the measurement-based feedback approach to purification under the Bloch vector length cost function, the optimal control is a discrete function $f:[0,T]\times[0,1]\rightarrow\{-1,0,1\}$, which significantly reduces the size of the search space. A further simplification comes from the observation that the control values $u=\pm1$ are physically equivalent. Therefore in the rest of this paper we need only consider the situations with $u=0$ and $u=1$ control values. Globally optimal control ======================== In this section we describe our numerical approach to finding the globally optimal strategy which maximizes $r$ at a final time $T$ in the presence of finite measurement inefficiency. For unit efficiency ($\eta=1$), the locally optimal strategy can be analytically solved (see Eq. above) and has been shown to be globally optimal by making use of the verification theorem [@Wiseman2008]. However, an analytic verification for inefficient measurements ($0< \eta <1$) is intractable, because in this regime neither $u=0$ nor $u=1$ is optimal, and integration of the equation of motion Eq. is difficult to solve for non-constant $u$. Nevertheless, the principle of the verification theorem is sufficiently powerful to warrant development of numerical approaches. Here we present a general numerical method to search for the globally optimal strategy. In the following we first provide a general description of our approach and then apply it to the purification problem with arbitrary measurement efficiency. We use a backwards propagation algorithm to iteratively minimize the cost function $C[u]$ to obtain the globally optimal protocol and the associated cost $C_{g}$. We discretize the continuous variable $t$ into $M$ time steps, which are labelled by the upper index $j$ ($t^{j}=j\Delta t + t^0$, $\Delta t = T/M$). Suppose that $u(r,t)$ is piecewise constant over each time step $\Delta t$, so that we may use the notation $u^{j}=u(r,t^j)$. Under this constraint, the globally optimized cost function $C_g$ over a set of functions $\{u^{j}(r)\}_{j=1,..,M}$ is given by $$\begin{aligned} \label{eq:GlobalCost} C_{g}&=&\min_{u^{1},...,u^{M}}\int C(r^{M})P(r^{M}\|r_{0},u^{1},...u^{M})d{\bf r}^{M}\nonumber\\ &=&\min_{u^{1},...,u^{M}}\iint C(r^{M})P(r^{M}\|r^{M-1},u^{M})\times \nonumber\\ &&\qquad\qquad\qquad P(r^{M-1}\|r_{0},u^{1},...u^{M-1})d{\bf r}^{M-1}dr^{M}\nonumber\\ &=&\min_{u^{1},...,u^{M-1}}\int \tilde{C}^{M-1}(r^{M-1}) \times \\ \nonumber &&\qquad\qquad\qquad P(r^{M-1}\|r_{0},u^1,...,u^{M-1})d{\bf r}^{M-1},\end{aligned}$$ where we have defined $d\mathbf{r}^n \equiv \prod_{i=1}^n dr^i$. We have also defined the cost-to-go recursively as $$\label{eq:CostToGo} \tilde{C}^{j-1}(r^{j-1}) \equiv\min_{u^{j}}\int \tilde{C}^{j}(r^{j})P(r^{j}\|r^{j-1},u^j)dr^{j}. $$ and take the base case as $\tilde{C}^{M}(r^M) \equiv C(r^M) = 1-r^M$. Here $C(r^{j})$ is the value of the cost function at time step $j$ and $P(r^{j}\|r_{0},u^1,...,u^{j})$ is the probability distribution of $r$ at time step $t^j$, conditioned on some initial value $r_0$ and on the controls. We use the notation $d{\bf r}^k \equiv dr^1...dr^{k-1}dr^k$ to simplify the form of the equations. The form of Eq. clearly illustrates the nature of the globally optimal control strategy, as a sequence of instantaneous control protocols $\{ u^{j} \}$ that applied sequentially in time will generate the optimal value of the cost function, $C_g$. The $u^{j}$ control fields are determined as a whole and are not in general equal to the locally optimal control fields at the corresponding times $t^j$. The numerical optimization of $C_g$ proceeds as follows. Starting with $j=M$, we first perform the minimization in the last line. For this special case, the minimization is actually equivalent to finding the locally optimal protocol $u^M$, since $\tilde{C}^M(r^M)$ is the cost-to-go evaluated over only a single time step. Since the cost function at the final time is known ($C(r^{M})=1-r^{M}$), using this together with the time evolving conditional probability distribution $P(r^{j}\|r_{0},u^1,...,u^j)$, we are then able to calculate the overall cost function in an iterative procedure. We emphasize that although we write $r$ here as a continuous variable, in order to implement the algorithm numerically we replace the integral of $r$ by a Riemann sum with interval $\Delta r$, resulting in a discrete representation of the configuration and control parameter spaces $r^j$ and $u^{j}$ respectively. Provided that $u$ is constant over each individual time interval $\Delta t$, the probability distribution function $P(r^{j}\|r^{j-1},u^{j})$ can be calculated analytically from Eq. using the Fokker-Planck equation. As we claimed before, due to Pontryagin’s maximum principle, we need only consider the cases $u=0$ or $1$. Within each time step we then optimize the cost function by evaluating the probability distribution of $r$ under the two different choices for $u$. When $u=0$, the evolution is deterministic and we find $$\label{r0} r^{j}=\sqrt{\eta-(\eta-(r^{j-1})^{2})e^{-2k\Delta t}}.$$ Since $r$ is discrete in the numerical implementation of our algorithm, we use a finite width $\delta$ function (implemented by a narrow Gaussian) to emulate the distribution $P(r^{j}\|r^{j-1})$ in presence of feedback control ($u=0$): $$\label{p1} P(r^j\|r^{j-1},0)=\frac{1}{\mathcal{N}}e^{-\frac{(r-r^{j})^{2}}{2\sigma^{2}}},$$ where $\mathcal{N}$ is the usual Gaussian normalization factor. Thus for $u=0$, we propagate a $\delta$ function centered at $r^{j}$ with finite but small width $\sigma=\Delta r$. The $u=1$ case is equivalent to measuring along the direction of the quantum state (which we assume is aligned along the $+z$ axis), without any feedback controls. The differential equation for the projection of the quantum state along the $z$ axis, $z(t)$, is solved in this case by [@Jacobs2004optimal] $$\label{zu=1} z(t)= \frac{\sinh(\sqrt{2k\eta}W)+z_{0}\cosh(\sqrt{2k\eta}W)}{\cosh(\sqrt{2k\eta}W)+z_{0}\sinh(\sqrt{2k\eta}W)},$$ where W is a random variable with distribution function $$\begin{aligned} \label{p2} P&(W\|z_{0}) \\ \nonumber &=\frac{1}{\sqrt{2\pi t}}e^{-\frac{W^{2}}{2t}-\eta kt}[\cosh(\sqrt{2k\eta}W)+\sinh(\sqrt{2k\eta}W)z_{0}].\end{aligned}$$ Changing variables from $W$ to $z$ in Eq. , we then obtain the probability distribution function of $z$, which we denote by $F(z,t\|z_{0})$. Since the state remains aligned along the $z$ axis, i.e., $r=\|z\|$, we arrive at the following probability distribution function for $r$: $$\begin{aligned} \label{eq:ru=1} P(&r^j\|r^{j-1},1) \\ \nonumber &= \begin{cases} F(r^j,\Delta t\|r^{j-1}) & \text{if $r^j=0$} \\ F(r^j,\Delta t\|r^{j-1})+F(-r^j,\Delta t\|r^{j-1}) & \text{if $r^j>0$}, \end{cases}\end{aligned}$$ where the $r^j=0$ case appears in order to avoid double-counting the probability of being at the origin, and the $F(-r^j,\Delta t\|r^{j-1})$ term in the second line derives from the contributions with $z<0$. We summarize the searching process for $u^j$ as follows: - Calculate $P(r^{j}\|r^{j-1},u^{j})$ from Eqs. - for all $r^{j-1}$ between 0 and 1, and both possible control values $u^{j}=0,1$. - Compute the cost-to-go for each control value $u^j$ at this time step as $\tilde{C}_{u^j}^{j-1}(r^{j-1})= \displaystyle{\sum_{r^{j}}} \tilde{C}^{j}(r^{j})P(r^{j}\|r^{j-1}, u^j)$. - Take $\tilde{C}^{j-1}(r^{j-1})=\min\limits_{u}\tilde{C}^{j-1}_{u^j}(r^{j-1})$ i.e., Eq. and begin the next round. This procedure allows us to propagate the searching process backwards in time, with the final outcome being an explicit numerical solution for the optimal control strategy $u(r,t)$, represented as a discrete, two-parameter lookup table. By construction, this lookup table will provide the optimal control strategy for all possible values of $r_0 = r(t_0)$. Results ======= ![Optimal control lookup table for measurement efficiency $\eta=0.3$. The control variable $u(r,t)$ takes the value $0$ (apply feedback) in the black region and the value $1$ (no-feedback applied) in the white region. The feedback control is turned on for large $t$ and small $r$. For any allowed total purification time $t=T<1.5$, the time range \[1.5 - T,1.5\] provides a complete description of the control protocol. During the purification process, the state will follow a stochastic trajectory with the optimal control operation fully determined by the values of $(r,t)$.[]{data-label="controleta3"}](globaleta03.jpg){width="50.00000%"} We choose an intermediate value of efficiency, $\eta=0.3$, as an example to illustrate and analyze our results. This is somewhat below the realistic values of measurement efficiency for superconducting qubit technologies today [@hacohen2016noncommuting]. Results for other efficiencies may be found in Appendix C. In Fig. \[controleta3\], we plot the optimal value of $u$ as a function of $r$ and $t$, for a total purification time $T=1.5$. This fully parametrizes our control strategy over the time interval $[0, 1.5]$ and provides a lookup table that can be used to generate the optimal control protocol for purification starting from any Bloch vector value $r(t_0)$ and evolving for a time $T = 1.5 - t_0 \leq 1.5$. The control values $u(r,t)$ are simply read off the table as time proceeds from $t=t_0$ to $t=t_0+T = 1.5$. For longer purification durations, the backwards recursion procedure would be made starting at a larger $T$ value. Fig. \[controleta3\] suggests that for all $T > 1.5$, the optimal strategy will be to omit feedback at early times, meaning that the lookup table takes a value of $u=1$ for all times earlier than $t=T-1.5$. Verifying this is however challenging, due to the numerical evaluation of the cost-to-go function, Eq. . We find a smooth boundary between regions in which one should apply feedback to align the state along the $x$ axis ($u=0$, black region), and regions where one merely measures parallel to the state ($u=1$, white region). The parameter region where feedback is required is centered around $r\rightarrow 0$, $t\rightarrow T$. This implies that to obtain the best possible value of the Bloch vector length $r$ at time $T$, feedback should be turned on only when we still have a small value of $r$, even if the system is already close to the final time. This derives from the fact that as $t$ approaches $T$, the globally optimal strategy in region $[t,T]$ approaches a locally optimal strategy. At $t\approx T$, the boundary between “feedback” and “non-feedback” regions is located exactly at $r = r^{} = \sqrt{\mu}$ in the equation for the locally optimal strategy, Eq. and global and local optimality in $[t,T]$ thus become asymptotically equivalent to each other as this special point is approached. This situation differs from the optimal protocol when using purity as the figure of merit, where for $\eta<0.5$ the globally optimal strategy is $u=1$, i.e., measuring the state without any feedback [@Li2013]. In contrast, Fig. \[controleta3\] indicates that feedback does indeed benefit the purification rates under low efficiency measurements when we instead use the Bloch vector length $r$ to measure the quality of the state. Furthermore, when the available purification time $T_a$ is restricted to be significantly less than the specified time for the globally optimal protocol (here $T=1.5$), e.g., for $T_a \leq 0.3$, feedback will be an essential component of the optimal procedure when starting from the mixed state $r(t_0)=0$. However, the final value of $r(T_a)$ will thereby be reduced. When a longer duration can be used, the optimal protocol will only require feedback if and when the Bloch vector length $r(t)$ crosses the boundary to the $u=0$ (black) region at small $r$ values as the final time is approached. In order to illustrate the benefits of the new control protocol, we simulate trajectories of $r$ using Monte Carlo method and evaluate $\langle r(t)\rangle$, where $\langle \cdot\rangle$ is the average over many trajectories. We compare the results achieved under three different strategies: $u=0$ (feedback always on), $u=1$ (measurement without feedback) and the globally optimal strategy. (Recall that the $u=0$ protocol is known to be globally optimal for the special case of perfect measurement efficiency, $\eta=1$ [@Wiseman2008]). The advantage of the globally optimal control strategy is clearly evident in Fig. \[r&t\], where we plot the results for a total purification time $T=1.5$. We see that until $t\approx 1.2$, the average value $\langle r(t)\rangle$ for the globally optimal strategy (red dotted line) is the same as the no feedback strategy (blue solid line). However, when the system reaches the critical time point $t=t^{}\approx 1.2$ in the control lookup table of Fig. \[controleta3\], the optimal control strategy will switch on feedback for those trajectories having small $r(t)$ values. This causes the globally optimal outcomes to increase faster than those for the no feedback $u=1$ strategy, resulting in a larger value of the Bloch vector length for the globally optimal strategy at the final time, $r(T)$. It is important to note that while Fig. \[controleta3\] specifies the globally optimal strategy for any final time $T < 1.5$, the curves in Fig. \[r&t\] are specific to the final time value $T=1.5$. For smaller values of $T$, feedback would be activated at an earlier time and the difference between protocols would be larger. \[tab:cg&cmc\] --------------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- $\eta$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 $C_{\text{MC}}$ 0.5763 0.4290 0.3310 0.2601 0.2048 0.1611 0.1263 0.0967 0.0681 0.0255 $\delta C_{MC}(\%)$ 0.56 0.83 1.05 1.23 1.38 1.49 1.53 1.46 0.82 0.84 $\Delta(\%)$ 0.19 0.29 0.20 0.21 0.05 0.07 0.07 0.03 0.01 0.03 --------------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- We can validate our globally optimal control protocol by undertaking Monte Carlo trajectory simulations of the global cost function under the numerically obtained protocol $u(r,t)$. We denote the cost function value generated by Monte Carlo as $C_{\text{MC}}=1-\langle r(T)\rangle$, where $ \langle \cdot \rangle$ denotes the average over Monte Carlo trajectories. When a large number of trajectories are used to compute $C_{\text{MC}}$, the value should approach the value $C_{g}$ obtained from the backwards propagation algorithm. We confirm this relation in Table I, which shows a comparison between the values of $C_{\text{MC}}$ and $C_{g}$ for $\eta$ values ranging from $0.1$ to $1$. We simulate ten thousand trajectories under the optimal control strategy to obtain $C_{MC}$ and the associated uncertainty $\delta C_{MC}$, where the latter is estimated by the standard error of the mean. Table I shows that the normalized difference $\Delta=\|C_{g}-C_{\text{MC}}\|/C_{\text{MC}}$ is consistently smaller than the uncertainty $\delta C_{MC}$, verifying the reliability of our results. To further demonstrate the benefits of the globally optimal protocol, we use the Monte Carlo trajectory simulation to calculate the final average Bloch vector value $\langle r(T)\rangle$ under different values of measurement efficiency for the three different control protocols, $u=0$, $u=1$, and the globally optimal protocol. A comparison of the three resultsover the range from $\eta =0.1$ to $\eta =1$ for purification time $T=1.5$ is shown in Fig. \[r&eta\]. We see that for finite but not perfect measurement efficiency, the globally optimal protocol always improves over both $u=0$ and $u=1$ control protocols. Furthermore, it approaches the no-feedback $u=1$ protocol as $\eta \rightarrow 0$ and approaches the always on $u=0$ feedback protocol as $\eta \rightarrow 1$. Thus at low $\eta$ values, the performance of the no-feedback protocol ($u=1$) is close to globally optimal, while the always on protocol ($u=0$) yields significantly lower values of $r(T)$. Indeed, from Eq. , it is straightforward to show that in the long time limit $r\rightarrow \sqrt{\eta}$, independent of the initial value of $r$ (see also Ref. [@Combes:2011wt]). In contrast, at high $\eta$ values, the no-feedback $u=1$ protocol becomes increasingly ineffective, while the always on $u=0$ protocol becomes increasingly effective, resulting in a crossover of the corresponding $\langle r(T) \rangle$ values at $\eta \simeq 0.8$. As $\eta$ further approaches unity, the globally optimal and always on ($u=0$) feedback protocols become increasingly close in value and converge at $\eta = 1$, as expected from the known equality of local and globally optimal protocols in the case of perfectly efficient measurements [@Wiseman2008] (see also Eq. and Fig. \[global04to1\] in Appendix C). In principle, it would now be possible to construct a plot of final Bloch vector length $\langle r(T)\rangle$ as a function of purification duration $T$, which could allow evaluation of the speedup of the globally optimal protocol relative to both the no feedback $u=1$ protocol and the always-on feedback $u=0$ protocol for finite efficiencies $\eta$. However, since the final time $T$ is finite, asymptotic speedup factors cannot be extracted from such numerical solutions. Nevertheless, Fig. \[r&eta\] indicates that for $\eta < 1$ the globally optimal strategy will show a speedup over the no-feedback $u=1$ strategy, since a larger value of $\langle r(T)\rangle$ is reached for a given time $T$. The figure similarly indicates that there will be a speedup relative to the always-on $u=0$ strategy. ![Average value of the Bloch vector length $\langle r \rangle$ vs. time for the globally optimal strategy (red dotted line), for the $u=1$ protocol (no-feedback - solid blue line) and for the $u=0$ protocol (feedback always on - dashed green line) for purification with measurement efficiency $\eta=0.3$ and measurement strength $k=1$, over a total purification time $T=1.5$. The initial condition here is the totally mixed state, $r_{0}=0$. Evolution under first two strategies is stochastic, according to Eq. , and we take the average over 10,000 trajectories. Evolution under the $u=0$ protocol is deterministic and is given by Eq. .[]{data-label="r&t"}](comparer.jpg){width="50.00000%"} ![Average value of the Bloch vector length $\langle r(T) \rangle$ at the final time $T=1.5$ for the globally optimal strategy (red triangles), the $u(r,t)=0$ protocol (always on feedback, green circles) and the $u(r,t) =1$ protocol (no feedback, blue triangles), with measurement efficiencies $\eta$ varying from $0.1$ to $1$. All calculations were carried out with measurement strength $k=1$, and initial condition equal to the totally mixed state, $r_{0}=0$. Evolution under first two strategies is stochastic, according to Eq. and we take the average over 10,000 trajectories. Evolution under $u(r,t)=0$ protocol is deterministic and is given by Eq. .[]{data-label="r&eta"}](compare_1.jpg){width="50.00000%"} Conclusion {#sec:Conclusion} ========== This work shows that under all finite measurement efficiency $\eta$, including routinely achieved experimental values, quantum feedback accelerates qubit purification relative to purification via measurement alone. Our results identify the globally optimal feedback strategy for the full range of quantum measurement efficiencies, and show that an observable purification speedup should be achievable with existing technology employing less than 50% measurement efficiencies. The globally optimal strategy approaches the no-feedback strategy as $\eta \rightarrow 0$ and the always on strategy as $\eta \rightarrow 1$, but performs better than both of these for all $0 < \eta < 1$. Even for finite purification duration times $T$, the globally optimal strategy is seen to show speedup relative to both of the constant control protocols in achieving purification to a given Bloch vector length $\langle r(T) \rangle$ close to unity. Since no measurement is truly instantaneous, these results apply to a wide range of physical systems, although the greatest gains would come in systems with slower intrinsic measurement rates and demonstrated higher quantum efficiencies, such as superconducting qubits. The present study indicates several clear goals for further research. Firstly, for a given physical architecture, a complete analysis of all imperfections should be undertaken to ensure that the demonstrated advantage of feedback outweighs the cost of implementing this. In superconducting circuits, the primary imperfections after the non-unit measurement efficiency are likely to be presence of feedback delay and also unwanted interactions between the feedback operations and the readout protocol, such as non-Markovian effects in circuit QED readout[@gambetta2008quantum]. Such non-Markovian effects are a rich and active topic of fundamental research, and our current work motivates a significant and novel application of this work to these settings. Secondly, the methods implemented here are applicable to many related problems in quantum control. Even larger gains are available for purifying systems larger than a single qubit [@Combes2008rapid; @combes2011maximum], although to our knowledge no work has addressed the effect of measurement inefficiency in the multi-qubit context. Measurement-based feedback for entanglement generation has also proven a fruitful avenue of research[@stockton2004dicke; @HPFPRA; @martin2017optimal; @zhang2018locally], and open problems regarding optimality remain even in the case of two qubit Bell state generation[@martin2017optimal]. Systems substantially more complex than a single qubit may require the use of approximations or restrictions on the feedback protocol, which can impede proof of global optimality. However these challenges should not discourage the search for better feedback protocols, particularly in cases where there is a large margin to gain over the corresponding no-feedback protocols. The fact that the benefits of measurement-based feedback persist in realistic scenarios with inefficient measurement bodes well for future applications. LM was supported by the National Science Foundation Graduate Fellowship Grant No. (1106400) and the Berkeley Fellowship for Graduate Study. KBW was supported by Laboratory Directed Research and Development (LDRD) funding from Lawrence Berkeley National Laboratory, provided by the U.S. Department of Energy, Office of Science under Contract No. DE-AC02-05CH11231. [16]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [**, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop []{}, Vol. (, ) in @noop []{}, Vol. (, ) pp. @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [*, ()]{} @noop [ ()]{} Appendix A: Stochastic extension of Pontryagin’s Maximum Principle {#appendix-a-stochastic-extension-of-pontryagins-maximum-principle .unnumbered} ================================================================== In this section, we show that the optimal control function $u(r,t)$ must take the values $0$ or $\pm 1$. The original version of Pontryagin’s Maximum Principle is for deterministic problems and thus can not be directly applied to our problem of purification by measurement-based feedback. Instead, we use the Stochastic Maximum Principle [@yong1999stochastic] to prove our claim. Consider the stochastic master equation for a variable $x(t)$, $$\label{eq:SME_Pontryagin} dx(t)=A(t,x(t),u(t))dt+B(t,x(t),u(t))dW,$$ with cost functional $$J(u)=\textbf{E}\left[ \int_{0}^{T}f(t,x(t),u(t))dt, +h(x(T)) \right],$$ where $dW$ is a stochastic increment, $u(t)$ the control function, and the cost functional depends on the time dependent function $f(t,x(t),u(t))$ and a function $h(x(T)$ dependent only on the final time $T$. The Stochastic Maximum Principle states that the optimal $u(t)$ must satisfy the condition $$\label{condition} \mathcal{H}(t,\overline{x}(t),\overline{u}(t))=\max_{u(t)}\mathcal{H}(t,\overline{x}(t),u(t))$$ where $\{\overline{x}(t),\overline{u}(t)\}$ defines the optimal protocol and the function $\mathcal{H}$ is defined below. Before giving the explicit form of $\mathcal{H}$, we first define two pairs of adjoint variables $\{P(t),Q(t),q(t),p(t)\}$. The variables $P(t), p(t)$ are solved for by the following set of terminal value stochastic differential equations [@yong1999stochastic] $$\label{solve1} \left\{ \begin{array}{lll} dp(t)&=&-\{A_{x}(t,\overline{x},\overline{u})^\intercal p(t) +\displaystyle{\sum_{j=1}^{m}}B^{j}_{x}(t,\overline{x},\overline{u})^\intercal q_{j}(t)\\ &&-f_{x}(t,\overline{x},\overline{u})\}dt+q(t)dW\\ \\ p(T)&=&-h_{x}(\overline{x}(T)), \end{array} \right.$$ $$\label{solve2} \left\{ \begin{array}{lll} dP(t)&=&-\{A_{x}(t,\overline{x},\overline{u})^\intercal P(t)+ P(t)A_{x}(t,\overline{x},\overline{u})\\ &&+\displaystyle{\sum_{j=1}^{m}}B^{j}_{x}(t,\overline{x},\overline{u})^\intercal P(t)B^{j}_{x}(t,\overline{x},\overline{u})\\ &&+\displaystyle{\sum_{j=1}^{m}}\{B^{j}_{x}(t,\overline{x},\overline{u})^\intercal Q_{j}(t)+Q_{j}(t)B^{j}_{x}(t,\overline{x},\overline{u}) \}\\ &&+H_{xx}(t,\overline{x},\overline{u},p(t),q(t))\}dt+\displaystyle{\sum_{j=1}^{m}}Q_{j}(t)dW^{j}\\ \\ P(T)&=&-h_{xx}(\overline{x}(T)), \end{array} \right.$$ where for notational convenience we have omitted the explicit time dependence of $\{\overline{x}(t),\overline{u}(t)\}$. We shall similarly drop the explicit time dependence in other functions below. The Hamiltonian $H$ in Eq. is given by $$H(t,x,u,p,q) \equiv \langle p,A(t,x,u)\rangle + \text{tr}[q^\intercal B(t,x,u)]-f(t,x,u).$$ Given the 6-tuple $x, u, P, Q, p, q$, we then define the function $\mathcal{H}$ as $$\begin{aligned} &\mathcal{H}(t,x,u)\\ \equiv &\frac{1}{2}\text{tr}[B(t,x,u)^\intercal P(t)B(t,x,u)]+\langle p(t),A(t,x,u)\rangle-f(t,x,u)\\ &+\text{tr}[q(t)^\intercal B(t,x,u)]-\text{tr}[B(t,x,u)^\intercal P(t)B(t,\overline{x},\overline{u})]. \end{aligned}$$ Since all we need here is to find the solutions to Eq. , it is unnecessary to get the complete solutions of these adjoint equations. We first note that any feedback controlled evolution must be “adapted" in the sense that the feedback terms cannot depend on the measurement noise $dW$. This implies that the terms $Q(t)$ and $q(t)$ in Eqs. - must be equal to zero. Then replacing $x$ by $r$, with $A, B$ given by the purification equation of motion for $r$ in Eq. with cost functions $f=0, h=1-r(T)$ corresponding to maximizing the length of the Bloch vector, we find $$\begin{aligned} \mathcal{H}(t,\overline{r},u)=2kP(t)\eta&(1-\overline{r}^{2})^{2}(\frac{u^{2}}{2}-u\overline{u})\\ &+p(t)k(\overline{r}-\frac{\eta}{\overline{r}}) (u^{2}-1) \end{aligned}$$ where $u\in[-1,1]$. Since $\mathcal{H}$ is a continuous function of $u$, the maximum value of $\mathcal{H}$ with respect to variations in $u$ can be reached at the boundary points $u = \pm 1$ and at any point where the derivative with respect to the control is zero, i.e., when $$\begin{aligned} \mathcal{H}_{u} & =2k\eta P(t)(1-\overline{r}^{2})^{2}(u-\overline{u})+2p(t)k(\overline{r}-\frac{\eta}{\overline{r}})u \\ \nonumber & = 0 \end{aligned}$$ From this it is evident that $\mathcal{H}_{u}$ is zero only when $\overline{u}=u=0$. Thus the optimal control values at each time $t$ can be drawn only from the discrete set $\overline{u}= \{0, \pm 1 \}$. Appendix B: Error Analysis {#sec:verification} ========================== In this section we analyze the magnitude of numerical errors and their effects on the final results. The numerical error of our protocol mainly comes from the discretization of $r$ and $t$. Since we already use analytical solutions of the underlying stochastic differential equation, error from discretization of $t$ should be negligible. However, the magnitude of $\Delta r$ affects the accuracy of our protocols due to the fact that we use a Riemann sum to approximate the integration of the cost function. The approximation leads to uncertainty in the cost function, which could affect the optimal control lookup tables. In this section, we show a method to obtain the uncertainty in the cost function and use it to calculate the error of our control protocol. Recall that the cost-to-go at time step $j$ is given by $$\tilde{C}^{j-1}(r^{j-1})= \displaystyle{\sum_{r^{j}}} \tilde{C}^{j}(r^{j})P(r^{j}\|r^{j-1})$$ where minimization over $u^j$ is taken to be implicit. In this formula, $P$ is exact and the error comes from cost-to-go $\tilde{C}$ and the approximation of the Riemann sum. We write the error in $\tilde{C}^{j-1}(r^{j-1})$ as $$\delta \tilde{C}^{2}(r^{j-1})=\displaystyle{\sum_{r^{j}}} \delta \tilde{C}^{2}(r^{j})P^{2}(r^{j})+\delta^{2}_{R}$$ where the first term describes the propagation of error from the previous time step and second term is the error caused by approximation of the Riemann sum at the current time step. It should be emphasized that at the first step $[T-dt,T]$, $\tilde{C}$ is exact, therefore the error all comes from $\delta_{R}$. We now explain the method for calculating $\delta_{R}$. When $u=1$, the error can be simply calculated using the standard formula for the error of a Riemann sum. When $u=0$, the distribution function $P$ will be a delta function and the standard formula fails. Instead, the error can be estimated via the fact that $\int C(r)\delta(r-r_{0})dr\approx C(r_{0})+\mathcal{O}(\sigma)$ where $\sigma$ is standard deviation of the approximate delta function used in practice. At each step we evaluate $\delta_{R}$ and thereby determine $\delta \tilde{C}_{u=0}(r,t)$ and $\delta \tilde{C}_{u=1}(r,t)$ completely. Recall that at each step we determine the optimal $u$ by evaluating $\Delta \tilde{C}(r,t)=\tilde{C}_{u=0}(r,t)-\tilde{C}_{u=1}(r,t)$. Thus the error of $r$ in the position of the boundary points between the $u=0$ and $u=1$ regions is given by $$\delta r= \pm \sqrt{\delta \tilde{C}_{u=0}^{2}+\delta \tilde{C}_{u=1}^{2}}\left(\frac{d\Delta \tilde{C}(r,t)}{dr}\right)^{-1}$$ The result is shown in Fig. \[er\], where we have plotted the ratio of $\delta r$ to the instantaneous radial increment $\Delta r$, i.e., v $\delta r / \Delta r$ as a function of time for the boundary points in Fig \[controleta3\]. Choosing $\delta r/\Delta r$ helps us measure the relative error of boundary points compared to the increment $\Delta r$. While the error of $r$ in most boundary points is less than the radial step increment $dr$, there are some exceptional points with error larger than $\Delta r$. One may worry that such points might lead to divergence and make the whole protocol unstable. To check this, we choose several different values of $\Delta r$. We find the position of boundary points are identical, confirming the accuracy of our optimal control protocols. The above error analysis is restricted to boundary regions. In other words, for all values of $\eta$, our method accurately provides a set of boundary points between feedback and non-feedback regions. However, due to numerical error, a complete error analysis of the control table within the feedback and non-feedback region is harder to predict, especially for high values of $\eta$. ![Relative error in the length of the Bloch vector, $\delta r/\Delta r$, in the boundary points between feedback and no-feedback regions of the control lookup table, shown as a function of time $t$, for controlled evolution under measurement efficiency $\eta=0.3$. The parameters are the same as those in Fig.](eboundary.jpg){width="50.00000%"} \[controleta3\]. \[er\] Appendix C: Optimal purification strategy {#sec:LookupTable} ========================================== We supplement the main text here with figures showing the globally optimal strategy lookup tables, i.e.*, $u(r,t)$, for different values of $\eta$. As is shown in Fig. \[global04to1\], for better measurement efficiency $\eta$, the feedback is activated earlier and covers a larger region. For perfect efficiency, $\eta=1$, the optimal control strategy found by our numerical optimization procedure is $u=0$ regardless of $r$ and $t$, which coincides with the analytical solution derived in previous work [@Wiseman2008]. ![Optimal control lookup tables for $u(r,t)$ at measurement efficiency values $\eta=0.4,0.6,0.8,1$. The black regions indicate feedback on, $u(r,t)=0$, and the white regions indicate [feedback off,]{} $u(r,t)=1$.[]{data-label="global04to1"}](globaleta04to1.jpg){width="50.00000%"} [^1]: These authors contributed equally to this work [^2]: These authors contributed equally to this work
--- abstract: \| As a first step toward circumventing the difficulty to obtain an attractive isospin-independent $NN$ spin-orbit force from Skyrme-type models involving only pions, we investigate an improved Skyrme Lagrangian that incorporates the scalar-isoscalar meson $\epsilon$ which can be viewed as the cause behind the enhancement of the $\pi \pi\ S$-wave. We find that at large distances, the main contribution to the spin-orbit potential comes from the scalar Lagrangian and it is found to be attractive. We briefly discuss how to pursue this work to finally obtain a medium-range attractive interaction. 0.5cm PACS numbers(s): 11.10 Lm, 12.39 Dc, 13.75 Cs. Report-no: MC/TH 96/17 hep-ph/9606270 address: \| Theoretical Physics Group, Department of Physics and Astronomy,\ University of Manchester, Manchester M13 9PL, England author: - 'Abdellatif Abada[^1]' title: ' Isoscalar $NN$ spin-orbit potential from a Skyrme model with scalar mesons[^2]' --- INTRODUCTION ============ Ten years before the advent of QCD, Skyrme proposed a model [@Sk] describing hadronic physics which involves only pion fields and where baryons emerge as topological solitons. This model is recognized as the simplest chiral realization of QCD at low energies and large $N_c$ [@Wi]. The corresponding Lagrangian contains, in addition to the well known nonlinear $\sigma$ model, an antisymmetric term of fourth order in powers of the derivatives of the pion field (the so-called Skyrme term). The latter has been added by Skyrme in order to avoid soliton collapse. Despite its relative success in describing some properties of baryons ([cf.]{} Ref. [@ZB] for a review), the Skyrme model presents several shortcomings such as its prediction of a very small value for the axial coupling constant [@ANW] and an almost zero nucleon mass when the Casimir effects are taken into account [@Mou]. But probably the most important crisis of the Skyrme model is that it does not allow for the formation of nuclei. Indeed, it predicts a repulsive $NN$ central potential [@JJP]. Because of these drawbacks, several extensions of the standard Skyrme model have been proposed in order to improve its prediction power. The most famous one consists in adding to the Skyrme Lagrangian a term of order six in powers of the derivatives of the pion field, proportional to the square of the baryon current as first proposed in [@Jac]. The extended Skyrme model thus obtained contains a second, fourth and sixth-order term, but still is an effective theory of pions. It has been shown recently [@Mou; @AM] that this extended model is much more realistic when describing low-energy hadronic physics than the standard Skyrme model. In contrast with the velocity-independent part of the $NN$ potential, the kinetic part and especially the $NN$ spin-orbit force has not been studied extensively within the framework of Skyrme-type models. However, all the authors who worked on this subject, e.g. [@autho; @WA], arrived at the disappointing result that the standard Skyrme model predicts a repulsive isospin-independent spin-orbit force while, according to the phenomenology [@PB], it should be attractive. A glimpse of hope has emerged from Ref. [@RS] where it was claimed that when considering the sixth-order term, an attractive isoscalar spin-orbit interaction could be obtained. However, we have shown recently [@Ab2] that this is not the case: the sixth-order term contributes with a positive sign as is the case for the Skyrme term. Indeed, the authors of Ref. [@RS] considered in their calculations only one piece of the force due to the sixth-order term and omitted the piece that stems from the baryon exchange current, which turns out to be not only repulsive, but the dominant one too [@Ab2]. Thus, despite its success in improving the predictions of 1-baryon data, even an extended Skyrme model including higher-order terms in powers of the derivatives of the pion field fails to predict an attractive $NN$ isoscalar spin-orbit interaction as it fails to predict the correct $NN$ central potential [@Ka]. Therefore, one has to consider more realistic low-energy models to describe the $NN$ interaction. The Skyrme term as well as the sixth-order term can both be regarded as an [infinite]{} mass limit, the so-called local approximation, of a model with $\rho$-meson [@Ike] and $\omega$-meson [@Jac] exchanges respectively. These findings have provided the ground for improving Skyrme-type models by constructing effective Lagrangians which include, in addition to the pions, the other low-lying mesons. The authors of Ref. [@KV] have investigated the $NN$ force by considering such an effective model incorporating the mesons $\pi, \rho, \omega, A_1$ and the scalar meson $\epsilon$. Unlike previous works, they have introduced the vector meson fields by gauging the linear $\sigma$-model instead of the nonlinear one. They have found that the different channels of the $NN$ interaction arising from the potential piece of the Lagrangian are very well described, in particular the central potential for which an attraction has been found in a quantitative agreement with the phenomenology [@PB]. These results lead us to think that, similarly to the central potential, it may be that the right way to circumvent the difficulty of finding an attractive isoscalar spin-orbit force with Skyrme-type models is to consider effective theories which incorporate low-mass mesons with [finite]{} masses. In order to check these claims, one has to investigate the kinetic energy of a two-nucleon system within an effective Lagrangian such as the one in [@KV]. This is of course not an easy task and goes beyond the scope of this short paper. As a first step, we rather focus here only on the scalar meson contribution to the isoscalar spin-orbit force. Indeed, it has been pointed out in Ref. [@KV] that the presence of scalar degrees of freedom plays the crucial role of recovering the missed central attraction, so one may think that this will also be the case for the isoscalar spin-orbit force. This idea to account for a scalar field in the spin-orbit channel has been investigated in Ref. [@KE]. There, a dilaton field is coupled to Skyrmions in order to mimic the scale breaking of QCD. We will not discuss here if a dilaton field is suitable or not to provide a good description of low-energy hadron physics; we refer the interested reader to Ref. [@Mk] for a detailed discussion on this subject. The fact is that a combination of the sixth-order term and dilaton coupling yields an attractive isoscalar spin-orbit force, as it has been claimed in [@KE]. Note though that this result remains questionable since these authors, like those of Ref. [@RS], ignored the baryon exchange current contribution in their calculations [@Ab2]. In any case, here we will follow a more transparent way in order to incorporate scalar degrees of freedom [@KV]. It consists in considering the linear $\sigma$ model in which the presence of the scalar field is intrinsic. In Sec.II, the model with the scalar-isoscalar $\epsilon$-meson is presented. Sec.III contains the derivation of the isoscalar spin-orbit force. Our results and discussions are presented in Sec.IV. THE MODEL WITH SCALAR MESONS ============================ The infinite-mass limit of the effective Lagrangian with $\pi, \rho, \omega$ and $\epsilon$ mesons [@KV] reads: $$\begin{array} {ll} \label{La} {\cal L} &{\displaystyle ={\cal L}_{{\rm NL}\sigma} +{\cal L}_4 +{\cal L}_6 +{\cal L}_{\chi {\rm SB}} }\\ &{\displaystyle =\frac {f_{\pi}^2 }{4} \hbox {Tr} ~(\partial_{\mu} U \partial^{\mu} U^{+}) + \frac {1}{32 e^2} \hbox {Tr} \left[ (\partial_{\mu} U) U^{+} , (\partial_{\nu} U) U^{+} \right]^2 - e_6^2 B_{\mu} B^{\mu} +\frac {f_{\pi}^2 m_{\pi}^2}{2}\hbox {Tr}(U -1)} \end{array}$$ where $B^{\mu} = \epsilon ^{\mu \nu \alpha \beta} \hbox {Tr} \left(~ (\partial_{\nu} U) U^+ (\partial_{\alpha} U) U^+ (\partial_{\beta} U) U^+~\right) /24 \pi^2$ is the baryon current [@Sk], and $U$ an SU(2) matrix which characterizes the pion field. The first term in Eq. (\[La\]) corresponds to the nonlinear $\sigma$ model, $f_{\pi}$ being the pion decay constant. The second term, parameterized by the coupling constant $e$, is the so-called Skyrme term. The third term is of order six in the derivatives of the pion field and corresponds to $\omega$-meson exchange in the case of an infinite $\omega$-meson mass [@Jac]. The coupling constant $e_6^2$ is a parameter related to the $\omega \to \pi \gamma$ width. The last term in Eq. (\[La\]) which is proportional to the square of the pion mass $m_{\pi}$ (139 MeV) implements a small explicit breaking of chiral symmetry. As compared to the standard Skyrme model, the extended model (\[La\]), when used with realistic parameters, provides a more accurate description of the 1-baryon properties (e.g., the nucleon quantum mass, the $\Delta$-$N$ mass splitting, the breathing mode energy the axial-vector coupling constant $g_A$,...) [@Mou; @AM]. However, as mentioned in the introduction, it does not describe properly neither the $NN$ isospin-independent spin-orbit force [@Ab2] nor the central potential [@Ka]. The scalar-isoscalar meson $\epsilon$ is viewed as the responsible for the $S$-wave attraction in the $\pi \pi$ interaction and, therefore, should be considered as an essential ingredient in the low-energy hadronic phenomenology. It is incorporated in the Lagrangian (\[La\]) by replacing the nonlinear $\sigma$ model ${\cal L}_{{\rm NL}\sigma}$ with the linear one. Namely, $$\label{Ls} \displaystyle {\cal L}_{{\rm NL}\sigma} \to {\cal L}_{{\rm L}\sigma} = \frac{1}{2} \partial_{\mu}\xi \partial^{\mu}\xi + \frac {\xi^2 }{4} ~ \hbox {Tr} ~(\partial_{\mu} U \partial^{\mu} U^{+}) - \lambda(\xi^2-f_{\pi}^2)^2$$ where $\xi$ is the isoscalar content of the quaternion field $\sigma+i\mbox{\boldmath $\tau.\pi$}$, $\sigma$ being the scalar chiral partner of the pion field. The scalar field $\epsilon$ is defined as $\epsilon({\bf r})=f_{\pi}-\xi({\bf r})$. The coupling constant $\lambda$ is related to the $\epsilon$-meson mass through $m_{\epsilon}^2=8 \lambda f_{\pi}^2$. It is straightforward to check that by taking $m_{\epsilon}\to \infty$ in Eq. (\[Ls\]), one recovers the nonlinear $\sigma$ model ${\cal L}_{{\rm NL}\sigma}$. Indeed, in that limit, $\lambda$ becomes infinite and so the $\xi$-field has to be frozen to its asymptotic value $f_{\pi}$ in order to keep the potential energy in Eq. (\[Ls\]) finite. Therefore, for a realistic finite $\epsilon$-meson mass, the Lagrangian density (\[La\]) is then replaced by the following one: $$\label{Lanew} {\cal L}'={\cal L}_{{\rm L}\sigma} +{\cal L}_4 +{\cal L}_6 +{\cal L}_{\chi {\rm SB}}$$ The one-soliton system is commonly solved by assuming the hedgehog ansatz for the pion field: $$\label{Uh} U({\bf r}) \equiv U_H({\bf r}) = \displaystyle \exp\left(~i\mbox{\boldmath $\tau$}.\hat {\bf r} F(r) ~\right)$$ where the $\tau_a$’s are the Pauli matrices and the notation $\hat {\bf r}$ means ${\bf r}/r$. The static Euler-Lagrange equations for the chiral field $F$ and the $\xi$-field corresponding to the Lagrangian (\[Lanew\]) read $$\begin{array} {cc}\label{eqs} \displaystyle \left( \left[~r^2\xi^2 + \frac{2}{e^2}\sin^2(F)+\frac{e_6^2} {2\pi^4~r^2}\sin^4(F)~\right]~ F'\right)'= \left(\xi^2+\frac{1}{e^2r^2}\sin^2(F)~\right)\sin(2F) ~+\\ \displaystyle ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left(\frac{1}{e^2}+\frac{e_6^2} {2\pi^4~r^2}\sin^2(F)~\right) \sin(2F)~F^{'2}~+ f_{\pi}^2m_{\pi}^2 r^2 \sin(F) ~,\\ \displaystyle (r\xi)''=\left(F^{'2}+\frac{2}{r^2} \sin^2(F) + 4 \lambda(\xi^2-f_{\pi}^2)~\right)r\xi \end{array}$$ where primes indicate radial-coordinate differentiation. In order to ensure a winding number one the chiral angle $F$ obeys the usual boundary conditions while the $\xi$-field fulfills the conditions $\xi'(0)=0$ and $\xi(\infty)=f_{\pi}$. Let us observe that the above Euler-Lagrange equations are solved with a set of parameters whose values have been fixed by fitting to the mesonic sector [@Mou; @AM; @KV]. Namely, $f_{\pi}=93$ MeV, $e=7.2$, $e_6=1.66$ fm, $m_{\epsilon}=650$ MeV. These parameters yield, e.g., a value of the nucleon mass of $\sim 1$ GeV after subtracting the Casimir energy [@Mou], and a $\Delta$-$N$ mass splitting of $\sim$ 267 MeV. We have plotted in Fig.1 the chiral function $F$ and the $\xi$-field, solutions to the equations of motion (\[eqs\]) obtained with the set of parameters given above. We will use below these two functions as an input in the numerical computation of the spin-orbit force. For reference, we have also shown in Fig.1 the chiral solution $F$ in the case of the model (\[La\]), i.e. $\xi=f_{\pi}$, [@AM]. THE SPIN-ORBIT FORCE ==================== So far, all the calculations on the $NN$ spin-orbit force within Skyrme-type models have been carried out in the framework of the product ansatz as suggested by Skyrme [@Sk2] ([cf.]{} Ref. [@autho; @WA] and references therein). It is worthwhile noticing that the latter is only a simple approximation to the two-baryon configuration and not a self-consistent solution. We will not discuss here the degree of validity of this approximation. It is commonly chosen because of its relative simplicity as compared to other two-baryon field configurations which can be found in the literature [@NW]. Furthermore, it becomes exact for large $NN$ separation[^3]. Thus, following common practice, we use it here to describe a system of two interacting solitons. Furthermore, in order to obtain the appropriate spin and isospin structure, we also introduce rotational dynamics [@ANW]. Hence, the field configuration of the two-nucleon system separated by a vector ${\bf r}$ reads: $$\begin{array}{cc}\label{pro} U(A_1,A_2,{\bf x}, {\bf r}) = U_1 U_2 = A_1 U_H( {\bf r}_1)~A_1^+A_2~U_H( {\bf r}_2) A_2^+ ~~,\\ {\bf r}_1 = {\bf x}-{\bf r}/2 ~~,~~~ {\bf r}_2 = {\bf x}+{\bf r}/2~~, \end{array}$$ where $A_1$ and $A_2$ are SU(2) matrices, and $U_H$ the hedgehog single soliton (\[Uh\]). For the scalar field, we use the configuration suggested in Ref. [@KV] $$\label{xi12} \displaystyle \xi_{\rm P}({\bf x}, {\bf r})=\frac{1}{f_{\pi}} \xi({\bf r}_1)\xi({\bf r}_2)$$ which obviously is the form the most compatible with the product-ansatz approximation (\[pro\]). To carry out a simultaneous quantization of the relative motion of the two nucleons and the rotational motion, we need to treat ${\bf r}, A_1$ and $A_2$ as collective coordinates. Therefore, we make all these parameters (${\bf r}, A_1, A_2$) time-dependent. The spin-orbit potential will emerge from a coupling between the relative motion and the spins of the two nucleons so that we have to calculate the kinetic energy corresponding to (\[Ls\]). As reported in Ref. [@WA; @Ab2], generally, one has to treat with care the conversion from velocities to canonical momenta before identifying and extracting the spin-orbit potential. Indeed, one has to start from a Lagrangian formalism, take its “classical” kinetic energy and extract from it the mass matrix and then invert it properly in order to move to a Hamiltonian formalism. However, for a large relative distance $r$, the region of validity of the product ansatz, this procedure [@WA] is equivalent to that of Refs. [@autho; @RS; @Ab2] which we will use here. In this latter one starts, in the case (\[Ls\]), directly from $$\label{Ks} \displaystyle K_{{\rm L}\sigma}(A_1,A_2,{\bf r})= (-1) \int \hbox {d}^3 x ~ \left(\frac{1}{4}\xi_{\rm P}^2 \hbox{Tr} (\partial_0U\partial_0U^+) + \frac{1}{2} (\partial_0\xi_{\rm P})^2 \right)~$$ where the minus sign in front of the integral is put in explicitly so that account is taken of the change in sign of the off-diagonal terms of the $2\times 2$ mass matrix under inversion [@WA]. Afterwards, one makes the usual identifications [@ANW; @Nym]: $$\label{subst} \displaystyle \dot {\bf r}_n \to \frac{{\bf p}^{(n)}}{M} ~~,~~ \mbox{\boldmath $\omega$}_n = -\frac{i}{2} \hbox{Tr} ( \mbox{\boldmath $\tau$} A_n^+\dot A_n) \to \frac{{\bf s}^{(n)}}{2\Lambda}~~,~~ n=1,2 ~~,$$ where ${\bf p}^{(n)}$ and ${\bf s}^{(n)}$ are respectively the radial momentum and the spin of the $n$-th nucleon while $M$ and $\Lambda$ are respectively the mass and the moment of inertia of the single soliton. Inserting the product ansätze (\[pro\],\[xi12\]) in the expression (\[Ks\]) gives $$\label{Ks2}\displaystyle K_{{\rm L}\sigma}= -\frac{1}{2f_{\pi}^2}\int \hbox {d}^3 x ~ \xi^2({\bf r}_1)\xi^2({\bf r}_2) ~(R_{0a}(U_1)+L_{0a}(U_2)~)^2 ~ +\cdots$$ where we have omitted to write the second term in Eq. (\[Ks\]) since it does not contain angular velocities and thus will not contribute to the spin-orbit force. In Eq. (\[Ks2\]), $R_{0a}$ and $L_{0a}$ are the time components of the right and left currents respectively: $$\begin{array} {ll}\label{currents} \displaystyle R_{0a} (U_1) = -\frac{i}{2} \hbox {Tr} (\tau_a U_1^+ ~\partial_{0} U_1 ) =D_{ab} (A_1) ~\left(T_{cb}^{(1)} ~\frac{s^{(1)}_c}{2\Lambda} + \frac{p^{(1)}_i}{M} R_{ib}^{(1)} ~\right) ~,\\ \displaystyle L_{0a} (U_2) = -\frac{i}{2} \hbox {Tr} (\tau_a \partial_{0} U_2 ~U_2^+ ) =D_{ab} (A_2) ~\left(-T_{bc}^{(2)} ~\frac{s^{(2)}_c}{2\Lambda} + \frac{p^{(2)}_i}{M} R_{bi}^{(2)} ~\right) ~. \end{array}$$ In the above equations, the sum from 1 to 3 on repeated indices is understood. The tensors $T_{ab}^{(n)}$ and $R_{ab}^{(n)}$ depend only on the position ${\bf r}_n$ of the $n$-th nucleon, and their explicit expressions can be found in Eq. (11) of Ref. [@Ab2]. The $D_{ab}$’s are the matrix elements of the $3\times3$ rotation matrix in the adjoint representation: $D_{ab}(A)=\hbox {Tr}~(\tau_a A \tau_b A^+)/2$. To obtain the right-hand sides of expressions (\[currents\]), we have used the quantization scheme (\[subst\]). The next step now is to expand the square in the expression of $K_{{\rm L}\sigma}$ (\[Ks2\]) after inserting expressions (\[currents\]) and making use of the $D$-matrix properties and the definitions of the tensors $T_{ab}^{(n)}$ and $R_{ab}^{(n)}$ [@Ab2]. Thus, by considering the isoscalar part[^4] and keeping only terms proportional to ${\bf L.S}$, where ${\bf S}={\bf s}^{(1)}+{\bf s}^{(2)}$ is the total spin and ${\bf L}={\bf r}\times {\bf p}$ the angular momentum (${\bf p}$ being the relative momentum, i.e., ${\bf p}={\bf p}^{(2)}=-{\bf p}^{(1)}$), we obtain: $$\begin{array}{ll}\label{spor} \displaystyle K_{{\rm L}\sigma} \to \frac{1}{M\Lambda}V_{{\rm L}\sigma}(r)~{\bf L.S}\\ \displaystyle V_{{\rm L}\sigma}(r)=-\frac{1}{f_{\pi}^2~r} \int {\rm d}^3x~\xi^2(r_1)\xi^2( r_2) \frac{1}{r_1} \sin^2(F(r_1)) ~\hat{\bf r}.\hat{\bf r}_1 ~. \end{array}$$ The total isoscalar spin-orbit potential of the model (\[Lanew\]) is then: $$\label{SO} \displaystyle V_{{\rm SO},I=0}=\frac{1}{M\Lambda}\left(V_{{\rm L}\sigma}(r) +V_4(r)+V_6(r)\right)\bf {L.S}$$ where the linear $\sigma$-model contribution $V_{{\rm L}\sigma}$ is given in Eq. (\[spor\]). The rather lengthy expressions of the Skyrme-term contribution $V_4$ and the sixth-order term contribution $V_6$ can be found explicitly in Refs. [@autho; @WA; @KE] and [@Ab2], respectively, and do not need to be rewritten here. Note that the analytical expressions of $V_4$ and $V_6$ are not affected by the presence of the scalar field and still depend explicitly only on the chiral field $F$. This is because the $\xi$-field does not couple to ${\cal L}_4$ nor ${\cal L}_6$, as it can be seen from Eq. (\[Lanew\]). But obviously their numerical values will change due to the dependence of $F$ on the scalar degrees of freedom. RESULTS AND DISCUSSION ====================== Before displaying our numerical results concerning $V_{{\rm L}\sigma}$, let us first discuss the analytical properties of the terms $V_4(r)$ and $V_6(r)$ at large distances \[[cf]{}. Eq. (\[SO\])\]. As already shown in Ref. [@Ab2], these potentials decrease as $e^{-2m_{\pi}r}$ a large $r$. Indeed, one can see from Eq. (\[eqs\]) that at large distances the chiral angle $F(r)$ falls off as $e^{-m_{\pi}r}/r$ (note that at large $r$, $\xi \to f_{\pi}$). Then, following the approximation of Ref. [@Nym] which consists of treating at large distances the field of one soliton as constant in the presence of the other, one obtains from [@autho; @WA; @KE] that $V_4$ decreases as $e^{-2m_{\pi}r}$ over a power of $r$. For the potential $V_6$ we have shown in Ref. [@Ab2] that it is a sum of two terms. The first term (the one which has been considered in [@RS; @KE]) decreases as $e^{-3m_{\pi}r}$. This behaviour is expected since the sixth-order term ${\cal L}_6$ in Eq. (\[La\]) can be regarded as the local approximation of an effective model of $\omega$ meson coupled to three pions [@Jac]. However the asymptotic expression of the second part of $V_6$, which arises from the baryon exchange current [@Ab2], contains both $e^{-2m_{\pi}r}$ and $e^{-3m_{\pi}r}$ terms so that it behaves at large $r$ as $e^{-2m_{\pi}r}$. This means that the Lagrangian ${\cal L}_6$ generates, in addition to the expected three-pion exchange piece, a two-pion exchange one (coming from the exchange current) similar to that of the the fourth-order Skyrme term ${\cal L}_4$. Thus we see that for large distance both potentials $V_4$ and $V_6$ are of Yukawa type with range $2m_{\pi}$. In fact, the main motivation of including a scalar-isoscalar meson field by hand in Skyrme-type models [@KV; @KE] is to mimic the two-pion exchange since the latter is not well addressed in these models with simple zero-mode quantization. A straightforward manner to calculate numerically the integrals giving the different contributions to the spin-orbit force consists in making the change in variable ${\bf r}'={\bf x}-{\bf r}/2$ and taking the $NN$ separation vector ${\bf r}$ along the $z'$ axis. E.g., with these changes, the expression of the linear $\sigma$ model potential $V_{{\rm L}\sigma}$ in Eq. (\[spor\]) becomes: $$\label{num} \displaystyle V_{{\rm L}\sigma}(r)=-\frac{2\pi}{f_{\pi}^2~r} \int_0^{\infty} {\rm d}r'~r'\sin^2(F(r'))\xi^2(r') \int_{-1}^{1} {\rm d}u~u ~\xi^2\left( \sqrt{r^{'2}+r^2+2r r' u}\right)$$ where $u=\hat{\bf r}.\hat{\bf r}'=z'/r'$. In Fig.2, we plot $V_{{\rm L}\sigma}$, $V_4$, $V_6$ together with the total sum $V_{{\rm L}\sigma}+V_4+V_6$ as functions of the relative distance $r$ between the two nucleons. As it was the case for the extended Skyrme model (\[La\]), $V_4$ and $V_6$ are still repulsive in the case of the model (\[Lanew\]). On the other hand, while the nonlinear $\sigma$ model ${\cal L}_{{\rm NL}\sigma}$ has a zero contribution to the isoscalar spin-orbit force[^5], the linear $\sigma$ model contributes and with a [negative]{} sign as it can be seen from the behaviour of $V_{{\rm L}\sigma}$ in Fig.2. This result is in agreement with the phenomenology [@PB]. The total potential (full line in Fig.2) is found to be repulsive for intermediate distances ($r\le 2$fm) and attractive for large distances ($r>2$fm). For short distances (not shown on Fig.2), neither the product ansatz nor the effective model should be trusted since this region corresponds to processes involving perturbative QCD. The total potential is attractive at large distances because in this region the scalar meson contribution (\[spor\]) is the dominant one as compared to $V_4$ and $V_6$. This can be checked straightforwardly by comparing the asymptotic behaviour of each expression in Eq (\[SO\]). We have also displayed in Fig.2 the potential related to the linear $\sigma$ model and the sixth-order term only, $V_{{\rm L}\sigma} +V_6$. We observe from that curve that the attraction starts at about 1.7 fm instead of the 2 fm in the case of the total potential discussed above. This result is in some way in agreement with the phenomenological isoscalar spin-orbit potential for which it is well known that scalar-isoscalar and $\omega$ mesons are the mesons which play the most important role [@PB]. Similar results have already been found in Ref. [@KE] where a different way has been used to introduce a scalar-isoscalar degrees of freedom. In Ref. [@KE], the scalar-isoscalar meson field is a dilaton, and it explicitly couples to the sixth-order term. However the contribution to the spin-orbit force coming from the sixth-order term was not complete in Ref. [@KE]. Indeed, these authors ignored the baryon exchange current contribution to the spin-orbit force in their calculations, and we have shown in Ref. [@Ab2] that this contribution is significant, with respect to the direct term [@RS; @KE], and should be taken into consideration. It is then legitimate to question whether the qualitative results of Ref. \[17\] remain valid if the baryon exchange current contribution is taken into account. In this work, by using a different way of including the scalar-isoscalar meson in the Skyrme Lagrangian \[14,16\], and by considering the [entire]{} contribution of the baryon current to the spin-orbit channel, we show that the main result of Ref. [@KE] (i.e., an attractive $N$-$N$ spin-orbit force at large distance from Skyrmions with scalar mesons) is still valid. In this sense, my results can be viewed as a confirmation of those of Kälbermann and Eisenberg [@KE]. Even though we have obtained an attractive force at large distances in the isoscalar spin-orbit channel, attraction is still missing at intermediate distances, and thus the problem remains unsolved within Skyrme-type models. This is not surprising since we have considered here only a finite-mass scalar field without the other vector mesons. However, the result obtained here, namely, the change from a zero contribution to the isospin-independent spin-orbit interaction to an attractive one when replacing the frozen $\epsilon$-meson field (nonlinear $\sigma$ model) with a realistic one (linear $\sigma$ model), is very encouraging. Indeed, it suggests that the right way to obtain an attractive isospin-independent $NN$ spin-orbit force within Skyrme-type models in the framework of the product ansatz is to replace the pion theory (\[La\]) with a realistic effective model including, in addition to the scalar-isoscalar $\epsilon$ field, all low-lying vector mesons and taking into account the finiteness of their masses. In a sense, the model considered here can be seen as a minimal and modest improvement of the pion Skyrme theory. Similarly, it has been shown in Ref. [@Ka] that the model (\[Lanew\]) gives rise to attraction in the $NN$ central potential but the attraction occurs at distances larger than required by phenomenology. The problem has been finally solved when the other vectors mesons were included [@Ka; @KV]. For this reason, we believe that in order to cure the problem of the isoscalar spin-orbit force, one has to consider effective models which incorporate the first mesonic resonances with finite masses. For instance, when considering a finite-mass $\omega$-meson model, the $\omega$-field couples directly to the nucleon via the baryon current (defined after Eq. (\[La\])) and generates the common three-pion exchange diagram, contrary to the case of the local approximation ${\cal L}_6$ ([cf.]{} Eq. (\[La\])). Indeed, in the latter, an unexpected two-pion exchange piece coming from the baryon exchange current arises in addition to the three-pion piece and contributes with a positive sign to the isoscalar spin-orbit force yielding a repulsive interaction [@Ab2]. This problem with the baryon exchange current is obviously avoided when the sixth order term ${\cal L}_6$ is replaced with a more realistic $\omega$-meson model. Finally, in addition to considering finite-mass mesons Lagrangians, we would like to mention a further way that might lead to the desired attractive isoscalar spin-orbit force within the framework of Skyrme-type models. It is concerned with the approximation of the product ansatz configuration (\[pro\]). In the latter, the two nucleons are supposed to keep their spherical shape without deformation even after overlapping. This is certainly not true in reality. Thus, while still using the product ansatz so that we keep benefiting from its simplicity and being able to perform analytical calculations, we may improve on the approximation (\[pro\]) by allowing the shape of the nucleon to deform when approaching the other nucleon ([cf]{}. Ref. [@ROMK] and references therein). 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Abada, “[The isospin independent spin-orbit force in the extended Skyrme model]{}”, hep-ph/9401341, unpublished. M. Lacombe [et al]{}, Phys. Rev. C[21]{}, 861 (1980);\ R. Machleidt, K. Holinde and Ch. Elster, Phys. Rep. [149]{}, 1 (1987). D. O. Riska and B. Schwesinger, Phys. Lett. B [229]{}, 339 (1989). A. Abada, J. Phys. G [22]{}, L57 (1996). D. Kalafatis, hep-ph/9406410, thesis, Univ. Paris XI. K. Iketani, Kyushu University preprint 84-HE-2 (1984), unpublished. D. Kalafatis and R. Vinh Mau, Phys. Rev. D [46]{}, 3903 (1992) and references therein. G. Kälbermann and J. M. Eisenberg, Phys. Lett. B [349]{}, 416 (1995). M. C. Birse, J. Phys. G [20]{}, 1287 (1994). T. H. R. Skyrme, Nucl. Phys. [31]{}, 556 (1962). N. R. Walet, Nucl. Phys. A [586]{}, 649 (1995) and references therein. E. M. Nyman, Phys. Lett. B [162]{}, 244 (1985). D. O. Riska and E. M. Nyman, Phys. Lett. B [183]{}, 7 (1987). A. Rahimov [et al]{}, Phys. Lett. B [378]{}, 12 (1996). A. Abada, work in progress. [ ]{} [ ]{} [^1]: Address after Feb. $1^{\rm st}$, 1997: BP Finance, BP International Ltd, Britanic House, 1 Finsbury Circus, London EC2M 7BA. [^2]: Supported by the EPSRC, UK. [^3]: The region of validity of the product ansatz corresponds to a relative distance $r$ much larger than 1 fm. [^4]: Due to the projection theorem, the terms proportional to $D(A_1^+A_2)$ contribute to the isospin dependent force. [^5]: Indeed, ${\cal L}_{{\rm NL}\sigma}$ contributes only to the isospin-dependent force [@RN].
--- abstract: \| We suggest a more general than quantum statistical mechanics ($QSM$) microdescription of objects in a heat bath taken into account a vacuum as an object environment - modification of quantum mechanics at finite temperatures; we call it $(\hbar, k)$-dynamics ($ \hbar kD$). This approach allows us in a new manner to calculate some important macroparameters and to modify standard thermodynamics. We create an effective apparatus for features description of nearly perfect fluids in various mediums. As an essentially new model of an object environment we suppose a quantum heat bath and its properties, including cases of cold and warm vacuums, are studied. We describe the thermal equilibrium state in place of the traditional density operator in term of a wave function the amplitude and phase of which have temperature dependence. We introduce a new generative operator, Schroedingerian, or stochastic action operator, and show its fundamental role in the microdescription. We demonstrate that a new macroparameter, namely the effective action , can be obtained through averaging of the Schroedingerian over the temperature dependent wave function. It is established that such different parameters as internal energy, effective temperature, and effective entropy and their fluctuations can be expressed through a single quantity - the effective action. PACS numbers: 05.40.-a, 03.75.Hh author: - 'A.D. Sukhanov$^1$' - 'O.N. Golubjeva $^2$' date: '15.06.2010' title: ' New version of quantum mechanics at finite temperatures as a ground for description of nearly perfect fluids\' --- Fundamental microtheory: ($\hbar, k$)- dynamics =============================================== We develop a version of the universal theory that is unlike to the particular theory proposed earlier for explaining of ratio "shear viscosity to specific entropy’ in quark-gluon plasma. We propose to formulate a quantum-thermal dynamics or, briefly, ($\hbar,k$)- dynamics ($\hbar kD$), as a modification of standard quantum mechanics taking thermal effects into account. The principal distinction of such a theory from QSM is that in it the state of a microobject under the conditions of contact with the quantum heat bath (QHB) is generally described not by the density matrix but by a temperature-dependent complex wave function. We note that this is not a “thechnical sleight-of-hand”. Using the wave function, we thereby suppose to consider pure and mixt states simultaneously in the frame of Gibbs’ ensemble. It is in principle differs from Boltzmann’ assembly used in QSM. A general idea of our investigation: to construct a theory it is necessary 1\. to change $\hat\rho(T) \Rightarrow \Psi_T(q);$ 2\. to introduce (except of Hamiltonian) a new operator - the stochastic action operator $\hat j;$ 3\. to use an idea of heat bath at $T=0$ (“cold” heat bath) also; 4\. to use an idea of vacuum at $T>0$ (“thermal” vacuum) also. This theory is based on a new microparameter, namely, the stochastic action operator. In this case, we demonstrate that averaging the corresponding microparameters over the temperature-dependent wave function, we can find the most important effective macroparameters, including internal energy, temperature, and entropy. They have the physical meaning of the standard thermodynamic quantities using in the phenomenological macrodescription. [The model of the QHB: a case of the “cold” vacuum]{} ----------------------------------------------------- To describe the environment with the holistic stochastic action that was previously called the thermal field vacuum by Umezawa, we introduce a concrete model, the QHB. According to this, the QHB is a set of weakly coupled quantum oscillators with all possible frequencies. The equilibrium thermal radiation can serve as a preimage of such a model in nature. The specific feature of our understanding of this model is that we assume that we must apply it to both the “thermal” ($T\neq 0$) and the “cold” ($T = 0$) vacua. Thus, in the sense of Einstein, we proceed from a more general understanding of the thermal equilibrium, which can, in principle, be established for any type of environmental stochastic action (purely quantum, quantum-thermal, and purely thermal). We begin our presentation by studying the “cold” vacuum and discussing the description of a single quantum oscillator from the number of oscillators forming the QHB model for $T= 0$ from a new standpoint. But we recall that the lowest state in the energetic ($\Psi_n(q)$) and coherent states (CS) is the same. In the occupation number representation, the “cold” vacuum in which the number of particles is $n=0$ corresponds to this state. In the $q$ representation, the same ground state of the quantum oscillator is in turn described by the real wave function $$\label{4} \Psi_0(q) =[2\pi (\Delta q_0)^2]^{-1/4}\exp \left\{-\frac{q^2} {4(\Delta q_0)^2}\right\}.$$ As is well known, CS are the eigenstates of the non-Hermitian particle annihilation operator $\hat a$ with complex eigenvalues. But they include one isolated state $\|0_a\rangle = \|\Psi_0(q)\rangle$ of the particle vacuum in which eigenvalue of $\hat a$ is zero $$\label{5} \hat{a}\|0_a\rangle=0\|0_a\rangle;\;\;\;\;\;\hat{a}\|\Psi_0(q)\rangle =0\|\Psi_0(q)\rangle.$$ In what follows, it is convenient to describe the QHB in the $q$ representation. Therefore, we express the annihilation operator $\hat a$ and the creation operator $\hat a^\dagger$ in terms of the operators $\hat p$ and $\hat q$ using the traditional method. We have $$\label{6} \hat a=\frac1 2 \left (\frac{\hat p}{\sqrt{\Delta p_0^2}}-i\frac{\hat q}{\sqrt{\Delta q_0^2}}\right);\;\;\;\hat a^\dag=\frac1 2 \left (\frac{\hat p}{\sqrt{\Delta p_0^2}}+i\frac{\hat q}{\sqrt{\Delta q_0^2}}\right).$$ The particle number operator then becomes $$\label{7} \hat N_a=\hat a^\dag\hat a=\left(\frac{\hat p^2}{\Delta p_0^2} -\frac 12\hat I +\frac{\hat q^2}{\Delta q_0^2}\right)=\frac{1}{\hbar\omega}\left(\frac{\hat p^2}{2m}-\frac{\hbar\omega}{2}\hat I + \frac{m\omega^2\hat q^2}{2}\right).$$ The sum of the first and third terms in the parentheses forms the Hamiltonian $\mathcal{H}$ of the quantum oscillator, and after multiplying relation (7) by $\hbar\omega$ on the left and on the right, we obtain the standard interrelation between the expressions for the Hamiltonian in the $q-$ and $n-$ representations: $$\label{8} \hat{\mathcal{H}}=\frac{\hat p\,^2}{2m}+\frac{m\omega^2\hat q^2}{2}=\hbar\omega(\hat N_a+\frac12\hat I),$$ where $\hat I$ is the unit operator. From the thermodynamics standpoint, we are concerned with the internal energy of the quantum oscillator in equilibrium with the “cold” QHB. Its value is equal to the mean of the Hamiltonian calculated over the state $\|0_a\rangle\equiv\|\Psi_0(q)\rangle$ : $$\label{9} U_0=\langle\Psi_0(q)\|\hat{\mathcal H}\|\Psi_0(q)\rangle=\hbar\omega \langle\Psi_0(q)\|\hat {N}_a\|\Psi_0(q)\rangle+\frac{\hbar\omega}{2}=\frac{\hbar\omega}{2} =\varepsilon_0.$$ It follows from formula (9) that in the given case, the state without particles coincides with the state of the Hamiltonian with the minimum energy $\varepsilon_0$. The quantity $\varepsilon_0$, traditionally treated as the energy of zero oscillations, takes the physical meaning of the internal energy $U_0$ of the quantum oscillator in equilibrium with the “cold” vacuum. [ Passage to the “thermal” vacuum]{} ------------------------------------- We can pass from the “cold” to the “thermal” vacuum in the spirit of Umezawa using the Bogoliubov $(u, v)-$ transformation with the temperature-dependent coefficients $$\label{11} u=\left(\frac12\coth\frac{\hbar\omega}{2k_BT}+\frac12\right)^{1/2}e^{\textstyle {i\frac{\pi}{4}}};\;\;\;\;v=\left(\frac12\coth\frac{\hbar\omega}{2k_BT}-\frac12\right)^{1/2}e^{-\textstyle {i\frac{\pi}{4}}}.$$ In the given case, this transformation is canonical but leads to a unitarily nonequivalent representation because the QHB at any temperature is a system with an infinitely number of freedom degrees. In the end, such a transformation reduces to passing from the set of quantum oscillator CS to a more general set of states called the thermal correlated coherent states (TCCS). They are selected because they ensure that the Schrodinger coordinate-momentum uncertainty relation is saturated at any temperature. From the of the second-quantization apparatus standpoint, the Bogoliubov $(u, v)-$ transformation ensures the passage from the original system of particles with the “cold” vacuum $\|0_a\rangle$ to the system of quasiparticles described by the annihilation operator $\hat b$ and the creation operator $\hat b^{\dag}$ with the “thermal” vacuum $\|0_b\rangle$. To obtain from “cold” vacuum to “thermal” one using $(u,v)$ - Bogolubov’s transformations it is necessary to pass: 1\. from CS to TCCS: $\Psi_0(q)\Rightarrow\Psi_T(q); \;\;\;\; \|0_a\rangle\Rightarrow\|\|0_b\rangle; $ 2. from particles to quasiparticles: $\hat a \Rightarrow \hat b = \hat b(T). $ In this case, the choice of transformation coefficients (10) is fixed by the requirement that for any method of description, the expression for the mean energy of the quantum oscillator in thermal equilibrium be defined by the Planck formula, which can be obtained from experiments: $$\label{11} {\mathcal{E}}_{Pl.}=\hbar\omega(\exp{\frac {\hbar\omega}{k_BT}}-1)^{-1}+\frac{\hbar\omega}{2} =\frac{\hbar\omega}{2}\coth\frac{\hbar\omega}{2k_BT}.$$ Earlier was shown by us, the state of the “thermal” vacuum $\|0_b\rangle \equiv\|\Psi_{\scriptscriptstyle{T}}(q)\rangle$ in the $q-$ representation corresponds to the complex wave function $$\label{12} \Psi_{\scriptscriptstyle{T}}(q)=[2\pi (\Delta q)^2]^{-1/4}\exp \left\{-\frac{q^2} {4(\Delta q)^2}(1-i\alpha)\right\},$$ where $$\label{13} (\Delta q)^2 = \frac{\hbar}{2m\omega} \coth\frac {\hbar\omega}{2k_BT};\;\;\;\;\alpha=\left[\sinh\frac{\hbar\omega}{2k_BT}\right]^{-1} ;\;\;\;\; (\Delta p)^2 = \frac{\hbar m\omega}{2} \coth\frac {\hbar\omega}{2k_BT}.$$ We note that the expressions for the probability densities $\rho_{\scriptscriptstyle{T}}(q)$ and $\rho_{\scriptscriptstyle{T}}(p)$ have already been obtained by Bloch, but the expressions for the phase that depend on the parameter $\alpha$ play a very significant role and were not previously known. It is also easy to see that as $T \rightarrow 0$, the parameter $\alpha \rightarrow 0$ and the function $\Psi_{\scriptscriptstyle{T}}(q)$ from the set of TCCS passes to the function $\Psi_{\scriptscriptstyle{0}}(q)$ from the set of CS. Of course, the states from the set of TCCS are the eigenstates of the non-Hermitian quasiparticle annihilation operator $\hat b$ with complex eigenvalues. They also include one isolated state of the quasiparticle vacuum in which eigenvalue of $\hat b$ is zero, $$\label{15} \hat{b}\|0_b\rangle=0\|0_b\rangle; \;\;\;\;\hat{b}\|\Psi_T(q)\rangle =0\|\Psi_T(q)\rangle.$$ Using condition (15) and expression (12) for the wave function of the “thermal” vacuum, we obtain the expression for the operator $\hat b$ in the $q-$ representation: $$\label{16} \hat b=\frac 12\sqrt{\coth \frac{\hbar\omega}{2k_BT}}\left[\frac{\hat p}{\sqrt{\Delta p_0^2}}-i\frac{\hat q}{\sqrt{\Delta q_0^2}} (\coth \frac{\hbar\omega}{2k_BT})^{-1}(1-i\alpha)\right].$$ The corresponding quasiparticle creation operator has the form $$\label{17} \hat b^{\dag}=\frac {1}{2}\sqrt{\coth \frac{\hbar\omega}{2k_BT}}\left[\frac{\hat p}{\sqrt{\Delta p_0^2}}+i\frac{\hat q}{\sqrt{\Delta q_0^2}} (\coth \frac{\hbar\omega}{2k_BT})^{-1}(1+i\alpha)\right].$$ We can verify that as $T\rightarrow 0$ , the operators $\hat b^{\dag}$ and $\hat b$ for quasiparticles pass to the operators $a^{\dag}$ and $\hat a$ for particles and $\|0_b\rangle\Rightarrow \|0_a\rangle;\;\;\;\;\Psi_T(q)\Rightarrow\Psi_0(q).$ Acting just as above, we obtain the expression for the quasiparticle number operator in the $q-$ representation $$\label{18} \hat N_b=\hat b^{\dag}\hat b=\frac 14\coth \frac{\hbar\omega}{2k_BT}\left[\frac {\hat p^2}{\Delta p_0^2}-2(\coth \frac{\hbar\omega}{2k_BT})^{-1} (\hat I+\frac \alpha \hbar\{\hat p,\hat q\})+\frac {\hat q^2}{\Delta q_0^2}\right],$$ where we take $1+\alpha^2=\coth^2\frac{\hbar\omega}{2k_BT}$ into account when calculating the last term. [ Hamiltonian in TCCS]{} ------------------------ Passing from the quasiparticle number operator to the original Hamiltonian and multiplying by $\hbar\omega$, we obtain $$\label{19} \hat{\mathcal{H}}=\hbar\omega(\coth \frac{\hbar\omega}{2k_BT})^{-1}\left[\hat N_b+\frac 12 (\hat I+ \frac\alpha \hbar\{\hat p,\hat q\})\right].$$ We stress that the operator $\{\hat p,\hat q\}$ in formula (19) can also be expressed in terms of bilinear combinations of the operators $\hat b^{\dag}$ and $\hat b$, but they differ from the quasiparticle number operator $N_b$. This means that the operators $\hat{\mathcal H}$ and $\hat N_b$ do not commute and that the wave function of form (12) characterizing the state of the “thermal” vacuum is therefore not the eigenfunction of the Hamiltonian. As before, we are interested in the thermodynamic quantity, namely, the internal energy $U$ of the quantum oscillator now in thermal equilibrium with the “thermal” QHB. Calculating it just as earlier, we obtain $$\label{20} U= \hbar\omega(\coth \frac{\hbar\omega}{2k_BT})^{-1}\left[\langle\Psi_{\scriptstyle T}(q)\|\hat N_b\|\Psi_{T}(q)\rangle+\frac{1}{2} +\frac{\alpha}{2\hbar} \langle\Psi_{\scriptstyle T}(q)\|\{\hat p,\hat q\} \|\Psi_{T}(q)\rangle\right]$$ in the $q-$ representation. Because we averaging over the quasiparticle vacuum in formula (20), the first term in it vanishes. At the same time, it was shown earlier by us that $ \langle\Psi_{\scriptstyle T}(q)\|\{\hat p,\hat q\}\|\Psi_{T}(q)\rangle=\hbar\alpha. $ As a result, we obtain the expression for the internal energy of the quantum oscillator in the “thermal” QHB in the $\hbar kD$: $$\label{22} U=\frac {\hbar\omega}{2 (\coth \frac{\hbar\omega}{2k_BT})}(1+\alpha^2) =\frac{\hbar\omega}{2}\coth \frac{\hbar\omega}{2k_BT}={\mathcal{E}}_{Pl.},$$ where ${\mathcal{E}}_{Pl.}$ is defined by Planck formula (11). This means that the average energy of the quantum oscillator at $T\neq 0$ has the thermodynamic meaning of its internal energy in the case of equilibrium with the “thermal” QHB. As $T\rightarrow 0$, it passes to a similar quantity corresponding to equilibrium with the “cold” QHB. [New fundamental operator - Schroedingerian ]{} ================================================ [ Schroedinger uncertainties relation]{} ---------------------------------------- Because the original statement of the $\hbar kD$ is the idea of the holistic stochastic action of the QHB on the object, we introduce a new operator in the Hilbert space of microstates to implement it. We recall the general expression of Schwartz inequality $ \|A\|^2\cdot\|B\|^2\geqslant \|A\cdot B\|^2. $ Saturated Schroedinger uncertainties relations (SUR) coordinate-momentum following from it is: $$\label {23} (\Delta p)^2(\Delta q)^2 = \|\tilde R_{qp}\|^2\equiv\sigma^2+\frac{\hbar^2}{4}.$$ In the absent of stochastic action $\tilde{R}_{qp}\equiv0.$ As leading considerations, hereinafter we use an analysis of the right-hand side of the saturated SUR coordinate-momentum. [The stochastic action operator (Schroedingerian)]{} ---------------------------------------------------- For not only a quantum oscillator in a QHB but also any object, the complex quantity in the right-hand side of (23) $$\label{24} \widetilde{R}_{p\,q}=\langle\Delta p\|\Delta q\rangle\qquad\mbox {or }\widetilde{R}_{p\,q}=\langle\,\|\Delta \widehat{p}\,\Delta \widehat{q}\,\|\,\rangle$$ has a double meaning. On one hand, it is the amplitude of the transition from the state $\|\Delta q\rangle $ to the state $\|\Delta p\rangle $; on the other hand, it can be treated as the Schroedinger quantum correlator calculated over an arbitrary state $\|\; \rangle$ of some operator. As is well known, the nonzero value of quantity (24) is the fundamental attribute of nonclassical theory in which the environmental stochastic action on an object plays a significant role. Therefore, it is quite natural to assume that the averaged operator in formula (24) has a fundamental meaning. In view of dimensional considerations, we call it the stochastic action operator or Schroedingerian, $$\label{25} \widehat{j}\equiv\Delta \widehat p\,\Delta \widehat{q}.$$ Of course, it should be remembered that the operators $\Delta \hat q $ and $\Delta\hat p $ do not commute and their product is a non-Hermitian operator. To analyze further, following Schroedinger, we can write the given operator $$\label{26} \widehat{j}= \frac{1}{2} \left \{\Delta\widehat{p}\Delta\widehat{q}+ \Delta\widehat{q}\Delta\widehat{p}\,\right\} +\frac{1}{2}\left[\Delta\widehat{p}\Delta\widehat{q}- \Delta\widehat{q}\Delta\widehat{p}\,\right]= \widehat{\sigma}-i\,\widehat{j}_{\scriptscriptstyle 0}.$$ It allows separating the Hermitian part of $\hat j$ from the anti-Hermitian one. Then the Hermitian operators $\widehat\sigma$ and $\hat j_0$ have the form $$\label{27} \widehat{\sigma}\equiv \frac12\{\Delta \hat p,\;\Delta \hat q\}; \;\;\;\;\hat{j}_{\scriptscriptstyle 0} \equiv \frac{ i}{2}\; [\hat{p}\,,\hat{q}\,]\, \,\ = \frac{\hbar}{2}\;\hat {I}.$$ It is easy to see that the mean $\sigma=\langle\,\|\widehat{\sigma}\|\,\rangle$ of the operator $\hat\sigma$ resembles the expression for the standard correlator of coordinate and momentum fluctuations in classical probability theory. It transforms into this expression if the operators $\Delta \hat q $ and $\Delta \hat p $ are replaced with $c$-numbers. It reflects the contribution to the transition amplitude $\tilde R_{pq}$ of the environmental stochastic action. Therefore, we call the operator $\hat{\sigma}$ the external action operator in what follows. At the same time, the operators $\hat j_0$ and $\hat j $ were not previously introduced. The operator $\hat j_0$ of form (27) reflects a specific peculiarity of the objects to be “sensitive” to the minimum stochastic action of the “cold” vacuum and to respond to it adequately regardless of their states. Therefore, it should be treated as a minimum stochastic action operator. Its mean $J_0 = \frac {\hbar}{2}$ is independent of the choice of the state over which the averaging is performed, and it hence has the meaning of the invariant eigenvalue of the operator $\hat j_0$. [Effective action as a fundamental generative macroparameter]{} ================================================================= [The mean of the operator $\widehat j$ ]{} ------------------------------------------ We now construct the macrodescription of objects using their microdescription in the $\hbar kD$. It is easy to see that the mean $\tilde J$ of the operator $\hat j$ of form (26) coincides with the complex transition amplitude $\widetilde R_{pq}$ or Schroedinger’s correlator and, in thermal equilibrium, can be expressed as $$\label{28} \tilde J=\langle \Psi_{\scriptstyle T}(q)\|\,\widehat{j}\,\|\Psi_{\scriptstyle T}(q)\rangle=\sigma - \,i\,J_0,$$ where $\sigma $ and $J_0$ are the means of the corresponding operators. In what follows, we regard the modulus of the complex quantity $\widetilde J$, $$\label{29} \|\tilde J\|=\sqrt{\sigma^2+J_0^2}=\sqrt{\sigma^2+\frac{\hbar^2}{4}}\equiv J_{ef.}$$ as a new macroparameter and call it the effective action. It has the form $$\label{30} J_{ef.}=\frac\hbar 2\coth\frac{\hbar\omega}{2k_BT}=\frac{U}{\omega}=\frac{k_BT_{ef}}{\omega}$$ for the quantum oscillator and coincides with a similar quantity previously postulated from intuitive considerations. [ Effective entropy in the $\hbar kD$ ]{} ----------------------------------------- The possibility of introducing entropy in the $\hbar kD$ is also based on using the wave function instead of the density operator. Using the dimensionless expressions for $\rho (q)= \|\Psi(q)\|^2$ and $\rho(p) = \|\Psi(p)\|^2$, we propose to define a formal coordinate - momentum entropy $S_{qp}$ by the equality $$\label{41} S_{qp}\,=-k_B\left\{\int\tilde{\rho}(\tilde{q}) \ln\tilde{\rho}(\tilde{q}) d\tilde{q}+\int\tilde{\rho}(\tilde{p})\ln\tilde{\rho}(\tilde{p}) d\tilde{p}\right\}.$$ Substituting the corresponding expressions for $\tilde{\rho} (\tilde q) $ and $\tilde{\rho}(\tilde p)$ in (41), we obtain $$\label{42} S_{qp}=k_B \left\{(1+\ln\frac{2\pi}{\delta}) +\ln\coth\frac{\hbar\omega}{2k_BT}\right\}.$$ Obviously, the final result depends on the choice of the constant $\delta$. Choosing $\delta = 2\pi$, we can interpret expression (42) as the quantum-thermal entropy or, briefly, the $QT-$ entropy $S_{QT}$ because it coincides exactly with the effective entropy $S_{ef.}$ obtained earlier by us in the macrotheory framework. $$\label{43} S_{QT}\equiv S_{ef.}=k_B\left\{1+\ln\frac{J_{ef.}}{J_0}\right\}=k_B\left\{1+\ln\Omega\right\},$$ where $\Omega$ according to Boltzmann is a number of microstates in the given macrostate. This ensures the consistency between the main results of our proposed micro- and macrodescriptions and their correspondence to experiments. [ Quantum Statistical Thermodynamics on the base of the effective action]{} ---------------------------------------------------------------------------- The above presentation shows that using the $\hbar kD$ developed here, we can introduce the effective action $J_{ef.} $ as a new fundamental macroparameter. The advantage of this quantity is that it has a microscopic preimage, namely, the stochastic action operator $\hat j$, or Schroedingerian, which has an obvious physical sense. Moreover, we can in principle express the main thermodynamic characteristics of objects in thermal equilibrium in terms of it. As is well known, temperature and entropy are the most fundamental of them. If the notion of effective action is used, these heuristic considerations can acquire an obvious meaning. For this, we turn to expression (34) for $T_{ef.}\sim J_{ef.}$. It follows from it that the effective action is also an intensive macroparameter characterizing the stochastic action of the “thermal” QHB. In view of this, the Zero Law of equilibrium quantum Statistical Thermodynamics can be rewritten as $$\label{48} J_{ef.}= J_{ef.}^{therm.} \pm \Delta J_{ef.},$$ where $J^{term.}_{ef.}$ is effective action of QHB, $J_{ef.}$ and $\Delta J_{ef.}$ are the means of the effective action of an object and the standard deviation from it. The state of thermal equilibrium can actually be described in the sense of Newton, assuming that “the stochastic action is equal to the stochastic counteraction” in such cases. [ Connection with experiment]{} ================================ Nowadays there is alot of papers on the theory of nearly perfect fluids where the ratio of shift viscosity to entropy volume density is the subject of interest. It was shown by us that this quantity can be given by $$\label{51} \frac{J^{ef.}}{S^{ef.}}=\frac{J^{ef.}_{min}}{S^{ef.}_{min}}\;\cdot\frac{\coth T^{ef.}_{min}/ T}{1+\ln \coth T^{ef.}_{min}/ T}= \varkappa\;\cdot\frac{\coth\varkappa\omega/T}{1+\ln\coth \varkappa\omega/T}\rightarrow \varkappa.$$ In this expression, $$\label{52} \varkappa\equiv\frac{J^{ef.}_{min}}{S^{ef.}_{min}}=\frac{\hbar}{2k_B}=3,82\cdot 10^{-12} \mbox{K c} .$$ is the limiting ratio for $T\ll T^{ef.}. $ In our opinion, the quality is not only the notation for one of the possible combinations of the world constants $\hbar$ and $k_B$. It also has its intrinsic physical meaning. We sure that the quantity $\varkappa$ plays the role of a constant essentially characterizing the holistic stochastic action on the object. The analogical with (51) relation in QSM, in contrast to one in $\hbar kD$, has the form $$\label{55} \frac{J}{S}\rightarrow\frac{\hbar\exp(-\hbar\omega/k_BT)} {k_B(\hbar\omega/k_BT)\exp(-\hbar\omega/k_BT)}=\frac{T}{\omega}\rightarrow 0.$$ Therefore it is now possible to compare two theories ($\hbar kD$ and QSM) experimentally by measuring the limiting value of this ratio. The first indication that the quantity $\varkappa$ plays an important role was obtained in Andronikashvili’s experiments (1948) on the viscosity of liquid Helium below the $\lambda-$ point. There are also another areas of Physics where the constant $\varkappa$ appears. Now the constant $\varkappa$ is also observed in cold atomic gases and different solids, including graphene, as a characteristic of the fundamentally new state of matter - nearly perfect fluids (Shaefer, Teaney, 2009). We hope that $\hbar kD$ can serve as an initial microtheory for constructing of modified thermodynamics fit for very small objects at ultra-low temperatures. This work was supported by the Russian Foundation for Basic Research (project No. 10-01-90408) [99]{} A.D. Sukhanov, O.N. Golubjeva. Theor. Math.Phys. 160,2, p.1177-1189 (2009) A.D. Sukhanov. Theor. Math. Phys. 154,1, p.153-164 (2008) A.D. Sukhanov. Theor. Math. Phys. 148,2, p.1123-1134 (2006) A.D. Sukhanov. ArXiv: 0509113 v1 \[cond-mat, stat-mech\] 16p (5 Sep 2005) A.D. Sukhanov. Theor. Math. Phys. 132,3, p.1277-1294 (2002) T.Schaefer, D. Teaney. ArXiv: 0904.3107 v2 \[hep-ph\], 76 p. (17 July, 2009)
--- abstract: 'We present a direct experimental investigation of the thermal ordering in an artificial analogue of an asymmetric two dimensional Ising system composed of a rectangular array of nano-fabricated magnetostatically interacting islands. During fabrication and below a critical thickness of the magnetic material the islands are thermally fluctuating and thus the system is able to explore its phase space. Above the critical thickness the islands freeze-in resulting in an arrested thermalized state for the array. Determining the magnetic state of the array we demonstrate a genuine artificial two-dimensional Ising system which can be analyzed in the context of nearest neighbour interactions.' author: - 'Unnar B. Arnalds' - Jonathan Chico - Henry Stopfel - Vassilios Kapaklis - Oliver Bärenbold - 'Marc A. Verschuuren' - Ulrike Wolff - Volker Neu - Anders Bergman - Björgvin Hjörvarsson title: 'A new look on the two-dimensional Ising model: thermal artificial spins' --- The Ising model invented by Wilhelm Lenz and solved in one dimension by Ernst Ising in 1924 is one of the pillars of statistical mechanics[@ising; @History_Ising_model_RevModPhys.39.883]. Although built on a simple basis, that of an interacting system composed of a chain of entities with only two discrete states, ${\bf s}=\{1,-1\}$, the Ising model still to this day is used to model magnetic systems and can be applied to a wealth of atomistic and mesoscopic experimental systems ranging from ferromagnetic ordering and atomic-scale antiferromagnets [@Loth:Science2012] to the ordering of binary colloidal structures[@Khalil_NatComm_2012] and thermal artificial spin systems [@Zhang_Nat_2013]. In the two dimensional case spins are arranged on a square lattice and each spin interacts with four nearest neighbours, as seen in Fig. \[fig:2d\_Ising\]. The total energy of such a two-dimensional Ising system can be described by the equation $$E = - J_h \sum_{i,j}^h {\bf s}_i\cdot {\bf s}_j - J_v \sum_{i,j}^v {\bf s}_i\cdot {\bf s}_j \label{eq:hamiltonian}$$ where $J_h$ and $J_v$ correspond to interaction energies in the horizontal \[10\] and vertical \[01\] lattice directions of the two dimensional crystal and the sum is taken over all pairs of nearest neighbour spins ${\bf s}_i$ and ${\bf s}_j$. As opposed to the one dimensional model which shows no spontaneous magnetization at temperatures $T>0$ a spontaneous magnetization appears in the two dimensional case[@peierls] with an order parameter given by[@Onsager_2d_Ising; @Yang:1952] $$M(T) = \biggl( 1 - \biggl[\sinh \biggl(\frac{2J_h}{k_{\mathrm{B}}T}\biggr) \sinh \biggl(\frac{2J_v}{k_{\mathrm{B}}T} \biggr) \biggr]^{-2} \biggr)^{\frac{1}{8}} \label{eq:MofT}$$ where $T$ corresponds to the temperature and $k_{\mathrm{B}}$ is the Boltzmann constant. The model is not complicated by the choice of different values or signs of $J_h$ and $J_v$ [@Onsager_2d_Ising]. For $J>0$ the interaction between neighbouring spins favors a parallel alignment (see Fig. \[fig:2d\_Ising\]) while in the case of $J<0$ spins have a preference for an antiferromagnetic alignment. ![The two dimensional Ising system. A two dimensional Ising system consists of an array of spins which are only capable of pointing in two opposite directions, ${\bf s}=\pm 1$ and interacting with their four nearest neighbours with interaction energies $J_h$ and $J_v$ along the \[10\] and \[01\] lattice directions. At $T=0$ K the ground state is doubly degenerate consisting of all spins pointing in the (+) or the (-) directions for the case of a symmetric Ising system. For $T>0$ K defects above the ground state occur enclosed by a domain wall separating the areas of spins in the all (+) or all (-) directions. []{data-label="fig:2d_Ising"}](Fig1_twod_Ising_schematic_reduced.pdf "fig:")\ Nano-patterned single-domain magnetic thin film islands have been a prominent candidate in recent years for creating artificial analogues of interacting systems. Using modern lithographic techniques it has become feasible to design directly the shape of such islands and their geometrical arrangement creating two dimensional arrays of interacting artificial spins. Artificial Ising-like spins can be realized by designing elongated islands of thin films materials, for which shape anisotropy confines the magnetization to only two possible orientations. By arranging such islands in different geometries a wealth of interacting systems can be studied including cellular automata [@Imre_Science_2006; @Cowburn_Science_2000] and frustrated systems such as artificial spin ice [@wang_nature; @Nisoli_Review]. The two dimensional nano-scale nature of these systems enables their state to be directly determined by imaging techniques such as magnetic force microscopy[@wang_nature; @Morgan_NPHYS], photoemission electron microscopy[@mengotti:NPHYS2011; @KagomeAPL; @Kapaklis:NatNano:2014] and Lorentz microscopy [@Daunheimer:PRL:2011]. In this paper we investigate a two dimensional array composed of elongated thin film islands in a square lattice (see Fig. \[fig:AFM\]). During fabrication the array goes through a dynamic phase enabling the system to thermally explore its phase space leading to a low energy ordered state [@Morgan_NPHYS]. After this dynamic phase the state of the system becomes frozen-in, generating a snapshot of an arrested thermalized state. The determination of the magnetization of individual islands in the array allows us to demonstrate that the dynamic phase leads to an ordering of the magnetic state of the array, which can be described by the two-dimensional Ising model. ![Nano-patterned artificial spins. Atomic force microscopy (AFM) image of part of the patterned array showing the structure and arrangement of the nano-patterned islands (see Methods). The shape of the islands confines the magnetization to point along the long axis of the islands and during thermalization at the onset of a frozen state they can be considered as thermally active superspins. In the superspin model the vertical coupling (\[01\] direction) prefers a ferromagnetic alignment of the spins ($J_v>0$) whereas the lateral coupling (\[10\] direction) prefers an antiferromagnetic alignment ($J_h<0$) resulting in the ground state ordering of the artificial spins illustrated to the right in the figure. []{data-label="fig:AFM"}](Fig2_AFM_Spins_small_reduced.pdf) The thickness regime wherein thermal dynamics of the spins are obtained occurs during the deposition of the magnetic material onto prepatterned substrates as shown by Morgan [et al.]{} [@Morgan_NPHYS; @Morgan_PRB_2013]. During the deposition, the island thickness (and thereby their volume) becomes gradually larger and the magnetization reversal energy barrier, $E_r$, associated with their shape anisotropy increases. In the initial stages of the island growth this barrier is smaller than the thermal energy enabling the magnetization to spontaneously fluctuate between the two low energy states defined by the shape anisotropy. As the thickness becomes larger the reversal energy increases, eventually overcoming the thermal energy, reaching a threshold where the superparamagnetic behaviour is suppressed as the scale of the combined shape anisotropy reversal energy barrier and the energy landscape of the array are sufficient to lock the magnetization in each of the islands (see Fig. \[fig:3dspins\]). Subsequently the array can be imaged by magnetic force microscopy (MFM) and the magnetization direction of each island can be determined (see Fig. \[fig:MFM\]). ![image](Fig3_3D_spins_SMALL_reduced.pdf) During the limited time window where fluctuations occur the fast dynamics allow the array to explore its phase space and achieve an equilibrium condition. During this dynamic phase before the magnetization becomes frozen the reversal energy barrier and the interaction energies between neighbouring islands is of the order of the thermal energy $k_{\mathrm{B}}T$. For the island sizes and array parameters used for this study the energy values involved correspond to room temperature values for film thicknesses in the $\sim$1 nm regime. Uniform thermal dynamics over the entire array therefore require a well defined, stable thickness of each island in order to minimize randomization effects due to variations in the island thickness and film roughness. Amorphous magnetic materials display soft magnetic properties and a high degree of structural uniformity and are thereby suitable for well defined layers below 1 nm in thickness [@ahlberg11]. For this study we therefore choose amorphous Co$_{68}$Fe$_{24}$Zr$_8$ as the island material previously used for creating ultra-thin magnetic layers[@Hase_PRB_CoFeZr_XMCD] and well defined nano-patterned multilayered structures [@Arnalds:XRMS_2012]. Furthermore, a field imprinted anisotropy can be induced in Co$_{68}$Fe$_{24}$Zr$_8$ enhancing the energy barrier for reversal as the magnetization has settled in a fixed direction [@Raanaei:JAP]. Considering magnetostatic interactions in the point dipole approximation between neighbouring islands reveals an interaction scheme which can be mapped to a ferromagnetic interaction in the vertical direction $J_v>0$ and antiferromagnetic in the horizontal direction $J_h<0$. The lowest energy state of the array is therefore composed of a staggered arrangement of ferromagnetically aligned chains (see Fig. \[fig:AFM\]). The lowest energy state of an asymmetric two dimensional Ising system is two degenerate, in our case corresponding to a ferromagnetic ordering in the vertical direction and an antiferromagnetic ordering in the horizontal direction. Excitations above the ground state occur through the reversal of a macrospin and can be viewed in the form of boundary walls separating the two possible ground states as illustrated in Fig. \[fig:MFM\](c) and the energy state of individual island can be categorized into 9 different energy states of varying degeneracy shown in Fig. \[fig:MFM\](d). ![Macrospin arrangements imaged by MFM. Analysis of a part of the MFM images recorded showing a portion of the $N=10487$ island array for which the direction of the magnetization was determined. The lateral extent of the entire nano-patterned array was $2\times2$ mm$^2$, corresponding to a total number of islands 40 million. [a]{} Magnetic force microscopy image showing part of the analyzed array. [b]{} Schematic highlighting of island contours and the excitation boundaries. [c]{} The energy state of the islands quantified into the nine different possible energy states of individual islands with respect to the orientation of their nearest neighbours shown in [d]{}. The energy states along with their respective degeneracies are listed with respect to the lowest energy ground state assuming that $J_h>J_v$. []{data-label="fig:MFM"}](Fig4_schematic_and_MFM_small_reduced.pdf) ![Macrospin energy state statistics. The number of observations of each of the 9 energy states illustrated in Fig. \[fig:MFM\](d) divided by their degeneracy. The inset shows the number of observed energy states composed exclusively of either vertical excitations (energy levels $2J_v$ and $4J_v$), blue line, or horizontal excitations (energy levels $2J_h$ and $4J_h$), red line. From the ratio of the slopes the ratio of the energy scale between the vertical and horizontal interaction energies is determined to be $\|J_v/J_h\|=0.2943$. Considering this ratio the number of observations of the 9 different energy states can be listed as a function of their individual energy in units of $2\|J_h\|$. The error bars correspond to the square root of the number of observations for each of the energy states. []{data-label="fig:logplot"}](Fig5_logProb_vs_Energy_split_model.pdf) Counting the abundances of excitations composed only of independent vertical or horizontal excitations the relative energy scale between the two directions can be attained. The observed probabilities of these excitations decreases exponentially with increasing energy (inset in Fig. \[fig:logplot\]) in accordance with a Boltzmann distribution of the states. Determining the ratio of the excitation energies from the inset the energies relating to all energy values of the individual islands can be listed in units of the energy involved with a horizontal excitation $\|J_h\|$. Within the combined plot (Fig. \[fig:logplot\]) the observed abundances decrease exponentially establishing the probability for a macrospin to be in an energy state $E$ to be given by a Boltzmann distribution $\sim \exp{(E/k_{\mathrm{B}}T)}$ and that the system can be sufficiently described by a nearest neighbour interaction model. Identifying the domain walls separating the two degenerate ground states of the array facilitates the mapping of the magnetic structure of the system as two ordering states, as illustrated in Fig. \[fig:3dspins\]. The mapping of these two states onto two domain colours for the entire array is shown in Fig. \[fig:domain\]. Counting the number of islands falling into each of these two domains the order parameter of the array can be obtained. The resulting order parameter for the array, defined by $M=(n_b-n_w)/(n_b+n_w)$ where $n_b$ and $n_w$ correspond to the domain populations of black and white domains, can then be obtained. Truncating the data array to a square shape the array size is reduced to $n = 9828$ out of which $n_b=5283$ islands fall into the [black]{} domain while $n_w = 4545$ islands fall into the [white]{} domain. The resulting value of $M = 0.075\pm0.015$ reveals a slightly higher population of the black domains. Although the statistics of this value are limited by the finite number of observed islands in the array it could be indicative of the array being in a state close to, or above $T_c$, since the lack of global order in the macroscopic arrangement can lead to a finite value of the order parameter. The listed uncertainty of $M$ is determined from the square root of the domain populations, $n_w$ and $n_b$. ![Macrospin array domain configuration. The macrospin configuration of the investigated arrays mapped to black and white domains corresponding to the two degenerate ground state of the system. []{data-label="fig:domain"}](Fig6_Domains_scaled.pdf) By calculating the pairwise correlation between spins, at a distance ${\bf{r}}$, within the experimental array the correlation function for the system, $G({\bf{r}})$, can be determined. The resulting correlation array is shown in Fig. \[fig:correlations\]. The preference for an antiferromagnetic arrangement of neighbouring spins in the \[10\] direction introduces the possibility of negative values in the correlation and alternating positive and negative values along the \[10\] direction. Figure \[fig:correlations\] therefore shows the absolute value of the pairwise correlation $\|G({\bf{r}})\|$ as determined from the array. As can be seen in Fig. \[fig:correlations\] the pairwise correlation follows an exponentially decreasing function as expected for an extended array of Ising like spins at $T>T_c$. Furthermore, the asymmetric nature of the array is revealed in the different values of the correlation lengths along the \[10\] and \[01\] directions of the array with $\xi_{\mathrm{[10]}} = 2.87$ and $\xi_{\mathrm{[01]}} = 1.02$. The correlation length in the \[11\] direction, $\xi_{\mathrm{[11]}} = 0.97$, is revealed to be similar to $\xi_{\mathrm{[01]}}$. ![Pairwise correlations $G$ for the array. The graph shows the absolute value, $\|G\|$, for easier mapping of the pairwise correlations for both the ferromagnetic and antiferromagnetic directions. From the slopes the resulting correlation lengths along the different directions is determined to be $\xi_{\mathrm{[10]}} = 2.87$, $\xi_{\mathrm{[01]}} = 1.02$, and $\xi_{\mathrm{[11]}} = 0.97$ []{data-label="fig:correlations"}](Fig7_correlations_10_01_11.pdf) The role of the ratio of the interaction strengths on the order parameter and specifically the ordering temperature of the array can be investigated using Onsager’s solution, as initially done by Chang [@Chang:1952] or utilizing numerical calculations such as Monte Carlo simulations, see Fig. \[fig:MC\_order\_parameter\]. In the case of an isotropic system, wherein the interaction strength between neighbours, $J$, is the same in the two main lattice directions (\[01\] and \[10\]), the ordering temperature is given by $T_c = \frac{2J/k_{\mathrm{B}} }{\ln(1+\sqrt{2})} \approx 2.269J/k_{\mathrm{B}}$. As $J_v$ decreases with respect to $J_h$ the relative $T_c$ of the system also decreases. Figure \[fig:MC\_order\_parameter\] shows results from Monte Carlo simulations for decreasing values of the ratio $\|J_v/J_h\|$. The change in $T_c$ corresponds well to what is expected in the two dimensional Ising model as the probability of observing two parallel spins in the \[01\] direction decreases. As can be seen in Fig. \[fig:MC\_order\_parameter\] the results of numerical simulations correspond well to the analytical solution with small deviations arising from the finite nature of the simulated array. ![Order parameter versus asymmetry. [a]{} Results of Monte Carlo simulations showing the order parameter $M$ as a function of temperature, in units of $J_h/k_{\mathrm{B}}$, for different values of $\|J_v/J_h\|$. Each point in the graph corresponds to an average of several realizations (see methods). As the ratio decreases from the symmetric value of the $\|J_v/J_h\| = 1$ the $T_c$ also decreases. The analytical Onsager solution given by equation (solid lines) fits quite well with the obtained numerical data. The inset shows the correlation length $\xi$ as a function of temperature for the ratio $\|J_v/J_h\| = 0.3$ for the three major crystallographic directions of the array, \[10\], \[01\], and \[11\]. [b]{} - [e]{} Representative snapshots of the numerical simulations at temperatures of $1.0J_h/k_{\mathrm{B}}$, $1.34J_h/k_{\mathrm{B}}$, $1.5J_h/k_{\mathrm{B}}$, and $2.0J_h/k_{\mathrm{B}}$, respectively, for an asymmetry ratio of $\|J_v/J_h\| = 0.3$. Due to the asymmetry the correlation length in the \[10\] direction is larger than in the \[01\] direction stretching out the domains in the \[10\] direction, similar to what is observed in the experimental data. []{data-label="fig:MC_order_parameter"}](Fig8_MC_drawing.pdf) Assuming a system in a perfectly ordered ground state at 0 K as the temperature is increased thermal energy is introduced into the system and spins will start to change their directions. The perfect initial order is therefore broken and the system moves towards a thermally disordered state with a reduced order parameter. From numerical simulations this transition can, furthermore, be observed through the pairwise correlation $G({\bf{r}})$ between spins. For the two dimensional Ising model the correlation function can be written as $G({\bf{r}}) \sim \exp (-\frac{{\bf{r}}}{\xi})$ for $T<T_c$ and $T>T_c$ where $\xi$ is the correlation length affected by temperature and the interaction strength between neighbouring spins. The inset in Fig. \[fig:MC\_order\_parameter\] shows the temperature dependence of $\xi$ for the three major directions of the array, \[10\], \[01\], and \[11\] for the asymmetry ratio $\|J_v / J_h\| = 0.3$ as determined from Monte Carlo calculations. As $J_h > J_v$ the corresponding correlation length in the \[10\] direction is larger than in the other directions while the correlation length in the \[01\] and \[11\] directions are similar. Considering the results of the Monte Carlo simulations for an interaction ratio of $\|J_v/J_h\| = 0.3$ one can see that for temperatures larger than $T_c$ the correlation lengths $\xi_{\mathrm{[10]}}$ and $\xi_{\mathrm{[01]}}$ follow the same trend as the experimental results, i.e. $\xi_{\mathrm{[10]}} > \xi_{\mathrm{[01]}}$ and $\xi_{\mathrm{[01]}}\sim\xi_{\mathrm{[11]}}$. In particular at $T=1.9 J_h/k_{\mathrm{B}}$ the values obtained from the simulations are $\xi_{\mathrm{[10]}} = 1.95$, and \[01\], $\xi_{\mathrm{[01]}} = 1.06$, $\xi_{\mathrm{[11]}} = 1.08$, similar to the values obtained experimentally. These results indicate that in the arrested state the experimental observations can be described with an effective nearest neighbour Ising model. Hence, long range dipolar interactions are not needed to model the present system in the arrested stated. However, dipolar interactions cannot be completely excluded if one desires a full description of the system in any non-arrested state. Further advances in materials science and magnetic imagining techniques will undoubtedly allow us to follow the thermal evolution of a multitude of different artificial spin systems[@Stamps_Heyderman_2013] such as artificial spin ice, one and two dimensional Ising arrays and different frustrated arrangements[@Morrison_NJP_2013; @Gilbert_Natphys_2014] as they explore their phase space. Such investigations do not merely offer a direct real space probe of the microstate of thermal systems at a local scale but also a direct determination of the dynamics involved. The possibility of determining directly the state of these systems enables the experimental probing of the effect of external variables such as temperature and applied magnetic field. This introduces the possibility of investigating e.g. the two-dimensional Ising model under applied external field directly for which an analytical solution does not exist. [Methods]{}\ The island array was prepared by magnetron sputtering thin film growth onto a pre-patterned fused silica substrate. The 2$\times$2 mm$^2$ pre-patterned array was fabricated by surface conformal nano-imprint lithography [@Pierret2010_SCIL; @verschuuren_SCIL_2011]. The array was composed of 470 nm $\times$ 160 nm islands arranged in a square lattice with a periodicity of 200 nm along the short axis of the islands (\[10\] direction) and a periodicity of 500 nm along the \[10\] direction. The amorphous film was deposited by magnetron sputtering from a compound target at room temperature. Argon was used as the sputtering gas at a pressure of 3.0 mTorr. Before growth the base pressure of the sputtering system was below $2\times10^{-7}$ Pa. Initially a 2 nm thick layer of Al$_{70}$Zr$_{30}$ was grown onto the pre-patterned substrate followed by the growth of a 7 nm thick Co$_{68}$Fe$_{24}$Zr$_{8}$ and finally a 2 nm thick Al$_{70}$Zr$_{30}$ layer to prevent degradation. After patterning the structural quality of the films was investigated by AFM carried out in contact mode using a Nanosurf Mobile S instrument.\ The thermally ordered state of the array was investigated by magnetic force microscopy using a digital instrument dimension 3100. Topography was imaged by height contrast in tapping mode and magnetic micrographs were scanned in lift mode with a lift scan height of 70 nm. In order to reduce the risk of the stray field from the MFM tip altering individual island states the images were recorded using a low magnetic moment Co alloy MFM tip (PPP-LM-MFMR). Repeated scanning of the same area did not alter the state (see supplementary material).\ The system was modeled as an Ising system, in which every nano-island is considered as a single macrospin with only two possible orientations ($+1$ and $-1$), the simulation cell consists of $150\times 150$ spins. Simulations with both periodic and open boundary conditions were performed. The interactions were only considered between nearest neighbours. The properties of the system were studied via Monte Carlo simulations using the Metropolis-Hastings algorithm. To simulate the process of the spins [freezing]{} at a given configuration and then evolving towards the most stable configuration before any measurement is performed the system is considered to be completely disordered at a temperature higher than its critical temperature. $T_c$. During this thermalization phase no measurements are performed, the system is allowed to evolve in accordance to the usual Metropolis-Hastings algorithm. After a certain number of Monte Carlo steps the system is cooled down and the process is repeated once more. This procedure is then repeated until the desired measurement temperature is reached. Acknowledgements ================ The authors acknowledge the support of the Knut and Alice Wallenberg Foundation, the Swedish Research Council, and the Swedish Foundation for International Cooperation in Research and Higher Education. UBA acknowledges funding from the Icelandic Research Fund grant nr. 141518-051. AB acknowledges the Swedish Research Council (VR) and eSSENCE. The computer simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre (NSC) and High Performance Computing Center North (HPC2N). [28]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [**, ()](\doibase 10.1007/BF02980577) [, ()](\doibase 10.1103/RevModPhys.39.883) [, ()](\doibase 10.1126/science.1214131), [ ()](http://www.scopus.com/inward/record.url?eid=2-s2.0-84860279506&partnerID=40&md5=256ecfc338ce7fd9ae774e422dc619c2) [, ()](http://www.scopus.com/inward/record.url?eid=2-s2.0-84883057884&partnerID=40&md5=5cc3cc71ea64c2560c2b0cb54ac074e1) @noop [, ()]{} @noop [, ()]{} [, ()](\doibase http://dx.doi.org/10.1103/PhysRev.85.808) [, ()](\doibase 10.1126/science.1120506) @noop [, ()]{} [, ()](\doibase 10.1038/nature04447) [, ()](\doibase 10.1103/RevModPhys.85.1473) [, ()](\doibase doi:10.1038/nphys1853) [, ()](\doibase 10.1038/nphys1794) [, ()](\doibase 10.1063/1.4751844) [, ()](\doibase doi:10.1038/nnano.2014.104) [ (), 10.1103/PhysRevLett.107.167201](\doibase 10.1103/PhysRevLett.107.167201) [, ()](\doibase 10.1103/PhysRevB.87.024405) [, ()](\doibase 10.1103/PhysRevB.83.224404) [, ()](\doibase 10.1103/PhysRevB.80.134402) [, ()](\doibase 10.1103/PhysRevB.86.064426) [, ()](\doibase http://dx.doi.org/10.1063/1.3169523) [, ()](\doibase http://dx.doi.org/10.1103/PhysRev.88.1422) [ ()](http://www.scopus.com/inward/record.url?eid=2-s2.0-84882734272&partnerID=40&md5=77223903ed6053fdcb9b4157c440fa85) [ ()](http://www.scopus.com/inward/record.url?eid=2-s2.0-84876748564&partnerID=40&md5=36c756ef69024fd6e0d7a7bc4eee8aa4), [, ()](\doibase 10.1038/NPHYS3037) [ ()](http://www.scopus.com/inward/record.url?eid=2-s2.0-75249091313&partnerID=40&md5=85fa6d96d17198ae3bf67f61203b91c1) [**, ()](\doibase doi:10.1088/0957-4484/22/50/505201)
--- abstract: 'We observed the edge-on Seyfert 1.9 galaxy NGC 2992 with the ACIS CCD array on the Chandra X-ray Observatory, and found several extranuclear ($r \gapprox$ 3$^{\prime\prime}$) X-ray nebulae within 40$^{\prime\prime}$ (6.3 kpc for our assumed distance of 32.5 Mpc) of the nucleus. The net X-ray luminosity from the extranuclear sources is $\sim$2$-$3 $\times$ 10$^{39}$ erg s$^{-1}$ in the 0.3$-$8.0 keV band. The X-ray core itself ($r \lapprox 1^{\prime\prime}$) is positioned at 9$^h$45$^m$41.$^s$95 $-$14$^\circ$19$^\prime$34.$^{\prime\prime}$8 (J2000) and has a remarkably simple power-law spectrum with photon index $\Gamma=$1.86 and intrinsic N$_H=$7 $\times$ 10$^{21}$ cm$^{-2}$. The near-nuclear ($3^{\prime\prime} \lapprox r \lapprox 18^{\prime\prime}$) Chandra spectrum is best modelled by three components: (1) a direct AGN component from the wings of the PSF, or an electron-scattered AGN component, with $\Gamma$ fixed at 1.86, (2) cold Compton reflection of the AGN component with intrinsic absorption N$_H \sim$ 10$^{22}$ cm$^{-2}$, with approximately the same 0.3$-$8.0 keV flux as the direct component, and (3) a 0.5 keV low-abundance ($Z < 0.03 Z_\odot$) thermal plasma, with $\sim$10% of the flux of either of the first two components. The X-ray luminosity of the 3rd component (the “soft excess") is $\approx$1.4 $\times$ 10$^{40}$ erg s$^{-1}$, or $\sim$5$\times$ that of all of the detected extranuclear X-ray sources. We suggest that most ($\sim$75$-$80%) of the soft excess emission originates from a region between radii of 1$^{\prime\prime}$ and 3$^{\prime\prime}$, which is not imaged in our observation due to severe CCD pile-up. We also require the cold reflector to be positioned at least 1$^{\prime\prime}$ (158 pc) from the nucleus, since there is no reflection component in the X-ray core spectrum. Much of the extranuclear X-ray emission is coincident with radio structures (nuclear radio bubbles and large-scale radio features), and its soft X-ray luminosity is generally consistent with luminosities expected from a starburst-driven wind (with the starburst scaled from L$_{FIR}$). However, the AGN in NGC 2992 seems equally likely to power the galactic wind in that object. Furthermore, AGN photoionization and photoexcitation processes could dominate the soft excess, especially the $\sim$75$-$80% which is not imaged by our observations.' author: - 'Edward J. M. Colbert' - 'David K. Strickland' - Sylvain Veilleux - 'Kimberly A. Weaver' title: 'Extranuclear X-ray Emission in the Edge-on Seyfert Galaxy NGC 2992' --- Introduction ============ Although active galactic nuclei (AGNs) usually dominate the X-ray luminosity in Seyfert galaxies, luminous extranuclear X-ray emission regions (hereafter “EXRs”) extending out to radii of $\sim$10$^{2-3}$ pc are fairly common. For example, of the six Seyfert galaxies studied with the ROSAT High Resolution Imager (HRI; spatial resolution $\sim$10$^{\prime\prime}$, energy range 0.2$-$2.4 keV) by Wilson and collaborators (Wilson 1994), four show spatially extended ($\gapprox$5$^{\prime\prime}$) soft X-ray emission. In all four cases, the extended X-ray emission is oriented along the same position angle as the nuclear radio structures or the extended narrow-line regions (ENLRs), suggesting a possible connection between the EXRs and the active nucleus. Typical soft X-ray luminosities of the extended emission are $\sim$10$^{40}$$-$10$^{41}$ erg s$^{-1}$. Soft X-ray [spectral]{} features with similar luminosities were also commonly observed in large-aperture X-ray spectra of Seyfert galaxies, and it is possible that they are in fact the EXRs. In the standard AGN paradigm (e.g., see review by Antonucci 1993), type-1 Seyfert nuclei, which emit a power-law X-ray spectrum, are observed directly (i.e. with little or no absorption), since the line-of-sight to the nucleus is roughly perpendicular to the plane of the obscuring torus. Type-2 Seyfert nuclei are observed at larger angles from the torus symmetry axis, so that the AGN is hidden behind the large absorbing column (N$_H \gapprox$ 10$^{22}$ cm$^{-2}$) of the obscuring torus. Thus, a power-law model with varying degrees of absorption serves as a generic “baseline” model for AGN X-ray spectra (e.g. Mushotzky et al. 1980, Mushotzky 1982). However, as shown by Turner & Pounds (1989) and Turner et al. (1991), $\gapprox$50% of the type-2 Seyfert galaxies observed with EXOSAT and [Einstein]{} show evidence for surplus X-ray emission in the soft band (E $\lapprox$ 2 keV), when the hard-band data are modelled reasonably well with a simple absorbed power-law. This surplus emission is known as “soft excess." Since spatially resolved X-ray spectroscopy on arcsecond scales was not possible before the launch of [Chandra]{}, the origin of the soft excess was not well understood, but some possibilities included (1) X-ray emission from the AGN that “leaks” through holes in the obscuring torus, or possibly around it – the so-called “partial covering” scenario, (2) X-ray emission from the AGN, scattered into our line of sight by electrons along the torus axis, or (3) emission from extranuclear X-ray sources, i.e. EXRs. Since EXRs can be detected with the ROSAT HRI and Chandra, it would be useful to know if they are as common as “soft excess,” and, in particular, if there is a direct relationship between the two. In specific cases, this seems to be true. For example, Weaver et al. (1995) conclude that extranuclear X-ray emission observed in a ROSAT HRI image of the Seyfert galaxy NGC 2110 also produces the soft excess that is observed in the total galaxy BBXRT X-ray spectrum. EXRs in some of the nearest Seyfert galaxies have already been studied with Chandra (e.g. NGC 4151: Ogle et al. 2000, Yang, Wilson, & Ferruit 2001; NGC 1068: Young, Wilson & Shopbell 2001; Cyg A: Young et al. 2002; NGC 3516: George et al. 2002; NGC 4388: Iwasawa et al. 2003). The extranuclear X-ray emission generally has a soft (E $\lapprox$ 2 keV) spectrum, and is typically spatially coincident with high-ionization optical ENLRs, such as \[O III\] $\lambda$5007 nebulae. Their X-ray spectra imply very low abundances ($Z \lapprox 0.1 Z_\odot$) when fit with a simple one-component thermal plasma model. Chandra grating spectra of the brightest regions are well modelled by photoionization and photoexcitation by the AGN (e.g. see Ogle et al. 2003), suggesting there may not be a need at all for collisional excitation (thermal emission) due to a nuclear outflow. Young et al. (2001) conclude that the EXRs in the composite Seyfert-starburst galaxy NGC 1068 are not dominated by starburst phenomena. However, some hot X-ray gas must be produced by the central starburst, and also by any AGN-driven outflows (jets or winds), if they are present. The key is to determine the balance between the outflow and the photoionization/photoexcitation components, since both should be present to some degree. A more systematic statistical study of EXRs in Seyfert galaxies is needed to understand their nature: wind shocks, jet shocks (e.g. NGC 4258, Cecil, Wilson & de Pree 1995), an expanding hot bubble of gas, or even discrete sources of X-ray emission located outside the nuclear region (e.g. NGC 1068, Wilson et al. 1992). Associating EXRs with the AGN is tempting, since the AGN can easily both provide enough energy to drive powerful shocks, and provide enough high-energy photons to ionize gas out to kpc-scales. However, many type-2 Seyfert galaxies also have nuclear starbursts that can produce very luminous X-ray point sources (e.g., Fabbiano, Zezas & Murray 2001, Colbert et al. 2004) and drive galactic superwinds that emit diffuse soft X-ray emission (e.g. Heckman 2004, Strickland et al. 2004a,b). By studying the EXRs with the excellent spatial and spectral resolution of Chandra, we can make great progress in understanding the nature of EXRs, and by studying many of them, determine if there is an exact relationship with the very common “soft excess” observed in of Seyfert galaxies. Here we report details from a single deep Chandra ACIS observation of an interesting case study. We observed the edge-on Seyfert galaxy NGC 2992 with Chandra, since it is known to have a galactic-scale outflow that has been well-studied at many wavelengths: optical (e.g., Colbert et al. 1996a, Allen et al. 1999, Veilleux, Shopbell, & Miller 2001, Garcia-Lorenzo, Arribas, & Mediavilla 2001), near infrared (e.g. Chapman et al. 2000), radio (Ward et al. 1980, Hummel et al. 1983, Colbert et al. 1996b), and X-ray (Colbert et al. 1998). This galaxy has a wide-angled galactic outflow (Veilleux et al. 2001) and a “diffuse” (as opposed to “linear”) sub-kpc radio structure. Since there is heavy obscuration in the nuclear region, it is difficult to tell whether NGC 2992 has a strong nuclear starburst or not. The galactic outflow in NGC 2992 may be quite different from those in Seyfert galaxies that have already been observed with [Chandra]{} (e.g. NGC 2110, M51, and NGC 4151), which have “linear” nuclear radio structures. Understanding its origin of the EXRs and the outflows may offer important clues to the “starburst-AGN connection.” Since NGC 2992 has very well studied ENLRs and also has a kiloparsec-scale outflow, it is an ideal galaxy to study with Chandra. In Figure 1, we show a $V$-band image of NGC 2992 and its companion galaxy NGC 2993. This interacting pair is also known as Arp 245. A comprehensive multi-wavelength study of the interacting system is presented in Duc et al. (2000). NGC 2992 has morphological type Sab, while NGC 2993 has type Sa (Sandage & Bedke 1994). The projected separation between the two galaxies is $\sim$3$^\prime$ (28 kpc, assuming a distance[^1] of 32.5 Mpc). Optically luminous tidal plumes extending from both galaxies are evident in deep optical images (e.g. Sandage & Bedke 1994, Duc et al. 2000), and some evidence of them can be seen in Figure 1. We describe the observations and data reduction in section 2, and give results in section 3. An interpretation of the X-ray emission as starburst-driven and AGN-driven winds are given in section 4. Observations and Data Reduction =============================== The edge-on Seyfert galaxy NGC 2992 and its companion NGC 2993 were observed with the Chandra X-ray Observatory (CXO) in the AO4 cycle, on UT dates 16$-$17 February 2003 (Obs. ID 3956). The ACIS-S CCD array was used, and the pointing center was chosen such that the two galaxies were centered in the back-illuminated S3 chip (CCD 7). As a result, the off-axis angle of the nucleus of NGC 2992 was 3.$^{\prime}$26. The satellite roll angle was constrained to 353$^\circ$ so that the CCD readout column would be oriented along the major axis of NGC 2992, and the anticipated readout “streak” (due to the very bright nuclear point source) would not interfere with X-ray emission extending along the galaxy minor axes. The total on-source EXPOSURE time of the level-1 events file was 50.23 ks, using a CCD frame time of 3.2 s. The CCD temperature during the observation was $-$120$^\circ$ C. The level-1 events data were re-processed with the [acis\_reprocess\_events]{} script using the CXO data reduction software package CIAO v3.0.2, and Chandra calibration database CALDB v2.26. No CTI correction was performed due to pile-up effects of the nuclear source. After re-processing, the filtered level-2 events had a total EXPOSURE time of 49.54 ks. Two separate level-2 event files were created. The first was made using the normal event filtering (i.e., keeping those events with “good” ASCA event grades of 0, 2, 3, 4, and 6). This standard filtering is appropriate for unpiled events, but piled events tend to migrate from “good” grades to “bad” grades (see Chandra POG[^2]), so we created another level-2 event file for photons with all event grades. The second events file is [only]{} used to illustrate which pixels near the X-ray core are contaminated by pile-up (see section 3.1.1). Spectra were extracted from the first (“good”) level-2 event file for various regions using the CIAO [ACISSPEC]{} script (v3.2), which creates weighted ancillary response files (ARFs) and response matrix files (RMFs) for extended spatial regions. Results ======= Spatial Analysis ---------------- In Figure 2, we show several images of the large-scale X-ray emission from NGC 2992/3. In 1992 and 1994, soft (0.2$-$2.4 keV) X-ray emission was detected from both galaxies with the ROSAT High-Resolution Imager (HRI) instrument (Figure 2a). For comparison with the HRI image, we constructed a soft X-ray image from the ACIS data, using photons with energies 0.3$-$2.4 keV (Figure 2b). The readout streak, due to CCD pile-up from the very bright X-ray nucleus of NGC 2992, is clearly seen in the ACIS images. ### X-ray Point Sources In addition to the very bright nucleus of NGC 2992, we detect twenty[^3] X-ray point sources in the ACIS CCD that includes the NGC 2992/3 complex. We list X-ray properties of the twenty point sources in Table 1, and display the positions of the sources nearest to NGC 2992/3 in Figure 3. There are four sources within the R$_{25}$ ellipses of the two galaxies: source 7 (L$_X \approx$ 10$^{39}$ erg s$^{-1}$) is 17$^{\prime\prime}$ (2.7 kpc) southeast of the nucleus of NGC 2992, and sources 9, 10, and 11 (L$_X \approx$ 5 $\times$ 10$^{39}$ erg s$^{-1}$) are all within $\sim$10$^{\prime\prime}$ (1.6 kpc) of the center of NGC 2993. There are three other sources within ellipses corresponding to R$=$2R$_{25}$: source 5 (NGC 2992), and sources 8 and 13 (NGC 2993). ### Position of the X-ray Nucleus The X-ray point source in the nucleus of NGC 2992 is obviously the dominant source of X-ray emission. Its X-ray luminosity of $\sim$10$^{42-43}$ erg s$^{-1}$ (e.g. Gilli et al. 2000) indicates that the emission is most likely coming from the Seyfert nucleus (Ward et al. 1980). We determined the position of the X-ray core using two independent methods. We first fit elliptical isophotes to the full-band X-ray image of the nuclear source, using the [ELLIPSE]{} task in STSDAS v3.2 (with IRAF v2.12.2). Next, we obtained very accurate constraints for one dimension of the X-ray position by finding the centroid of the CCD readout streak. The X-ray nucleus of NGC 2992 is positioned off-axis by $\approx$3.26$^{\prime}$, so that the ACIS point-spread function is asymmetric, and approximately elliptical. In the top two panels of Figure 4, we show grayscale maps of the raw X-ray counts for all (“good” and “bad”) event grades, and also for just the “good” event grades (see section 2). Grade-migration due to pile-up is not noticeable in the lowest three contours 30, 60, and 90 counts pixel$^{-1}$, where the pixel size is 0.492$^{\prime\prime}$), but is very noticeable at the fourth contour level (120 counts pixel$^{-1}$). For comparison, the three ellipses in Figure 4d are for best-fit surface brightnesses of 52, 75, and 110 counts pixel$^{-1}$. The centroiding uncertainty for each of the three ellipses, as reported by the [ELLIPSE]{} routine, is very small ($\approx$0.1 pixel, or $\sim$0.05$^{\prime\prime}$). The three positions differ by $\lapprox$0.25$^{\prime\prime}$, and we assume this to be the net uncertainty of this method. In Figure 4d, we show three $r=$0.25$^{\prime\prime}$ circles, labeled $A$, $B$, and $C$, for the inner, middle, and outer ellipses, respectively. Crosses are marked centers of the three circles. As mentioned, we also used the readout streak in the ACIS image to constrain the position of the X-ray core. For very bright X-ray point sources, a significant number of events are detected during the very brief CCD readout of the CCD (40 $\mu$s of each 3.2 s frame; see Chandra POG). At the end of the readout time, these events are left scattered along the CCD column of the X-ray point source. By rotating our ACIS image by the roll angle (7.05$^{\circ}$) and finding the best-fit image column for the center of the readout streak, we were able to construct a line in the original (unrotated) image that accurately estimates one linear dimension of the position of the X-ray core (see Figure 4, lower two panels). We used IRAF [IMEXAMINE]{} to fit Gaussian profiles across the streak, for positions $\approx$30$-$100$^{\prime\prime}$ above and below the X-ray point source. The standard deviation for the best-fit column values was 0.3 pixels. As can be seen in the lower left panel of Figure 4, both our ellipse fitting and our readout streak method give consistent results. A position mid-way between the three ellipse centroids and the readout streak marks our best estimate for the position of the NGC 2992 X-ray peak (9$^h$45$^m$41.95$^s$ $-$14$^\circ$19$^{\prime}$34.8$^{\prime\prime}$ \[J2000\]). This position was estimated by eye; however, the uncertainty is a fraction of a pixel. We mark this position with a diamond symbol in Figure 4d. The estimated nominal uncertainty in absolute astrometry for Chandra is $\approx$0.5$^{\prime\prime}$ (e.g., Aldcroft 2002). Thus, we also show an error circle for the position of the X-ray core, as a dashed circle of radius 0.5$^{\prime\prime}$. In section 4, we compare our measured position of the X-ray core with positions of features in other wavebands. ### Near-nuclear X-ray Emission Here we concentrate on extra-nuclear emission within $\sim$10$^{\prime\prime}$ of the X-ray nucleus of NGC 2992 ($\lapprox$1.58 kpc at a distance of 32.5 Mpc). In Figure 5, we show grayscale plots of the near-nuclear X-ray emission in four different energy ranges: very soft (0.3$-$0.5 keV), soft (0.3$-$1.0 keV), hard (4.0$-$8.0 keV), and full-band (0.3$-$8.0 keV). We also show a smoothed image of the CHART/MARX PSF in the full band, and compare it with a smoothed image of our full-band ACIS observation. The slight elongation of the PSF along PA $\sim$ 45$^\circ$ is due to the source being off-axis by $\sim$3.3$^{\prime}$. This elongation is noticeable in all of the full-band images, and in the hard-band image. There is no obvious extended (i.e., asymmetric with respect to the elongated PSF) hard X-ray emission (Figure 5e), but there [is]{} extended (asymmetric) soft X-ray emission from $\sim$2$-$4$^{\prime\prime}$ (0.3$-$0.6 kpc), toward the SE (see Figures 5c and 5f). Throughout the rest of the paper, we refer to this extended X-ray source as source ‘SE.’ Due to the severity of the CCD pile-up $\lapprox$2.5$^{\prime\prime}$ (see section 3.1.2), estimating the flux of this extended soft X-ray source within $\sim$2.5$^{\prime\prime}$ is not straightforward. The total number of 0.3$-$1.0 keV counts between 2.5 and 5.0$^{\prime\prime}$ is 242$\pm$16, 360$\pm$19, 126$\pm$11, and 60$\pm$8 for the four quadrants 1$-$90$^\circ$, 91$-$180$^\circ$, 181$-$270$^\circ$, and 271$-$360$^\circ$, respectively. Under the assumption that the first and fourth quadrants are background plus AGN contamination, source SE (i.e., the second and third quadrants) has $\sim$100$-$200 net counts in the 0.3$-$1.0 band. This equates to an observed 0.3$-$8.0 keV luminosity of $\sim$1$-$2 $\times$ 10$^{39}$ erg s$^{-1}$ for a 0.5 keV bremsstrahlung model with Galactic absorption. Using the [Einstein]{} HRI (E $\approx$ 0.1$-$3.5 keV), Elvis et al. (1990) claimed to detect extended emission within a radius of 10$^{\prime\prime}$, with an X-ray luminosity of $\approx$ 4 $\times$ 10$^{41}$ erg s$^{-1}$ (adjusted for our distance, and using a 0.25 keV Bremsstrahlung model with Galactic absorption). The [Einstein]{} emission was detected in the quandrant centered at PA $\sim$ 23$^\circ$, but no emission was found in the other three quadrants. Since this conflicts with our Chandra results, we suggest that the [Einstein]{} detection was either due to a variable X-ray source, or was spurious. ### Kiloparsec-Scale X-ray Emission Next, we investigate extended X-ray emission on scales $\gapprox$10$^{\prime\prime}$ ($\gapprox$1.6 kpc at 32.5 Mpc). To investigate the presence of any extended emission in our Chandra image, we computed X-ray counts in radial bins as a function of angle. We used eight 45$^\circ$ angular wedges with the first octant starting at PA $-$15.5$^\circ$ (see Figure 6). Octants 1 and 5 were centered on the CCD readout streak and were ignored in the rest of the analysis. We modelled the PSF of the X-ray core in a number of different ways, none of which were ideal. We first tried to simulate the PSF with the Chandra Ray Tracer[^4] (ChaRT), using the spectrum of the X-ray core (section 3.2.1) as an input model. The ChaRT simulation was performed with the same off-axis angle and telescope roll angle as our ACIS observation. The output SAOsac ray-tracing files were converted to event files with the MARX[^5] Chandra simulator. We plot radial profiles for the CHART/MARX PSF in Figure 6a. The background level (0.1 counts pixel$^{-1}$) was added to the CHART/MARX PSF. While this experiment was useful to show that no large-scale azimuthal variation is expected for our observational parameters, the CHART/MARX PSF falls off more steeply than the data for all octants (see Figure 6a; see also Smith, Edgar & Shafer 2002). We attribute this difference to the fact that MARX does not account for aspect dither, which makes the simulated PSF narrower than the actual one (K. Ishibashi, priv. comm.). We first tried to use radial profiles from three octants that appeared free of excess emission (regions 2, 6 and 8 in Figure 6b). We found no significant excess emission for hard X-ray emission with energies $\ge$ 2.5 keV, but did find excess emission in the softer bands. We first analyzed the “ROSAT" band from 0.3$-$2.4 keV, and later experimented with smaller energy ranges in the soft band, and with harder bands. Octant 6 appeared to have some possible excess X-ray emission, so we downgraded our radial profile to include only octants 2 and 8, both of which did not show significant excess after subsequent PSF subtraction (see Figure 6d). The most significant excess extends eastward along PA $\sim$ 190$^\circ$, out to $\sim$40$^{\prime\prime}$, roughly perpendicular to the galaxy disk (see Figure 6c). Using the combined radial profiles from regions 2 and 8 for a PSF, we calculate an excess of 90$\pm$22 X-ray counts in this octant, from 20$-$40$^{\prime\prime}$ (3.2$-$6.3 kpc). We found three other regions which had significant ($>$3$\sigma$) excess over our model PSF (see Figure 6d): a point-like source in region 4 (source 7 in Table 1) with 82$\pm$19 counts (from 15$-$25$^{\prime\prime}$), diffuse emission with 75$\pm$22 counts in region 6, from 15$-$30$^{\prime\prime}$, and an extended source nearer to the nucleus (region 7, 10$-$20$^{\prime\prime}$), with 74$\pm$24 counts. We then constructed an image from our model PSF and subtracted it from from the Chandra image to show the structure of these sources (see Figure 6d). The 0.3$-$2.4 keV Chandra image was first smoothed with a Gaussian kernel of $\sigma=$1.5$^{\prime\prime}$. The PSF was scaled by trial and error so that after subtraction, the net counts for extra-nuclear regions within $\sim$2$^{\prime}$ were zero. This was done by examining radial and horizontal cuts across the image and ensuring that the pixel values in source-free regions far away from the nucleus fluctuated about 0.0 counts pixel$^{-1}$. For comparison, we also show the (unsubtracted) 0.3$-$2.4 keV Chandra image Figure 6b. No excess emission was found from our radial profile analyses of the hard (2.4$-$8.0 keV) X-ray emission. While not enough counts were typically available in narrow energy bands, we did notice that the point-like source in region 4 (point source 7 in Table 1) was significantly harder than expected for thermal plasma emission with kT of $\lapprox$1 keV, with a significant number of photons with $E \gapprox$1.5 keV. We list excess counts in several other energy bands in Table 2. Since the angular breaks that define our octants often pass through well-defined X-ray sources, we also list X-ray counts and 0.3$-$2.4 keV luminosities for the most well-defined clumps in the PSF-subtracted image. These sources are labelled $e3a^$, $e3b$, $e6^$, and $e7^$ in Table 2, and their defining regions are shown in Figure 9. Those sources with an asterisk fall outside the angular range defined by their octants: for example, $e7^$ is primarily in octant 7, but also extends into octant 6. Spectroscopy ------------ ### Nuclear Source We extracted a spectrum of the X-ray core from the CCD readout streak following a prescription from the CXO calibration team (R. Smith, priv. comm.). We note here that the events in the readout streak do not suffer from CCD pile-up. Source photons from the readout streak were taken from two thin rectangular regions extending $\sim$0.75$-$4$^{\prime}$ above and below the nuclear source. Background counts were subtracted, using four thin rectangular regions, one both sides of the readout streak. Ancillary Response Files and Response Matrices were generated for a small circular region enclosing the X-ray core. The source spectrum was then grouped so that it had a minimum of 20 counts per spectral bin. Since the fraction of the total counts that are detected in the streak is not well calibrated, the absolute normalization obtained from spectral fitting is not useful. However, the shape of the spectrum is very useful. Restricting the energy range to 0.3$-$8.0 keV, we obtain a spectrum with 6744 source counts. We use XSPEC v11.3.0 for spectral fitting. An external Hydrogen column was fixed at the Galactic value of 5.26 $\times$ 10$^{20}$ cm$^{-2}$ (Dickey & Lockman 1990). An additional intrinsic Hydrogen column was allowed to vary during the fit. With a simple absorbed power-law model, we obtain a very good fit ($\chi^2/dof =$ 197.0/205), with a power-law model with $\Gamma =$1.86$\pm$0.08[^6],and N$_H =$ 0.71$\pm$0.05 $\times$ 10$^{22}$ cm$^{-2}$. Adding a neutral (E $=$ 6.4 keV) Fe K$\alpha$ line as a Gaussian line only improves the fit by $\Delta\chi^2 \approx$ 0.2. This power-law slope is consistent with the 1997$-$1998 large-aperture SAX spectrum ($\Gamma\approx$1.7; Gilli et al. 2000), but is not consistent with the much flatter ASCA spectrum taken in 1994 ($\Gamma\approx$1.2; Weaver et al. 1996). Photon indices of $\Gamma\approx$1.9 are typical in type-1 Seyfert nuclei (e.g. Nandra & Pounds 1994; Nandra et al. 1997), where the X-ray AGN is viewed directly. Since we know the X-ray core streak spectrum comes from a very small region $\approx$1$-$2 pixels across ($\approx$75$-$150 pc), and its spectrum is not consistent with Compton reflection, we conclude that it must be either emission directly from the AGN, as in type-1 Seyfert galaxies, or gray scattering of the AGN emission from electrons within $\sim$150 pc ($\approx$1$^{\prime\prime}$) of the AGN. In the next section, we assume the direct spectrum of the AGN is given by the streak spectrum, and use this to constrain the AGN component of the spectrum within $\sim$3 kpc of the nucleus. ### Near-Nuclear X-ray Emission Using the [ACISSPEC]{} CIAO tool, we extracted a spectrum of the X-ray emission within a radius of 18$^{\prime\prime}$ (2.8 kpc), centered on the X-ray core (see section 3.1.2). We excluded the very central region (with piled events) by rejecting all events within the “middle” ellipse in the lower left panel of Figure 4 (semi-major and semi-minor axes 3.4$^{\prime\prime} \times$ 2.7$^{\prime\prime}$). We also excluded events from a faint point source (source 7, Table 1), and from a thin rectangular region enclosing the CCD readout streak. This extraction region thus includes a fraction of the X-ray core (section 3.1.2), all of source SE (section 3.1.3), some of the emission from extended sources e6 and e7 (section 3.1.4), and the accumulation of any other “undetected” extended X-ray sources within 2.8 kpc of the nucleus. Background counts were taken from four remote source-free rectangular regions adjacent to the nuclear region. The $\approx$14,600-count 0.3$-$8.0 keV source spectrum was binned to $>$20 counts bin$^{-1}$ and analyzed with XSPEC. We list the results from spectral fitting in Table 3. All models include an additional neutral Hydrogen absorption component, fixed at 5.26 $\times$ 10$^{20}$ cm$^{-2}$ (Galactic value from Dickey & Lockman 1990). Since much of the emission from the AGN point-source has been rejected due to pile-up and much of the remaining emission has been omitted by our extraction region, the absolute fluxes we obtain from the AGN spectral components are not useful for comparison with previous results (e.g. ASCA). The near-nuclear spectrum is fairly well fit ($\chi_\nu^2 =$ 1.13) by a shallow ($\Gamma \approx$ 0.9) absorbed power-law. However, since some of the photons in source region are from the PSF wings of the X-ray core ($\Gamma =$ 1.86; section 3.2.1), we must reject this simple model. We thus included a power-law component with the slope fixed at $\Gamma =$ 1.86, which was the value determined from the spectrum of the CCD readout streak (i.e. one or two pixels from the X-ray core). Since the PSF wings from the X-ray core will contribute in our near-nuclear region ($r >$ 18$^{\prime\prime}$), a $\Gamma =$ 1.86 power-law component should be present. The absolute normalization of the streak spectra is not well calibrated (R. Smith, priv. comm.), so we left the normalization to the $\Gamma =$1.86 power-law free while fitting. We did not correct for the difference in the PSF wings for soft and hard energies (e.g., Allen et al. 2004), but we note that the PSF wings are quite similar for energies $\sim$4.0$-$8.0 keV, and this is the energy range for which much of the events from the power-law component reside. A simple absorbed power-law, with $\Gamma$ and $N_H$ fixed at the same values as for the X-ray core, is a very poor fit ($\chi_\nu^2 \approx$ 3.7). However, if a second absorbed power-law is added (“partial covering"), a good fit ($\chi_\nu^2 =$ 1.05; $\chi^2/\nu =$ 372/354) is obtained, with $\Gamma \approx$ 0.5, and N$_H \sim$ 10$^{20}$ cm$^{-2}$ for the second component. In this model, the flat power-law component has an observed 0.3$-$8.0 keV flux $\sim$2.5 times that of the first (AGN) component. While we do not view this partial covering model as physically reasonable, it does show the importance of components other than direct or electron-scattered AGN emission. If the second component is replaced with a [MEKAL]{} thermal plasma model, the fit is very poor: $\chi^2_\nu \approx$ 2.8, for either fixed (solar) or variable abundances (best fit at Z $\approx$ 0.1Z$_\odot$). We next tried a cold reflection model ([PEXRAV]{}) for the second component, fixing the input power-law spectrum at $\Gamma=$1.86. This is again a fairly good fit ($\chi_\nu^2=$1.18), although it is still significantly poorer than the partial covering model, with a $\Delta\chi^2$ of 37. Adding a third component to the fit (a Mekal thermal plasma model) significantly improves the fit ($\chi_\nu^2=$1.03). Our best fit model (PL$+$PEXRAV$+$Mekal, see Table 3, model 3b) is obtained for a low-abundance plasma with kT$\approx$0.5 keV and Z$\approx$0.01Z$_\odot$. Note that the best-fit PEXRAV components are almost completely reflection-dominated, i.e. they have very large $R$ values. We discuss the issue of low abundances further in section 4.2.3. Compared with the partial-covering model, $\chi^2$ is lower for this model by a difference of 10. We take model 3b as our best model and measure observed 0.3$-$8.0 keV luminosities of 2.3 $\times$ 10$^{41}$ erg s$^{-1}$ for the reflected component, and 1.4 $\times$ 10$^{40}$ erg s$^{-1}$ for the “thermal” component. In Figure 7, we show the near-nuclear spectrum, together with the various model components. We find no evidence for an Fe K line in the near-nuclear spectrum. If we add in a Gaussian line fixed at 6.4 keV to model 1a, 1b (fixed PL models), or our best-fit model (3b), an equivalent width $\approx$50 eV line is fit best, but $\Delta\chi^2$ $\lapprox$ 1. Although our proposed reflection component in the near-nuclear spectrum may well produce a strong Fe K line, there is also significant direct AGN emission in the composite spectrum. For example, for our best-fit model (3b), $\sim$ 50% of the 0.3$-$8.0 keV flux comes from the direct (PL) component (see Table 3). Thus, any Fe K line emission produced by cold Compton reflection may just be washed out. To test this hypothesis, we used a single PEXRAV model with $\Gamma =$ 1.86 for the AGN component, and added a narrow Gaussian line fixed at 6.4 keV. The fit was quite reasonable ($\chi^2 =$ 378.1) compared with other models in Table 3. The best-fit value of $R$ was 15.87. The Gaussian line was best fit with an equivalent width of 24 eV and an upper limit ($\Delta\chi^2 =$ 2.7) of 674 eV. If only the reflection component is allowed by the PEXRAV model, the equivalent width increases by a factor of 1.63. Therefore, the upper limit one would expect if the direct component was not present is $\sim$1.1 keV, which is more consistent with a reflection-dominated X-ray spectrum. Properties of the Soft Excess Emission -------------------------------------- As mentioned in section 3.1.4, the extended X-ray sources e6 and e7 with net 0.3$-$8.0 keV luminosity L$_X =$ 1.2 $\times$ 10$^{39}$ erg s$^{-1}$ (Table 2) comprise part ($\sim$10%) of the “thermal” soft excess (L$_X \approx$ 1.4 $\times$ 10$^{40}$ erg s$^{-1}$), as does the source SE with L$_X \sim$ 1$-$2 $\times$ 10$^{39}$ erg s$^{-1}$ (section 3.1.3). It is quite possible that the unaccounted ($\sim$75$-$80%) soft excess emission could be explained by undetected or unresolved hot, extranuclear gas clouds, although diagnosing this with the existing ACIS data is not very practical, due to the strong X-ray core and uncertainties in the PSF. Since the AGN core spectrum extracted from the readout streak does not show any evidence for soft excess, an alternative explanation is that the unaccounted soft excess originates from very luminous (L$_X \sim$ 10$^{40}$ erg s$^{-1}$) EXRs in the region between $\approx$1$-$2.5$^{\prime\prime}$. A fraction of these photons would then be repositioned to larger radii by the Chandra PSF and so would have been included in our near-nuclear spectrum ($r \gapprox$ 3$^{\prime\prime}$). Discussion ========== As we have shown, we detect $\sim$2$-$3 $\times$ 10$^{39}$ erg s$^{-1}$ of extended soft X-ray emission in NGC 2992, and find evidence for $\sim$1 $\times$ 10$^{40}$ erg s$^{-1}$ in a “low-abundance thermal" spectral component. The AGN nucleus ($r \lapprox$ 1$^{\prime\prime}$) is well fit by a power-law, but a larger aperture spectrum of the near-nuclear region requires a significant cold reflection component. In this section, we discuss possible scenarios for the emission processes producing the X-ray emission from the nuclear source, the extended sources e3, e6, e7 and SE, and the soft excess. The Nuclear Point Source ------------------------ Large-aperture 2.5$^{\prime}$ ASCA observations of the X-ray nucleus in 1994 by Weaver et al. (1996) yielded similar [observational]{} results to our near-nuclear model 1a (i.e., when both datasets are modelled with a simple absorbed power law; $\Gamma \sim$ 1 and N$_H \sim$ 2 $\times$ 10$^{21}$ cm$^{-2}$). This [observational]{} model is obviously an inadequate description of the emission components, but it shows general similarity in the spectral shape. Weaver et al. model the ASCA spectrum with both Compton reflection and gray scattering (partial-covering model) of a $\Gamma =$ 1.7 nuclear source that was hidden behind an absorbing column of $\sim$10$^{22}$ cm$^{-2}$. However, BeppoSAX observations in 1997 and 1998, with a similar aperture, were well fit by a simple power-law model with $\Gamma \approx$ 1.7 and N$_H \approx$ 10$^{22}$ cm$^{-2}$ (Gilli et al. 2000). Thus, as shown by Weaver et al. (1996), the cold (Compton) reflection component is historically variable on the scale of a few years. The fact that we do not see evidence for cold reflection in the nuclear streak spectrum but we do see evidence for it in the near-nuclear ($r \gapprox$ 3$^{\prime\prime}$) spectrum implies that the reflection region is located outside of the 1$-$2 pixels that are clocked along the readout streak. Thus, we argue that the cold reflection region is located at radii $r \gapprox$ 0.5$-$1.0$^{\prime\prime}$, or $r \gapprox$ 75$-$150 pc. We show the positions of the X-ray core in optical and radio images of the nuclear region in Figure 8. The V-band, H$\alpha$, and \[O III\] peaks (panels a, b, and d) are displaced from the X-ray core position by 1$-$2$^{\prime\prime}$ to the northwest. This displacement could be an obscuration effect, or it could be due to alignment uncertainties between the optical and X-ray frame. Unfortunately, CCD pile-up prevents reliable imaging of any of the ACIS X-ray emission near the peak of the NW “ionization cone” (i.e., the NW H$\alpha$ and \[O III\] peak) and we cannot diagnose if there is another X-ray source there, such as the cold reflector. Veilleux et al. (2001) and Garcia-Lorenzo et. al (2001) find that the stellar kinematic center is positioned $\approx$1$^{\prime\prime}$ [southwest]{} of the \[O III\] peak. Thus, the supermassive black hole, as marked by the X-ray and radio core, may not yet have settled into the center of the galaxy. However, since gas motions are quite complex in the center of NGC 2992 (e.g., Veilleux et al. 2001), and the galaxy disk is viewed nearly edge-on, perhaps the X-ray/kinematic center displacement is consistent with uncertainties in modelling the optical data. The X-ray core is, however, coincident with the 6 cm radio core – the apex of the two diffuse “radio bubbles”, a.k.a. “figures-of-eight” (see Figure 8c). This suggests that the radio bubbles may be inflated by the AGN (e.g. Wehrle & Morris 1988), and not a starburst, since NIR observations suggest that the youngest star-forming region is located in the dust lane (Chapman et al. 2000), a few arcseconds northwest of the X-ray core and the radio core. Extended X-ray Emission and Soft Excess --------------------------------------- ### Near-nuclear X-ray Sources The diffuse soft X-ray emission within 5$^{\prime\prime}$ is displayed against the optical and radio emission in Figure 8. The SE X-ray source, which has a centroid along PA $\sim$ 160$^\circ$, is coincident with the SE radio bubble, but there is no luminous optical line or continuum emission there. It is thus possible that the X-rays from the SE source are collisionally excited emission produced by outflowing gas in the bubble. The apparent “low abundance” of this X-ray gas is discussed in section 4.2.3. Allen et al. (1999) and Veilleux, Shopbell, & Miller (2001) find very high velocities for the ENLR gas in the SE cone, as high as 670 km s$^{-1}$ (Allen et al. 1999). Shock-model fits to the emission line ratios from that region imply shock velocities of $\sim$300$-$500 km s$^{-1}$ (Allen et al. 1999). A shock velocity of 500 km s$^{-1}$ will produce thermal X-rays with temperature $\approx$0.3 keV, which is certainly consistent with the results from our spectral fitting (kT $=$ 0.3$-$0.8 keV, model 3b, Table 3). Based on kinematic modelling of the H$\alpha$-emitting gas, Veilleux et al. (2001) estimate a kinetic energy luminosity of L$_{KE} \approx$ 10$^{40}$ (n$_e$/100 cm$^{-3}$)$^{-1}$ erg s$^{-1}$. Starburst-driven winds yield X-ray luminosities of $\approx$0.03 L$_{KE}$ (e.g. Strickland 2004), implying $\sim$3 $\times$ 10$^{38}$ erg s$^{-1}$ of soft X-rays from the nuclear outflow in NGC 2992. This is an order of magnitude lower than L$_X$ of the SE X-ray source; however, since L$_X \propto$ n$_e^2\phi$, where n$_e$ is the electron density and $\phi$ is the volume filling factor, it is possible that the physical conditions in the SE region could produce a significantly larger X-ray luminosity in NGC 2992. Compared with typical ENLR gas, the hot X-ray emitting gas is quite tenuous. Assuming a 0.2$-$2.4 keV emissivity of 10$^{-22.4}$ erg cm$^{3}$ s$^{-1}$ (Suchov et al. 1994), we derive gas densities $n \sim$ $\phi_X^{-1/2} \times$ (0.3$-$2) $\times$ 10$^{-2}$ cm$^{-3}$ for the four well-defined X-ray sources in Table 2 ($e3a^, e3b, e6^$, and $e7^$; see Colbert et al. 1998 for detailed description of similar calculations). Here $\phi_X$ is the volume filling factor of the X-ray emitting gas, and is distinct from $\phi$ for the optical line emitting gas. ENLR clouds have much larger electron densities: $\sim$10$^{2-4}$ cm$^{-3}$ (e.g. Robinson 1989, 1994). The implied pressure $p = 2nkT$ of the X-ray gas, assuming kT=0.5 keV (as in Table 2), is $p \sim \phi_X^{-1/2} \times$ (0.5$-$3) $\times$ 10$^{-11}$ dyne cm$^{-2}$, which is reasonably close to $n_e kT$ of ENLR clouds, suggesting there may be pressure equilibrium between the two gas phases. The cooling-time of the X-ray gas can also estimated (cf. Colbert et al. 1998), and we find $t_{cool} \sim \phi_X^{+1/2} \times$ (3$-$20) $\times$ 10$^{7}$ yr. ### Kiloparsec-scale X-ray Sources As mentioned in section 3.1.4, we find three diffuse extranuclear X-ray sources (e3, e6 and e7), each with L$_X \approx$ 6 $\times$ 10$^{38}$ erg s$^{-1}$ (Table 2). These sources are located at larger distances ($r =$ 10$-$40$^{\prime\prime}$, or 1.6$-$6.3 kpc; see Figure 6d). In Figure 9, we show the relative positions of these sources with respect to the kpc-scale optical and radio emission. We also label the well-defined X-ray clumps in the image as $e3a^$, $e3b$, $e6^$, and $e7^$ (see section 3.1.4). A kpc-scale 20 cm radio “arm” extends eastward from the nucleus in the same PA as source e3, and reaches the inner part of this X-ray source. Thus both could be generated by a kpc-scale nuclear outflow. Source e3 is also positioned within the opening angle of the SE ionization cone (see Figure 9d), and thus AGN photoionization is also a possible emission mechanism. ### Soft Excess As summarized by Ogle et al. (2003), many of the extended X-ray sources found in Seyfert galaxies are well modelled with pure photoionization and photoexcitation models — collisional excitation is not required. The characteristic observational spectral signature of soft excess in non-grating Chandra and XMM observations is a one- or two-temperature thermal model [with near-zero abundances]{} (e.g. Bianchi et al. 2003, Schurch et al. 2002, Done et al. 2003), which is precisely the same as our results for NGC 2992. The spectra of the brightest extranuclear X-ray nebulae in NGC 1068 and NGC 4151 are well fit by AGN photoionization and photoexcitation models, with no need for collisional excitation (e.g. Sako et al. 2000, Sambruna et al. 2001, Kinkhabwala et al. 2002, Ogle et al. 2003). However, the exact same “low-abundance” phenomenon occurs for X-ray emission from starburst-driven superwinds when they are fit with a one-temperature thermal model (e.g. Ptak et al. 1997). A high-quality grating spectra of the starburst galaxy M82 is much better represented with multi-temperature abundances (Read & Stevens 2002). Strickland et al. (2004a,b) find that a [physically reasonable]{} solution for Chandra CCD (non-grating) spectra of superwinds is a two-temperature model with low Fe abundances. Thus, the low metallicity implied by our simple single-temperature “thermal plasma” component (e.g. model 3b, Table 3) does not necessarily imply that the soft excess (which includes e3, e6, e7 and SE) is produced solely (or even partially) by AGN photoionization. The result is ambiguous. AGN-driven Outflow [vs.]{} Starburst-driven Outflow ----------------------------------------------------- Does NGC 2992 have a nuclear starburst that could indeed power a galactic wind? It has a total FIR luminosity of 7 $\times$ 10$^{43}$ erg s$^{-1}$ (Moshir et al. 1997), which suggests a SFR of $\approx$4 M$_\odot$ yr$^{-1}$, assuming the FIR emission from the AGN can be neglected and the SFR $=$ 5.7 $\times$ 10$^{-44}$ L$_{FIR}$ (cf. Colbert et al. 2004). The IRAS colors of NGC 2992 are more similar to those of starbursts than to “pure" (not composite starburst-Seyfert) Seyfert galaxies (e.g., Colbert 1997). Strickland et al. (2004b) show empirically that X-ray halo emission from a starburst L$_{halo} \approx$ 4 $\times$ 10$^{-5}$ L$_{FIR}$, which suggests NGC 2992 should emit $\approx$ 3 $\times$ 10$^{39}$ erg s$^{-1}$ in soft X-rays from the hypothetical 4 M$_\odot$ yr$^{-1}$ starburst. Based on the range in L$_{halo}$/L$_{FIR}$ for the starburst galaxies studied by Strickland et al. (2004b), the starburst wind component could be as much as a factor of 2 more: L$_{halo}^{max} \approx$ 6 $\times$ 10$^{39}$ erg s$^{-1}$. This is certainly consistent with the 20$-$25% of the soft excess that we image with Chandra (L$_X^{net} \approx$ 3$-$4 $\times$ 10$^{39}$ erg s$^{-1}$ from sources e3, e6, e7, and SE). However, detecting a nuclear starburst in NGC 2992 is difficult, due to heavy obscuration and the bright X-ray AGN. It is quite possible that the outflow (and thus the extranuclear X-ray sources) may be energized by the AGN instead. Soifer et al. (2004) find that $\approx$50% of the 12$\mu$m IRAS emission from NGC 2992 is positioned within 0.$^{\prime\prime}$34 (50 pc) and conclude that this compact 10$\mu$m emission is produced by an AGN instead of by a nuclear starburst. It is not clear, however, if this is also true of the higher wavelength IRAS fluxes (e.g. 60$\mu$m and 100$\mu$m), which are usually used to estimate the star-formation rate of the putative starburst. The FIR/radio flux ratio $\mu$ for NGC 2992 is quite low (2.09), compared with starburst galaxies, for which $\mu$ clusters tightly near 2.5 (cf. Colbert et al. 1996b), suggesting significant AGN radio emission. Thus, the radio loops may be more consistently explained by an AGN-driven outflow. Chapman et al. (2000) do, however, find very red ($R-H$ as high as 4.8) “starburst” regions within $\sim$1$^{\prime\prime}$ of the NIR core, in the dust lane. Hence, it is possible that a strong starburst does exist, but remains hidden behind the dust lane (e.g. see V-band image in Figures 8a and 9c). It is certainly possible that AGN photoionization could be powering the 75$-$80% of the soft excess that was [not]{} imaged in our Chandra observations. For example, this emission could be positioned closer ($r \lapprox$ 5$^{\prime\prime}$) to the AGN and thus invisible in the Chandra image due to pile-up effects and/or the tremendous intensity of the AGN X-ray core. 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A. 2002, ApJ, 564, 176 [rrrrcccl]{} 1 & 09:45:30.07 & $-$14:18:42.1 & 2.4 & 0.8$\pm$0.2 & 4.2 &\ 2 & 09:45:33.74 & $-$14:15:26.4 & 17.5 & 5.6$\pm$0.4 & 30.2 &\ 3 & 09:45:34.33 & $-$14:16:36.1 & 4.8 & 1.5$\pm$0.2 & 8.3 &\ 4 & 09:45:35.57 & $-$14:17:42.7 & 4.5 & 1.4$\pm$0.2 & 7.8 &\ 5 & 09:45:37.25 & $-$14:21:20.7 & 3.9 & 1.2$\pm$0.2 & 6.7 & (38.9)\ 6 & 09:45:39.10 & $-$14:23:21.4 & 2.1 & 0.7$\pm$0.1 & 3.7 &\ 7 & 09:45:42.72 & $-$14:19:48.0 & 5.7 & 1.8$\pm$0.4 & 9.8 & 39.1 & ULX in NGC 2992\ 8 & 09:45:44.76 & $-$14:22:20.7 & 2.2 & 0.7$\pm$0.1 & 3.8 & (38.7)\ 9 & 09:45:47.88 & $-$14:22:01.5 & 25.7 & 8.2$\pm$0.5 & 44.4 & 39.7 & (NGC 2993)\ 10 & 09:45:48.17 & $-$14:22:06.5 & 22.0 & 7.0$\pm$0.7 & 38.0 & 39.7 & (NGC 2993), (SN2003ao)\ 11 & 09:45:48.43 & $-$14:22:05.2 & 30.1 & 9.6$\pm$0.7 & 52.1 & 39.8 & (NGC 2993), (SN2003ao)\ 12 & 09:45:49.95 & $-$14:19:05.7 & 4.5 & 1.4$\pm$0.2 & 7.8 &\ 13 & 09:45:50.46 & $-$14:21:38.3 & 20.5 & 6.6$\pm$0.4 & 35.4 & (39.7)\ 14 & 09:45:51.25 & $-$14:19:45.5 & 6.6 & 2.1$\pm$0.2 & 11.4 &\ 15 & 09:45:51.83 & $-$14:20:10.6 & 2.4 & 0.8$\pm$0.1 & 4.1 &\ 16 & 09:45:53.29 & $-$14:17:05.4 & 9.1 & 2.9$\pm$0.3 & 15.7 &\ 17 & 09:45:53.46 & $-$14:23:55.5 & 2.1 & 0.7$\pm$0.1 & 3.7 &\ 18 & 09:45:53.77 & $-$14:23:57.1 & 2.2 & 0.7$\pm$0.1 & 3.9 &\ 19 & 09:45:55.96 & $-$14:21:14.4 & 7.7 & 2.5$\pm$0.2 & 13.4 &\ 20 & 09:45:56.64 & $-$14:24:18.2 & 4.6 & 1.5$\pm$0.2 & 7.9 &\ [lccccccccl]{} 1 & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & CCD readout streak\ 2 & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & no significant excess\ 3/20$-$40$^{\prime\prime}$ & [e3]{} & $<$22 & 26$\pm$12 & 45$\pm$15 & 43$\pm$15 & 90$\pm$22 & $<$74 & 38.86 &\ 4/15$-$25$^{\prime\prime}$ & [Src7]{} & 17$\pm$7 & 23$\pm$11 & 51$\pm$13 & 35$\pm$14 & 82$\pm$19 & $<$60 & 38.82 & point-like (source 7, Table 1)\ 5 & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & CCD readout streak\ 6/15$-$30$^{\prime\prime}$ & [e6]{} & $<$24 & $<$35 & $<$44 & 57$\pm$16 & 75$\pm$22 & $<$70 & 38.78 &\ 7/10$-$20$^{\prime\prime}$ & [e7]{} & 42$\pm$12 & $<$40 & 64$\pm$18 & $<$49 & 74$\pm$24 & $<$72 & 38.78 &\ 8 & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & [...]{} & no significant excess\ & $e3a^$ & [...]{} & [...]{} & [...]{} & [...]{} & 22$\pm$6 & [...]{} & 38.25 & see Fig 9\ [...]{} & $e3b$ & [...]{} & [...]{} & [...]{} & [...]{} & 31$\pm$7 & [...]{} & 38.40 & see Fig 9\ [...]{} & $e6^$ & [...]{} & [...]{} & [...]{} & [...]{} & 83$\pm$10 & [...]{} & 38.83 & see Fig 9\ [...]{} & $e7^*$ & [...]{} & [...]{} & [...]{} & [...]{} & 94$\pm$11 & [...]{} & 38.88 & see Fig 9\ [llrcccllll]{} 1a & PL & 1.13 & 401.1/355 & 41.55 & $-$11.29 & $\Gamma =$0.91$\pm$0.02 & 0.19$\pm$0.04 & 3.21$^{+0.17}_{-0.15}$ &\ 1b & PL(fixed) & 3.74 & 1331.6/356 & 41.70 & $-$11.36 & $\Gamma =$1.86f & 0.73 & 9.33 &\ 2a & PL$+$PL & 1.05 & 371.7/354 & 41.16 & $-$11.68 & $\Gamma_1 =$1.86f & 0.71f & 3.71$^{+0.40}_{-0.83}$ &\ & [ (Part Cov)]{} & & & 41.56 & $-$11.53 & $\Gamma_2 =$0.52$^{+0.12}_{-0.05}$ & 0.008 ($<$0.059) & 1.25$^{+0.39}_{-0.12}$ &\ 2b & PL$+$ & 1.18 & 419.0/354 & 41.37 & $-$11.47 & $\Gamma =$1.86f & 0.71f & 5.93$\pm$0.33 &\ & [ PEXRAV]{}& & & 41.42 & $-$11.65 & & 0.061$\pm$0.028 & 0.77$^{+0.18}_{-0.17}$ & R$=$ 198$^{+38}_{-35}$\ 2c & PL$+$ & 2.82 & 997.4/354 & 41.54 & $-$11.30 & $\Gamma =$1.86f & 0.71f & 8.99 &\ & [ Mekal$_\odot$]{} & & & 39.82 & $-$13.12 & kT$=$0.25 & 0 & 0.36 & Z$=$1.0fZ$_\odot$\ 2d & PL$+$ & 2.80 & 989.6/353 & 41.54 & $-$11.30 & $\Gamma =$1.86f & 0.71f & 8.96 &\ & [ Mekal$_{\rm VarAb}$]{} & & & 39.86 & $-$13.06 & kT$=$0.27 & 0 & 2.25 & Z$=0.12$Z$_\odot$\ 3a & PL$+$ & 1.11 & 390.0/351 & 41.43 & $-$11.43 & $\Gamma =$1.86f & 0.71f & 6.85$^{+0.15}_{-0.23}$ &\ & [ PEXRAV$+$]{} & & & 41.35 & $-$11.74 & & 0 ($<$0.034) & 1.83$^{+1.56}_{-0.98}$ & R$=$ 785$^{+875}_{-355}$\ & [ Mekal$_\odot$]{} & & & 39.66 & $-$13.21 & kT$=$0.31$^{+0.04}_{-0.03}$ & 0.04 ($<$0.08) & 0.27$^{+0.30}_{-0.11}$ & Z$=$1.0fZ$_\odot$\ 3b & PL$+$ & 1.03 & 361.4/350 & 41.39 & $-$11.45 & $\Gamma =$1.86f & 0.71f & 6.3$^{+0.9}_{-2.7}$ &\ & [ PEXRAV$+$]{} & & & 41.36 & $-$11.48 & & 2.90$^{+2.33}_{-1.09}$ & 3.7$^{+1.5}_{-1.1}$ & R$=$ 28$^{+16}_{-10}$\ & [ Mekal$_{\rm VarAb}$]{} && & 40.16 & $-$12.53 & kT$=$0.51$^{+0.26}_{-0.19}$ & 0.12$^{+0.27}_{-0.08}$ & 8.8$^{+21}_{-5.2}$ & Z$=$0.013 ($<$0.026) Z$_\odot$\ [^1]: Assuming a Hubble constant H$_0 =$ 71 km s$^{-1}$ Mpc$^{-1}$ (Spergel et al. 2003) and a recessional velocity of 2311 km s$^{-1}$ (Keel 1996), NGC 2992 is located at a distance of 32.5 Mpc, so that 1$^{\prime\prime}$ corresponds to 158 pc. [^2]: The Chandra Proposer’s Observatory Guide (POG), http://asc.harvard.edu/proposer/POG [^3]: The twenty sources were detected by the XASSIST (see URL [www.xassist.org]{}) data reduction package. XASSIST first identifies a larger list of sources using the CIAO WAVDETECT program. These WAVDETECT sources are then rigorously tested using image fitting of a Gaussian model with a sloping background. Those sources that pass the spatial fitting test and have a signal-to-noise (S/N) ratio $>$2.0 are then kept. Here, S/N is $N(src) / \sqrt{N(bck) \times A(src)/A(bck)}$, where $N$ and $A$ are the number of counts and areas for the source and background regions. For CCD 7, this S/N ratio corresponds to $\sim$3.5 $\times$ 10$^{-15}$ erg s$^{-1}$ cm$^{-2}$. [^4]: URL [http://cxc.harvard.edu/chart]{} [^5]: URL [http://space.mit.harvard.edu/CXC/MARX]{} [^6]: Unless otherwise specified, all errors are for 90% confidence for one degree of freedom ($\Delta\chi^2 =$ 2.7).
--- abstract: 'Hyperon pairs bound in deuteron like states are obtained within the SU(3) Skyrme model in agreement with general expectations from boson exchange models. The central binding from the flavor symmetry breaking terms increases with the strangeness contents of the interacting baryons whereas the kinetic non-linear $\sigma$-model term fixes the spin and isospin of the bound pair. We give a complete account of the interactions of octet baryons within the product approximation to baryon number $B=2$ configurations.' address: - ' $^1$Institute of Theoretical Physics, University of Stellenbosch, Stellenbosch 7600, South Africa ' - ' $^2$Siegen University, Fachbereich Physik, 57068 Siegen, Germany ' author: - 'B. Schwesinger$^{1,2}$[^1], F.G. Scholtz$^1$[^2] and H.B. Geyer$^1$[^3]' title: 'Loosely bound hyperons in the SU(3) Skyrme model' --- Introduction {#sec1} ============ It seems to be generally recognized that the relative scarcity of experimental data on hyperon-nucleon and especially hyperon-hyperon interactions calls for major theoretical input to arrive at a satisfactory description of the data[@Nijm92]. From the boson exchange point of view such a description is typically approached by “upgrading" a reliable description of the $N\!N$-interaction with SU(3) flavor symmetry, assuming of course that the formulation of the $N\!N$-interaction admits such a generalization. An example of this approach can be found in the construction of the Nijmegen potentials (see Ref.[@Nijm92] and references therein). The one-boson exchange potentials obtained in this way generally predict the existence of loosely bound hyperons, while ruling out deeply bound hyperons. An alternative approach which is well tailored to address the construction of hyperon-nucleon and hyperon-hyperon potentials is one based on the Skyrme model[@HS86; @MZ86; @Russians]. It has already been successfully implemented in the construction of the nucleon-nucleon potential, as reviewed recently by Walhout and Wambach[@WW92]. In the Skyrme model deeply bound hyperons also seem to be excluded once Casimir energies are properly taken into account[@SSG93], but deuteron-like states, for which a Skyrmion product ansatz description seems to suffice, remain possible. An analysis of such configurations to determine hyperon-nucleon and hyperon-hyperon potentials have in fact recently been carried out by Kälbermann and Eisenberg[@KE90; @KE92] where much of the focus is on the central interactions. No conclusive statements about the existence or otherwise of bound states seem to have emerged from these two studies. Before we indicate below that the SU(3) Skyrme model construction of hyperon-nucleon and hyperon-hyperon potentials can indeed lead to qualitative and quantitative results, it is perhaps worthwhile to point out that such an approach has the further merit that it ties together the mesonic sector ($B=0$), the baryonic sector ($B=1$) and the $B=2$ sector in the single comprehensive framework of a dynamically closed model with only a few (typically 5) parameters. Furthermore the SU(3) flavor symmetry breaking part is fixed by meson masses and weak decay constants, while in turn it then reproduces the hyperon spectrum in the $B=1$ sector and will also be responsible for two hyperons forming loosely bound deuteron-like states. We now turn to some of the more salient aspects of our analysis. It is rather satisfactory to find that, apart from the known problems associated with the central attraction in the Skyrme model (see Sec.\[sec4\] for further discussion of this point), many of the more robust predictions from the one-boson exchange approach to the problem also emerge from the Skyrme model analysis. In this regard we mention that we extract unambiguously an attractive $\Lambda\Lambda$ interaction in the $^1S_0$ state, a result also required from a number of complementary analyses[@DMG+91a]. Secondly we find e.g. an $N\Sigma$ potential with relative components of the various spin and isospin interactions which are comparable with those found in the Nijmegen potentials[@DG84; @DMG+91b]. As another specific example, the spin dependence of the $N\!\Lambda$ potential we obtain is the same as that favoured from phenomenological and potential model estimates in Ref.[@MGD+85]. Furthermore it is possible to predict from our analysis the spin and isospin of bound deuteron type hyperon states. We also extract a rather prominent correlation of an increased binding of baryons with growing strangeness. A short outline of the presentation is as follows. In Secs. \[sec2\] and \[sec3\] we introduce the SU(3) lagrangian, recall aspects of the framework in which it is typically employed and discuss our strategy of identifying a set of independent relative orientations of two skyrmions which eventually facilitates the identification of the various interaction potentials. This strategy, which generalises the known approach for the $N\!N$-potential[@WW92; @VM85], seems to offer advantages over the approach of Refs.[@KE90; @KE92] where a Monte Carlo calculation is used to evaluate matrix elements[@Kal94]. (See also Sec. \[sec4\].) Detail of this rather technical part is collected in Appendix \[secappA\]. In Sec.\[sec4\] we consider the question of missing intermediate range attraction in the Skyrme model and possible resolutions, argue to which extent various aspects of our present analysis of hyperon-hyperon interactions would retain their validity should this be resolved satisfactorily, and then discuss general and specific results for the various interactions. Tables for the various interaction potentials as function of the separation $R$ are collected in Appendix \[secappB\]. Lagrangian and $B=1$ sector {#sec2} =========================== The lagrangian we use is the standard SU(3) Skyrme-model lagrangian given by[@JJPSW89; @PSW91] $$\label{su3lag} {\cal L} = {\cal L}^{({\rm SYM})} + {\cal L}^{({\rm FSB})}$$ In terms of the chiral field $U$ and with the standard notation for the left current $$L_{\mu} = U^\dagger \partial_{\mu} U = i \lambda_a L_{\mu}^a\; ,$$ and the topological current $$B^{\mu} = \frac{\varepsilon^{\mu \nu \alpha \beta}}{24 \pi^2} \;{\rm tr}\; L_\nu L_{\alpha} L_{\beta}\; ,$$ the flavor symmetric and flavor symmetry breaking parts are given by $$\begin{aligned} \label{lsym} {\cal L}^{({\rm SYM})} (U) &=& - \frac{f^2_\pi}{4} \;{\rm tr}\; L_{\mu} L^{\mu} +\frac{1-\chi}{32e^2} \;{\rm tr}\; [L_{\mu}, L_{\nu}][L^{\mu} , L^\nu] \nonumber\\ & & + \frac{\chi}{16e^2} \left\{ (\;{\rm tr}\; L_{\mu} L^{\mu})^2 - \;{\rm tr}\; L_{\mu}L_{\nu} \cdot \;{\rm tr}\; L^{\mu} L^\nu \right\} \nonumber\\ & & - \frac{\varepsilon_6^2}{2} B_{\mu} B^{\mu} + \frac{m_{\pi}^2 f_{\pi}^2}{4} \;{\rm tr}\;( U + U^\dagger - 2) \nonumber\\ &\equiv&{\cal L}^{({2})}+{\cal L}^{({\rm 4A})}+{\cal L}^{({\widetilde{\rm 4A}})}+{\cal L}^{({6})}+{\cal L}^{({\rm CSB})}\end{aligned}$$ and $$\begin{aligned} \label{lfsb} {\cal L}^{({\rm FSB})} (U) & = & - \frac{f_K^2 - f_{\pi}^2}{4} \;{\rm tr}\; { \renewcommand{\arraystretch}{.2} \renewcommand{\arraycolsep}{2 pt} \mbox{$ \left( \begin{array}{ccc} {\scriptstyle 0} & {\scriptstyle } & {\scriptstyle } \\ {\scriptstyle } & {\scriptstyle 0} & {\scriptstyle } \\ {\scriptstyle } & {\scriptstyle } & {\scriptstyle 1} \end{array} \right)$}} ( U L_{\mu} L^{\mu} + L_{\mu} L^{\mu} U^\dagger)\nonumber\\ &&+ \frac{f_K^2 m_K^2 - f_{\pi}^2 m_{\pi}^2}{2} \;{\rm tr}\; { \renewcommand{\arraystretch}{.2} \renewcommand{\arraycolsep}{2 pt} \mbox{$ \left( \begin{array}{ccc} {\scriptstyle 0} & {\scriptstyle } & {\scriptstyle } \\ {\scriptstyle } & {\scriptstyle 0} & {\scriptstyle } \\ {\scriptstyle } & {\scriptstyle } & {\scriptstyle 1} \end{array} \right)$}} ( U + U^\dagger - 2),\end{aligned}$$ where we have provided for the possibility of different stabilizing terms at most quadratic in the time derivatives. Among the stabilizers of fourth order the extension to SU(3) allows for an alternate term ${\cal L}^{({\widetilde{\rm 4A}})}$[@GL85] which reduces to the Skyrme term ${\cal L}^{({\rm 4A})}$ in the SU(2)-limit. It’s effects in the $B=1$ sector[@PSW91] have been small, whereas here they are shown to be large if $\chi$ is chosen accordingly. The lagrangian (\[su3lag\]) succesfully describes the baryon splittings in the $B=1$ sector when empirical meson masses and decay constants are used. Fig. \[fig1\] shows the baryon spectrum calculated by treating the flavor symmetry breaking in two different ways: for rigidly rotated solitons (RRA)[@YA88; @WSP90; @PSW91] the chiral angle minimizing the soliton in the isospin subgroup is rotated without changes into flavor directions of decreasing hypercharge. In case of the slow rotator approach (SRA)[@SRA1; @SRA2] the soliton is allowed to change its shape according to the flavor symmetry breaking forces excerted at a given strangeness angle. These forces mainly reduce the one pion tail of the SU(2) soliton to a more rapid decay corresponding to the higher meson masses present in the cloud. Product ansatz, energy functional and asymptotic interaction in the $B=2$ sector {#sec3} ================================================================================ In order to obtain a hyperon-hyperon potential (generally baryon-baryon potential) we follow the standard Skyrme model procedure, namely evaluate the energy of a $B=2$ configuration and subtract the energies of the two $B=1$ configurations which constitute that $B=2$ configuration. As the extremely short distance behaviour of the potentials to be calculated is not of immediate concern, we content ourselves with the following hedgehog product ansatz for the $B=2$ configuration $$U ({\bbox{r}};A,C) = A U_{\rm H} ( x, y, z - \case{R}{2}) C \, U_{\rm H} ( x, y, z + \case{R}{2}) C^\dagger A^\dagger$$ where $U_{\rm H}(\bbox{r}) = e^{i \bbox{\tau}\cdot \hat{r} \chi (r)}$. $A$ and $C=A^\dagger B$ are constant flavor rotations, $C$ denoting the relative orientation of the two SU(3) skyrmions in flavor space. As a matter of convenience we chose the two skyrmions to be separated by a distance $R$ along the $z$-axis. The profile function for the hedgehog is obtained by a numerical minimization of the energy functional in the $B=1$ sector. Using the product ansatz we can calculate the energy of the $B=2$ configuration for any given orientations $A$, $C$ and separation $R$ by performing a numerical integration over the spatial coordinates of the hedgehog configuration. One notes from translation and flavor symmetry that the energy functional resulting from ${\cal L}^{({\rm SYM})}$ depends only on the relative separation $R$ and orientation $C$ of the two skyrmions. The general structure of the energy functional is therefore established in the form $$\begin{aligned} V^{({\rm SYM})} (R;A,C) &=& U^{(0)} (R) + U_S^{(1)}(R) D_{ii}(C) + \Bigl( 3 D_{33} (C) - D_{ii}(C)\Bigr) U_T^{(1)}(R)\nonumber\\ & & + \sum_{(I) = (4A), (\widetilde{4A}), (6)} \Bigl\{ U_{S}^{(2,I)} (R) \, Q^{(I)} \case{1}{9} D_{ij}(C)D_{ij}(C) \nonumber\\ & & \qquad +U_{S'}^{(2,I)} (R) \, Q^{(I)} \case{1}{4} \Bigl(D_{ii}(C)D_{jj}(C) -D_{ij} (C) D_{ji} (C) \Bigr) \nonumber\\ & & \qquad + U_{T}^{(2,I)} (R) \, Q^{(I)} \case{1}{4} \Bigl(6D_{11}(C)D_{22}(C) - 6 D_{12} (C) D_{21} (C)\nonumber\\ & & \qquad -D_{ii} (C) D_{jj} (C) + D_{ij} (C) D_{ji} (C) \Bigr) \Bigr\}\\ & & +U_{S}^{(3)} (R) D_{88} (C)\quad .\nonumber\end{aligned}$$ Here we have used the convention $(i,j) \in \{1,2,3\}$ and the notation $$Q^{(I)} = \left\{ \begin{array}{cl} 3 - \case{1}{2}L_a^2\quad , & (I) = (4A)\\ 1\quad , & (I) = ( \widetilde{4A}),\; (6) \end{array} \right.$$ where $L_a^2$ is the Casimir operator of SU(3) and $D_{ab}$ the SU(3) $D-$functions in the adjoint representation. The occurence of the Casimir operator of SU(3) is due to a manipulation which allows us to keep the indices of the $D$-functions in the SU(2)-subgroup: the energy density from the Skyrme term ${\cal L}^{({\rm 4A})}$ has contributions proportional to $$\begin{aligned} f_{abc} D_{cc'} f_{ade} D_{ee'} &=& - \Bigl( L_a D_{bc'} \Bigr)\Bigl( L_a D_{de'} \Bigr)\nonumber\\ &=& -\case{1}{2} L_a^2 \Bigl( D_{bc'} D_{de'}\Bigr) +\case{1}{2} \Bigl( L_a^2 D_{bc'} \Bigr) D_{de'} + \case{1}{2} D_{bc'} \Bigl(L_a^2D_{de'}\Bigr) \; ,\end{aligned}$$ where $L_a$ are the left Euler angular momentum operators. The explicit expressions stemming from the action of the Casimir operators are given in Appendix \[secappA\]. In Eq. (7) we do not show those terms which contain combinations of $D-$functions which are of second rank in the spin in the energy functional, since their matrix elements between the spin one-half baryon states vanish. In principle they do, however, occur and due care has to be exercised, when the different components of the interaction are evaluated. For completeness we list these combinations ordered according to the spin-tensorial decomposition of the left and right indices of the two $D$-functions which refer to the Euler angles in $A$ and $B$ respectively. $$\begin{aligned} \label{dfuncs} \Bigl[0 \times 2\Bigr]_2\;:\qquad&&D_{i3} D_{i3}-\case{1}{3}D_{ij} D_{ij}\; ,\nonumber\\ \Bigl[2 \times 0\Bigr]_2\;:\qquad&&D_{3i} D_{3i}-\case{1}{3}D_{ij} D_{ij}\; ,\nonumber\\ \Bigl[2 \times 2\Bigr]_0\;:\qquad &&\case{1}{2}D_{ii} D_{jj}+\case{1}{2}D_{ij} D_{ji}-\case{1}{3}D_{ij} D_{ij} \; ,\nonumber\\ \Bigl[2 \times 2\Bigr]_2\;:\qquad &&\case{1}{2}D_{ii} D_{33}+\case{1}{2}D_{i3} D_{3i}-\case{1}{3}D_{i3} D_{i3} -\case{1}{3}D_{3i} D_{3i}+\case{1}{9}D_{ij} D_{ij}\; ,\nonumber\\ \Bigl[2 \times 2\Bigr]_4\;:\qquad &&D_{33} D_{33}-\case{1}{3}D_{i3} D_{i3}-\case{1}{3}D_{3i} D_{3i} +\case{1}{9}D_{ij} D_{ij}\; .\end{aligned}$$ The seven different components $U^{(0)}(R),U^{(1)}_S(R)\ldots U^{(2)}_T(R)$ of the interaction appearing in $V^{({\rm SYM})} (R;A,C)$ can now be calculated. We note that if we evaluate the energy of the $B=2$ hedgehog-hedgehog configuration for twelve orientations, $C$, such that the seven combinations of $D-$functions appearing in $V^{({\rm SYM})} (R;A,C)$ and the five combinations of Eq. (\[dfuncs\]) are linearly independent, we obtain a system of linear equations from which the interaction components can be solved by inversion of the coefficient matrix. The set of orientations $$\begin{aligned} && 1\; ,\qquad e^{i \frac{\pi}{2} \lambda_2}\; ,\qquad e^{i \frac{\pi}{2} \lambda_3} \; ,\qquad e^{i \frac{\pi}{2} \lambda_4} e^{i \frac{\pi}{4} \lambda_2}\; ,\quad e^{i \frac{2\pi}{3} \lambda_2}e^{i \frac{\pi}{2} \lambda_4} \; ,\qquad e^{i \frac{\pi}{4} \lambda_4}e^{i \frac{\pi}{2} \lambda_2} e^{-i \frac{\pi}{4} \lambda_4}\; ,\nonumber\\ && e^{i \frac{\pi}{4} \lambda_4}e^{i \frac{\pi}{2} \lambda_3} e^{-i \frac{\pi}{4} \lambda_4} \; ,\qquad e^{i \frac{\pi}{4} \lambda_1}e^{i \frac{\pi}{4} \lambda_4} e^{i \frac{\pi}{4} \lambda_2}\; ,\qquad e^{i \frac{\pi}{6} \lambda_4}e^{i \frac{\pi}{5} \lambda_6} e^{i \frac{\pi}{5} \lambda_4} \; ,\qquad e^{i \frac{\pi}{6} \lambda_1}e^{i \frac{\pi}{5} \lambda_6} e^{i \frac{\pi}{5} \lambda_4}\; , \nonumber\\ && e^{i \frac{\pi}{6} \lambda_2}e^{i \frac{\pi}{5} \lambda_6} e^{i \frac{\pi}{5} \lambda_4} \; ,\qquad e^{i \frac{\pi}{6} \lambda_3}e^{i \frac{\pi}{5} \lambda_6} e^{i \frac{\pi}{3} \lambda_4}\; .\end{aligned}$$ can be checked to give the required linear independence and are subsequently used to determine the interaction components as already outlined. Translation invariance implies that for the flavor symmetry breaking part ${\cal L}^{({\rm FSB})}$, the energy functional depends on $R$ alone, but since flavor symmetry is broken, it depends on the orientations $A$ and $B$ of the individual skyrmions in flavor space. Neglecting the numerically small and cumbersome terms proportional to $f_K^2-f_\pi^2$, a simple calculation shows the energy functional to be $$\begin{aligned} V^{({\rm FSB})} (R;A,B) &=& U_0^{({\rm FSB})} (R) \Bigl[ \Bigl(1 - D_{88} (A)\Bigr) \Bigl(1 - D_{88} (B)\Bigr) + \case{3}{4} D_{k8} (A) D_{k8} (B) \Bigr] \nonumber \\ &+& U_1^{({\rm FSB})} (R) \Bigr[ D_{8i} (A) D_{8i} (B)+\case{3}{4} D_{ki} (A) D_{ki} (B) \Bigr] \nonumber\\ &+& U_2^{({\rm FSB})} (R) \Bigl[ \Bigl( 3 D_{83} (A) D_{83} (B) - D_{8i} (A) D_{8i} (B)\Bigr) \nonumber \\ & &\phantom{U_2^{({\rm FSB})} (R) \Bigl[ } +\case{3}{4} \Bigl( 3 D_{k3} (A) D_{k3} (B) - D_{ki} (A) D_{ki} (B)\Bigr) \Bigr]\; ,\end{aligned}$$ where $k \in \{4,5,6,7\},\, i \in \{1,2,3\}$. The radial functions in the flavor symmetry breaking part of the energy functional, $V^{({\rm FSB})} (R;A,B)$, are given in terms of the integrals $$\begin{aligned} U_0^{({\rm FSB})} (R)&=&\frac{4}{9}\Bigl( f_K^2 m_K^2 -f_\pi^2 m_\pi^2 \Bigr) \int -(1-c_1)(1-c_2)\,d^3r \\ U_1^{({\rm FSB})} (R)&=&\frac{4}{9}\Bigl( f_K^2 m_K^2 -f_\pi^2 m_\pi^2 \Bigr) \int \hat r(1)\cdot \hat r(2) s_1 s_2 \, d^3r \nonumber \\ U_2^{({\rm FSB})} (R)&=&\frac{4}{9}\Bigl( f_K^2 m_K^2 -f_\pi^2 m_\pi^2 \Bigr) \int \case{1}{2}\Bigl(3\hat r_3(1) \hat r_3(2) -\hat r(1) \cdot \hat r(2)\Bigr) s_1 s_2 \, d^3r \nonumber \; , \\ & & {\bbox r}(i) = {\bbox r}\pm\frac{R}{2} {\bbox e}_3,\quad c_i= \cos \chi\bigl(r(i)\bigr),\quad s_i= \sin \chi\bigl(r(i)\bigr) \quad i=1,2 \nonumber \; .\end{aligned}$$ These integrals are evaluated numerically. Finally, knowing the interaction components, we have to evaluate the matrix elements of the energy functional between the spin one-half baryon wavefunctions diagonalizing the rotational hamiltonian with the symmetry breaking term, see [@YA88; @WSP90; @PSW91] and Eq. (A1) to extract the potentials in the various interaction channels. We have to distinguish direct and exchange matrix elements, see Fig. \[fig2\]. The single particle degrees of freedom of the interacting baryons are given by the collective rotations $A$ and $B$ respectively, which appear in all the $D$-functions of the potential energy. In such an interaction term $D(A)$, say, can transfer strangeness $0 \leq \vert S \vert \leq 1$ to the incoming baryon state depending on the Euler angles $A$. For the direct terms the outgoing baryon also carrying the Euler angles $A$ has the same hypercharge as the incoming one, whereas the exchange terms allow for different strangeness of the outgoing baryon. Since total strangeness is, of course, conserved, the baryon described by the Euler angles $B$ must compensate for the hypercharge difference. Speaking in terms of a one boson exchange description, the latter processes thus contain the kaon exchange terms. The details of the calculation of the matrix elements and the various interaction potentials are collected in Appendix \[secappA\]. We must emphasize, however, that generally, we will only use wave functions diagonalizing the rigid rotator hamiltonian, because the use of a chiral angle which changes in response to the flavor symmetry breaking forces[@SRA1; @SRA2] is technically too complicated, here. For testing purposes we will also use baryon wavefunctions which are SU(3) symmetric as well as those for the strong symmetry breaking limit. Unfortunately, the restriction to the rigidly rotating soliton directly leads to the fact that the asymptotic forces will always have the one-pion exchange tail, even for kaon exchange terms. This statement can be verified by examination of the asymptotic interaction, which is related to the lower order terms in the lagrangian after insertion of the equations of motion for the individual solitons. Within SU(2) this has been shown by Yabu and Ando [@YA85], and the extension to SU(3) is straight-forward: $$\begin{aligned} &&V(R;A,B) \stackrel{R\rightarrow \infty}{\longrightarrow} \\ &&\qquad-\case{1}{2} \int {\rm tr} \; \Bigl( f_\pi^2 \nabla \lambda_a \Phi_a(1) \cdot \nabla \lambda_b \Phi_b(2) +f_\pi^2 { \renewcommand{\arraystretch}{.2} \renewcommand{\arraycolsep}{2 pt} \mbox{$ \left( \begin{array}{ccc} {\scriptstyle m_\pi^2} & {\scriptstyle } & {\scriptstyle } \\ {\scriptstyle } & {\scriptstyle m_\pi^2} & {\scriptstyle } \\ {\scriptstyle } & {\scriptstyle } & {\scriptstyle 2 m_K^2-m_\pi^2} \end{array} \right)$}} \case{1}{2} \{ \lambda_a \Phi_a(1),\lambda_b \Phi_b(2) \} \Bigr) \; d^3r \; .\nonumber\end{aligned}$$ The eight asymptotic pseudoscalar soliton fields $\Phi_a$ solving the Euler Lagrange equations should have the form[^4] $$\begin{aligned} \Phi_a = D_{ab}\hat r_b \left\{ \begin{array}{ll} \chi^{\phantom K}_\pi & a\in \{1,2,3\}\\ \chi^{\phantom K}_K & a\in \{4,5,6,7\}\\ \chi^{\phantom K}_\eta & a=8 \end{array} \right.\end{aligned}$$ with $$\begin{aligned} \hat r \chi^{\phantom K}_C = -\nabla A_C \frac{{\rm e}^{-m_Cr}}{r}\end{aligned}$$ leading to $$\begin{aligned} &&V(R;A,B) \stackrel{R\rightarrow \infty}{\longrightarrow} \\ &&\qquad 4 \pi f_\pi^2 \nabla_i \nabla_j \left\{ D_{ki}(A)D_{kj}(B) A_K^2 \frac{{\rm e}^{-m_K R}}{R} \right. \nonumber\\ && \qquad \qquad \qquad \qquad \left. +D_{8i}(A)D_{8j}(B) A_\eta^2 \frac{{\rm e}^{-m_\eta R}}{R} +D_{i'i}(A)D_{i'j}(B) A_\pi^2 \frac{{\rm e}^{-m_\pi R}}{R} \right\} \; . \nonumber\end{aligned}$$ However, due to the rigid rotator approximation all masses are set to $m_\pi$ and the asymptotic constants all equal $$A_{\rm SU(2)}= \frac{g_{\pi NN}(-m_\pi^2)}{g_{\pi NN}(0)} \; \frac{3 g_A^{\rm SU(2)}}{8 \pi f_\pi^2} \; .$$ For SU(3)-symmetry the matrix elements of the $D$-functions in Eq. (17) for nucleons are summarized by $$\begin{aligned} D_{ij}(C)=D_{ai}(A)D_{aj}(B) \rightarrow (\case{7}{30})^2 \sigma_i(A) \sigma_j(B) \Bigl[ {\bbox \tau}(A) \cdot {\bbox \tau}(B) +\case{3}{49}\Bigr]\end{aligned}$$ and we recover the conventional one pion exchange interaction together with a small term from one $\eta$ exchange since for SU(3)-symmetry $g_A^{\rm SU(3)}=\case{7}{10} g_A^{\rm SU(2)}$ in the Skyrme model. Numerically and fortunately, the long range part of the exchange matrix elements will turn out to be small, so the impact of the wrong tail is not important. Results {#sec4} ======= The main emphasis of the present investigation is on the question of whether strangeness is loosely bound. Accessing the problem within the framework of the Skyrme model has some advantageous and some problematic aspects. The problematic aspects stem from uncertainties in the higher order terms of the lagrangian, inadequacies of the product ansatz and, ultimately, the disturbing feature that an intermediate range attraction in the nucleon-nucleon system does not emerge as simply and naturally as one might have hoped , see e.g. [@WW92; @VM85; @JJP85; @YSH89] and references therein. The obvious advantages lie in the fact that the lower order terms of the lagrangian are unambiguously fixed in the mesonic winding number zero sector. Now, the same lagrangian governs the higher winding number configurations, so that conclusions based solely on the lower order terms, should also be reliable for the case of higher winding numbers. One example for this is the baryon spectrum in SU(3), where the lower order SU(3) symmetry breaking in the lagrangian, Eq. (\[lfsb\]), is sufficient to describe the mass splittings, as recalled in Sec. \[sec2\]. Another example is given in the case of meson-baryon scattering where the use of mesonic weak decay constants and masses for this winding number one configuration is mandatory[@Wal91; @S92]. Provided that the problem of the missing intermediate range attraction finds its resolution in higher order terms and/or in improved two-baryon configurations[@WW92] beyond the product ansatz, we can confidently focus on the longer range part of the interaction: it does not involve the higher order terms explicitely and also the product ansatz gradually becomes exact. The longer range components of the baryon-baryon forces are due to the flavor symmetric and flavor symmetry breaking mass terms ${\cal L}^{({\rm CSB})}$ and ${\cal L}^{({\rm FSB})}$, and the non-linear $\sigma$-model term ${\cal L}^{({2})}$, see Eq. (14), where the latter does not contribute at all to the central interaction. If we accept the line of reasoning above, the central interactions from the flavor and chirally symmetric terms of the lagrangian are viable to modifications, the others not. In Figs. 3-4 we show the central interactions from the flavor symmetric part of the lagrangian for the $N\!N$-, $\Lambda \Lambda$-, $\Sigma \Sigma$- and $\Xi \Xi$-systems. The general absence of any attraction is as conspicuous as the fact, that the variation with respect to decreasing hypercharge of the system is small. This suggests, that any improvement on the central attraction in the $N\!N$-system will lead to very similar attractions for the other baryons. We explore this hypothesis in Figs. 3-4 by varying the higher order stabilizing terms of the lagrangian: Fig. 3 shows the changes when the usual Skyrme term ${\cal L}^{({\rm 4A})}$, i.e. $\chi=0$ in Eq. (\[lsym\]), is replaced by its alternate form ${\cal L}^{({\widetilde{\rm 4A}})}$, i.e. $\chi=1$ in Eq. (\[lsym\]). This substitution does not lead to any changes within SU(2), but in SU(3) the situation changes drastically as may be seen from the figure. Since the $B=1$ solitons in the SU(2)-subgroup are unaffected by variations of $\chi$ it is clear that large negative values of $\chi$ will yield arbitrary attraction for $B=2$-configurations in SU(3). But it seems very likely to us, that the $B=1$ soliton would not be stable towards deformations out of the SU(2)-subgroup then, to which it was constrained by ansatz. We therefore refrain from such manipulations. In any case, from the mesonic sector, ${\cal L}^{({\widetilde{\rm 4A}})}$ seems to be small[@GL85] since it violates the Zweig rule. Fig. 4 shows the changes, when the fourth order stabilizer is replaced by a sixth order one and when scalar degrees of freedom are taken into account. For the latter, we choose a form inspired by, but simpler than, a local approximation to the dilatons[@Sch86; @GJJS86; @YSH89], namely $$\begin{aligned} {\cal L}^{(2)} (U) & \rightarrow & \widetilde {\cal L}^{(2)} (U) \nonumber\\ & = & - \frac{f^2_\pi m_\pi^2}{4\kappa^2} \; (1-\exp\bigl({-\frac{\kappa^2}{m_\pi^2}{\rm tr}\; L_{\mu} L^{\mu}})\bigr),\end{aligned}$$ which effectively scales down $f_\pi$ in the interior of the soliton. Expanding the exponential to second order gives the non-linear $\sigma$-model and the usual attractive symmetric fourth order term which destabilizes the soliton, but here there are higher order terms present in the exponential to counteract this destabilization now. Within SU(2) the inclusion of dilatons was shown to lead to an intermediate range central attraction[@YSH89] when these fields were made strong enough. In SU(3), however, strong dilaton fields lead to a problem since not only $f_\pi$ is scaled down in the interior but also all mass terms resulting in too small hyperon splittings in the $B=1$ sector[@KSS90]. As already mentioned, the rationale of all these different higher order terms is not only to investigate further mechanisms for an intermediate range attraction, but also to demonstrate that the central interaction from the flavor symmetric terms is fairly independent of the hypercharge of the two-baryon configuration. Thus, if the $N\!N$-system acquires it’s intermediate range attraction from the flavor symmetric terms, this attraction will also be present for the hyperons. It remains to be said, that the interactions in Figs. 3-4 have all been determined with baryon wavefunctions diagonalizing the rotational motion in the presence of SU(3) symmetry breaking, see Eq. (A1). The distortions in the baryon wavefunctions away from SU(3), however, do not play a decisive role here, which means, that SU(3) symmetric wavefunctions or wavefunctions in the strong symmetry breaking limit always lead to very small variations of the flavor symmetric central interaction with respect to hypercharge. Fig. 5 shows the central interactions for identical baryons as they arise from those terms, which explicitly break the flavor symmetry. These contributions arise solely from the Euler angle matrix elements of $$\begin{aligned} U_0^{({\rm FSB})} (R) \Bigl(1 - D_{88} (A)\Bigr) \Bigl(1 - D_{88} (B)\Bigr)\end{aligned}$$ in Eq. (12) and are directly proportional to the product of the strangeness contents of the two interacting baryons, since $\langle Y \vert \bar s s \vert Y \rangle = \langle Y \vert \case{1}{3}\Bigl(1 - D_{88} \Bigr) \vert Y \rangle$. From Fig. 5 we see that $U_0^{({\rm FSB})} (R)$ is attractive and large, such that binding is enhanced with growing strangeness content. Since this additional binding comes from a term fixed in the mesonic sector, we judge this to be a reliable conclusion, which also summarises one of the main points of the present work. In Appendix \[secappB\] we give a complete list of all interaction components for $N\!N$, $N\!\Lambda$, $N\Sigma$, $N\Xi$, $\Lambda\Lambda$, $\Sigma\Sigma$ and $\Xi\Xi$ systems using only the normal Skyrme term, ${\cal L}^{({\rm 4A})}$, as stabilizer adjusting the $N\Delta$-splitting. Again, the longer-range spin-isospin dependence is due to terms fixed in the mesonic sector and thus insensitive to any reasonable modifications of the lagrangian. For the $N\!N$-case we exploit one of the advantages of our approach to the SU(3)-interactions, namely that we are able to switch continuously from SU(3)-symmetry to the SU(2)-limit by simply changing the baryon wavefunctions, see Eq. (A1). Comparing these cases for the $N\!N$-interaction, we find that there is a smooth transition with small changes in the longer range parts of the interaction to the well known results in SU(2). This finding apparently is in contrast to other work published recently on the baryon-baryon interactions in the SU(3) Skyrme model[@KE90; @KE92] and similar discrepancies persist for other baryon pairs, too, which could be[@Kal94] due to difficulties in the Monte Carlo integrations used in ref.[@KE90; @KE92]. From the fact that two nucleons are loosely bound and well separated in an isospin $I=0$, spin $J=1$ state (the deuteron), and that no additional repulsion arises in the longer-range central interactions when hypercharge is decreased, we are able to predict the spin and isospin of bound deuteron-like states of hyperons. To this end we just have to examine the sign of the different spin-isospin components of the long-range interaction, which, as already has been stated, is due to well established terms in the lagrangian. Explicitly (see Appendix \[secappB\]) we find that an $N\!\Lambda$-pair should be bound in a ($I=\case{1}{2},\; J=1$)-state, an $N\Sigma$-pair in a ($I=\case{1}{2},\; J=1$)-state, an $N\Xi$-pair in a ($I=0,\; J=0$)-state, a $\Lambda\Lambda$-pair in a ($I=0,\; J=0$)-state, a $\Sigma\Sigma$-pair in a ($I=2,\; J=0$)-state, and a $\Xi\Xi$-pair in a ($I=1,\; J=0$)-state. In SU(2) the most attractive configuration, the torus, can be found numerically by initializing a general minimization algorithm with the most attractive product configuration[@KS87; @VWWW86; @WW92]. This most attractive product configuration occurs at a relative orientation of $C=\exp(i\frac{\pi}{2}\lambda_2)$ for the two hedgehogs. With the tools set up for the present investigation we are in a position to determine the most attractive product configuration in SU(3) by minimizing the flavor symmetric part of the interaction energy, Eq. (7) with respect to the 8 Euler angles in $C$ at all separations. Interestingly, the most attractive orientation is unchanged, i.e. it resides in the SU(2) subgroup, hinting that the torus will also be the minimum in the $B=2$-sector of SU(3). In case of flavor symmetry with all meson masses equal to the pion mass the torus is known[@SSG93] to be lower in energy than the SO(3)-soliton found by Balachandran [et al.]{}[@BLRS85]. Conclusions {#sec5} =========== In the present work we have explored to which extent it is possible to make reliable statements about whether hyperons form bound deuteron like states. To this end we have used the SU(3) Skyrme model and compared the $N\!N$-interaction in the product ansatz to different $NY$- and $YY$-interactions which emerge from the same ansatz. ($Y$ represents any octet baryon with non-zero strangeness.) For loosely bound deuteron like configurations the product ansatz is sufficient if the only purpose is the investigation of whether there is more binding in the $NY$-system relative to the $N\!N$-case in the longer range part of the interaction. Such a comparison within the Skyrme model unfortunately also meets two other obstructions: the well-known problem of the missing intermediate range central attraction and the incomplete knowledge on the correct combination of higher order stabilizing terms. The latter statement, of course, reflects the belief that the lower order terms are fixed by their empirical values in the mesonic sector for which evidence has been given in several places[@WSP90; @Wal91; @S92; @M93]. Fortunately, it turns out that the higher order stabilizers, which exclusively make up the chiral and flavor symmetric contributions to the central interaction, show little variation with respect to different baryon pairs considered. Therefore, the uncertainties associated with these terms are eliminated when we look at differences between $N\!N$- $NY$- and $YY$-interactions: if some combination of higher order terms – most likely a coupling to scalar degrees of freedom – leads to a central attraction in the $N\!N$-case, it will also do so for the other cases. There remain the longer range contributions from the known lower order terms: the non-linear $\sigma$-model is responsible for the spin and flavor dependent components of the interaction, whereas the chiral and flavor symmetry breaking terms introduce further central interactions which become increasingly attractive as the strangeness contents of the participating baryons grow. The conclusion, that hyperons must be bound in deuteron like states, therefore seems inescapable. Although our findings confirm or extend what is commonly expected, we think that we actually have added some weight to such expectations. Since the contribution of the non-linear $\sigma$-model to the potential is also considered a reliable quantity here, we were actually in a position to predict the spins and isospins of the most attractive baryon-baryon configurations. Again, the Skyrme model result seems to agree with conclusions drawn from one boson exchange models. Acknowledgements {#sec6} ================ We thank A. Hayashi for allowing us to upgrade his SU(2) $N\!N$-interaction program for the present investigation and for useful discussions. One of us, B.S., wishes to thank the ITP at the University of Stellenbosch for its hospitality and a very enjoyable stay. {#secappA} The interaction terms depend on the Euler angles of the participating baryons, with the angles in $A$ corresponding to the degrees of freedom of the first baryon, the angles in $B$ to the second. The wavefunction for an individual spin one-half baryon can be expressed as follows: $$\begin{aligned} \langle A \vert {i\;y \choose i_3}{\case{1}{2} \, 1 \choose j_3} \rangle= \sum_{\{m\}}\sqrt{m} \;X^{iy}_{\{m\}}\; (-)^{\case{1}{2} +j_3} D^{\{m\}^}_{{i\;y \choose i_3}{{\frac{1}{2}}\;1 \choose -j_3}}(A) \; , \end{aligned}$$ where the $X_{\{m\}}^{iy}$ are real amplitudes that diagonalize the rotational hamiltonian of the baryon considered [@YA88; @WSP90; @PSW91]. We begin with the direct matrix elements between an incoming baryon $X(A)$ interacting with a baryon $Y(B)$. The relative rotations $C=A^\dagger B$ contained in the $D$-functions of all flavor symmetric interaction terms translate into the following expression: $$\begin{aligned} & &\langle X'\, Y' \vert \; D^{\{n\}}_{{j_1\;y_1 \choose m_1} {j_2\;y_2 \choose m_2}}(C)\; \vert X\, Y \rangle = \\& & \qquad \qquad\qquad \qquad \sum_{I,y,M} \langle X \vert \; D^{\{n\}}_{{I\;y \choose M}{j_1\;y_1 \choose m_1}}(A)\; \vert X' \rangle^ \, \langle Y' \vert \; D^{\{n\}}_{{I\;y \choose M}{j_2\;y_2 \choose m_2}}(B) \; \vert Y \rangle \; \nonumber.\end{aligned}$$ With the conventions of de Swart[@DeSwart] individual angular integrals are actually real, so we may omit the complex conjugation later. For the direct matrix elements the transferred hypercharge $y$ vanishes, and $I,j_1,j_2$ are the possible isospins and spins in the irreducible representation $\{n\}$ corresponding to hypercharge $y,y_1,y_2$ respectively. The angular integrals of the baryon-baryon interaction may all be decomposed to integrals over three $D$-functions, where isospin and spin dependence factorizes via SU(2) Clebsch-Gordan coefficients leaving isoscalar SU(3)-factors[@DeSwart]. We define reduced SU(3) matrix elements in terms of these isoscalar factors: $$\begin{aligned} & &\langle (i'\;y') (\case{1}{2}\;1) \Vert D_{(IY)(J0)}^{\{n\}} \Vert (i\;y) ( \case{1}{2}\;1) \rangle = \\ & & \qquad \qquad \quad \sum_{ \{m\},\{m'\} } \sqrt{ \frac{m}{m'} } X_{\{m\}}^{iy} X_{\{m'\}}^{i'y'} \sum_{\gamma} \left( \begin{array}{cc\|c} \{m\} & \{n\}& \{m'\}_{\gamma}\\[-2pt] iy &IY & i'y' \end{array} \right) \left( \begin{array}{cc\|c} \{m\} & \{n\}& \{m'\}_{\gamma}\\[-2pt] \case{1}{2}\;1 & J0 & \case{1}{2}\;1 \end{array} \right) \, ,\nonumber\end{aligned}$$ and abbreviate the SU(2)-Clebsch-Gordan coefficients by isospin and spin operators of rank $I=1,2$, $J=1$. For $I=1$ we adopt the usual notation $$\langle b \vert T_a \vert c \rangle = (T_a)_{bc} = \left\{ \begin{array}{lll} 0&{\rm for} & \Lambda\\[1mm] (\tau_a)_{bc} & {\rm for} & N {\rm ~and~} \Xi\\[1mm] i \varepsilon_{abc} & {\rm for} & \Sigma \end{array} \right.$$ For $I=2$ it is also convenient to define an isospin quadrupole operator which will appear in the $\Sigma\Sigma$-interactions $$\left( \begin{array}{cc\|c} 1 & 2 & 1\\[-2pt] c & a & b \end{array} \right) = (Q_a)_{bc} \, .$$ Since all baryons considered have spin one-half, the Pauli-matrices $\sigma_b$ suffice to reproduce all $J=1$ matrix elements. The different non-vanishing Euler angle matrix elements of the interaction which now can be expressed by the reduced matrix elements given in (A3) and by isoscalar factors are listed below. The spin and isospin operators embody the different possibilities for spin or isospin flips between incoming and the outgoing baryons $X'(A)$ and $Y'(B)$. Note that the implicit summations run over $i,j\in\{1,2,3\}$. $$\begin{aligned} & &\langle X' \, Y' \vert \Bigl(1-\lambda D_{88}(A)\Bigr) \Bigl( 1-\lambda' D_{88}(B) \Bigr) \vert X \, Y \rangle \rightarrow \\& &\qquad \qquad \qquad \qquad \qquad \Bigl(1-\lambda\langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(00)(00)}^{\{ 8 \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle\Bigr) \, \nonumber \\ & & \qquad \qquad \qquad\qquad \qquad \qquad\qquad \qquad \Bigl( 1-\lambda'\langle (i_Yy_Y) (\case{1}{2} 1) \Vert D_{(00)(00)}^{\{ 8 \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle \Bigr) \nonumber\end{aligned}$$ for arbitrary coefficients $\lambda, \lambda'$. $$\begin{aligned} \langle X' \, Y' \vert D_{j8}(A) D_{j8}(B) \vert X \, Y \rangle \rightarrow & & {\mbox {\boldmath $T$}}_X \cdot {\mbox {\boldmath $T$}}_Y \frac{\sqrt{(2i_X + 1)(2i_Y +1)}}{6}\nonumber \\ & & \qquad \langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(10)(00)}^{\{ 8 \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle \, \nonumber \\ & & \qquad \qquad \qquad \langle (i_Yy_Y) (\case{1}{2} 1) \Vert D_{(10)(00)}^{\{ 8 \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle\end{aligned}$$ $$\begin{aligned} \langle X' \, Y' \vert D_{8i}(A) D_{8i}(B) \vert X \, Y \rangle \rightarrow & & \case{1}{3} {\mbox {\boldmath $\sigma$}}_X \cdot {\mbox {\boldmath $\sigma$}}_Y \langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(00)(10)}^{\{ 8 \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle \, \nonumber \\ & & \qquad \qquad \qquad \langle (i_Yy_Y) (\case{1}{2} 1) \Vert D_{(00)(10)}^{\{ 8 \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle\end{aligned}$$ $$\begin{aligned} \langle X' \, Y' \vert D_{ji}(A) D_{ji}(B) \vert X \, Y \rangle \rightarrow & & \case{1}{3} {\mbox {\boldmath $\sigma$}}_X \cdot {\mbox {\boldmath $\sigma$}}_Y \; {\mbox {\boldmath $T$}}_X \cdot {\mbox {\boldmath $T$}}_Y \frac{\sqrt{(2i_X + 1)(2i_Y +1)}}{6}\nonumber \\ & & \qquad \langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(10)(10)}^{\{ 8 \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle \, \nonumber \\ & & \qquad \qquad \qquad \langle (i_Yy_Y) (\case{1}{2} 1) \Vert D_{(10)(10)}^{\{ 8 \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle\end{aligned}$$ The combination of $D$-functions where the right indices are coupled to a second rank tensor $$\begin{aligned} 3 D_{83}(A) D_{83}(B) - D_{8i}(A) D_{8i}(B) \\ 3 D_{j3}(A) D_{j3}(B) - D_{ji}(A) D_{ji}(B)\end{aligned}$$ lead to the same expressions as (A8) and (A9) respectively apart from the replacement ${\mbox {\boldmath $\sigma$}}_X \cdot {\mbox {\boldmath $\sigma$}}_Y \rightarrow S_{XY} $ where $$\begin{aligned} S_{XY}= 3{\mbox {\boldmath $\sigma$}}_X \cdot \hat R \; {\mbox {\boldmath $\sigma$}}_Y \cdot \hat R -{\mbox {\boldmath $\sigma$}}_X \cdot {\mbox {\boldmath $\sigma$}}_Y\end{aligned}$$ is the tensor force operator and $\hat R ={\bf e}_3$ has been used. $$\begin{aligned} & &\langle X' \, Y' \vert \case{1}{9} Q^{(I)} D_{ji}(C) D_{ji}(C) \vert X \, Y \rangle \rightarrow \case{1}{3} \sum_{\{ n \}{\gamma}} \left( \begin{array}{cc\|c} \{ 8 \} & \{ 8 \} & \{ n \}_{\gamma}\\[-2pt] 10 & 10 & 00 \end{array} \right)^2 q_{\{n\}}^{(I)}\times \\ & & \quad \Bigl\{ \langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(00)(00)}^{\{ n \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle \langle (i_Yy_Y) (\case{1}{2} 1) \Vert D_{(00)(00)}^{\{ n \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle \nonumber \\ & & \qquad + {\mbox {\boldmath $T$}}_X \cdot {\mbox {\boldmath $T$}}_Y \frac{\sqrt{(2i_X + 1)(2i_Y +1)}}{6} \langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(10)(00)}^{\{ n \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle \nonumber \\ & & \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad \langle (i_Yy_Y) (\case{1}{2}1) \Vert D_{(10)(00)}^{\{ n \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle \nonumber \\ & & \qquad +{\mbox {\boldmath $Q$}}_X \cdot {\mbox {\boldmath $Q$}}_Y \langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(20)(00)}^{\{ n \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle \langle (i_Yy_Y) (\case{1}{2}1) \Vert D_{(20)(00)}^{\{ n \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle \Bigr\} \nonumber\end{aligned}$$ $$\begin{aligned} & &\langle X' \, Y' \vert \case{1}{4} Q^{(I)} \Bigl(D_{ii}(C) D_{jj}(C) - D_{ij}(C) D_{ji}(C) \Bigr) \vert X \, Y \rangle \rightarrow \\ & &\quad \case{1}{3} {\mbox {\boldmath $\sigma$}}_X \cdot {\mbox {\boldmath $\sigma$}}_Y \case{1}{2} \sum_{\{ n \}{\gamma}} \left( \begin{array}{cc\|c} \{ 8 \} & \{ 8 \} & \{ n \}_{\gamma}\\[-2pt] 10 & 10 & 10 \end{array} \right)^2 q_{\{n\}}^{(I)}\times \nonumber \\ & &\qquad \Bigl\{ \langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(00)(10)}^{\{ n \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle \langle (i_Yy_Y) (\case{1}{2}1) \Vert D_{(00)(10)}^{\{ n \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle \nonumber \\ & & \qquad \qquad+ {\mbox {\boldmath $T$}}_X \cdot {\mbox {\boldmath $T$}}_Y \frac{\sqrt{(2i_X + 1)(2i_Y +1)}}{6} \langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(10)(10)}^{\{ n \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle \nonumber \\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad \langle (i_Yy_Y) (\case{1}{2}1) \Vert D_{(10)(10)}^{\{ n \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle \nonumber \\ & & \qquad\qquad +{\mbox {\boldmath $Q$}}_X \cdot {\mbox {\boldmath $Q$}}_Y \langle ( i_Xy_X) (\case{1}{2} 1) \Vert D_{(20)(10)}^{\{ n \}}\Vert (i_Xy_X) (\case{1}{2}1) \rangle \langle (i_Yy_Y) (\case{1}{2}1) \Vert D_{(20)(00)}^{\{ n \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle \Bigr\} \nonumber\end{aligned}$$ As has been explained in section 3. the evaluation of antisymmetric fourth order terms from ${\cal L}^{(4A)}$ is greatly simplified by the use of the operator $Q^{(I)}$, Eq. (8), for which we now give the explicit eigenvalues: $$\begin{aligned} q^{(I)}_{\{n\}}=\left\{ \begin{array}{cc} 1 & (I) =(\widetilde{4A}),(6) \\ 3-\case{1}{6}(p^2+q^2+3p+3q+pq)& (I)=(4A) \end{array} \right. \, ,\end{aligned}$$ $p$ and $q$ are related to the dimension of the representation $\{n\}$ via $n=\case{1}{2}(p+1)(q+1)(p+q+2)$. The combination of $D$-functions where the spin indices are coupled to a second rank tensor $$\begin{aligned} \case{1}{4}\Bigl(6 D_{11}(C) D_{22}(C) - 6 D_{12}(C) D_{21}(C) - D_{ii}(C) D_{jj}(C) + D_{ij}(C) D_{ji}(C) \Bigr)\end{aligned}$$ leads to the same expressions as (A12) apart from the replacement ${\mbox {\boldmath $\sigma$}}_X \cdot {\mbox {\boldmath $\sigma$}}_Y \rightarrow S_{XY} $. For the sake of brevity, and because no other interactions will be considered, we give the exchange matrix elements only for the case where one of the interacting baryons is a nucleon $N$ the other then is a hyperon $Y$ the hypercharge of which we also designate by $Y$. For $C=A^\dagger B$ we make use of the relation $$\begin{aligned} & &\langle Y'\, N' \vert D^{\{n\}}_{{i_1\;y_1 \choose m_1}{i_2\;y_2 \choose m_2}}(C) \vert N\, Y \rangle = \\& & \qquad \qquad\qquad \qquad \sum_{I,y,M} \langle N \vert D^{\{n\}}_{{I\;y \choose M}{i_1\;y_1 \choose m_1}}(A) \vert Y' \rangle^* \, \langle N' \vert D^{\{n\}}_{{I\;y \choose M}{i_2\;y_2 \choose m_2}}(B) \vert Y \rangle \; . \nonumber\end{aligned}$$ Again, with the conventions of de Swart[@DeSwart] individual angular integrals are real, so we may omit the complex conjugation. Non-vanishing matrix elements must now transfer hypercharge $y=1-Y$ and this also restricts the possible isospins $I$ occurring in the sum of (A14). We can give a compact representation of the non-vanishing exchange matrix elements once the different isospin cases occurring are abbreviated by $$\begin{aligned} {\bf T}^I_{NY }=\left\{ \begin{array}{l} % \left[ \begin{array}{ccc} 1 & {\rm ~~~~~~~~}Y=\Lambda & \\ \case{1}{3}(1+ {\mbox {\boldmath $\tau$}}_N \cdot {\mbox {\boldmath $T$}}_\Sigma)& {\rm ~~~for~}Y=\Sigma&{\rm ~~~for~} I=\case{1}{2}\\ 0&{\rm ~~~~~~~~}Y=\Xi & \end{array} \right. \nonumber \\ \nonumber \\ % \left[ \begin{array}{ccc} 0 & {\rm ~~~~~~~~}Y=\Lambda & \\ \case{1}{3}(1-\case{1}{2} {\mbox {\boldmath $\tau$}}_N \cdot {\mbox {\boldmath $T$}}_\Sigma)& {\rm ~~~for~}Y=\Sigma&{\rm ~~~for~} I=\frac{3}{2}\\ 0&{\rm ~~~~~~~~}Y=\Xi & \end{array} \right. \nonumber \\ \nonumber \\ % \left[ \begin{array}{ccc} 0 & {\rm ~~~~~~~~}Y=\Lambda & \\ 0&{\rm ~~~for~}Y=\Sigma&{\rm ~~~for~} I=0\\ \case{1}{2}(1+ {\mbox {\boldmath $\tau$}}_N \cdot {\mbox {\boldmath $\tau$}}_\Xi)& {\rm ~~~~~~~~}Y=\Xi & \end{array} \right. \nonumber \\ \nonumber \\ % \left[ \begin{array}{ccc} 0 & {\rm ~~~~~~~~}Y=\Lambda & \\ 0&{\rm ~~~for~}Y=\Sigma&{\rm ~~~for~} I=1\\ \case{1}{2}(1-\case{1}{3} {\mbox {\boldmath $\tau$}}_N \cdot {\mbox {\boldmath $\tau$}}_\Xi)& {\rm ~~~~~~~~}Y=\Xi & \end{array} \right. \nonumber \end{array} \right.\end{aligned}$$ As usual, we follow our conventions for implicit sums: $i,j\in\{1,2,3\}$ and $k\in\{4,5,6,7\}$. $$\begin{aligned} \langle Y'\, N' \vert D_{ii}(C) \vert N \, Y \rangle \rightarrow & & \case{1}{2}(1-\case{1}{3} {\mbox {\boldmath $\sigma$}}_N \cdot {\mbox {\boldmath $\sigma$}}_Y)\, {\bf T}^I_{NY} \, \langle ( \case{1}{2} 1) (\case{1}{2} 1) \Vert D_{(Iy)(10)}^{\{ 8 \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle^2\end{aligned}$$ The combination of $D$-functions where the spin indices are coupled to a second rank tensor $$\begin{aligned} 3 D_{33}(C) - D_{ii}(C)\end{aligned}$$ leads to the same expression as (A15) apart from the replacement $\case{1}{2}(1-\case{1}{3} {\mbox {\boldmath $\sigma$}}_N \cdot {\mbox {\boldmath $\sigma$}}_Y) \rightarrow \case{1}{3} S_{NY} $. $$\begin{aligned} \langle Y'\, N' \vert D_{88}(C) \vert N \, Y \rangle \rightarrow \case{1}{2}(1+ {\mbox {\boldmath $\sigma$}}_N \cdot {\mbox {\boldmath $\sigma$}}_Y)\, {\bf T}^I_{NY} \, \langle ( \case{1}{2} 1) (\case{1}{2} 1) \Vert D_{(Iy)(00)}^{\{ 8 \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle^2\end{aligned}$$ $$\begin{aligned} & &\langle Y'\, N' \vert \case{1}{9} Q^{(I)} D_{ij}(C)D_{ij}(C) \vert N \, Y \rangle \rightarrow \\ & &\quad \case{1}{6}(1+ {\mbox {\boldmath $\sigma$}}_N \cdot {\mbox {\boldmath $\sigma$}}_Y)\, \sum_{\{ n \}{\gamma}} \left( \begin{array}{cc\|c} \{ 8 \} & \{ 8 \} & \{ n \}_{\gamma}\\[-2pt] 10 & 10 & 00 \end{array} \right)^2 q_{\{n\}}^{(I)} \sum_I \langle ( \case{1}{2} 1) (\case{1}{2} 1) \Vert D_{(Iy)(00)}^{\{ 8 \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle^2 \, {\bf T}^I_{NY} \nonumber\end{aligned}$$ $$\begin{aligned} & &\langle Y'\, N' \vert \case{1}{4} Q^{(I)} (D_{ii}(C)D_{jj}(C)-D_{ij}(C)D_{ji}(C)) \vert N \, Y \rangle \rightarrow \\ & &\quad \case{1}{4}(1-\case{1}{3} {\mbox {\boldmath $\sigma$}}_N \cdot {\mbox {\boldmath $\sigma$}}_Y)\, \sum_{\{ n \}{\gamma}} \left( \begin{array}{cc\|c} \{ 8 \} & \{ 8 \} & \{ n \}_{\gamma}\\[-2pt] 10 & 10 & 10 \end{array} \right)^2 q_{\{n\}}^{(I)} \sum_I \langle ( \case{1}{2} 1) (\case{1}{2} 1) \Vert D_{(Iy)(10)}^{\{ 8 \}}\Vert (i_Yy_Y) (\case{1}{2}1) \rangle^2 \, {\bf T}^I_{NY} \nonumber\end{aligned}$$ The combination of $D$-functions where the spin indices are coupled to a second rank tensor $$\begin{aligned} \case{1}{4}\Biggl( 6 D_{11}(C) D_{22}(C)-6 D_{12}(C) D_{21}(C) - D_{ii}(C) D_{jj}(C)+ D_{ij}(C) D_{ji}(C)\Biggr)\end{aligned}$$ leads to the same expression as (A18) apart from the replacement $\case{1}{4}(1-\case{1}{3} {\mbox {\boldmath $\sigma$}}_N \cdot {\mbox {\boldmath $\sigma$}}_Y) \rightarrow \case{1}{6} S_{NY} $. The flavor symmetry breaking terms finally introduce other new structures which may, however, be related to existing ones: $$\begin{aligned} \langle Y'\, N' \vert D_{k8}(A)D_{k8}(B) \vert N \, Y \rangle = \langle Y'\, N' \vert D_{88}(C) \vert N \, Y \rangle\end{aligned}$$ $$\begin{aligned} \langle Y'\, N' \vert D_{ki}(A)D_{ki}(B) \vert N \, Y \rangle = \langle Y'\, N' \vert D_{ii}(C) \vert N \, Y \rangle\end{aligned}$$ $$\begin{aligned} \langle Y'\, N' \vert 3D_{k3}(A)D_{k3}(B)-D_{ki}(A)D_{ki}(B) \vert N \, Y \rangle = \langle Y'\, N' \vert 3 D_{33}(C)-D_{ii}(C) \vert N \, Y \rangle \; .\end{aligned}$$ {#secappB} In this appendix we list all interaction components of octet baryon interactions for the lagrangian given by Eqs. (1-5,20 ). Only the cumbersome but numerically small contributions from the term proportional to $f_K^2-f_\pi^2$ have been dropped. The parameters are those fitting the $B=0$ sector, $f_\pi=93.0{\rm MeV}$, $m_\pi=138.0{\rm MeV}$, $f_K=113.5{\rm MeV}$, $m_K=495.0{\rm MeV}$, with the higher order terms, $e= 4.12$, $\chi= 0$, $\kappa=0$, $\epsilon_6=0$, adjusted to the $N\Delta$-split in the $B=1$-sector. The Euler angle wave functions are obtained by diagonalizing the rotational and flavor symmetry breaking terms in the rigid rotator approximation. The tables list the different radial functions subscripted by the commonly used mnemonic notation indicating the two-body spin or isospin operator combination according to (A4, A5, A10) which must be multiplied. In order to maintain uniformity of the tables there often are superfluous columns in the tables, e.g. the isoquadrupole-isoquadrupole interactions $U_{QQ}$, $U_{\sigma\sigma QQ}$, $U_{SQQ}$ which can only be non-zero for the $\Sigma \Sigma$-system. For the $\Sigma \Sigma$-system they then turn out to be numerically small. [cdccdccccc]{}\ & & & & & & & & &\ 0.10& 722.1& 0.3& 0.3& 15.9& 0.0& 0.2& 0.0& 0.0& 0.0\ 0.20& 680.9& 0.3& 0.3& 15.2& 0.0& 0.6& 0.0& 0.0& 0.0\ 0.30& 618.1& 0.3& 0.2& 14.1& 0.0& 1.2& 0.0& 0.0& 0.0\ 0.40& 540.9& 0.3& 0.2& 12.8& 0.0& 1.8& 0.0& 0.0& 0.0\ 0.50& 457.0& 0.2& 0.2& 11.3& 0.0& 2.5& 0.0& 0.0& 0.0\ 0.60& 373.2& 0.2& 0.1& 9.8& 0.0& 3.0& 0.0& 0.0& 0.0\ 0.70& 295.1& 0.2& 0.1& 8.3& 0.0& 3.3& 0.0& 0.0& 0.0\ 0.80& 226.3& 0.1& 0.1& 6.9& 0.0& 3.5& 0.0& 0.0& 0.0\ 0.90& 168.5& 0.1& 0.1& 5.5& 0.0& 3.5& 0.0& 0.0& 0.0\ 1.00& 122.2& 0.1& 0.0& 4.4& 0.0& 3.4& 0.0& 0.0& 0.0\ 1.10& 86.6& 0.1& 0.0& 3.4& 0.0& 3.1& 0.0& 0.0& 0.0\ 1.20& 60.1& 0.1& 0.0& 2.6& 0.0& 2.8& 0.0& 0.0& 0.0\ 1.30& 41.1& 0.1& 0.0& 2.0& 0.0& 2.5& 0.0& 0.0& 0.0\ 1.40& 27.8& 0.0& 0.0& 1.5& 0.0& 2.2& 0.0& 0.0& 0.0\ 1.50& 18.6& 0.0& 0.0& 1.1& 0.0& 1.9& 0.0& 0.0& 0.0\ 1.60& 12.5& 0.0& 0.0& 0.9& 0.0& 1.7& 0.0& 0.0& 0.0\ 1.70& 8.3& 0.0& 0.0& 0.7& 0.0& 1.4& 0.0& 0.0& 0.0\ 1.80& 5.6& 0.0& 0.0& 0.5& 0.0& 1.2& 0.0& 0.0& 0.0\ 1.90& 3.8& 0.0& 0.0& 0.4& 0.0& 1.1& 0.0& 0.0& 0.0\ 2.00& 2.6& 0.0& 0.0& 0.3& 0.0& 0.9& 0.0& 0.0& 0.0\ 2.10& 1.7& 0.0& 0.0& 0.3& 0.0& 0.8& 0.0& 0.0& 0.0\ 2.20& 1.2& 0.0& 0.0& 0.2& 0.0& 0.7& 0.0& 0.0& 0.0\ 2.30& 0.8& 0.0& 0.0& 0.2& 0.0& 0.6& 0.0& 0.0& 0.0\ 2.40& 0.6& 0.0& 0.0& 0.2& 0.0& 0.5& 0.0& 0.0& 0.0\ 2.50& 0.4& 0.0& 0.0& 0.1& 0.0& 0.4& 0.0& 0.0& 0.0\ 3.00& 0.1& 0.0& 0.0& 0.1& 0.0& 0.2& 0.0& 0.0& 0.0\ 3.50& 0.0& 0.0& 0.0& 0.0& 0.0& 0.1& 0.0& 0.0& 0.0\ \[table1\] [cdccdccccc]{}\ & & & & & & & & &\ 0.10& 689.8& -7.5& 0.0& 0.0& 0.0& 0.0& 0.0& 0.0& 0.0\ 0.20& 650.8& -7.4& 0.0& 0.0& -0.1& 0.0& 0.0& 0.0& 0.0\ 0.30& 591.2& -7.3& 0.0& 0.0& -0.3& 0.0& 0.0& 0.0& 0.0\ 0.40& 518.0& -7.1& 0.0& 0.0& -0.6& 0.0& 0.0& 0.0& 0.0\ 0.50& 438.4& -6.8& 0.0& 0.0& -0.9& 0.0& 0.0& 0.0& 0.0\ 0.60& 358.9& -6.4& 0.0& 0.0& -1.2& 0.0& 0.0& 0.0& 0.0\ 0.70& 284.7& -5.9& 0.0& 0.0& -1.6& 0.0& 0.0& 0.0& 0.0\ 0.80& 219.1& -5.4& 0.0& 0.0& -1.9& 0.0& 0.0& 0.0& 0.0\ 0.90& 164.0& -4.9& 0.0& 0.0& -2.2& 0.0& 0.0& 0.0& 0.0\ 1.00& 119.6& -4.3& 0.0& 0.0& -2.5& 0.0& 0.0& 0.0& 0.0\ 1.10& 85.4& -3.8& 0.0& 0.0& -2.7& 0.0& 0.0& 0.0& 0.0\ 1.20& 59.8& -3.2& 0.0& 0.0& -2.8& 0.0& 0.0& 0.0& 0.0\ 1.30& 41.4& -2.7& 0.0& 0.0& -2.9& 0.0& 0.0& 0.0& 0.0\ 1.40& 28.4& -2.3& 0.0& 0.0& -2.9& 0.0& 0.0& 0.0& 0.0\ 1.50& 19.4& -1.9& 0.0& 0.0& -2.9& 0.0& 0.0& 0.0& 0.0\ 1.60& 13.3& -1.6& 0.0& 0.0& -2.9& 0.0& 0.0& 0.0& 0.0\ 1.70& 9.1& -1.3& 0.0& 0.0& -2.8& 0.0& 0.0& 0.0& 0.0\ 1.80& 6.3& -1.1& 0.0& 0.0& -2.7& 0.0& 0.0& 0.0& 0.0\ 1.90& 4.4& -0.9& 0.0& 0.0& -2.6& 0.0& 0.0& 0.0& 0.0\ 2.00& 3.1& -0.7& 0.0& 0.0& -2.4& 0.0& 0.0& 0.0& 0.0\ 2.10& 2.2& -0.6& 0.0& 0.0& -2.3& 0.0& 0.0& 0.0& 0.0\ 2.20& 1.6& -0.5& 0.0& 0.0& -2.2& 0.0& 0.0& 0.0& 0.0\ 2.30& 1.2& -0.4& 0.0& 0.0& -2.1& 0.0& 0.0& 0.0& 0.0\ 2.40& 0.9& -0.3& 0.0& 0.0& -1.9& 0.0& 0.0& 0.0& 0.0\ 2.50& 0.6& -0.2& 0.0& 0.0& -1.8& 0.0& 0.0& 0.0& 0.0\ 3.00& 0.1& 0.0& 0.0& 0.0& -1.3& 0.0& 0.0& 0.0& 0.0\ 3.50& 0.0& 0.0& 0.0& 0.0& -0.9& 0.0& 0.0& 0.0& 0.0\ \[table2\] [cdccdccccc]{}\ & & & & & & & & &\ 0.10& 628.7& 12.3& 94.5& 100.5& 0.0& 0.1& 0.0& 0.0& 0.0\ 0.20& 592.6& 11.5& 89.2& 94.9& 0.0& 0.5& 0.0& 0.0& 0.0\ 0.30& 537.6& 10.4& 81.1& 86.4& 0.0& 1.0& 0.0& 0.0& 0.0\ 0.40& 470.0& 9.1& 71.1& 76.0& -0.1& 1.5& 0.0& 0.0& 0.0\ 0.50& 396.6& 7.6& 60.3& 64.6& -0.1& 2.0& 0.0& 0.0& 0.0\ 0.60& 323.5& 6.2& 49.4& 53.3& -0.1& 2.4& 0.0& 0.0& 0.0\ 0.70& 255.4& 4.8& 39.3& 42.6& -0.2& 2.7& 0.0& 0.0& 0.0\ 0.80& 195.4& 3.7& 30.3& 33.1& -0.2& 2.8& 0.0& 0.0& 0.0\ 0.90& 145.1& 2.7& 22.8& 25.0& -0.3& 2.8& 0.0& 0.0& 0.0\ 1.00& 104.8& 1.9& 16.7& 18.4& -0.3& 2.7& 0.0& 0.0& 0.0\ 1.10& 73.9& 1.3& 12.0& 13.3& -0.3& 2.5& 0.0& 0.0& 0.0\ 1.20& 51.0& 0.9& 8.4& 9.5& -0.4& 2.2& 0.0& 0.0& 0.0\ 1.30& 34.6& 0.6& 5.9& 6.6& -0.4& 2.0& 0.0& 0.0& 0.0\ 1.40& 23.2& 0.4& 4.1& 4.6& -0.4& 1.7& 0.0& 0.0& 0.0\ 1.50& 15.4& 0.2& 2.8& 3.2& -0.4& 1.5& 0.0& 0.0& 0.0\ 1.60& 10.2& 0.1& 1.9& 2.2& -0.4& 1.2& 0.0& 0.0& 0.0\ 1.70& 6.7& 0.1& 1.3& 1.6& -0.4& 1.1& 0.0& 0.0& 0.0\ 1.80& 4.4& 0.0& 0.9& 1.1& -0.4& 0.9& 0.0& 0.0& 0.0\ 1.90& 2.9& 0.0& 0.7& 0.8& -0.4& 0.7& 0.0& 0.0& 0.0\ 2.00& 2.0& 0.0& 0.5& 0.6& -0.3& 0.6& 0.0& 0.0& 0.0\ 2.10& 1.3& 0.0& 0.3& 0.4& -0.3& 0.5& 0.0& 0.0& 0.0\ 2.20& 0.9& 0.0& 0.3& 0.3& -0.3& 0.4& 0.0& 0.0& 0.0\ 2.30& 0.6& 0.0& 0.2& 0.3& -0.3& 0.4& 0.0& 0.0& 0.0\ 2.40& 0.4& 0.0& 0.1& 0.2& -0.3& 0.3& 0.0& 0.0& 0.0\ 2.50& 0.3& 0.0& 0.1& 0.2& -0.3& 0.3& 0.0& 0.0& 0.0\ 3.00& 0.0& 0.0& 0.0& 0.1& -0.2& 0.1& 0.0& 0.0& 0.0\ 3.50& 0.0& 0.0& 0.0& 0.0& -0.1& 0.1& 0.0& 0.0& 0.0\ \[table3\] [cdccdccccc]{}\ & & & & & & & & &\ 0.10& 552.2& -2.9& 2.7& -4.1& 0.0& 0.0& 0.0& 0.0& 0.0\ 0.20& 520.3& -2.8& 2.5& -4.0& 0.0& -0.2& 0.0& 0.0& 0.0\ 0.30& 471.7& -2.6& 2.3& -3.7& 0.0& -0.3& 0.0& 0.0& 0.0\ 0.40& 412.0& -2.4& 2.0& -3.3& 0.0& -0.5& 0.0& 0.0& 0.0\ 0.50& 347.3& -2.1& 1.7& -3.0& 0.0& -0.7& 0.0& 0.0& 0.0\ 0.60& 282.8& -1.8& 1.4& -2.6& 0.0& -0.8& 0.0& 0.0& 0.0\ 0.70& 222.8& -1.6& 1.1& -2.2& 0.0& -0.9& 0.0& 0.0& 0.0\ 0.80& 170.0& -1.3& 0.9& -1.8& 0.0& -0.9& 0.0& 0.0& 0.0\ 0.90& 125.8& -1.1& 0.6& -1.5& 0.1& -0.9& 0.0& 0.0& 0.0\ 1.00& 90.5& -0.9& 0.5& -1.2& 0.1& -0.9& 0.0& 0.0& 0.0\ 1.10& 63.5& -0.7& 0.3& -0.9& 0.1& -0.8& 0.0& 0.0& 0.0\ 1.20& 43.5& -0.6& 0.2& -0.7& 0.1& -0.8& 0.0& 0.0& 0.0\ 1.30& 29.3& -0.5& 0.2& -0.5& 0.1& -0.7& 0.0& 0.0& 0.0\ 1.40& 19.4& -0.4& 0.1& -0.4& 0.1& -0.6& 0.0& 0.0& 0.0\ 1.50& 12.7& -0.3& 0.1& -0.3& 0.2& -0.5& 0.0& 0.0& 0.0\ 1.60& 8.3& -0.3& 0.1& -0.2& 0.2& -0.4& 0.0& 0.0& 0.0\ 1.70& 5.3& -0.2& 0.0& -0.2& 0.2& -0.4& 0.0& 0.0& 0.0\ 1.80& 3.4& -0.2& 0.0& -0.1& 0.2& -0.3& 0.0& 0.0& 0.0\ 1.90& 2.2& -0.1& 0.0& -0.1& 0.2& -0.3& 0.0& 0.0& 0.0\ 2.00& 1.4& -0.1& 0.0& -0.1& 0.2& -0.2& 0.0& 0.0& 0.0\ 2.10& 0.9& -0.1& 0.0& -0.1& 0.1& -0.2& 0.0& 0.0& 0.0\ 2.20& 0.6& -0.1& 0.0& -0.1& 0.1& -0.2& 0.0& 0.0& 0.0\ 2.30& 0.4& -0.1& 0.0& -0.1& 0.1& -0.2& 0.0& 0.0& 0.0\ 2.40& 0.2& 0.0& 0.0& 0.0& 0.1& -0.1& 0.0& 0.0& 0.0\ 2.50& 0.1& 0.0& 0.0& 0.0& 0.1& -0.1& 0.0& 0.0& 0.0\ 3.00& 0.0& 0.0& 0.0& 0.0& 0.1& -0.1& 0.0& 0.0& 0.0\ 3.50& 0.0& 0.0& 0.0& 0.0& 0.1& 0.0& 0.0& 0.0& 0.0\ \[table4\] [cdccdccccc]{}\ & & & & & & & & &\ 0.10& 600.8& 12.6& 0.0& 0.0& 0.0& 0.0& 0.0& 0.0& 0.0\ 0.20& 566.2& 12.1& 0.0& 0.0& 0.0& 0.0& 0.0& 0.0& 0.0\ 0.30& 513.4& 11.3& 0.0& 0.0& 0.1& 0.0& 0.0& 0.0& 0.0\ 0.40& 448.5& 10.3& 0.0& 0.0& 0.1& 0.0& 0.0& 0.0& 0.0\ 0.50& 378.1& 9.2& 0.0& 0.0& 0.1& 0.0& 0.0& 0.0& 0.0\ 0.60& 308.0& 8.1& 0.0& 0.0& 0.0& 0.0& 0.0& 0.0& 0.0\ 0.70& 242.8& 7.0& 0.0& 0.0& -0.1& 0.0& 0.0& 0.0& 0.0\ 0.80& 185.3& 5.9& 0.0& 0.0& -0.1& 0.0& 0.0& 0.0& 0.0\ 0.90& 137.3& 4.9& 0.0& 0.0& -0.2& 0.0& 0.0& 0.0& 0.0\ 1.00& 98.8& 4.0& 0.0& 0.0& -0.4& 0.0& 0.0& 0.0& 0.0\ 1.10& 69.4& 3.3& 0.0& 0.0& -0.5& 0.0& 0.0& 0.0& 0.0\ 1.20& 47.7& 2.7& 0.0& 0.0& -0.5& 0.0& 0.0& 0.0& 0.0\ 1.30& 32.1& 2.1& 0.0& 0.0& -0.6& 0.0& 0.0& 0.0& 0.0\ 1.40& 21.3& 1.7& 0.0& 0.0& -0.7& 0.0& 0.0& 0.0& 0.0\ 1.50& 14.0& 1.4& 0.0& 0.0& -0.7& 0.0& 0.0& 0.0& 0.0\ 1.60& 9.1& 1.1& 0.0& 0.0& -0.7& 0.0& 0.0& 0.0& 0.0\ 1.70& 5.9& 0.9& 0.0& 0.0& -0.7& 0.0& 0.0& 0.0& 0.0\ 1.80& 3.8& 0.7& 0.0& 0.0& -0.7& 0.0& 0.0& 0.0& 0.0\ 1.90& 2.5& 0.6& 0.0& 0.0& -0.7& 0.0& 0.0& 0.0& 0.0\ 2.00& 1.6& 0.5& 0.0& 0.0& -0.7& 0.0& 0.0& 0.0& 0.0\ 2.10& 1.0& 0.4& 0.0& 0.0& -0.7& 0.0& 0.0& 0.0& 0.0\ 2.20& 0.7& 0.3& 0.0& 0.0& -0.6& 0.0& 0.0& 0.0& 0.0\ 2.30& 0.4& 0.2& 0.0& 0.0& -0.6& 0.0& 0.0& 0.0& 0.0\ 2.40& 0.3& 0.2& 0.0& 0.0& -0.6& 0.0& 0.0& 0.0& 0.0\ 2.50& 0.2& 0.2& 0.0& 0.0& -0.5& 0.0& 0.0& 0.0& 0.0\ 3.00& 0.0& 0.0& 0.0& 0.0& -0.4& 0.0& 0.0& 0.0& 0.0\ 3.50& 0.0& 0.0& 0.0& 0.0& -0.3& 0.0& 0.0& 0.0& 0.0\ \[table5\] [cdccdccccc]{}\ & & & & & & & & &\ 0.10& 555.7& 7.0& 23.4& 10.0& 0.0& 0.1& 0.0& 0.0& 0.0\ 0.20& 523.4& 6.7& 22.1& 9.5& 0.0& 0.4& 0.0& 0.0& 0.0\ 0.30& 474.2& 6.3& 20.1& 8.8& 0.0& 0.7& 0.0& 0.0& 0.0\ 0.40& 413.8& 5.8& 17.6& 8.0& 0.0& 1.2& 0.0& 0.0& 0.0\ 0.50& 348.3& 5.2& 14.9& 7.1& 0.0& 1.6& 0.0& 0.0& 0.0\ 0.60& 283.2& 4.5& 12.2& 6.1& 0.0& 1.9& 0.0& 0.0& 0.0\ 0.70& 222.6& 3.9& 9.7& 5.2& 0.0& 2.1& 0.0& 0.0& 0.0\ 0.80& 169.4& 3.3& 7.4& 4.3& -0.1& 2.2& 0.0& 0.0& 0.0\ 0.90& 124.9& 2.7& 5.6& 3.5& -0.1& 2.2& 0.0& 0.0& 0.0\ 1.00& 89.5& 2.3& 4.0& 2.8& -0.2& 2.1& 0.0& 0.0& 0.0\ 1.10& 62.4& 1.8& 2.9& 2.1& -0.3& 2.0& 0.0& 0.0& 0.0\ 1.20& 42.4& 1.5& 2.0& 1.6& -0.3& 1.8& 0.0& 0.0& 0.0\ 1.30& 28.3& 1.2& 1.4& 1.2& -0.3& 1.6& 0.0& 0.0& 0.0\ 1.40& 18.5& 1.0& 1.0& 0.9& -0.4& 1.4& 0.0& 0.0& 0.0\ 1.50& 11.9& 0.8& 0.6& 0.7& -0.4& 1.2& 0.0& 0.0& 0.0\ 1.60& 7.6& 0.6& 0.4& 0.5& -0.4& 1.0& 0.0& 0.0& 0.0\ 1.70& 4.8& 0.5& 0.3& 0.4& -0.4& 0.9& 0.0& 0.0& 0.0\ 1.80& 3.0& 0.4& 0.2& 0.3& -0.4& 0.8& 0.0& 0.0& 0.0\ 1.90& 1.8& 0.3& 0.1& 0.3& -0.4& 0.7& 0.0& 0.0& 0.0\ 2.00& 1.1& 0.3& 0.1& 0.2& -0.4& 0.6& 0.0& 0.0& 0.0\ 2.10& 0.7& 0.2& 0.1& 0.2& -0.4& 0.5& 0.0& 0.0& 0.0\ 2.20& 0.4& 0.2& 0.0& 0.1& -0.4& 0.4& 0.0& 0.0& 0.0\ 2.30& 0.2& 0.1& 0.0& 0.1& -0.3& 0.4& 0.0& 0.0& 0.0\ 2.40& 0.1& 0.1& 0.0& 0.1& -0.3& 0.3& 0.0& 0.0& 0.0\ 2.50& 0.0& 0.1& 0.0& 0.1& -0.3& 0.3& 0.0& 0.0& 0.0\ 3.00& 0.0& 0.0& 0.0& 0.0& -0.2& 0.1& 0.0& 0.0& 0.0\ 3.50& 0.0& 0.0& 0.0& 0.0& -0.2& 0.1& 0.0& 0.0& 0.0\ \[table6\] [cdccdccccc]{}\ & & & & & & & & &\ 0.10& 508.0& 25.2& 23.1& 1.1& 0.0& 0.0& 0.0& 0.0& 0.0\ 0.20& 477.9& 24.2& 21.8& 1.1& 0.1& 0.0& 0.0& 0.0& 0.0\ 0.30& 432.2& 22.7& 19.8& 1.0& 0.1& 0.1& 0.0& 0.0& 0.0\ 0.40& 376.3& 20.7& 17.3& 0.9& 0.1& 0.1& 0.0& 0.0& 0.0\ 0.50& 315.7& 18.5& 14.7& 0.8& 0.1& 0.2& 0.0& 0.0& 0.0\ 0.60& 255.6& 16.2& 12.0& 0.7& 0.0& 0.2& 0.0& 0.0& 0.0\ 0.70& 199.8& 14.0& 9.5& 0.6& -0.1& 0.2& 0.0& 0.0& 0.0\ 0.80& 151.0& 11.8& 7.3& 0.5& -0.3& 0.3& 0.0& 0.0& 0.0\ 0.90& 110.3& 9.9& 5.5& 0.4& -0.5& 0.3& 0.0& 0.0& 0.0\ 1.00& 78.1& 8.1& 4.0& 0.3& -0.7& 0.2& 0.0& 0.0& 0.0\ 1.10& 53.6& 6.6& 2.8& 0.2& -0.9& 0.2& 0.0& 0.0& 0.0\ 1.20& 35.7& 5.3& 2.0& 0.2& -1.1& 0.2& 0.0& 0.0& 0.0\ 1.30& 23.2& 4.3& 1.4& 0.1& -1.2& 0.2& 0.0& 0.0& 0.0\ 1.40& 14.6& 3.5& 0.9& 0.1& -1.3& 0.2& 0.0& 0.0& 0.0\ 1.50& 9.0& 2.8& 0.6& 0.1& -1.4& 0.1& 0.0& 0.0& 0.0\ 1.60& 5.4& 2.3& 0.4& 0.1& -1.4& 0.1& 0.0& 0.0& 0.0\ 1.70& 3.1& 1.8& 0.3& 0.0& -1.5& 0.1& 0.0& 0.0& 0.0\ 1.80& 1.7& 1.5& 0.2& 0.0& -1.4& 0.1& 0.0& 0.0& 0.0\ 1.90& 0.9& 1.2& 0.1& 0.0& -1.4& 0.1& 0.0& 0.0& 0.0\ 2.00& 0.4& 1.0& 0.1& 0.0& -1.4& 0.1& 0.0& 0.0& 0.0\ 2.10& 0.1& 0.8& 0.1& 0.0& -1.3& 0.1& 0.0& 0.0& 0.0\ 2.20& -0.1& 0.6& 0.0& 0.0& -1.3& 0.0& 0.0& 0.0& 0.0\ 2.30& -0.2& 0.5& 0.0& 0.0& -1.2& 0.0& 0.0& 0.0& 0.0\ 2.40& -0.2& 0.4& 0.0& 0.0& -1.2& 0.0& 0.0& 0.0& 0.0\ 2.50& -0.2& 0.3& 0.0& 0.0& -1.1& 0.0& 0.0& 0.0& 0.0\ 3.00& -0.1& 0.1& 0.0& 0.0& -0.8& 0.0& 0.0& 0.0& 0.0\ 3.50& 0.0& 0.0& 0.0& 0.0& -0.6& 0.0& 0.0& 0.0& 0.0\ \[table7\] Th. 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The crosses indicate the experimental splittings, the dashed lines the spectrum calculated in the slow rotator approach, dots the results from rigidly rotated solitons. For the rigid rotator we used the parameters $f_\pi=93.0{\rm MeV}$, $m_\pi=138.0{\rm MeV}$, $f_K=113.5{\rm MeV}$, $ m_K=495.0{\rm MeV}$, $e= 4.12$, $\chi= 0$, $\kappa=0$, $\epsilon_6=0$ and for the slow rotator $f_\pi=93.0{\rm MeV}$, $m_\pi=138.0{\rm MeV}$, $f_K=113.5{\rm MeV}$, $m_K=495.0{\rm MeV}$, $e= 3.45$, $\chi= -0.07$, $\kappa=0$, $\epsilon_6=0$. (14.,15.)(0.,-4.) (-.2,0)[(0,1)[9.5]{}]{} (-.35,1.)(0,1.0)[8]{}[(1,0)[0.15]{}]{} (-1.2,2.)[200]{} (-1.2,4.)[400]{} (-1.2,6.)[600]{} (-1.3,8.5)[MeV]{} (0,0)(.3,0.)[3]{}[$\, _\times$]{} (1.5,1.77)(.3,0.)[3]{}[$\, _\times$]{} (3.0,2.54)(.3,0.)[3]{}[$\, _\times$]{} (4.5,3.79)(.3,0.)[3]{}[$\, _\times$]{} (6.5,2.97)(.3,0.)[3]{}[$\, _\times$]{} (8.0,4.46)(.3,0.)[3]{}[$\, _\times$]{} (9.5,5.92)(.3,0.)[3]{}[$\, _\times$]{} (11.0,7.33)(.3,0.)[3]{}[$\, _\times$]{} (0,0)(.3,.0)[4]{}[(1,0)[.1]{}]{} (1.5,1.76)(.3,.0)[4]{}[(1,0)[.1]{}]{} (3.0,2.87)(.3,.0)[4]{}[(1,0)[.1]{}]{} (4.5,3.82)(.3,.0)[4]{}[(1,0)[.1]{}]{} (6.5,3.00)(.3,.0)[4]{}[(1,0)[.1]{}]{} (8.0,4.75)(.3,.0)[4]{}[(1,0)[.1]{}]{} (9.5,6.14)(.3,.0)[4]{}[(1,0)[.1]{}]{} (11.0,7.25)(.3,.0)[4]{}[(1,0)[.1]{}]{} (0.3,0)(.15,.0)[4]{} (1.8,1.45)(.15,.0)[4]{} (3.3,2.50)(.15,.0)[4]{} (4.8,3.50)(.15,.0)[4]{} (6.8,3.17)(.15,.0)[4]{} (8.3,4.38)(.15,.0)[4]{} (9.8,5.55)(.15,.0)[4]{} (11.3,6.65)(.15,.0)[4]{} (0.5,-1.5)[$N$]{} (2.0,-1.5)[$\Lambda$]{} (3.5,-1.5)[$\Sigma$]{} (5.0,-1.5)[$\Xi$]{} (7.0,-1.5)[$\Delta$]{} (8.5,-1.5)[$\Sigma ^$]{} (10.0,-1.5)[$\Xi ^$]{} (11.5,-1.5)[$\Omega$]{} Figure 2\ Direct and exchange matrix elements for non-identical baryons. (16.5,15.)(1.5,3.) (1.5,3)[(1,2)[1.5]{}]{} (3,7)[(-1,2)[1.5]{}]{} (3,6)[(0,1)[1]{}]{} (3,6)[(1,0)[1.5]{}]{} (4.5,6)[(0,1)[1]{}]{} (3,7)[(1,0)[1.5]{}]{} (3.1,6.4)[$D(A)$]{} (1.9,3.2)[$N$]{} (1.9,9.8)[$N'$]{} (4.7,6.5) (8.,3)[(-1,2)[1.5]{}]{} (6.5,7)[(1,2)[1.5]{}]{} (6.5,6)[(0,1)[1]{}]{} (6.5,6)[(-1,0)[1.5]{}]{} (5.,6)[(0,1)[1]{}]{} (6.5,7)[(-1,0)[1.5]{}]{} (5.1,6.4)[$D(B)$]{} (7.,3.2)[$Y$]{} (7.,9.8)[$Y'$]{} (11.5,3)[(1,2)[1.5]{}]{} (13,7)[(-1,2)[1.5]{}]{} (13,6)[(0,1)[1]{}]{} (13,6)[(1,0)[1.5]{}]{} (14.5,6)[(0,1)[1]{}]{} (13,7)[(1,0)[1.5]{}]{} (13.1,6.4)[$D(A)$]{} (11.9,3.2)[$N$]{} (11.9,9.8)[$Y'$]{} (14.7,6.5) (18.,3)[(-1,2)[1.5]{}]{} (16.5,7)[(1,2)[1.5]{}]{} (16.5,6)[(0,1)[1]{}]{} (16.5,6)[(-1,0)[1.5]{}]{} (15.,6)[(0,1)[1]{}]{} (16.5,7)[(-1,0)[1.5]{}]{} (15.1,6.4)[$D(B)$]{} (17.,3.2)[$Y$]{} (17.,9.8)[$N'$]{} [^1]: Electronic address: schwesinger@hrz.uni-siegen.d400.de [^2]: Electronic address: fgs@sunvax.sun.ac.za [^3]: Electronic address: hbg@sunvax.sun.ac.za [^4]: We thank A. Hayashi for supplying us the asymptotic solution of the Euler Lagrange equations when flavor symmetry is broken.
--- abstract: 'Charge and spin stripe order is a type of electronic crystal observed in certain layered cuprates associated with high-temperature superconductivity. Quantum-disordered stripes could be relevant for understanding the superconductivity. Here I discuss recent experimental characterizations of the stripe-ordered state in , and compare them with properties of superconducting compositions.' address: 'Brookhaven National Laboratory, Upton, NY 11973-5000, USA' author: - 'J. M. Tranquada' title: 'Charge stripes in cuprate superconductors: The middle way' --- Introduction ============ For two-dimensional (2D) electronic systems, there are two limiting ground states. When kinetic energy dominates, one obtains a Fermi liquid, with delocalized quasiparticles. On the other hand, when Coulomb repulsion dominates, one ends up with electrons localized in a Wigner crystal [@baer05]. The parent compounds of cuprate superconductors are Mott-Hubbard charge-transfer insulators, where neither of the limiting ground states is applicable. The doped antiferromagnetic insulator becomes metallic and superconducting, with a normal state that is not well understood. There have been attempts to describe various aspects of doped cuprates starting with either a Fermi-liquid picture [@chub03] or a Wigner crystal description [@fran04; @rast04; @komi05], though there are features of each that are not entirely satisfactory. As discussed at this workshop by Boris Spivak, there is also a “middle way”. A proper treatment of the transition from a Fermi liquid state to a Wigner crystal indicates that there should be intermediate “micro-emulsion” phases [@jame05]. In the case of hole-doped antiferromagnets such as cuprates, these are predicted to be stripe phases [@kive03; @zaan01; @mach89; @sach91]. Indeed, ordered charge and spin stripes have been observed in certain cuprates (especially near a hole concentration of $\frac18$ per Cu site) [@ichi00; @fuji04] and in layered nickelates over a broad range of doping [@yosh00]. Static stripe order competes with superconductivity [@ichi00], but quantum-disordered stripes could be compatible, and perhaps responsible for, high-temperature superconductivity [@emer97; @arri04]. In this paper, I will discuss some recent results for , a compound that exhibits static stripe order at temperatures below $\approx 50$ K [@fuji04]. The magnetic excitation spectrum in the ordered state [@tran04] looks very much like that in superconducting [@hayd04]. New measurements on in the paramagnetic state indicate that the magnetic excitations change relatively little, consistent with maintaining fluctuating stripe correlations [@xu05]. New optical conductivity [@dord05] and photoemission measurements [@vall05] on the same material indicate that stripes are compatible with the so-called nodal-metal state, but also suggest a charge-density-wave gap within the stripes that is presumably responsible for the suppression of superconductivity [@kive98]. This leads to a new view of stripes and metallic transport. To appreciate some of these results, I will first discuss some characteristics of the “pseudogap” phase in the next section. Characteristics of the pseudogap phase ====================================== It is common to describe the normal state of underdoped cuprate superconductors as a pseudogap phase, at least for temperatures below some crosssover $T^$ [@timu99]. There are two types of features associated with the pseudogap concept. One is the general depression of the density of states near the Fermi level, $E_{\rm F}$, in underdoped cuprates compared to what one might expect for weakly interacting electrons. A second is the temperature-dependent depression of signal seen first in optical conductivity measured with the polarization along the $c$-axis (perpendicular to the CuO$_2$ planes) [@home93] and soon after in angle-resolved photoemission (ARPES) measurements in the vicinity of $(\pi,0)$ (the “antinodal” region) [@loes96; @norm98]. That the depressed density of states, on the one hand, and the temperature-dependent effects, on the other, are distinct phenomena becomes very clear when one considers studies of the optical conductivity parallel to the CuO$_2$ planes, $\sigma_{ab}$. Undoped, insulating cuprates have a charge transfer gap of 1.5 to 2 meV. As shown by Uchida [et al.]{} [@uchi91] for the case of , hole doping introduces finite conductivity into the gap, and this conductivity grows with doping up to $x\sim0.2$. (The growth in integrated conductivity is opposite to what one would expect if the conduction electrons were noninteracting.) This effect is visible at room temperature; on cooling, $\sigma_{ab}$ shows no loss of density of states in the normal state [@sant04]. (Below the superconducting transition temperature, $T_c$, the contribution from electrons that participate in the superfluid move to zero frequency, leaving a gap-like feature in $\sigma_{ab}$.) In contrast to the 1-eV energy scale for the “depressed” density of states, the temperature-dependent pseudogaps seen in ARPES and $\sigma_c$ have an energy scale on the order of 50 meV, comparable to the the superconducting gap maximum. Clearly, the temperature-dependent effect occurs on an energy scale that is much smaller than that associated with the general depression of the density of states. While no temperature-dependent pseudogaps are seen in $\sigma_{ab}$, there is nevertheless an interesting variation in $\sigma_{ab}(\omega)$ as $T$ is reduced towards $T_c$. At high temperatures, the conductivity of underdoped cuprates is rather flat as a function of frequency [@take03; @lee05]. On cooling, the conductivity below $\sim0.1$ eV narrows into a Drude peak, while that at higher energies changes little. At the lower temperatures it becomes possible to decompose the spectrum into two components: a narrow Drude peak at low frequency and a broad Lorentzian oscillator (the “mid-IR” peak) with a doping-dependent mid-point in the range 0.2–0.5 eV [@lee05; @padi05; @rome92]. In the superconducting state, the superfluid density comes almost entirely from the Drude peak [@lee05]. It should be noted that the integrated conductivity in the Drude peak is only a quarter of that in the mid-IR [@lee05; @padi05]. The Drude peak has to come from states close to $E_{\rm F}$. In the underdoped cuprates, these states are along the Fermi arc [@norm98; @yosh03], centered on the nodal point \[the point at which the $d$-symmetry superconducting gap goes to zero, close to $(\frac{\pi}2,\frac{\pi}2)$\]. The regime, below $T^$, where the Drude peak can be resolved has been called a nodal-metal state [@lee05]. I believe that this is a more useful description than the more ambiguous “pseudogap” phase. It was originally noted by Ido [et al.]{} [@ido91] that the energy, magnitude, and doping dependence of the mid-IR peak in is similar to that in the nickelates, . It was only later that the existence of diagonal stripe order was identified in the nickelates [@tran94a]. An interpretation of the mid-IR peak in the nickelates in terms of the electronic structure of stripes has recently been proposed [@home03]. The main difference between the cuprates and nickelates is the absence of a Drude component in the latter. It should be noted, however, that a Drude peak has been observed in the cuprate La$_{1.275}$Nd$_{0.6}$Sr$_{0.125}$CuO$_4$ [@dumm02], which should exhibit stripe order as in similar compositions with slightly less Nd [@ichi00]. Lorenzana and Seibold [@lore03] have obtained both Drude and mid-IR contributions in a calculation for cuprate stripes. Universal magnetic excitation spectrum ====================================== As mentioned before, exhibits static charge and spin stripe order for $T<50$ K. The magnetic excitation has been measured at $T=12$ K using the MAPS spectrometer at the ISIS Facility [@tran04]. The dispersion along a line through the positions of a pair of magnetic superlattice peaks is shown in Fig. 1. At low energies, the excitations disperse upwards out of the superlattice peaks as expected for spin waves [@fuji04]. With increasing energy, we would expect to begin to resolve cones of spin waves, as observed in stripe-ordered [@woo05]; in contrast, we can only identify inwardly dispersing excitations that merge at about 50 meV. Above that energy the excitations disperse outward again, forming an hour-glass shape. ![ \[fig:disp\] Symbols: experimentally measured dispersion[@tran04] of magnetic excitations along ${\bf Q}=(0.5+q,0.5,l)$ in stripe-ordered La$_{1.875}$Ba$_{0.125}$CuO$_4$. Horizontal bars indicate the fitted half-width of the scattering. The solid line is the calculated[@barn94] dispersion along a two-leg ladder with a superexchange energy of $J=100$ meV.](fig_disp.eps){height="10cm"} While the observed dispersion differs from the simplest spin-wave predictions, it turns out to be quite similar to what is observed in superconducting [@arai99; @bour00; @hayd04; @rezn04; @stoc04; @pail04] and [@chri04]. The main difference among these systems is that there is a spin gap in the superconducting state [@tran05a], with a pile up of weight above the spin gap [@chri04; @tran04b]. The magnitude of the spin gap is roughly proportional to $T_c$ [@tran05b]. Guangyong Xu has recently extended the magnetic excitation measurements for to temperatures above the stripe-ordering transition [@xu05]. At 65 K, just above the transition, the main differences are at energies below 10 meV [@fuji04; @xu05], and are of the type expected in a magnetic system that goes through a disordering transition. At higher energies, the main effect seems to be a broadening of the spectrum in terms of momentum widths. The general features of the spectrum are still recognizable at 300 K. The smooth evolution of the magnetic spectrum suggests that the high-temperature state is a dynamically-disordered version of the stripe-ordered phase; in other words, we have evidence for a fluctuating stripe phase. The similarity of the spectrum with that of and then suggests that quantum-disordered stripes may be common to the superconducting cuprates. Electronic nature of the stripe-ordered state ============================================= A remaining question is: are the electronic properties of a static or fluctuating stripe state compatible with the nodal-metal phase discussed earlier? As a first test, Dordevic and Homes [@dord05] have measured the in-plane optical conductivity on a cleaved crystal of . The results are generally quite similar to those in the normal state of with $x=0.125$ [@dumm02]. In particular, the conductivity below 50 cm$^{-1}$ ($\sim6$ meV) grows with cooling, even in the stripe-ordered state. Those low-energy excitations must be associated with the nodal states along the Fermi arc [@yosh03]. A new behavior is the gap-like loss of weight from the range 150–300 cm$^{-1}$ on cooling into the stripe-ordered phase. This loss of weight is likely to occur in the antinodal region of reciprocal space. Tonica Valla, working with Alexei Federov, has very recently succeeded in measuring angle-resolved photoemission from [@vall05]. Although the measurements were obtained with an energy resolution of 30–35 meV, they provide clear evidence for a Fermi arc and nodal quasiparticles. This indicates that stripes are compatible with the nodal metal; in fact, the main conduction path is at 45$^\circ$ to the stripes. The higher-energy gap-like behavior seen in the optical conductivity suggests that there might be a charge-density wave (CDW) along the stripes in the ordered phase. The competition between CDW and superconducting correlations within the stripes has been discussed by Kivelson, Fradkin, and Emery [@kive98], and the nesting of Fermi-surface segments associated with stripes was noted by Zhou [et al.]{} [@zhou99], with $2k_{\rm F} = a^\ast/4$. The CDW would require electron-phonon coupling, and hence one might expect to see a phonon anomaly at ${\bf q}=(2k_{\rm F},0,0)$ (for stripes running along the $x$ direction). Intriguingly, Reznik [et al.]{} [@rezn05] have observed a 10-15 meV energy broadening of the longitudinal-optical bond-stretching branch at just this wave vector. (Because of the nature of the crystal structure, the measurement averages over directions parallel and perpendicular to the stripes, so that it is not possible to tell directly in which orientation the anomaly occurs.) Further measurements are in progress. New view of stripes and metallic transport ========================================== The new experimental results are changing the picture of metallic transport in a striped phase. Previously, the simplest assumption was that charge would move more easily along a stripe than perpendicular to it. 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--- abstract: \| Using a hadron-string cascade model LUCIAE, the $\phi$ meson production in heavy ion collisions ($Pb+Pb$) and elementary collisions ($p+p$) were studied systematically. Within the framework of the model, the experimentally measured $\phi$ enhancement in $Pb+Pb$ over $p+p$ collisions can be mostly explained by the collective effects in the gluon string emission and the reduction of the $s$-quark suppression.\ [PACS numbers: 25.75.Dw, 24.10.Lx, 24.85.+p, 25.75.Gz]{} address: \| $^1$ China Institute of Atomic Energy, P. O. Box 275 (18), Beijing, 102413 China\ $^2$ Institute for Theoretical Physics, University of Tuebingen, Auf der Morgenstelle 14, D-72076 Tuebingen Germany\ $^3$ Department of Physics and Astronomy, University of California, at Los Angeles, Los Angeles, CA 90095 USA\ $^4$ Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA\ $^5$ CCAST (World Lab.), P. O. Box 8730 Beijing, 100080 China\ $^6$ Institute of Theoretical Physics, Academia Sinica, Beijing, 100080 China author: - \| Ben-Hao Sa$^{1,2,5,6}$, Amand Faessler$^2$, C. Fuchs$^2$, An Tai$^3$, Xiao-Rong Wang, Nu Xu$^4$\ and E. Zabrodin$^2$ title: 'Enhancement of $\phi$ mesons in $Pb+Pb$ collisions at 158A GeV/c' --- Already suggested in the early eighties [@raf1] strangeness enhancement is presently considered as one of the most promising signatures for the creation of a Quark-Gluon Plasma (QGP) phase in relativistic nuclear collisions. At the CERN SPS the WA97 Collaboration has measured a clear enhancement of multi-strange baryons ($\Lambda , \Xi , \Omega$) in 158A GeV/c $Pb+Pb$ collisions relative to $p+Pb$ collisions [@wa97]. As the mesonic counterpart, also an enhancement of the $\phi$ meson production in relativistic nuclear collisions was suggested as an evidence of the QGP formation in Ref. [@shor], since in the environment of a QGP the copious strange and antistrange quarks originating from gluon annihilation would be very likely to coalesce forming $\phi$ mesons during the hadronization period. Due to the small cross sections of $\phi$ mesons interacting with non-strange hadrons [@raf1; @shor], penetrating $\phi$ mesons are also messengers of the early stage of the colliding system. Thus, the $\phi$ meson is not only a promising signature for the QGP formation but also a good probe to study the reaction dynamics. Strangeness enhancement in relativistic nucleus-nucleus collisions has in the meantime been investigated by various types of models, besides LUCIAE [@tai1; @sa2]. These are: thermal models assuming an equilibrated quark gluon plasma phase [@raf; @pet; @red], the non-equilibrium hadron gas model with a hadronic strangeness saturation factor [@gaz], the RQMD [@sor] and SFM [@ame] models including the fusion of overlapping strings, HIJING [@wan] and HIJING with modifications of the baryon junction exchange mechanism [@gyu], UrQMD with a reduction of the constituent quark masses [@soff] or with a strong color field effect [@nu], the diquark breaking model [@cap], and the model of strangeness content in nucleon [@liu], etc. Recently, NA49 measured the $\phi$ yield, the rapidity and transverse mass distributions in $p+p$ and $Pb+Pb$ collisions at 158A GeV/c [@na49]. The model studies on the $\phi$ meson enhancement in relativistic nucleus- nucleus collisions are rare and to our knowledges there exists up to now no theoretical description of the full set of NA49 data on the $\phi$ production. In this letter we use a hadron and string cascade model, LUCIAE [@sa], in order to investigate their data and the enhancement mechanism especially. We have successfully used LUCIAE to study the enhanced production of multi-strange baryons ($\Lambda , \Xi , \Omega$) and determined the model parameters related to production of strang particles [@tai1; @sa2]. Therefore, there is no additional free parameter in the present calculations for $\phi$ meson production. The LUCIAE model is based on FRITIOF [@pi], which is an incoherent hadron multiple scattering and string fragmentation model. In FRITIOF, the nucleus-nucleus collision is depicted simply as a superposition of nucleon-nucleon collisions. What characterizes LUCIAE beyond FRITIOF are the following features: First of all, the rescattering between the participant and spectator nucleons and the produced particles from the string fragmentation processes are generally taken into account [@sa1]. However, as proposed in [@raf1; @shor] we, in this work, assume that the final state interaction plays no significant role for the $\phi$ production. Thus, effects of the final state interactions on the $\phi$ meson production and propagation are neglected. Secondly, the collective effect in the gluon emission of strings is considered by so-called firecracker model [@tai]. In relativistic heavy ion collisions the string density can be quite high such that some strings might form a collective state. Such a string state may emit gluons using its larger common energy density. Thirdly, a phenomenological mechanism for the reduction of the $s$ quark suppression in the string fragmentation process [@tai1] is introduced. It is well known that the $s$ quark suppression factor (the suppression of $s$ quark pair production with respect to $u$ or $d$ pair production in the string fragmentation), i. e. the parameter $parj(2)$ in JETSET which runs together with FRITIOF and deals with the string fragmentation, is not a constant but energy dependent in hadron-hadron collisions [@tai1; @gaz]. In $p+A$ and $A+A$ collisions $parj(2)$ depends even on the size and centrality of collision system as a result of mini-jet (gluon) production stemming from the string-string interactions. The phenomenological mechanism introduced in [@tai1] considers all of the above facts via the effective string tension and therefore the pertained JETSET parameters. The extra model parameters introduced were fixed by fitting to $p+p$ data [@tai1]. [cccccc]{}\ \ & &n$_{ch}$ &n$_{\pi}$ &n$_{\phi}$ &n$_{\phi}$/n$_{\pi}$\ $p+p$ &NA49 &7.2 &2.87$^$ &0.012$\pm$0.0015 &0.00418$\pm$0.00053\ & & &2.61$^@$ & &0.00460$\pm$0.00053$^@$\ &LUCIAE &7.82 &2.67 &0.0141 &0.00528\ $Pb+Pb$ &NA49 & &611 &7.6$\pm$1.1 &0.0124$\pm$0.0018\ &LUCIAE & &679 &7.89 &0.0116\ \ \ [cccc]{}\ \ &n$_{\pi}$ &n$_{\phi}$ &n$_{\phi}$/n$_{\pi}$\ LUCIAE &679 &7.89 &0.0116\ w/o ’s’$^$ &687 &6.28 &0.00914\ w/o ’f’$^@$ &679 &4.29 &0.00632\ w/o s and f &643 &5.48 &0.00852\ \ \ [ccccc]{}\ \ &$parj(1)$ &$parj(2)$ &$parj(3)$ &$parj(21)$\ LUCIAE &0.116 &0.313 &0.409 &0.373\ w/o s &0.100 &0.300 &0.400 &0.320\ w/o f &0.0497 &0.215 &0.313 &0.318\ w/o s and f &0.100 &0.300 &0.400 &0.320\ In Table. 1 the LUCIAE results for the $\phi$ meson yield and the average multiplicities of $\pi^+$ and $\pi^-$, etc. are compared to the NA49 data. It should be mentioned here that the pion multiplicities were quoted by NA49 from [@ros] where the experiment triggers on the total inelastic reaction cross section while only 91% of this cross section was measured in the NA49 experiment. Thus, a correction must be made which is referred to as ’after correction’ in Table. 1 . The experimental result for the $\phi$-enhancement factor ($\displaystyle \frac{<\phi>/<\pi> (Pb+Pb \hspace{0.2cm} central)}{<\phi>/<\pi>(p+p \hspace{0.2cm} inelastic)}$) [@na49] in $Pb+Pb$ relative to $p+p$ after correction is 2.7$\pm$0.7 and the corresponding\ LUCIAE result is 2.2 . The transverse mass distributions and the rapidity distributions of $\phi$ mesons in $p+p$ and $Pb+Pb$ collisions at momentum 158 GeV/c per nucleon are compared in Fig. 1 . Fitting rapidity distributions obtained from LUCIAE with a Gaussian, $f(y)=c\times\exp{(-(y-y_{cm})^2/2/\sigma^2)}$, one obtains $\sigma$=0.967 ($p+p$) and 1.05 ($Pb+Pb$), which should be compared with the NA49 results of 0.89$\pm$0.06 and 1.22$\pm$0.16, respectively. Since the inverse slope parameter T extracted from the transverse mass distributions is very sensitive to the details of the fitting procedure we fit the highest four $m_t$ data points both for the NA49 and LUCIAE transverse mass distributions of $Pb+Pb$ collisions with an exponential of form $f(m_t)=c\times\exp{(-m_t/T)}$. We obtain then $T_{NA49}$=289 MeV and $T_{LUCIAE}$=212 MeV. For $p+p$, if one fits the highest three $m_t$ data points both for the NA49 and LUCIAE results one obtains nearly the same inverse slope parameter T=189 MeV. To further improve the agreement between the data and LUCIAE results one might need to invoke the intrinsic transverse momentum broadening in string fragmentation [@lar] provided the rescattering of $\phi$ meson is not important. However, one sees from Table. 1 and Fig. 1 that employing the mechanisms of collective string effects in the gluon emission and the reduction of the $s$ quark suppression in the string fragmentation process, LUCIAE is able to describe, to certain extent, both the data of $p+p$ and $Pb+Pb$ collisions consistently. The roles of the mechanisms of the collective effect in the gluon emission of strings and the reduction of the $s$ quark suppression in string fragmentation in the $\phi$ enhancement are investigated in Table. 2. In order to understand the results shown in Table. 2 the JETSET parameters relevant to the effective string tension are given in Table. 3 . These are the parameters $parj(1)$, $parj(3)$ and $parj(21)$, besides $parj(2)$. $parj(1)$ stands for the suppression of diquark-antidiquark pair production compared to the quark-antiquark pair production in the string fragmentation, $parj(2)$ is the suppression of $s$ quark pair production with respect to $u$ or $d$ pair production, $parj(3)$ refers to the extra suppression of strange diquark production compared to the normal suppression of the strange quark pair. Finally $parj(21)$ is the width of the Gaussian transverse momentum distribution of $q\bar{q}$ pairs in the string fragmentation. The values of these parameters given in the second or fourth line of Table. 3 are the default values in JETSET. The mechanism of the reduction of $s$ quark suppression is considered in a phenomenological way where the effective string tension is linked mainly to the transverse momentum of the hardest gluon on a string [@tai1]. In case without the firecracker mechanism, but with the reduction of $s$ quark suppression, the transverse momentum of gluons on a string is small and thus the values of the JETSET parameters are smaller than the default values correspondingly (cf. lines three and four of Table. 3). That is the reason why the $\phi$ meson yield in case without firecracker but with the reduction of $s$ quark suppression is even lower than in the case without both, firecracker and the reduction of $s$ quark suppression (cf. lines three and four of Table. 2). A note is in order here, LUCIAE calculations with the default JETSET parameters which are determined using $e^+e^-$ data overestimate production of strange particles in $p+p$ collisions [@tai1], which is the very reason that we proposed a phenomenological mechanism to investigate how the string tension varies as a function of collision energy in $p+p$ collisions. One can see from Table. 1 and 2 that the firecracker model plays the major role and the reduction of $s$ quark suppression is significant only in combination with the firecracker model. It is interesting to compare LUCIAE [@tai1] with UrQMD [@soff; @nu] in the mechanism of strangeness enhancement. Both of them start from the quantum tunneling probability $$\exp{(\frac{-\pi m^2}{\kappa})}\exp{(\frac{-\pi p_t^2}{\kappa})}$$ for the production of $q\bar{q}$ pair with the quark mass $m$ and the transverse momentum $p_t$ from a string of string tension $\kappa$ [@sch]. Thus, the suppression factor of the $s\bar{s}$ pair production with respect to $u$ or $d$ pair, for instance, can be expressed as $$parj(2)=\exp{(\frac{-\pi(m_s^2-m_u^2)}{\kappa})}.$$ In [@soff; @nu] it was then argued that in the relativistic $A+A$ collisions the string tension should be three times larger than that in $p+p$ collisions at the same energy due to the higher string density. The increase of the string tension is further attributed to the reduced quark mass stemming possibly from a transition of chiral restoration [@soff]. On the other hand, in [@tai1] an effective string tension was introduced and is phenomenologically related to the multigluon string in comparison with the pure $q\bar{q}$ string. Consequently the effective string tension and the pertained JETSET parameters are increasing with the energy, the size and centrality of collision system. Therefore, strangeness production in relativistic $p+p$, $p+A$ and $A+A$ collisions might be investigated consistently within a hadron-string model without introducing the QGP formation explicitly. An interesting issue arised here is worthy to be studied further. We also plan to improve the agreement between the experimental rapidity and transverse mass distributions and the LUCIAE results via transverse excitation of string and the intrinsic transverse momentum broadening in string fragmentation. The investigation for the role played by the hard and semi-hard processes, such as $gg\rightarrow s\bar{s}$ and $q\bar{q}\rightarrow s\bar{s}$ on strangeness enhancement is needed as well. In summary, the experimentally found $\phi$ enhancement in $Pb+Pb$ relative to $p+p$ collisions is described consistently by the hadron-string cascade model LUCIAE. In this model, $\phi$ mesons are exclusively produced from string fragmentation processes without any further rescattering interactions. However, LUCIAE has employed the mechanisms of the string collective effect (firecracker model) in the gluon emission and of the reduction of $s$-quark suppression in the string fragmentation. This implies that, at the CERN SPS energy, the $\phi$-mesons are mostly produced in primordial collisions and final state interactions at the hadronic stage do not play a significant role. Finally, the financial supports from NSFC in China, DFG in Germany, and DOE in USA are acknowledged. [10]{} B. Müller, J. 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--- abstract: 'In this paper, we give infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[-1-\sqrt2, -2)$ and also infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[\alpha_1, -1-\sqrt2)$ where $\alpha_1$ is the smallest root$(\approx -2.4812)$ of the polynomial $x^3+2x^2-2x-2$. From these results, we determine the largest and second largest limit points of smallest eigenvalues of regular graphs less than $-2$. Moreover we determine the supremum of the smallest eigenvalue among all connected 3-regular graphs with smallest eigenvalue less than $-2$ and we give the unique graph with this supremum value as its smallest eigenvalue.' author: - \| Hyonju Yu\ \ title: On the limit points of the smallest eigenvalues of regular graphs --- Introduction ============ Line graphs are examples of graphs with smallest eigenvalue at least $-2$ (where the eigenvalues of a graph are the eigenvalues of its adjacency matrix). In 1976, P. J. Cameron et al. characterized the graphs with smallest eigenvalue at least $-2$ [@180]. They showed (using root lattices): (Cf. [@bcn Theorem 3.12.1]) A connected graph $G$ with smallest eigenvalue at least $-2$ is a generalized line graph (for definition, see next section) or the number of vertices of $G$ is at most 36. A. J. Hoffman studied graphs with smallest eigenvalue less than $-2$. He showed that the largest limit point smaller than $-2$ for the smallest eigenvalues of graphs is $-1-\sqrt 2$ [@Hoff]. More precisely, he showed: \[Hoff\] Let $\widehat{\theta}_k$ be the supremum of the smallest eigenvalues of graphs with minimal valency $k$ and smallest eigenvalue $<-2$. Then $(\widehat{\theta}_k)_k$ forms a monotone decreasing sequence with limit $-1-\sqrt 2$. A consequence of the results of R. Woo and A. Neumaier is that $\alpha_1$ (where $\alpha_1$ is the smallest root$(\approx -2.4812)$ of the polynomial $x^3+2x^2-2x-2$) is the largest limit point of the smallest eigenvalues of graphs with smallest eigenvalues less than $-1-\sqrt 2$ [@woo]. More precisely: \[woolim\] Let $\widehat{\sigma}_k$ be the supremum of the smallest eigenvalues of graphs with minimal valency $k$ and smallest eigenvalue $<-1-\sqrt 2$. Then $(\widehat{\sigma}_k)_k$ forms a monotone decreasing sequence with limit $\alpha_1$. We will give proofs of Theorems \[Hoff\] and \[woolim\] in Section 2. In this paper, we study the limit points of the smallest eigenvalues of regular graphs. Clearly $-2$ is a limit point for regular graphs. Also $\frac{-3-\sqrt 5}{2}$ is a limit point for regular graphs as the cartesian product of a pentagon and a complete graph with at least 2 vertices has smallest eigenvalues $\frac{-3-\sqrt 5}{2}$. But it is not known whether this is the largest limit point smaller then $-2$ for regular graphs. In this paper, we will construct $k$-regular graphs which have the smallest eigenvalues in $[-1-\sqrt2, -2)$ for all $k$. And as a corollary, we will show that for regular graphs, the largest limit point of regular graphs with smallest eigenvalue less than $-2$ is also $-1-\sqrt 2$. Our results are: \[infimany\] Let $k$ be a positive integer at least 3. Then there are infinitely many $k$-regular graphs which have smallest eigenvalue in $[-1-\sqrt2, -2)$. Also, there are infinitely many $3$-regular triangle free graphs which have the smallest eigenvalue in $[-1-\sqrt2, -2)$. \[short\] Let $\hat{\eta}_k$ be the supremum of the smallest eigenvalues of $k$-regular graphs with smallest eigenvalue $<-2$. Then $$\lim_{k\rightarrow\infty}(\hat{\eta}_k)_k=-1-\sqrt{2}.$$ Furthermore, for enough large $k$, we will construct $k$-regular graphs which have smallest eigenvalues in $[\alpha_1, -1-\sqrt2)$ where $\alpha_1$ is the smallest root$(\approx -2.4812)$ of the polynomial $x^3+2x^2-2x-2$. And as a corollary, we will show that the largest limit point of regular graphs with smallest eigenvalue less than $-1-\sqrt 2$ is also $\alpha_1$, as shown for general graphs by R. Woo and A. Neumaier. \[infimanyhh\] There exist $N$ such that there are infinitely many $k$-regular graphs which have smallest eigenvalue in $[\alpha_1, -1-\sqrt2)$ for $k\geq N$. \[short1\] Let $\hat{\xi}_k$ be the supremum of the smallest eigenvalues of $k$-regular graphs with smallest eigenvalue $<-1-\sqrt 2$. Then $$\lim_{k\rightarrow\infty}(\hat{\xi}_k)_k=\alpha_1$$ where $\alpha_1$ is the smallest root$(\approx -2.4812)$ of the polynomial $x^3+2x^2-2x-2$. Theorems \[infimany\] disproves the following conjecture by R. Woo and A. Neumaier. [@woo p.590, Conjecture (ii)]\[conj\] A regular graph with smallest eigenvalue $\geq -1-\sqrt 2$ and sufficiently large valency is a line graph or a cocktail party graph. In order to show Theorems \[infimany\] and \[infimanyhh\], we will use Hoffman graphs, which were introduced by R. Woo and A. Neumaier[@woo]. In the next section we will give definitions and preliminaries. Also in this section we will recall Hoffman graphs and some of their basic theory. In Section 3, we will show Theorems \[infimany\], \[infimanyhh\] and Corollary \[short\], \[short1\]. In Section 4, we will determine $\hat{\eta}_3$. In the last section, we will give concluding remarks. Definitions and preliminaries ============================= All the graphs considered in this paper are finite, undirected and without multiple edges. A graph $G$ is an ordered pair $(V(G), E(G))$ of vertex set $V=V(G)$ and edge set $E=E(G)$, where $E(G)$ consists of unordered pairs of two adjacent vertices, so we can consider $E(G)\subseteq\binom{V(G)}{2}$. For two graph $G$ and $H$ such that $V(H)\subseteq V(G)$ and $\{x,y\}\in E(H)$ if and only if $\{x,y\}\in E(G)$ for all $x, y \in V(H)$, we call that $H$ is induced subgraph of $G$ and $G$ is super graph of $H$. The adjacency matrix $A(G)$ of a graph $G$ is the $(0,1)$-matrix whose rows and columns are indexed by the vertex set $V(G)$ and the $(x,y)$-entry is $1$ whenever $x$ is adjacent to $y$ and $0$ otherwise. The eigenvalues of $G$ are the eigenvalues of its adjacency matrix. A graph $G$ is $k$-regular if for any vertex $x$, the number of edges which contain $x$, we call valency of $x$, is $k$. A graph $G$ is regular if it is $k$-regular for some integer $k$. A graph $G$ is $(k_1, k_2)$-semiregular if there is a partition $\{A\neq\emptyset, B\neq\emptyset\}$ of vertex set such that every vertex in $A$ has valency $k_1$ and every vertex in $B$ has valency $k_2$. We denote $\delta(G)$ the minimum valency of $G$. For a given graph $G$, the line graph of $G$, $\L(G)$, has vertex set $E(G)$ and $\{x, y\}$ is adjacent to edge $\{v, w\}$ if and only if $\|\{x, y\}\cap \{v, w\}\|=1$. A graph $G$ is called a line graph if $G=L(H)$ for some graph $H$. For a line graph $G$, say $G=L(H)$, we obtain the relation $C^T C=A(G)+2I$ where $C$ is a vertex-edge incidence matrix of $H$ and $I$ is an identity matrix. This relation implies that the smallest eigenvalue of a line graph is at least $-2$. Line graphs can be recognized combinatorially as J. Krausz [@kra] showed: (Cf. [@line Theorem 2.1.1])\[linechar\] A graph is a line graph if and only if its edge set can be partitioned into non-trivial cliques such that: - two cliques have at most one vertex in common; - each vertex is in at most two cliques. In this paper, we are mainly interested in regular graphs. P. J. Cameron et al. [@180] characterized the connected regular graphs with smallest eigenvalue at least $-2$. Their result: (Cf. [@bcn Theorem 3.12.2])\[cla1\] Let $G$ be a connected regular graph with $\nu$ points, valency $k$, and smallest eigenvalue at least $-2$. Then one of the following holds: - $G$ is the line graph of a regular or a bipartite semiregular connected graph. - $\nu=2(k+2)\leq 28$ and $G$ is an induced subgraph of $E_7(1)$. - $\nu=\frac{3}{2}(k+2)\leq 27$ and $G$ is an induced subgraph of Schläfli graph. - $\nu=\frac{4}{3}(k+2)\leq 16$ and $G$ is an induced subgraph of Clebsch graph. - $\nu=k+2$ and $G\cong K_{m\times 2}$ for some $m\geq3$. This theorem means that there are only finitely many connected regular graphs with smallest eigenvalue at least $-2$ which are neither line graphs nor Cocktail party graphs. Hoffman graphs -------------- Hoffman graphs can be used as a tool to construct graphs with smallest eigenvalue at least a fixed number. In this subsection, we will define Hoffman graphs (as introduced by R. Woo and A. Neumaier) and discuss their basic theory [@woo]. A Hoffman graph $\mathfrak{H}$ is a pair $(H=(V, E), \ \mu : V\rightarrow \{f, s\})$ satisfying the following conditions: - H is a graph; - every vertex with label $f$ is adjacent to at least one vertex with label $s$; - vertices with label $f$ are pairwise non-adjacent. We call a vertex with label $s$ a slim vertex, and a vertex with label $f$ a fat vertex. We denote by $V_s=V_s(\mathfrak H)$ (resp. $V_f=V_f(\mathfrak H)$) the set of slim (resp. fat) vertices of $\mathfrak H$. The subgraph of a Hoffman graph $\mathfrak H$ induced on $V_s(\mathfrak H)$ is called the slim graph of $\mathfrak H$. Let $\mathfrak{H}$ be a Hoffman graph and let $\mathfrak{H}^1$ and $\mathfrak{H}^2$ be two non-empty induced Hoffman subgraphs of $\mathfrak{H}$. The Hoffman graph $\mathfrak{H}$ is said to be the sum of $\mathfrak{H}^1$ and $\mathfrak{H}^2$, written as $\mathfrak{H}=\mathfrak{H}^1\oplus\mathfrak{H}^2$, if the following conditions are satisfied: - $V(\mathfrak{H})=V(\mathfrak{H}^1)\cup V(\mathfrak{H}^2)$; - $\{V_s(\mathfrak{H}^1), V_s(\mathfrak{H}^2)\}$ is a partition of $V_s(\mathfrak{H})$; - if $x\in V_s(\mathfrak{H}^1)$ and $y\in V_s(\mathfrak{H}^2)$, then $x$ and $y$ have at most one common fat neighbor and they have one if and only if they are adjacent; - if $x\in V_s(\mathfrak{H}^i)$ and $F\in V_{f}(\mathfrak{H})$ are adjacent in $\mathfrak{H}$, then $F\in V_{f}(\mathfrak{H}^i)$ and $x$ and $F$ are adjacent in $\mathfrak{H}^i$ for $i=1, 2$. It easily follows that the sum defined above is associative, in the sense that if $\mathfrak{H}=\mathfrak{H}^1\oplus (\mathfrak{H}^2\oplus \mathfrak{H}^3)$, then $\mathfrak{H}=(\mathfrak{H}^1\oplus \mathfrak{H}^2)\oplus \mathfrak{H}^3$ and vice versa. Instead of $\mathfrak{H}=((\ldots(\mathfrak{H}^1\oplus \mathfrak{H}^2)\oplus \mathfrak{H}^3) \ldots \mathfrak{H}^n)$, we write $\mathfrak{H}=\bigoplus_{i=1}^n \mathfrak{H}^i$. Let $\mathfrak{A}$ be a family of Hoffman graphs. An $\mathfrak{A}$-line graph is an induced subgraph of the slim graph of a Hoffman graph $\mathfrak{H}=\bigoplus_{i=1}^s \mathfrak{H}^i$ for some $s$. In particular, it can be easily checked that a line graph is a $\{\mathfrak{H_2}\}$-line graph where $\mathfrak{H_2}$ is as shown in [Figure 1]{} at the end of this subsection. We define a generalized line graph as a $\{\mathfrak{H_2}, \mathfrak{H_3}\}$-line graph where $\mathfrak{H_3}$ is also shown in [Figure 1]{}. This definition is equivalent with original definition in [@bcn p. 106]. Now, we define eigenvalues of Hoffman graphs. For a Hoffman graph $\mathfrak H=(H, \mu)$, let $A$ be the adjacency matrix of $H$, $$A=\left[ \begin{array}{cc} A_s & C \\ C^T & 0 \end{array} \right]$$ where $A_s$ is the adjacency matrix of the slim graph of $\mathfrak{H}$. Eigenvalues of $\mathfrak H$ are defined as the eigenvalues of the real symmetric matrix $B=B(\mathfrak{H}):= A_s - C C^T$. By $\lambda_{min}(\mathfrak H)$, we denote the smallest eigenvalue of $\mathfrak H$. From above definitions, we can regard any ordinary graph $H$ as a Hoffman graph $\mathfrak H=(H, \mu)$ satisfing $\mu(x)=s$ for all $x\in V(H)$, and then the eigenvalues of $H$ are exactly the same as the eigenvalues of $\mathfrak{H}$. We denote by $\lambda_{min}=\lambda_{min}(G)$ the smallest eigenvalue of a given graph $G$. [Figure 1]{} shows that the smallest eigenvalues of some Hoffman graphs where $\alpha_0 = -1-\sqrt2$ and $\alpha_1\approx -2.4812$ is the smallest root of $x^3 +2x^2-2x-2$. The following theorem shows the relation between the smallest eigenvalue of a Hoffman graph and any of its induced subgraph, in particular, its slim graph. (Cf. [@woo Theorem 3.2 and 3.7])\[sinter\] - If $\mathfrak G$ is an induced subgraph of a Hoffman graph $\mathfrak H$, then $\lambda_{min}(\mathfrak{G})\geq \lambda_{min}(\mathfrak{H})$ holds. - If $\mathfrak{H}=\mathfrak{H_1}\oplus\mathfrak{H_2}$, then $\lambda_{min}(\mathfrak{H})=\min\{\lambda_{min}(\mathfrak{H_1}), \lambda_{min}(\mathfrak{H_2})\}$. To consider the limit points of smallest eigenvalues, we discuss the following proposition which was shown by A. J. Hoffman and A. M. Ostrowski. (For a proof, see [@JKMT]). (Cf. [@woo Proposition 5.3])\[Hoffthm\] Let $\mathfrak{H}$ be a Hoffman graph and denote by $\mathfrak{H}^{(n)}$ the clique extension obtained by replacing all fat vertices by slim $n$-cliques, all of whose vertices are adjacent to the neighbors of the corresponding fat vertices. Then $$\lim_{n\rightarrow \infty} \lambda_{min}(\mathfrak{H}^{(n)})=\lambda_{min}(\mathfrak{H}).$$ The following theorem gives a structure theory for graph with smallest eigenvalue a little bit smaller than $-2$. \[Hoff1wootheorem\] - (Cf. [@Hoff Theorem 1.1]) For $-1-\sqrt2 <\lambda \leq -2$, there exists a number $f(\lambda)$ such that if $G$ is a connected graph with minimum valency of $G$ is at least $f(\lambda)$ and $\lambda_{min}(G)\geq \lambda$, then $\lambda_{min}(G)=-2$ and $G$ is a generalized line graph. - (Cf. [@woo Theorem 5.1]) Let $\lambda\leq -1$ be a real number larger than $\alpha_1\approx-2.4812$, the smallest root of the polynomial $x^3+2x^2-2x-2$. Then there exists a number $\kappa_{\lambda}$ such that every graph with minimum valency at least $\kappa_{\lambda}$ such that its smallest eigenvalue $\lambda_{min}$ at least $\lambda$ is an $\{{\mathfrak H}_2, {\mathfrak H}_5, {\mathfrak H}_7, {\mathfrak H}_9\}$-line graph. In particular, $\lambda_{min}\geq -1-\sqrt 2$. [Proofs of Theorem \[Hoff\] and Theorem \[woolim\]]{}\ Proof of Theorem \[Hoff\]: By Theorem \[Hoff1wootheorem\](i), we have $\lim_{k\rightarrow \infty}\widehat{\theta}_k\leq -1-\sqrt2$. By Proposition \[Hoffthm\] and the Hoffman graph $\mathfrak{H}_{9}$ in [Figure 1]{}, we construct a sequence of graphs $(G_n)_{n=2}^{\infty}$ such that $(\lambda_{min}(G_n))_n$ is a monotone decreasing sequence and $\lim_{n\rightarrow\infty}\delta(G_n)=\infty$. This shows Theorem \[Hoff\].\ Proof of Theorem \[woolim\]: The proof is similar except that we need to change Theorem \[Hoff1wootheorem\](i) to Theorem \[Hoff1wootheorem\](ii), $\mathfrak{H}_{9}$ to $\mathfrak{H}_{WN}$ and $-1-\sqrt2$ to $\alpha_1$. This shows Theorem \[woolim\]. [\ ]{} ![image](1.JPG){width="80.00000%"} Proofs of the main theorems =========================== In this section, we will give the proofs for Theorems \[infimany\], \[infimanyhh\] and Corollaries \[short\], \[short1\]. Proof of the Theorem \[infimany\] {#3} --------------------------------- Let $k$ be a positive integer at least 3. \[semi\] There exist a connected $(k-1, k)$-semi-regular graph with $a(2k-1)$ vertices for all positive integer $a$. Let $B_{k, k-1}$ be a semi-regular bipartite graph of partition $R$ with valency $k$ and $Y$ with valency $k-1$. For the graph $B_{k, k-1}$, we are going to construct connected $k$-regular graph $G(B_{k, k-1})=:G_k$ with $\lambda_{min}(G_k))\geq -1-\sqrt2$. The graph $G_k$ will be the slim graph of the Hoffman graph $\mathfrak{G}(B_{k, k-1})=:\mathfrak{G}_k$ as defined below. Let $V_s(\mathfrak{G}_k)=\{x_{t}^{(i,j)_k}\mid \{i, j\}\in E(B_{k, k-1}), i\in R , j\in Y$ and $t=1, 2, 3\}$\ and $V_f(\mathfrak{G}_k)=\{F_i, E_i\mid i\in R\}\cup \{D_j\mid j\in Y\}$.\ So, $\sharp V_s(\mathfrak{G}_k)=3k(\sharp R)=3(k-1)(\sharp Y)$ and $\sharp V_f(\mathfrak{G}_k)=2\sharp R +\sharp Y$. Define ![image](2.JPG){width="80.00000%"} and let $G_k$ be the slim graph of $\mathfrak{G}_k$. Then the following claim holds: - $G_k$ is a $k$-regular graph; - $\lambda_{min}(G_k)\geq \lambda_{min}(\mathfrak{G}_k)=\lambda_{min}(\mathfrak{H}_{8})=-1-\sqrt2$ for all $\{i, j\}\in E(B_{k, k-1})$ where $i\in R$ and $j\in Y$; - $G_k$ is neither a line graph or a Cocktail party graph. [[Proof: ]{}]{}$(i)$ : By construction of $G_k$.\ $(ii)$ : By Theorem \[sinter\].\ $(iii)$ : For any two edges $\{i_1, j\}$ and $\{i_2, j\}$ of $B_{k, k-1}$ with common vertex $j$ in $Y$, the induced subgraph on $\{x_1^{(i_1,j)_k}, x_2^{(i_1,j)_k}, x_3^{(i_1,j)_k}, x_2^{(i_2,j)_k}\}$ is 3-claw. By Theorem \[linechar\], this implies $(iii)$. [\ ]{} By Theorem \[cla1\] and Remark \[semi\], we obtain infinitely many $k$-regular graphs which have smallest eigenvalue in the half open interval $[-1-\sqrt2 , -2)$.\ Now, we consider 3-regular triangle free graphs. Let $C_{2n}$ be a $2n$-gon with vertex set $\{x_{2n+1}=x_1, x_2, \ldots , x_{2n}\}$ and edge set $\{\{x_i, x_{i+1}\}\mid i=1, 2, \ldots , 2n\}$. For the graph $C_{2n}$, we are going to construct $3$-regular triangle free graph $G(C_{2n})=:G'$ with $\lambda_{min}(G')\geq -1-\sqrt2$. The graph $G'$ will be the slim graph of the Hoffman graph $\mathfrak{G}(C_{2n})=:\mathfrak{G}'$. Let $V_s(\mathfrak{G}')=\{x_{t}^{(2i-1,2i)}, x_{t}^{(2i-1,2i-2)}\mid i=1, 2, \ldots , n$ and $t=1, 2, 3,4\}$\ and $V_f(\mathfrak{G}')=\{F_{2i-1}, E_{2i-1}\mid i=1, 2, \ldots , n\}\cup \{D_{2i}\mid i=1, 2, \ldots , n\}$. Define $\mathfrak{G}'=\bigoplus_{i\in\{1, 2, \ldots , n\}} \left(\mathfrak{G}^{(2i-1, 2i)}\bigoplus \mathfrak{G}^{(2i-1, 2i-2)}\right)$ where ![image](3.JPG){width="80.00000%"} and let $G'$ be the slim graph of $\mathfrak{G}'$. Then the following claim holds: - $G'$ is a $3$-regular graph; - $\lambda_{min}(G')\geq \lambda_{min}(\mathfrak{G}')=\lambda_{min}(\mathfrak{H}_{9})=-1-\sqrt2$; - Neighbors of a fixed vertex are not adjacent each other in $G'$; - $G'$ is neither a line graph nor a Cocktail party graph; - $G'$ is triangle free. [[Proof: ]{}]{}$(i)$ : By construction of $G'$.\ $(ii)$ : By Theorem \[sinter\].\ $(iii)$ : By symmetry, we only need to consider the neighbors of $x_1^{(1,2)}$ and $x_2^{(1,2)}$, those are $\{x_2^{(1,2)}, x_3^{(1,2)}, x_1^{(1,2n)}\}$ and $\{x_1^{(1,2)}, x_2^{(3,2)}, x_4^{(3,2)}\}$.\ And $(iii)$ implies $(iv)$ and $(v)$ . [\ ]{} By Theorem \[cla1\], we obtain infinitely many $3$-regular triangle free graphs which have smallest eigenvalue in the half open interval $[-1-\sqrt2 , -2)$. [\ ]{} Proof of Theorem \[infimanyhh\] {#2} ------------------------------- By Proposition \[Hoffthm\], we can take a positive integer $N$ such that $\lambda_{min}(\mathfrak{H}_{WN}^{(n-3)})< -1-\sqrt2$ for all $n\geq N$.\ Let $k\geq N$ and $a$ be positive integers.\ We are going to construct a $k$-regular graph $G_k$ such that $\alpha_1 \leq\lambda_{min}(G_k)< -1-\sqrt 2$. The graph $G_k$ will be the slim graph of a Hoffman graph $\mathfrak{G}_k$ with $2a(2k^2-4k+1)$ fat vertices and $4ak(k-1)(k-2)$ slim vertices.\ Let $P(k)$, $Q(k)$ and $R(k)$ be three partitions of $\{1, 2, \ldots , ak(k-1)(k-2)\}=:[ak(k-1)(k-2)]$ such that ${P(k)}=\{{p(k)}_i \mid i=1, 2, \ldots , ak(k-1) \}\subseteq \binom{[ak(k-1)(k-2)]}{k-2}$\ ${Q(k)}=\{{q(k)}_j \mid j=1, 2, \ldots , a(k-1)(k-2)\}\subseteq \binom{[ak(k-1)(k-2)]}{k}$\ ${R(k)}=\{{r(k)}_l \mid l=1, 2, \ldots , ak(k-2)\}\subseteq \binom{[ak(k-1)(k-2)]}{k-1}$. And let ${I(k)}=\{(i,j,l)_m \mid m\in {p(k)}_i\cap {q(k)}_j\cap {r(k)}_l$ for $1\leq m\leq ak(k-1)(k-2)\}$. Now, we define Hoffman graph $\mathfrak{G}_k$ with fat vertex set $V_f(\mathfrak{G}_k)=\{F_i \mid i=1, \ldots , ak(k-1) \}$\ $\cup \{ E_j \mid j=1, \ldots , a(k-1)(k-2)\} \cup\{ D_l, C_l \mid l=1, \ldots , ak(k-2)\}$ and slim vertex set $V_s(\mathfrak{G}_k)=\{x_{t}^{(i,j,l)_m}\mid (i,j,l)_m\in I(k), \ t=1, 2, 3, 4\}$ as following: ![image](4.JPG){width="80.00000%"} and let $G_k$ be the slim graph of $\mathfrak{G}_k$. For $p(k)_i=\{\alpha(1), \alpha(2), \ldots , \alpha(k-2)\}$, there exist $j(s)$ and $l(s)$ such that $\alpha(s)\in q(k)_{j(s)}\cap r(k)_{l(s)}$. This means that $(i, j(s), l(s))_{\alpha(s)}\in I(k)$ for $s=1, 2, \ldots , k-2$ and there are $k-2$ slim neighbors of fat vertex $F_i$. By a similar argument, there are $k$, $k-1$ and $k-1$ slim neighbors of fat vertex $E_j$, $D_l$ and $C_l$, respectively. Moreover we have the following claim: - $G_k$ is a $k$-regular graph; - $\lambda_{min}(G_k)\geq \lambda_{min}(\mathfrak{G}_k)=\lambda_{min}(\mathfrak{H}_{WN})=\alpha_1$; - $\lambda_{min}(\mathfrak{G}_k)< -1-\sqrt2$. [[Proof: ]{}]{}$(i)$ : By construction of $G_k$.\ $(ii)$ : By Theorem \[sinter\].\ $(iii)$ : Consider the induced subgraph on $x_1^{(i,j,k)_m}, x_2^{(i,j,k)_m}, x_3^{(i,j,k)_m}, x_4^{(i,j,k)_m}$ and its neighbors for some fixed $(i,j,k)_m \in I(k)$. Then this induced subgraph contains $\mathfrak{H}_{WN}^{(k-3)}$. So, $\lambda_{min}(G_k)\leq \lambda_{min}(\mathfrak{H}_{WN}^{(k-3)})<-1-\sqrt2$ as $k\geq N$. [\ ]{} It is not so difficult to see that without loss of generality we may assume that $G_k$ is connected for any positive integer $a$. We have constructed connected $k$-regular graphs on $4ak(k-1)(k-2)$ vertices which have the smallest eigenvalue in the half open interval $[\alpha , -1-\sqrt2)$ for $k\geq N$. Therefore, our result is proved. [\ ]{} Proofs of Corollary \[short\] and \[short1\] {#3} -------------------------------------------- By definition of $\hat{\theta_k}$ and $\hat{\eta_k}$, $\hat{\theta_k}\geq \hat{\eta_k}$. This implies that $\lim_{k\rightarrow\infty}\hat{\eta_k}\leq -1-\sqrt2$ as Theorem \[Hoff\]. Since Theorem \[infimany\] means that $\lim_{k\rightarrow\infty}\hat{\eta_k}\geq -1-\sqrt2$, we obtain Corollary \[short\]. As similar argument, Theorem \[woolim\] and Theorem \[infimanyhh\] imply Corollary \[short1\]. The value $\hat{\eta}_3$ ======================== In order to characterize $\hat{\eta}_3$, we use following lemma which follows immediately from Theorem \[linechar\]. \[line\] (i) A graph with a 3-claw as an induced subgraph is not a line graph.\ (ii) Let $G$ be a 3-regular graph which is not $K_4$. Then $G$ is a line graph if and only if $N(x)\cong K_1 \cup K_2$ for all $x$ in $V(G)$. The following theorem determines $\hat{\eta}_3$ and gives the graph such that the smallest eigenvalue is $\hat{\eta}_3$. \[limit\] Let $\beta\approx-2.0391$ be the smallest root of $x^6-3x^5-7x^4+21x^3+13x^2-35x-4$. Then $\hat{\eta}_3=\beta$ and ![image](5.JPG){width="20.00000%"} is the unique connected 3-regular graph with smallest eigenvalue $\beta$. [[Proof: ]{}]{}As the graph, shown in the theorem, has smallest eigenvalue $\beta\approx-2.0391$, we only need to show that the only connected $3$-regular graph with $\beta\leq\lambda_{min}<-2$ is the graph of this theorem. Let $G$ be a connected $3$-regular graph with smallest eigenvalue $\beta\leq \lambda_{min}(G)<-2$. Then $G$ contains a $3$-claw or a $K_{2,1,1}$ as induced subgraph by Lemma \[line\]. We start with the assumption that $G$ contains a $K_{2,1,1}$, that is the graph $A1$ of Appendix 1. Now we proceed by adding vertices to a vertex that does not have degree $3$ and consider all possible cases . If such a graph has smallest eigenvalue smaller than $\beta$, then we know that it can not be an induced subgraph of $G$ by Theorem \[sinter\]. In the tree below, we will show how we proceed from graph to supergraph, by adding vertices. We will underline a letter-number combination if the corresponding graph of Appendix 1 has the smallest eigenvalue less than $\beta$. From $C$ to $D$ we use the fact that $C1$, $C3$ to $C6$ of Appendix 1 are not possible. ![image](6.JPG){width="90.00000%"} So, this shows this theorem is true under the assumption that it contains a $K_{2,1,1}$ as an induced subgraph. From now on, we will consider only connected $3$-regular graphs that do not contain $K_{2,1,1}$. From $R1$ to $S$ of Appendix 1, we use the fact that $A1$ and $Q2$ of Appendix 1 do not occur. From there on, we assume that $G$ can not contain $Q2$ as an induced subgraph. ![image](7.JPG){width="90.00000%"} [\ ]{} $T5$ and $T7$ are isomorphic. Remarks and questions ===================== - Consider $K_{2,3}$ as the $(2,3)$-semi-regular graph in Remark \[semi\]. Then the graph $G_3$ has $-1-\sqrt2$ as the smallest eigenvalue. This implies that there is a $k$-regular graph with smallest eigenvalue $-1-\sqrt2$ for all $k\geq 3$. - Consider three partitions of $\{1, 2, \ldots , 12\}$: $P=\{p_i \mid i=1, 2, \ldots , 6\}$ where $p_i=\{i, i+6\}$;\ $Q=\{q_j \mid j=1, 2, 3\}$ where $q_j=\{j, j+3, j+6, j+9\}$;\ $R=\{r_l \mid l=1, 2, 3, 4\}$ where $r_l=\{3l-2, 3l-1, 3l\}.$ Then the smallest eigenvalue of the graph which is obtained by a similar argument of construction of $G_k$ in Section \[2\] is $\alpha_1$. This means that there is a $k$-regular graph with smallest eigenvalue $\alpha_1$ for all $k\geq 4$. <!-- --> - It is not known that whether $\hat{\theta_k}$ and $\hat{\eta_k}$ are monotone decreasing sequences or not. - It is not known that whether there are infinitely many regular graph with $\lambda_{min}\in [-1-\sqrt2 ,-2)$ which are not $\{\mathfrak{H}_2, \mathfrak{H}_5, \mathfrak{H}_7, \mathfrak{H}_9\}$-line graph. Acknowledgements {#acknowledgements .unnumbered} ================ Part of this work was done while visiting the Graduate School of Information Sciences(GSIS) at Tohoku University. The author greatly appreciates the hospitality of Profs. Munemasa and Obata. And also, the author would like to thank Prof. Koolen and Jongyook Park for the careful reading they did. [99]{} A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular graphs, Springer-Verlag, Berlin, 1989. P. J. Cameron, J.-M. Goethals, J. J. Seidel, E. E. Shult, Line graphs, root systems, and elliptic geometry, J. Algebra $\mathbf{43}$ (1976), no. 1, 305-327. D. Cvetković, P. Rowlinson, S. Simić, Spectral Generalizations of Line Graphs (On Graphs with Least Eigenvalue -2), London Mathematical Society Lecture Note Series, No. 314, 2004. A. J. Hoffman, On graphs whose least eigenvalue exceeds $-1-\sqrt 2$, Lin. Alg. Appl. $\mathbf{16}$ (1977), 153-166. H. Jang, J. Koolen, A. Munemasa, T. Taniguchi, On fat Hoffman graphs with smallest eigenvalue at least $-3$, in preparation. J. Krausz, Demonstration nouvelle d’une th$\acute{\mathrm e}$or$\grave{\mathrm e}$me de Whitney sur les reseaux, Mat. Fiz. Lapok 50(1943), 75-89. R. Woo, A. Neumaier, On graphs whose smallest eigenvalue is at least $-1-\sqrt2$, Linear Algebra Appl. $\mathbf{226/228}$ (1995), 577-591. Appendix 1 {#appendix-1 .unnumbered} ========== ![image](8.JPG){width="90.00000%"} ![image](12.JPG){width="90.00000%"} ![image](11.JPG){width="90.00000%"}
--- abstract: 'Recent high energy observations of Cas A suggests the presence of synchrotron radiation, implying acceleration of cosmic rays by young supernova remnants. We detect synchrotron emission from Cas A in the near-infrared using Two Micron All Sky Survey (2MASS) and Palomar 200 inch PFIRCAM observations. The remnant is detected in J, H, and K$_s$ bands using 2MASS: K$_s$ band is the brightest, H is moderate and J is faint. In the J and H bands, bright \[Fe II\] lines (1.24$\mu$m and 1.64$\mu$m) are detected spectroscopically. The Palomar observations include K$_s$ continuum, narrow-band 1.64 $\mu$m (centered on \[Fe II\]) and 2.12$\mu$m (centered on H$_2$(1-0)) images. While the narrow-band 1.64 $\mu$m image shows filamentary and knotty structures, similar to the optical image, the K$_s$ image shows a relatively smooth, diffuse shell, remarkably similar to the radio image. The H$_2$ image is identical to the K$_s$ continuum image with surface brightness reduced as expected for the ratio of filter bandwidths, showing no contribution of H$_2$ lines to the K$_s$-band image. The broad-band near-infrared fluxes of Cas A are generally consistent with, but a few tens of percent higher than, an extrapolation of the radio fluxes. The hardening to higher frequencies is possibly due to nonlinear shock acceleration and/or spectral index variation across the remnant. We show evidence of spectral index variation across Cas A using the ‘spectral tomography’ technique. The presence of near-infrared synchrotron radiation requires the roll-off frequency to be higher than 1.5 $\times$ 10$^{14}$ Hz, implying that electrons are accelerated to energies of at least $E = 0.3 B_{\rm mG}^{-1/2}$ erg or 0.2 TeV. The morphological similarity in diffuse emission between the radio and K$_s$ band images implies that synchrotron losses are not dominant, or we would expect to see a greater concentration in knots. We also show dust continuum is not significant in the near-infrared emission of Cas A. Our observations show unambiguous evidence that the near-infrared K$_s$ band emission of Cas A is from synchrotron emission by accelerated cosmic-ray electrons.' author: - Jeonghee Rho - 'Stephen P. Reynolds[^1]' - 'William T. Reach, Tom H. Jarrett' - 'Glenn E. Allen' - 'John C. Wilson[^2]' title: 'Near-Infrared Synchrotron Emission from Cas A' --- [Subject headings:]{} Supernova remnants: Cas A – Near-infrared –Particle acceleration Introduction ============ The origin of cosmic rays has been unclear since their discovery (Elster et al. 1969). It has been widely thought that their energy is derived from supernovae, which are naturally associated with particle acceleration because they are the primary energy input for interstellar gas. Shocks are thought to be primary acceleration sites of cosmic rays. Some evidence that cosmic-ray electrons are accelerated in supernova remnant shocks has been recently found using X-ray and gamma-ray observations. SN 1006 is a notable example. Its X-ray emission shows featureless spectra in the limb, strong evidence that the emission is synchrotron radiation (Koyama et al. 1995). The detection of TeV gamma-rays, probably cosmic-ray background photons upscattered by $\sim 100$ TeV electrons, strengthened the claim of synchrotron emission in SN 1006 (Tanimori et al. 1998). There are a few other SNRs including G347.3-05 [@koyama97; @slane99], and G266.2-1.2 [@slane01], which show featureless X-ray spectra, possibly due to synchrotron emission. Other examples of synchrotron emission are less clear, complicated by mixture with thermal emission, such as Cas A and RCW 86 (Allen et al. 1997; Borkowski et al. 2000; Rho et al. 2002). However, even if synchrotron radiation commonly contributes to the X-ray emission of SNRs, it does so at a level below the extrapolation from radio frequencies, by factors of 3 to 100 for 14 Galactic SNRs [@reynolds99] and 11 SNRs in the LMC [@hendrick01]. That is, the electron distribution responsible for radio synchrotron emission must steepen at energies below 100 TeV in all those 25 cases. This is problematic if SNRs are to accelerate cosmic rays up to the slight steepening around 3000 TeV in the cosmic ray ion spectrum known as the ‘knee.’ While relativistic electrons may be subject to radiative losses during the acceleration process, this should not be a problem for ions, so perhaps in all those cases electron acceleration is limited by losses, while ion acceleration invisibly extends to much higher energies. Alternatively only certain remnants (for instance, much older and larger ones than in the @reynolds99 study) may produce the highest energy cosmic rays. Another possibility is that a few remnants, such as Cas A, have very much stronger magnetic fields, increasing both the acceleration rate and the importance of losses so that the required steepening in the electron spectrum can be due to losses while ion acceleration can extend to $10^{15}$ eV and beyond [@ellison99]. Perhaps Cas A is a ‘super-accelerator’ of cosmic rays. While the observed X-rays from Cas A fall below the radio extrapolation by a smaller factor than any other Galactic SNR in the Reynolds & Keohane sample, this of course does not demonstrate that a portion of those X-rays are due to synchrotron emission. Most of soft X-rays (below 7 keV) are clearly thermal (e.g., Gotthelf et al. 2001) but emission is seen up to energies $> 10$ keV (Allen et al. 1997) with the [Rossi X-ray Timing Explorer]{} (RXTE), and to $\sim 100$ keV with OSSE on the [Gamma Ray Observatory]{} [@the96]. The latter study shows a power-law photon index of $-3.1 \pm 0.4$. A spectrum this hard to such high energies cannot be produced by synchrotron emission from shock-accelerated electrons in any reasonable model. The most likely explanation is nonthermal bremsstrahlung (NTB) (see, for instance, Asvarov et al. 1989; Laming 2001) from the lowest-energy nonthermal electron population, the beginning of the distribution that extends up to radio-emitting electrons at around 1 – 10 GeV. Evidence for at least some X-ray synchrotron emission near shock fronts was reported by @gotthelf01, but the total emission in the apparent synchrotron component is a small fraction of the total flux in the 4 – 6 keV range, and @bleeker01 cite observations from [XMM-Newton]{} to assert only a small role for synchrotron emission in the total X-ray flux. Thus the question of the highest frequency at which Cas A shows unambiguous synchrotron emission remains open. Recently, a $5 \sigma$ detection of Cas A has been reported at $\gamma$-ray energies above 1 TeV with the HEGRA Stereoscopic [Č]{}erenkov Telescope System (Aharonian et al. 2001). The emission could be due to leptonic processes (NTB or inverse-Compton upscattering of cosmic microwave background photons) or hadronic (inelastic cosmic-ray proton collisions with ambient gas producing neutral pions which decay to gamma-rays). NTB would require electrons of energies of a few TeV; $\pi^0$-decay gamma rays would require similar energies for protons. Inverse-Compton emission from the cosmic microwave background, however, would indicate the presence of electrons of energies of order 100 TeV. However, until this detection is confirmed and the nature of the processes producing the gamma rays clarified, the highest particle energies present in Cas A are still uncertain. The EGRET limits on emission above 100 MeV (Esposito et al. 1996) can be used to put a lower limit on the magnetic-field strength of at least 0.4 mG (Atoyan et al. 2000); if the field were weaker, the required relativistic electron density would have to be higher to explain the radio emission, and those electrons would emit NTB at levels higher than the upper limit in the EGRET band. Thus Cas A seems to have a much larger magnetic field than inferred for other remnants (e.g., $\sim 10\ \mu$G for SN 1006; Tanimori et al. 1998, Dyer et al. 2001). The total flux of Cas A at millimeter wavelengths (Liszt & Lucas 1999; Mezger et al. 1986) lies very close to the extrapolation from centimeter wavelengths. Shortward of 1.2 mm, however, there is very little information. Synchrotron continuum is required to fit the mid-infrared ISOCAM and ISO-SWS (6-30$\mu$m) spectra and the spectral shape of synchrotron emission (i.e. power law) dominates between 6–8$\mu$m (Douvion, Lagage, & Pantin 2001). @tuffs97 suspected from ISO data that emission in a line-free region near 6 $\mu$m could be synchrotron. Gerardy & Fesen (2001) report that a K-band image resembles the radio, but gave no quantitative flux determinations. Here we search for synchrotron emission from Cas A in the infrared, and show a definitive detection of synchrotron radiation at near-infrared wavelengths. We discuss the implications of our result for theories of particle acceleration and cosmic-ray origin in supernova remnants. Observations and Results ======================== 2MASS ------ The 2MASS images of Cas A were obtained as part of the routine observations for the Two Micron All Sky Survey (2MASS; Skrutskie et al. 1997; Cutri et al. 2002) using the northern telescope at Mt. Hopkins, Arizona, on 1998 December 26. The wavelength coverages of the bands are 1.11-1.36$\mu$m (J), 1.5-1.8$\mu$m (H) and 2-2.32$\mu$m (K$_s$). The spatial resolution of 2MASS is 3$''$. The detailed calibration of diffuse emission is explained in Rho et al. (2001). Here we used updated zero-point fluxes of 1592, 1024, and 668.8 Jy for J, H, and K$_s$, respectively. Figure \[casa2mass\] shows the 2MASS three-color image of Cas A, revealing a shell-like morphology for the near-infrared emission. The K$_s$ and H band images are brighter than the J band image, and the J band emission is barely detected in 2MASS. The differences in morphology among the three bands are not very noticeable except in the southwestern shell. The three color image in Figure \[casa2mass\] contrasts the difference: K$_s$ emission (reddish in color) dominates for most of Cas A, and H band emission (greenish in color) is more noticeable in the southwestern shell, some inner part of the northern shell, and sparsely distributed knots. We obtained a few near-infrared spectra of Cas A using the CorMASS instrument [@wilson] on the Palomar 60$''$ telescope. The observations took place on Sep. 10-12 and Nov. 14-15, 2000. The CorMASS spectra towards the northwestern and southwestern shell show bright lines of \[Fe II\] in the J and H bands (at 1.26$\mu$m and 1.64$\mu$m), which is consistent with the spectra of @gerardy01, and no lines in the K$_s$ band were detected. We also took the spectra towards possibly synchrotron dominated positions (positions D and C in Hughes et al. (2000) where no or weak X-ray emission lines are detected), and \[Fe II\] was not detected. This indicates the line emitting region is spatially limited across the remnant. The line contribution to each filter is estimated based on the near-infrared spectra from @gerardy01, which is described in the next section. Palomar 200 inch PFIRCAM images ------------------------------- We observed Cas A on July 12–13, 2001 using the Prime Focus Infrared Camera (PFIRCAM) on the Hale 200-inch telescope on Palomar Mountain. The PFIRCAM has a $256\times 256$ pixel array, with a pixel scale of 0.494$''$ at the f/3.3 prime focus of the 200-inch telescope. We took K$_s$ broad band ($\Delta\lambda=0.32\mu$m), \[\] narrow filter ($\lambda=1.64\mu$m; $\Delta\lambda=0.016\mu$m) and H$_2$ ($\lambda=2.12\mu$m, $\Delta\lambda=0.02\mu$m) images. Our goal was to distinguish the structures between the line and continuum as well as to measure the remnant fluxes. The seeing was 1$''$ to 1.5$''$. We used the observational technique described in Reach et al. (2002). The integration times per sky pixel for the K$_s$, , and H$_2$ images are 30, 60, and 60 sec, respectively. The PFIRCAM images were calibrated by comparing photometry of sources in the field to the 2MASS catalog (Cutri et al. 2002). The K$_s$ image was calibrated using 68 stars with 2MASS K$_s$ magnitudes ranging from 12.5 to 14.5, the H$_2$-band image was calibrated using 43 stars with 2MASS K$_s$ magnitudes ranging from 11.0 to 14.5, and the \[\] image was calibrated using 52 stars with 2MASS H-band magnitudes ranging from 12.6 to 14.6. Bright stars were excluded from the fit because of non-linearity in the PFIRCAM data; the diffuse emission is equivalent to stars fainter than 18$^{th}$ magnitude and is well within the linear regime of the detector. The total remnant fluxes were measured from the near-infrared images after subtracting stars using the DAOPHOT package. The regions (9$''$ radius) of two bright stars in the west were excluded from the estimation of the total fluxes. The narrow-band mosaiced image of \[\] in Figure \[pfirmcam\] (in blue) is globally similar to the optical image [@douvion01; @fesen87], showing sharp filamentary and knotty structures of Fe-rich ejecta. However, while \[Ne\] and \[\] optical maps show brighter emission in the north, the \[\] image shows much brighter emission in the southwest. Figures \[pfirmcam\] (in red) and \[kbandradioeach\]a show the Palomar PFIRCAM K$_s$ band image, which is noticeably different from the -band image. The K$_s$-band structure is much smoother than the ejecta structure that dominates the \[\] and optical images. The H$_2$-band image is essentially identical to the K$_s$ image, with surface brightness reduced (and noise increased) as expected for the ratio of filter bandwidths; thus the H$_2$ band appears to be completely due to the same continuum that generates the the K$_s$-band image. We show the 1.4 GHz radio image [@anderson91] in Figure \[kbandradioeach\]b for comparison with the K$_s$ band image, because they are remarkably similar. The radio image of [@anderson91] has a pixel size of 0.4$''$ and a spatial resolution of 1.3$''$, which is comparable to our Palomar image. The similarity between the K$_s$ band and radio images suggests that both the K$_s$ and radio images are due to the same physical process: synchrotron radiation. Infrared Flux of Cas A ====================== Total flux from near-infrared images ------------------------------------ The total remnant flux from the PFIRCAM K$_s$ band image is 0.32$\pm0.03$ Jy, where the error includes the uncertainty in calibration. In comparison, the 2MASS flux in the K$_s$ band is 0.31$\pm$0.02 Jy. Although the 2MASS diffuse emission may include some faint stars due to lower spatial resolution, the K$_s$ band flux is consistent between PFIRCAM and 2MASS. The statistical errors of the measurement are less than 1%, but a larger error is induced by the method of star subtraction because sometimes the sampling of the fluxes for the stars is insufficient or it is unclear if the bright blobs are faint stars or diffuse emission, especially in the \[\] image. We experimented with different thresholds of star identification, and we believe the error would not be larger than 20% of the measured flux (a conservative estimate). Extinction varies across the remnant, with A$_v$ ranging typically from 4.3 to 6.2 mag [@searle71; @fesen87], and as high as 8 mag in the western boundary [@hurford96]. The extinction-corrected K$_s$ flux is 0.53$^{+0.12}_{-0.04}$ Jy, using the near-infrared extinction law of @rieke85; the error includes the possible range of extinction. The CorMASS spectra show no lines in the K$_s$ band, and @gerardy01 also detected no lines within the K$_s$ band (2-2.3 $\mu$m) from FMK (fast moving knot) positions in the north and northeast of the remnant, while they detected faint lines of \[Fe II\], He I, and H II Br$\gamma$ from QSF (quasi-stationary flocculi) positions in the southwest. It is likely the line contribution is very small in the K$_s$ band image, considering that (1) QSF covers a small portion of the remnant, (2) the morphology of the K$_s$ image is remarkably similar to the radio image, and (3) the narrow filter 2.12$\mu$m image is very similar to the K$_s$-band image. The total flux of the PFIRCAM \[\] image is 0.37$\pm$0.06 Jy, and the extinction-corrected flux is 0.76 Jy using A$_v$ of 4.5 mag. This total flux is somewhat uncertain, because the \[\] emission is knotty and filamentary and some of the bright peaks of diffuse emission may have been subtracted during subtraction of stars. The 2MASS total flux in the H band is 0.27 Jy, and the extinction-corrected flux is 0.56 Jy. The 2MASS H band flux may be somewhat overestimated due to contribution from unresolved faint stars. Experimenting with different star-subtraction methods shows that neither of these effects is significant. Emission lines contribute a small fraction of the flux observed in the wide K$_s$ band. We estimate the line contribution as follows. We assume that the the emission in each band arises from a combination of a featureless continuum and spectral lines with a total brightness in the H-band that is a factor $f_L$ brighter than the \[\] 1.64$\mu$m line. Using the observed fluxes and the filter widths, the line contribution to the H band is $\sim 2 f_L$%, and even in the narrow \[\]-centered filter, is only $\sim 30$% due to the 1.64$\mu$m line. Spectroscopically, the 1.64 $\mu$m line is the dominant line in the H band as found in observations by @gerardy01 from FMK positions. While the 1.64 $\mu$m line is dominant ($f_L\simeq 1$) from FMK positions, the QSF have a wide range of lines, with $f_L\sim 2$. Thus the H band is dominated by continuum emission, with lines contributing less than 4% of the in-band flux. In the K$_s$ band, the conclusion is even more clear, because of the paucity of spectral lines. For most positions, there are no spectral lines detected in the K$_s$ band (in our CorMASS data and the FMK positions from Gerardy and Fesen \[2001\]). Even using the line-rich QSF spectra, the total line brightness in the K$_s$ band is 10% of the total line brightness in the H band, and spectral lines would contribute $< 1$% of the K$_s$-band flux. The conclusion that continuum dominates the wide bands is robust, and even after accounting for the uncertainties from star-subtraction methods, the line contribution would be less than 8% in the H bands, and less than 2% in K$_s$ band. Measuring the J band flux was challenging, because 2MASS barely detected the diffuse emission from the remnant. The 2MASS J band flux is 0.12 Jy, and the extinction-corrected flux is 0.39 Jy using A$_v$ of 4.5 mag, which may be highly uncertain. @gerardy01 show a higher resolution J band image, which is similar to our \[\] band image. Their J band spectra show a number of bright lines such as \[\], \[\] and \[\] such that the total flux of lines in the J band is 7-9 times greater than in H-band. Using the line fraction derived above for the H-band, we estimate that that lines contribute $\sim 20-40$% of the J-band flux. The reason the J-band and the narrow Fe-band images are similar is that they contain similar fractions of line emission. Possible dust contribution to Cas A flux ---------------------------------------- The mid-infrared spectrum of Cas A revealed dust newly formed from the ejecta material, as well as gas lines (Douvion, Lagage & Pantin 2001; DLP01). A model for the mid-infrared spectrum using dust composed of pyroxene, quartz, and aluminum oxide, with cold (90 K) and warm (350 K) components can reasonably approximate the spectrum longward of 8 $\mu$m (DLP01). We estimated the dust contribution to the near-infrared brightness by calculating the ratio of the 2.2 $\mu$m to 8 $\mu$m brightness of small pyroxene (Dorschner et al. 1995) particles at 350 K, then scaling to the observed 8 $\mu$m brightness (DLP01). In Figure 3 of DLP01 it is evident that dust dominates at 8 $\mu$m and above, and the synchrotron component only begins to contribute significantly at 6 $\mu$m. We find that the synchrotron emission is 3-4 orders of magnitude higher than the extrapolated dust model at 2.2 $\mu$m. We also estimated the brightness of starlight scattered by the Cas A dust, using the interstellar radiation field (Mathis, Mezger, and Panagia 1983), an albedo of 0.6, and the same optical depth as inferred from the mid-infrared thermal emission model (mostly from the colder component). The scattering contribution is 4-5 orders of magnitude smaller than that of synchrotron emission. Therefore, at 2.2 $\mu$m, synchrotron emission dominates over dust emission and scattering. Recent submillimeter observations of Cas A show no dust is present at 850 $\mu$m (Loinard 2002), which strengthens our conclusion of no dust contribution at 2 $\mu$m. Near-Infrared Synchrotron Radiation from Cas A ============================================== We show the broad-band spectrum of Cas A, from radio to TeV energies, in Figure \[sed\]. The data from Baars et al. (1977) for 10–35 MHz, Liszt & Lucas (1999) for 86 and 140 GHz, and @mezger86 for 1.2mm data, are marked. The high energy data of RXTE [@allen97], OSSE [@the96], EGRET [@esposito96], and Whipple [@lessard95] are also indicated. Our near-infrared points are above the fit to the radio data (solid line) with $\alpha$ = –0.77 (where log S= $\beta$ + $\alpha$ log $\nu$, $\beta$ = 5.745$\pm$0.025 for $\nu$ in MHz \[Baars et al. 1977\], which is equivalent to $\beta$ =10.365 for $\nu$ in Hz). This shows that the K$_s$ flux before correcting for extinction is comparable to the lower-frequency extrapolation, and the extinction-corrected K$_s$ flux is $\sim$ 70 % higher than that extrapolation. When we performed a weighted fit to the radio, mm and near-IR data, we obtain an index that is not significantly different from –0.77 and the value of the constant $\beta$ is sightly higher than that in Baars et al. (10.409$\pm$0.029, see thick solid line in Fig. \[sed\]). The extrapolation of this spectrum fell below the near-IR data points, suggesting positive (concave-up) curvature. A flatter index than –0.77 has been suggested by Mezger et al. (1986); they found –0.65. When we fit the 86 and 140 GHz, 1.2mm and 2$\mu$m data alone, $\alpha$ = –0.709$\pm$0.040 and $\beta$ = 9.762$\pm$0.444, showing a hardening to higher frequencies, although we cannot completely rule out the possibility that the excess near-IR flux above the extrapolation is due to contamination from line emission. It is possible that the concave spectrum between 140 GHz and near-IR is at least partially due to spectral-index variations across the remnant. If different shocks produce slightly different spectral indices, perhaps because of differences in Mach number, shocks with harder spectra will come to dominate at higher frequencies, and the spatially integrated spectrum will be concave. Spectral-index variations between 1.4 and 5 GHz were studied by [@anderson91], who obtained values from –0.64 to –0.92. Figure \[kbandradio\] is a two-color image with the radio image (1.4 GHz; Anderson et al. 1991) in red and the star-subtracted K$_s$-band image in green, and shows spectral-index variations. Radio-dominated regions (red in Figure \[kbandradio\]) have a steeper spectral index, and K$_s$-band dominated regions (green in Figure \[kbandradio\]) have a flatter spectral index between the radio and K$_s$ frequencies. The noticeable steep spectral-index regions (red in Figure \[kbandradio\]) are the southeastern shell and knots, and a blob in the west outside of the shell (at RA of 23$^{\rm h}$ 23$^{\rm m}$ 8.5$^{\rm s}$, $\delta$ = 58$^{\circ}$48$^{\prime}$44$''$). By contrast, the southwest rim appears to have a relatively flat spectrum. This image is indeed very similar to the radio spectral-index map between 1.4 and 5 GHz, which is shown in Figure 3 of [@anderson91]. Figure \[casaprofile\] shows K$_s$-band and radio radial profiles averaged over $5^\circ -- 10^\circ$ sectors toward the north, northeast, southeast, and southwest directions (each position angle is given in the Figure caption). While the K$_s$ band and radio profiles are similar in the N and NE (except the radio is more sharply peaked in the shell), the K$_s$ image is noticeably fainter in the SE shell and brighter in the SW shell, which is consistent with what we see in Figure \[kbandradio\]. We have also constructed images using the ‘spectral tomography’ technique [@katzstone97], in which scaled versions of one image are subtracted from another. That is, for a series of trial values $\alpha_t (< 0)$, we subtract from the radio image the K$_s$-band image multiplied by $(\nu_{\rm radio}/\nu_{\rm IR})^{\alpha_t}$. In such images, emission with a spectrum steeper than $\alpha_t$ appears positive, while flatter-spectrum emission appears negative. In Figure 7, we show images for $\alpha_t = -0.61, -0.67, -0.73,$ and $-0.77$. Since the –0.61 image is essentially all positive, and the –0.77 image all negative, the radio-near IR spectral index of different regions in Cas A is bracketed by these values. For $\alpha_t = -0.67$ and –0.73 we see both positive and negative structure. The disappearance of much of the emission along the bright southwestern and northern rims in the –0.67 image indicates that the average radio-near IR spectral index in these regions is about –0.67. Similarly, the southeastern rim mostly vanishes in the –0.73 image shows that the average spectral index along the southeastern rim is about that value. These results are consistent with the two-color radio/K$_s$ image, in which the SE rim appears steeper, while most of the rest of the remnant is flatter. There is a great deal of other small-scale structure in these images whose more detailed investigation might provide useful information on the location and properties of electron acceleration. However, it is not clear that the observed integrated-spectrum concavity can be produced by simply adding contributions from regions with straight spectra with different slopes. For instance, a region with a 1.4-5 GHz radio spectral index flatter than the average by 0.13 would be about 5 times brighter at K$_s$ band than a region of comparable brightness at 1.4 GHz with the average spectrum. Such large variations do not appear to be the case. But another explanation for the concavity is available. Such concave-up curvature of the spectrum is predicted by nonlinear shock-acceleration models in which shock transitions are broadened by pre-acceleration from cosmic rays diffusing ahead of the shock, if particles’ diffusion lengths increase with energy. In this case, more energetic particles feel a higher velocity jump between upstream and downstream fluids, and the spectrum is flatter at those energies [@eichler84; @ellison91]. Hints of this effect are seen in Tycho’s and Kepler’s SNRs [@reynolds92] as well as in Cas A. If such concavity can be shown to be true for localized regions as well as for the spatially integrated spectrum, it will strongly support the case for such nonlinear effects. Other theories of particle acceleration, such as stochastic (second-order Fermi) acceleration in magnetohydrodynamic turbulence, predict a steepening spectrum (e.g., Cowsik & Sarkar 1984), so that our observation supports diffusive (first-order Fermi) shock acceleration instead. A recent paper by Jones et al. (2003) also showed concavity of the spectrum from a small portion of the bright shell with the detection of polarized flux at 2.2$\mu$m, which reinforces presence of concavity of the spectrum, i.e. presence of non-linear shocks in Cas A. The synchrotron spectrum should eventually roll off due to one of several limitations on electron acceleration: radiative losses, finite acceleration time, or particle escape (Reynolds 1998; Sturner et al. 1997; Baring et al. 1999). For any of these, the electron distribution is likely to cut off no faster than exponentially (Reynolds 1998) with an e-folding energy $E_{\rm max}$. For a homogeneous source, the rolloff frequency is related to $E_{\rm max}$ by $${\nu_{\rm rolloff}} =4.7 \times 10^{15}\,\, \biggl({{B_\perp\over{\rm mG}}}\biggr) \biggl({{E_{\rm max}\over{\rm TeV}}}\biggr)^2 \,\, {\rm Hz}$$ where $B_\perp$ is the sky-plane component of the magnetic field. A lower limit for the roll-off frequency can be estimated directly from the high-frequency end of the K$_s$ band, which is 1.5 $\times$ 10$^{14}$ Hz. In a magnetic field $B$, synchrotron radiation at this frequency is produced by electrons with energies $E = 0.29 B_{\rm mG}^{-1/2}$ erg or $\sim$0.2 TeV, where $B_{\rm mG}$ is the mean magnetic field in milliGauss. These values assume that $B$ lies entirely in the plane of the sky, since this gives us the most conservative lower limit on $E$. Our claim of the presence of electrons at these energies makes it plausible that some portion of the X-ray emission is synchrotron as well. Reynolds & Keohane (1999) obtained an upper limit to the rolloff frequency of 3.2$\times 10^{17}$ Hz, by assuming all the X-ray emission to be synchrotron. Since the X-ray emission obviously contains thermal emission with a rich line spectrum, this is a very conservative upper limit. Allen et al. (1997) used a cut-off frequency of 2.4$\times 10^{17}$ Hz (1 keV). However, the cut-off frequency is very sensitive to the values of spectral index from the radio. The half-life of electrons with Lorentz factor $\gamma$ radiating in a magnetic field $B$ is given by $$t_{1/2} = {{ 5.1\times 10^{8} }\over {\gamma B^2}} \ {\rm s} = { {1.81 \times 10^{16} } \, {\nu^{-1/2} \, B_{mG}^{-3/2} } } \ {\rm s}$$ where we have averaged over magnetic-field directions. The electrons radiating in the K$_s$ band ($\nu = 1.5 \times 10^{14}$ Hz) have lifetimes of only about 47 $B_{\rm mG}^{-3/2}$ yr. With typical velocities in Cas A of several thousand km s$^{-1}$, electrons in such strong fields could travel distances of order tens of arcsec (for $d = 3.4$ kpc; Reed et al. 1995) in a loss time. The synchrotron emissivity varies roughly as $n_{\rm els} B^{1 + \alpha}$, where $n_{\rm els}$ is the energy density in relativistic electrons and $\alpha$ is the spectral index. If the magnetic field evolves by flux freezing and the relativistic electrons at these energies have relatively short diffusion lengths, we have $n_{\rm els}$ is proportional to n, the thermal gas density, and $B$ proportional to a power of $n$ between 0 and 1.8. Then the synchrotron emissivity varies as roughly $n^2$, like the optical emissivity; both optical emission and synchrotron emission would be prominant in high density regions, showing the morphological difference between the radio and optical is not due to density difference. The smoothness of the K$_s$ band and radio images, relative to optical images that more directly trace the shock fronts and high thermal-gas densities, is due to the relatively large diffusion distances of the extremely high-energy synchrotron emitting electrons, resulting in a generally smoother distribution of those electrons than that of thermal gas. Even though the synchrotron loss time for the $K_s$-band-emitting electrons is substantially less than the age of Cas A, it is much longer than the time required for shock acceleration to produce the electrons. We can estimate the acceleration time based on the general expression in Forman & Morfill (1979), for the particular case of electron mean free path proportional to gyroradius, $\lambda_{\rm mfp} = \eta r_g$ with $r_g = E/eB$ for extreme-relativistic electrons, and making other reasonable assumptions detailed in Reynolds (1998). The acceleration time for parallel shocks (shock normal parallel to the upstream magnetic field) is $$\tau_{\rm acc} = 1.3 \ \eta \left( B_{\rm mG} \right)^{-1} u_8^{-2} E \ {\rm yr}$$ where $u_8$ is the shock speed in units of $10^8$ cm s$^{-1}$, and $E$ is in erg. For perpendicular shocks, in which we assume a compression ratio of 4, the acceleration time is $$\tau_{\rm acc} = 0.53 \ \eta^{-1} \left( B_{\rm mG} \right)^{-1} u_8^{-2} E \ {\rm yr.}$$ We infer that shocks can produce the K$_s$-band emitting electrons in times of order one year. This very short timescale suggests that variability in acceleration, if not depletion, of these electrons could be observed over timescales of the same order. Now the radio morphology of Cas A makes it clear that there is no single acceleration site like the outer shock wave in SN 1006; instead, particle acceleration appears to take place in many small knots all over the remnant [@anderson91]. This picture can be supported by detailed theoretical analysis (Atoyan et al. 2000). In their model, the magnetic field in the knots where acceleration takes place is much larger than that in the general remnant volume, so one would expect longer synchrotron lifetimes once electrons diffused out of the acceleration zones. In any case, to the extent that our $K_s$ band image resembles the radio, synchrotron losses are evidently not dominant, or we would expect to see less diffuse emission and a greater concentration in knots. But small differences between the images of Figure \[kbandradio\] shows that it is not impossible that this greater concentration has already begun to occur. While most authors agree that shock acceleration produces the relativistic electrons in Cas A, it is not impossible that some form of turbulent or stochastic acceleration is responsible (e.g., Cowsik & Sarkar 1984). One must ask if this mechanism, designed to describe radio emission, can in the time available produce the $\sim 1$ erg electrons required by our near-infrared image. Melrose (1974) gives expressions for acceleration rates in magnetohydrodynamic turbulence which show that once electrons are substantially suprathermal (‘injected;’ a necessity for shock acceleration as well), they can reach relativistic energies in a fraction of a year. Subsequent acceleration of relativistic electrons in MHD turbulence with amplitude $\delta B/B \equiv \epsilon$ takes place on a timescale $$\tau_{\rm acc} \sim {1 \over \epsilon^2} {r_g \over c} \left({v_A \over c} \right)^{-2}$$ where $v_A$ is the Alfvén speed and $r_g$ is the electron’s gyroradius. Strictly speaking, this result is true in the quasilinear approximation ($\epsilon \ll 1$), but it should provide at least a rough estimate for $\epsilon = 1$ to give us a conservative idea of the fastest possible acceleration. In the above expression, $\epsilon$ refers to the amplitude of waves resonant with the particular particle, which must be presumed to exist (perhaps generated by the particles themselves). Again assuming that the electron mean free path is a multiple of its gyroradius, $\lambda_{\rm mfp} \equiv \eta r_g$, quasilinear theory then gives $\eta = (\delta B/B)^{-2} = \epsilon^{-2}$, so we can rewrite the acceleration time as $\tau_{\rm acc} \sim (\lambda_{\rm mfp}/c) (v_A/c)^{-2}$. Since in the extreme-relativistic limit, $r_g = E/eB$, with $E$ the particle energy, the acceleration time to a particular energy is just proportional to that energy. For the strong fields we anticipate in Cas A, $v_A = 2.2 \times 10^8 \ B_{\rm mG} (\mu n_H)^{-1/2}$ cm s$^{-1}$, where the matter density is $\rho = \mu m_p n_H$. So $v_A/c \sim 10^{-2}$, and with $r_g = 2 \times 10^{12} E B_{\rm mG}^{-1}$ cm, we estimate $\tau_{\rm acc} \sim 10^6$ s to reach energies adequate to produce infrared synchrotron radiation (in these strong fields) – comparable to the rate of shock acceleration. This can occur because the strong magnetic fields in Cas A can give Alfven speeds comparable to the current, decelerated outer blast-wave speed. So stochastic acceleration can certainly proceed rapidly enough to explain our observations. The spectrum produced by stochastic acceleration will depend on the properties of the turbulence and it is not clear if the required remarkably flat, slightly hardening spectrum can in fact be produced. However, neither shock acceleration nor stochastic acceleration can be ruled out on the basis of acceleration-time arguments. Summary and conclusions ======================= The 2MASS near-infrared images reveal emission from Cas A in the J, H, and K$_s$ bands. The images reveal a shell-like morphology shown in Figure \[casa2mass\]. The K$_s$ and H band images are brighter than the J band image. The three-color image demonstrates the difference: K$_s$ emission is dominant for the majority of Cas A, and H band emission is noticeable in the southwestern shell. Near-infrared spectra showed \[\] lines in J (1.25$\mu$m) and H (1.64$\mu$m), and no lines were detected within the K$_s$ band. A high-resolution Palomar PFIRCAM narrow-band image of \[\] (1.64$\mu$m) (see Figure \[pfirmcam\]) is similar to the optical image of Cas A, showing sharp filamentary and knotty structures of Fe-rich ejecta. The PFIRCAM K$_s$ band image is much smoother and more diffuse than the ejecta structure that dominates the optical images (and is present in the \[\] image). Instead, the K$_s$ image is remarkably similar to the radio image of Cas A, suggesting that the physical mechanism that produces radio emission–synchrotron radiation–is also responsible for the K$_s$ band emission. We measure a total flux of Cas A in the K$_s$ band of 0.32$\pm0.03$ Jy, implying a flux of 0.53$^{+0.12}_{-0.04}$ Jy after correcting for extinction. The spectral energy distribution of Cas A shows that the near-infrared flux is close to an extrapolation of the radio fluxes, supporting our interpretation of the near-IR flux as synchrotron radiation. The near-infrared K$_s$ band fluxes are actually a few tens of percent higher than an extrapolation of the radio fluxes. Such positive (concave-up) curvature of the spectrum (hardening to higher frequencies) is predicted by nonlinear shock-acceleration models in which shock transitions are broadened by pre-acceleration from cosmic rays diffusing ahead of the shock, if particles’ diffusion lengths increase with energy. The synchrotron spectrum may roll off due to any of several possible limitations on electron acceleration including radiative losses, acceleration time, or particle escape; this is measured as the rolloff frequency. Our near-infrared emission measurement gives us a lower limit to the roll-off frequency of 1.5 $\times$ 10$^{14}$ Hz. Synchrotron radiation at this frequency is produced by electrons with energies $E = 0.3 B_{\rm mG}^{-1/2}$ erg or about 0.2 TeV, where $B_{\rm mG}$ is the mean magnetic field in milliGauss. The synchrotron loss time for electrons radiating in the K$_s$ band is only about 47 $B_{\rm mG}^{-3/2}$ yr. The fact that such electrons can thus travel only distances on the order of tens of arcsec implies particle acceleration takes place in many sites all over the remnant. Our $K_s$ band image resembles the radio, so synchrotron losses evidently do not dominate the small-scale structure. The acceleration time scale for electrons with energy high enough to produce the near-infrared emission from Cas A is of order a year (or less), so either shock or stochastic acceleration is rapid enough to accelerate electrons to the required energies. 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--- abstract: 'In this paper we study the regularity properties of two maximal operators of convolution type: the heat flow maximal operator (associated to the Gauss kernel) and the Poisson maximal operator (associated to the Poisson kernel). In dimension $d=1$ we prove that these maximal operators do not increase the $L^p$-variation of a function for any $p \geq 1$, while in dimensions $d>1$ we obtain the corresponding results for the $L^2$-variation. Similar results are proved for the discrete versions of these operators.' address: 'IMPA - Estrada Dona Castorina, 110, Rio de Janeiro, RJ, Brazil 22460-320' author: - 'Emanuel Carneiro\* and Benar F. Svaiter' title: On the variation of maximal operators of convolution type --- [^1] Introduction ============ Background ---------- Let $\varphi \in L^1({\mathbb{R}}^d)$ be a nonnegative function such that $$\int_{{\mathbb{R}}^d} \varphi(x)\,{\text{\rm d}x}=1.$$ We let $\varphi_t(x) = t^{-d} \varphi(t^{-1}x)$ and consider the maximal operator ${\mathcal}{M}_{\varphi}$ associated to this approximation of the identity $$\label{Conv_type_max} {\mathcal}{M}_{\varphi}f(x) = \sup_{t>0}\big(\|f\|\varphi_t\big)(x).$$ The Hardy-Littlewood maximal function, henceforth denoted by $M$, occurs when we consider $\varphi(x) = (1/m(B_1)) \chi_{B_1}(x)$, where $B_1$ is the $d$-dimensional ball centered at the origin with radius $1$ and $m(B_1)$ is its Lebesgue measure. In a certain sense, one could say that $M$ controls other such maximal operators of convolution type. In fact, if our $\varphi$ admits a radial non-increasing majorant in $L^1({\mathbb{R}}^d)$ with integral $A$, from [@S Chapter III, Theorem 2] we know that $$\label{orig_bound} {\mathcal}{M}_{\varphi}f(x) \leq A \, Mf(x)$$ for all $x \in {\mathbb{R}}^d$ and thus we obtain the boundedness of ${\mathcal}{M}_{\varphi}$ from $L^p({\mathbb{R}}^d)$ to $L^p({\mathbb{R}}^d)$ if $p>1$, and from $L^1({\mathbb{R}}^d)$ to $L^{1,\infty}({\mathbb{R}}^d)$ in the case $p=1$. Over the last years there has been considerable effort in understanding the effects of the Hardy-Littlewood maximal operator $M$, and some of its variants, in Sobolev functions. In [@Ki] Kinnunen showed that $M:W^{1,p}({\mathbb{R}}^d) \to W^{1,p}({\mathbb{R}}^d)$ is bounded for $p>1$. The paradigm that an $L^p$-bound implies a $W^{1,p}$-bound was later extended to a local version of $M$ in [@KL], to a fractional version in [@KiSa] and to a multilinear version in [@CM]. The continuity of $M:W^{1, p}({\mathbb{R}}^d) \to W^{1, p}({\mathbb{R}}^d)$ for $p>1$ was established by Luiro in [@Lu1]. When $p=1$ the issues become more subtle. The question on whether the operator $f\mapsto \nabla M f$ is bounded from $W^{1,1}({\mathbb{R}}^d)$ to $L^1({\mathbb{R}}^d)$, posed by Hajłasz and Onninen in [@HO], remains open in its general case (see also [@HM]). Partial progress was achieved in the discrete setting in the work [@BCHP] for dimension $d=1$ and in the work [@CH] for general dimension $d>1$. In the continuous setting the only progress has been in dimension $d=1$. For the right (or left) Hardy-Littlewood maximal operator, which we call here $M_r$ (corresponding to $\varphi(x) = \chi_{[0,1]}(x)$ in ) Tanaka [@Ta] was the first to observe that, if $f \in W^{1,1}({\mathbb{R}})$, then $M_rf$ has a weak derivative and $$\label{Intro2.1} \\|(M_rf)'\\|_{L^1({\mathbb{R}})} \leq \\|f'\\|_{L^1({\mathbb{R}})},$$ which led to the bound for the [non-centered]{} Hardy-Littlewood maximal operator $\widetilde{M}$, $$\label{Intro2} \\|(\widetilde{M}f)'\\|_{L^1({\mathbb{R}})} \leq 2 \\|f'\\|_{L^1({\mathbb{R}})}.$$ This was later refined by Aldaz and Perez-Lázaro [@AP] who obtained, under the assumption that $f$ is of bounded variation on ${\mathbb{R}}$, that $\widetilde{M}f$ is absolutely continuous and $$\label{Intro3.1} V(\widetilde{M}f) \leq V(f),$$ where $V(f)$ denotes here the total variation of $f$. More recently, in the remarkable work [@Ku], Kurka showed that if $f$ is of bounded variation on ${\mathbb{R}}$, then $$\label{Intro3} V(Mf) \leq C \,V(f),$$ for a certain $C>1$ (see [@Te] for the discrete analogue). In this paper we turn our attention to understanding the action of a general maximal operator of convolution type in a Sobolev function. One can show that the original argument of Kinunnen [@Ki], that proves that $M: W^{1,p}({\mathbb{R}}^d) \to W^{1,p}({\mathbb{R}}^d)$ is a bounded operator for $p>1$, can be applied to a general ${\mathcal}{M}_{\varphi}$ of type that admits a radial non-increasing integrable majorant. In this case, if $f \in W^{1,p}({\mathbb{R}}^d)$ with $p>1$, we have that $\nabla {\mathcal}{M}_{\varphi}f$ exists in the weak sense and $$\label{Intro4} \\|\nabla {\mathcal}{M}_{\varphi}f \\|_p \leq C\, \\|\nabla f \\|_p,$$ for a certain constant $C>1$. Therefore, from and , we already know that ${\mathcal}{M}_{\varphi}:W^{1, p}({\mathbb{R}}^d) \to W^{1, p}({\mathbb{R}}^d)$ is bounded for $p>1$. We want to explore here the question on whether a maximal operator of convolution type can increase the variation (or $L^p$-variation) of a function. In other words, can we prove inequalities like (for $d=1$) and (for $p\geq 1$) with the constant $C=1$? To our knowledge, the only known [variation diminishing]{} bounds for maximal operators are the ones given in and , and the ones given in the recent work of Aldaz, Colzani and Pérez-Lázaro [@ACP] that show that the Lipschitz constant of a function (or Hölder constant) actually decreases under the action of the non-centered Hardy-Littlewood maximal function. Here we give a pool of affirmative answers to this question for two classical maximal operators of convolution type: the heat flow maximal operator (associated to the Gauss kernel) and the Poisson maximal operator (associated to the Poisson kernel). We shall obtain positive results for these two maximal operators in dimension $d=1$ for any $p\geq1$, and in dimensions $d>1$ for $p=2$ or $\infty$. We consider also the discrete versions of these operators and prove similar results. We start by reviewing the definitions and main properties of these convolution kernels, and as we move on the proofs, we shall see that the key idea to achieve these results is to explore the nice interplay between the maximal function analysis and the structure of the differential equations associated to these kernels (heat equation for the Gaussian and Laplace’s equation for the Poisson kernel). Main results ============ The continuous setting ---------------------- We start by reviewing the definitions and stating the results in the context of the Euclidean space ${\mathbb{R}}^d$. All the results presented here hold for complex-valued functions, but for simplicity (since the maximal operators only see the absolute value of a function) we will work with real-valued functions. ### Heat flow maximal operator Given $u_0 \in L^{p}({\mathbb{R}}^d)$, $1\leq p \leq \infty$, we define $u:{\mathbb{R}}^d \times (0,\infty) \to {\mathbb{R}}$ by $$u(x,t) = (u_0K_t)(x)\,,$$ where $K_t$ is the heat kernel given by $$\label{heat} K_t(x) = \frac{1}{(4\pi t)^{d/2}} e^{-\|x\|^2/4t}.$$ In this case we know that $u \in C^{\infty}({\mathbb{R}}^d \times (0,\infty))$ and solves the heat equation $$\partial_t u - \Delta u = 0 \ \ \textrm{in} \ \ {\mathbb{R}}^d \times (0,\infty),$$ with $$\lim_{t \to 0^+} u(x,t) = u_0(x) \ \ {\rm a.e.}\ x \in {\mathbb{R}}^d.$$ We consider here the maximal function associated to this heat flow, henceforth denoted by $$ to facilitate the notation, $$\label{def_u} u^(x)= \sup_{t >0} \,(\|u_0\| K_t)(x).$$ We shall prove the following regularity results for this maximal operator. \[thm\_cont\] Let $u^$ be the heat flow maximal function defined in . The following propositions hold. - Let $1 < p \leq \infty$ and $u_0 \in W^{1,p}({\mathbb{R}})$. Then $u^{} \in W^{1,p}({\mathbb{R}})$ and $$\\|(u^)'\\|_p \leq \\|u_0'\\|_p.$$ - Let $u_0 \in W^{1,1}({\mathbb{R}})$. Then $u^{} \in L^{\infty}({\mathbb{R}})$ and has a weak derivative $(u^)'$ that satisfies $$\\|(u^)'\\|_1 \leq \\|u_0'\\|_1.$$ - Let $u_0$ be of bounded variation on ${\mathbb{R}}$. Then $u^$ is of bounded variation on ${\mathbb{R}}$ and $$V(u^) \leq V(u_0).$$ - Let $d>1$ and $u_0 \in W^{1,p}({\mathbb{R}}^d)$, for $p =2$ or $p= \infty$. Then $u^{} \in W^{1,p}({\mathbb{R}}^d)$ and $$\\|\nabla u^\\|_p \leq \\|\nabla u_0\\|_p.$$ ### Poisson maximal operator Given $u_0 \in L^{p}({\mathbb{R}}^d)$, $1\leq p \leq \infty$, we now define $u:{\mathbb{R}}^d \times (0,\infty) \to {\mathbb{R}}$ by $$u(x,y) = (u_0P_y)(x)\,,$$ where $P_y$ is the Poisson kernel for the upper half-space given by $$\label{Poisson} P_y(x) = c_d \,\frac{y}{(\|x\|^2 + y^2)^{(d+1)/2}},$$ with $$c_d = \frac{\Gamma\left(\frac{d+1}{2}\right)}{\pi^{(d+1)/2}}.$$ In this case we know that $u \in C^{\infty}({\mathbb{R}}^d \times (0,\infty))$ and that it solves Laplace’s equation $$\Delta u = 0 \ \ \textrm{in} \ \ {\mathbb{R}}^d \times (0,\infty),$$ where here we take the Laplacian with respect to the $(d+1)$ coordinates of $(x,y) = (x_1,x_2,...,x_d,y)$, as opposed to the notation used in the heat flow case, when we wrote the Laplacian only on the $x$-variable. Note also that $$\lim_{y \to 0^+} u(x,y) = u_0(x) \ \ {\rm a.e.}\ x \in {\mathbb{R}}^d.$$ In other words, $u(x,y)$ is the harmonic extension of $u_0$ to the upper half-space. We consider here the maximal function associated to the Poisson kernel (henceforth denoted by the star $\star$, slightly different from the heat flow case) $$\label{def_u_star} u^{\star}(x)= \sup_{y>0} \,(\|u_0\| P_y)(x).$$ With respect to this maximal operator we shall prove the following regularity results, in analogy with Theorem \[thm\_cont\]. \[thm\_Poisson\] Let $u^\star$ be the Poisson maximal function defined in . The following propositions hold. - Let $1 < p \leq \infty$ and $u_0 \in W^{1,p}({\mathbb{R}})$. Then $u^{\star} \in W^{1,p}({\mathbb{R}})$ and $$\\|(u^\star)'\\|_p \leq \\|u_0'\\|_p.$$ - Let $u_0 \in W^{1,1}({\mathbb{R}})$. Then $u^{\star} \in L^{\infty}({\mathbb{R}})$ and has a weak derivative $(u^\star)'$ that satisfies $$\\|(u^\star)'\\|_1 \leq \\|u_0'\\|_1.$$ - Let $u_0$ be of bounded variation on ${\mathbb{R}}$. Then $u^\star$ is of bounded variation on ${\mathbb{R}}$ and $$V(u^\star) \leq V(u_0).$$ - Let $d>1$ and $u_0 \in W^{1,p}({\mathbb{R}}^d)$, for $p =2$ or $p= \infty$. Then $u^{\star} \in W^{1,p}({\mathbb{R}}^d)$ and $$\\|\nabla u^\star\\|_p \leq \\|\nabla u_0\\|_p.$$ The discrete setting -------------------- Again, all the results presented in this section hold for complex-valued functions, but for simplicity (since the discrete maximal operators only see the absolute value of a function) we will keep working with real-valued functions. For a bounded discrete function $f:{\mathbb{Z}}^d \to {\mathbb{R}}$ we define $$\partial_{x_i} f(n) := f(n+e_i) - f(n),$$ where $n \in {\mathbb{Z}}^d$ and $e_i = (0,. . .,1,. . . ,0)$ is the canonical $i$-th base vector with $1$ in the position $i$. The discrete gradient is then the vector $$\nabla f(n) = \left(\partial_{x_i} f(n), \partial f_{x_2}(n), . . . , \partial_{x_d} f(n)\right).$$ We let $$\\|f\\|_p = \left(\sum_{n \in {\mathbb{Z}}^d} \|f(n)\|^p\right)^{1/p},$$ if $1 \leq p < \infty$ and $$\\|f\\|_{\infty} = \sup_{n\in {\mathbb{Z}}^d} \|f(n)\|.$$ We define the $l^p({\mathbb{Z}}^d)$-norms of the discrete function $n \mapsto \|\nabla f(n)\|$ in an analogous way (here $\|\cdot\|$ will always denote the usual Euclidean norm in ${\mathbb{R}}^d$). For a discrete function $f:{\mathbb{Z}}^d \to {\mathbb{R}}$ we define its [discrete Laplacian]{} $\Delta f:{\mathbb{Z}}^d \to {\mathbb{R}}$ as $$\Delta f(n) = \frac{1}{2d} \sum_{\\|m-n\\|_{1}=1} \big[f(m)- f(n)\big],$$ where $\\|n\\|_1 = \|n_1\| + \|n_2\| + . . . + \|n_d\|$, if $n = (n_1, n_2,. . ., n_d) \in {\mathbb{Z}}^d$. ### Discrete heat flow maximal operator For a bounded function $u_0:{\mathbb{Z}}^d \to {\mathbb{R}}$, the [heat flow]{} in ${\mathbb{Z}}^d$ with initial condition $\|u_0\|$ is the unique (bounded and $C^1$ in time) solution $u: {\mathbb{Z}}^d \times [0,\infty) \to {\mathbb{R}}$ of the differential equation $$\left\{ \begin{array}{rl} \partial_t u (n,t) &= \Delta u(n,t);\\ \\ u(n,0)&= \|u_0(n)\|. \end{array} \right.$$ This solution can be given in terms of a convolution with the discrete heat kernel (see for instance [@KN] and the references therein) $$u(n,t) = \big(\|u_0\|{\mathcal}{K}_t \big)(n) = \sum_{m\in {\mathbb{Z}}^d} \|u_0(n-m)\| \,{\mathcal}{K}_t(m),$$ where $${\mathcal}{K}_t(m) = e^{-t}\,\prod_{i=1}^{d} I_{m_i}(t/d),$$ for $m = (m_1, m_2, . . . , m_d) \in {\mathbb{Z}}^d$, and where $I_k$, for an integer $k\geq 0$ and complex $z$, is the $I$-Bessel function defined by $$I_k(z) = \sum_{j=0}^{\infty} \frac{(z/2)^{2j + k}}{j! \, (j +k)!} = \frac{1}{\pi} \int_0^{\pi} e^{z \cos \theta}\,\cos k\theta\, {\rm d} \theta,$$ and for a negative integer $k$ we put $I_k: = I_{-k}$. We note here that $I$ satisfies the differential equation $$I_{k+1}(z) + I_{k-1}(z) = 2 I_{k}'(z),$$ which plainly implies that $$\partial_t {\mathcal}{K}_t(m) = \Delta {\mathcal}{K}_t(m).$$ Moreover, for any $\theta \in {\mathbb{R}}$ and $z \in {\mathbb{C}}$, we also have [@KN Lemma 7] $$\sum_{k = -\infty}^{\infty} \big[e^{-z}\,I_k(z)\big]\, e^{-i \theta k} = e^{z(\cos \theta - 1)},$$ which gives $$\\|{\mathcal}{K}_t\\|_{l^1({\mathbb{Z}}^d)} = \sum_{m \in {\mathbb{Z}}^d} {\mathcal}{K}_t(m) = 1.$$ We then define the discrete heat flow maximal operator by $$\label{def_u_disc} u^(n) = \sup_{t >0} u(n,t) = \sup_{t>0} (\|u_0\|{\mathcal}{K}_t \big)(n).$$ Our next result shows that this maximal operator does not increase the $l^p$-variation in two situations: (i) in dimension $d=1$, for any $1\leq p\leq\infty$; (ii) in dimension $d>1$, for $p=2$. \[thm\_disc\] Let $u^$ be the discrete heat flow maximal function defined in . The following propositions hold. - Let $1 \leq p \leq \infty$ and $u_0:{\mathbb{Z}}\to {\mathbb{R}}$ be a bounded discrete function such that $\\|u_0'\\|_p < \infty$. Then $$\\|(u^)'\\|_p \leq \\|u_0'\\|_p.$$ - Let $u_0:{\mathbb{Z}}^d \to {\mathbb{R}}$ be a bounded discrete function such that $\\|\nabla u_0\\|_2 < \infty$. Then $$\\|\nabla u^\\|_2 \leq \\|\nabla u_0\\|_2.$$ ### Discrete Poisson maximal operator Recall that the continuous Poisson kernel $P_y(x)$ for the upper half-space defined in satisfies the semigroup property $$\label{DP_sem} P_{y_1} P_{y_2} = P_{y_1 + y_2}$$ for any $y_1, y_2>0$, and its Fourier transform verifies $$\widehat{P_y}(\xi) = \int_{{\mathbb{R}}^n} P_y(x)\,e^{-2\pi i x \xi}\,{\text{\rm d}x}= e^{-2\pi y \|\xi\|} = \left(\widehat{P_1}(\xi) \right)^y.$$ Given a bounded discrete function $u_0:{\mathbb{Z}}^d \to {\mathbb{R}}$, we aim to lift this function to a harmonic function on the discrete upper half-space, i.e. we want to construct $u: {\mathbb{Z}}^d \times {\mathbb{Z}}^{+} \to {\mathbb{R}}$ such that $$\label{DP_eq-1} \left\{ \begin{array}{rl} \Delta u(n,y) = 0 & {\rm in } \ \ {\mathbb{Z}}^d \times {\mathbb{N}};\\ \\ u(n,0)&= u_0(n). \end{array} \right.$$ Observe that here we use the $(d+1)$-dimensional discrete Laplacian, and that the parameter $y$ is now also discrete. We aim to accomplish this by convolving the initial datum $u_0$ with a certain integrable discrete kernel ${\mathcal}{P}_y$ (also denoted below by ${\mathcal}{P}(\cdot, y)$) that satisfies the semigroup property . Let us proceed with a formal derivation of the kernel ${\mathcal}{P}_y$ first. The function ${\mathcal}{P}: {\mathbb{Z}}^d \times {\mathbb{Z}}^{+} \to {\mathbb{R}}$ would have to satisfy $$\label{DP_eq0} \left\{ \begin{array}{rl} \Delta {\mathcal}{P}(n,y) = 0 & {\rm in } \ \ {\mathbb{Z}}^d \times {\mathbb{N}};\\ \\ {\mathcal}{P}(n,0)&= \delta_0(n), \end{array} \right.$$ where $\delta_0$ is the function that is $1$, if $n=0$, and zero otherwise. Writing the harmonicity condition at the level $y=1$ we have $$2(d+1) {\mathcal}{P}_1(n) = {\mathcal}{P}_0(n) + {\mathcal}{P}_2(n) + \sum_{\\|m - n\\|_1=1} {\mathcal}{P}_1(m).$$ We now multiply the last expression by $e^{-2\pi i n \xi}$ and sum over $n \in {\mathbb{Z}}^d$ to get $$\begin{aligned} 2(d+1) \widehat{{\mathcal}{P}_1}(\xi) & = 1 + \widehat{{\mathcal}{P}_2}(\xi) + \widehat{{\mathcal}{P}_1}(\xi) \sum_{k=1}^d \big(e^{2\pi i \xi_k} + e^{-2\pi i \xi_k}\big),\end{aligned}$$ where $\xi = (\xi_1, \xi_2,..., \xi_d)$ belongs to the torus $\mathbb{T}^d = [-\tfrac12,\tfrac12]^d$. From the semigroup property we have $\widehat{{\mathcal}{P}_2}(\xi) = \widehat{{\mathcal}{P}_1}(\xi)^2$ and thus $$\label{DP_eq1} \widehat{{\mathcal}{P}_1}(\xi)^2 - \widehat{{\mathcal}{P}_1}(\xi)\left( 2(d+1) - 2\sum_{k=1}^d \cos(2 \pi \xi_k)\right)+ 1 = 0.$$ From we conclude that $$\label{DP_choice} \widehat{{\mathcal}{P}_1}(\xi) = \left( (d+1) - \sum_{k=1}^d \cos(2 \pi \xi_k)\right) \pm \left(\left((d+1) - \sum_{k=1}^d \cos(2 \pi \xi_k)\right)^2 - 1 \right)^{1/2},$$ where the choice of signs could (at least in principle) be taken in any measurable way. We shall pick the negative sign in and [define]{} our discrete Posson kernel by the following three expressions: $$\label{DP_eq2} \widehat{{\mathcal}{P}_1}(\xi) := \left( (d+1) - \sum_{k=1}^d \cos(2 \pi \xi_k)\right) - \left(\left((d+1) - \sum_{k=1}^d \cos(2 \pi \xi_k)\right)^2 - 1 \right)^{1/2},$$ $$\label{DP_eq3} \widehat{{\mathcal}{P}_y}(\xi) := \left(\widehat{{\mathcal}{P}_1}(\xi) \right)^y,$$ for $y \geq 0$, and $$\label{DP_eq4} {\mathcal}{P}_y(n) = \int_{\mathbb{T}^d} \widehat{{\mathcal}{P}_y}(\xi) \,e^{2 \pi i n \xi}\,{\text{\rm d}\xi}.$$ Observe from that $0 < \widehat{{\mathcal}{P}_1}(\xi)\leq 1$ for all $\xi \in \mathbb{T}^d$, with $\widehat{{\mathcal}{P}_1}(\xi) = 1$ if and only if $\xi=0$. Thus, for $\xi \in \mathbb{T}^d \setminus \{0\}$ the function $y \mapsto \widehat{{\mathcal}{P}_y}(\xi)$ is decreasing (and goes to $0$ as $y \to \infty$). From , and it is clear that ${\mathcal}{P}: {\mathbb{Z}}^d \times {\mathbb{Z}}^{+} \to {\mathbb{R}}$ satisfies , and therefore the maximum principle holds. From we have $\|P_y(n)\| \leq 1$ for all $(n,y) \in {\mathbb{Z}}^d \times {\mathbb{Z}}^{+}$. We want to show now that we also have ${\mathcal}{P}_y(n) \geq 0$ for all $(n,y) \in {\mathbb{Z}}^d \times {\mathbb{Z}}^{+}$. For this we use the maximum principle. Given $\varepsilon >0$ observe first that, by Lebesgue dominated convergence, there is a $y_0 >0$ such that $$\big\|{\mathcal}{P}_{y_0}(n)\big\| \leq \int_{\mathbb{T}^d} \widehat{{\mathcal}{P}_{y_0}}(\xi) \,{\text{\rm d}\xi}< \varepsilon$$ for any $n \in {\mathbb{Z}}^d$. Now, by the Riemann-Lebesgue lemma, there exists a radius $r_0 >0$ such that $$\big\|{\mathcal}{P}_{y}(n)\big\| < \varepsilon$$ for all $0 \leq y \leq y_0$ and $\|n\| \geq r_0$. When we consider the cylindrical contour of radius $r_0$, delimited by the hyperplanes $y=0$ and $y=y_0$, by the maximum principle we have $$\label{DP_eq5} {\mathcal}{P}_{y}(n) > -\varepsilon$$ within this region. Since we could take $y_0$ as large as we wanted (and then $r_0$ large as well) we conclude that actually holds for all $(n,y) \in {\mathbb{Z}}^d \times {\mathbb{Z}}^{+}$. Since $\varepsilon>0$ is arbitrary we have $$\label{DP_eq6} {\mathcal}{P}_{y}(n) \geq 0$$ for all $(n,y) \in {\mathbb{Z}}^d \times {\mathbb{Z}}^{+}$ as desired. Since $\widehat{{\mathcal}{P}_{y}}(0)=1$ for all $y \geq 0$, it follows from (and a standard approximation argument using the smoothing Féjer kernel) that ${\mathcal}{P}_y$ is integrable and $$\sum_{n\in {\mathbb{Z}}^d} {\mathcal}{P}_y(n) = 1$$ for all $y \geq 0$. Therefore, given a bounded discrete function $u_0:{\mathbb{Z}}^d \to {\mathbb{R}}$ the function $$u(n,y) = u_0 * {\mathcal}{P}_y(n) = \sum_{m \in {\mathbb{Z}}^d} u_0(n-m){\mathcal}{P}_y(m)$$ is well defined, and one now clearly sees that it satisfies . We now define our discrete Poisson maximal operator (keeping the notation $\star$) by $$\label{DP_eq7} u^{\star}(n) = \sup_{y \in {\mathbb{Z}}^{+}} (\|u_0\| * {\mathcal}{P}_y)(n).$$ By a classical result of Stein [@S1 Theorem 1] the maximal operator $\star$ verifies the $l^p$-boundedness $$\label{Stein_lp_bound} \\|u^{\star}\\|_{l^p({\mathbb{Z}}^d)} \leq C_p \\|u\\|_{l^p({\mathbb{Z}}^d)}$$ for $1 < p \leq \infty$. With respect to this maximal function we prove the following regularity properties, in analogy with Theorem \[thm\_disc\]. \[thm\_disc\_Poisson\] Let $u^\star$ be the discrete Poisson maximal function defined in . The following propositions hold. - Let $1 \leq p \leq \infty$ and $u_0:{\mathbb{Z}}\to {\mathbb{R}}$ be a bounded discrete function such that $\\|u_0'\\|_p < \infty$. Then $$\\|(u^\star)'\\|_p \leq \\|u_0'\\|_p.$$ - Let $u_0:{\mathbb{Z}}^d \to {\mathbb{R}}$ be a bounded discrete function such that $\\|\nabla u_0\\|_2 < \infty$. Then $$\\|\nabla u^{\star}\\|_2 \leq \\|\nabla u_0\\|_2.$$ We now move on to the proofs of these results. We shall start with the discrete cases, that are technically simpler but already give a good flavor of the main ideas that shall be used in the continuous cases. The key insight here is that all of these maximal functions have the property of being subharmonic in the set where they disconnect from the original function. This shall be obtained by exploiting the structure of the underlying partial differential equations. Proof of Theorem \[thm\_disc\] - Discrete heat kernel ===================================================== From now on we assume, without loss of generality, that $u_0 \geq0$, since $\\|\nabla \|u_0\|\\|_p \leq \\|\nabla u_0\\|_p$ for any $1\leq p \leq \infty$. Preliminaries ------------- The essence of the following lemma is the fact that $u^$ is subharmonic in the set where it disconnects from $u_0$. The statement in the following format will be more convenient later in the proof. \[lem3\] Let $u_0 \in l^{\infty}({\mathbb{Z}}^d)$ and $u^$ be its discrete heat flow maximal function. Let $f \in l^{\infty}({\mathbb{Z}}^d)$ be such that $f \geq u_0$. Let $I = \{n \in {\mathbb{Z}}^d;\, u^(n)\leq f(n)\}$ and suppose that $\Delta f(n) \leq 0$ for all $n \in I^c$. Then $I = {\mathbb{Z}}^d$. Suppose without loss of generality $u_0$ is not identically zero. Let $u: {\mathbb{Z}}^d \times [0,\infty)$ be the heat flow in ${\mathbb{Z}}^d$ with initial condition $u_0$. It suffices to prove that $$\sup_{t \geq 0} u(n,t) \leq f(n),$$ for all $n \in I^c$. It is easy to see that $\\|u(\cdot,t)\\|_{\infty} \leq \\|u_0\\|_{\infty}$ for any $t\geq0$. Therefore, $$\label{lem_Sec2.1} \|\partial_t u(n,t)\|\leq \\|\Delta u\\|_{\infty}\leq 2 \\|u_0\\|_{\infty}.$$ Take $\varepsilon >0$ and define $$T_{\varepsilon} = \sup\{t \geq 0;\ u(n,t) \leq f(n) + \varepsilon,\ \forall n \in I^c\}.$$ In view of we have $$T_{\varepsilon} \geq \frac{\varepsilon}{2\\|u_0\\|_{\infty}}> 0.$$ Suppose $T_{\varepsilon} < \infty$. For any $n \in I^c$ and $0 \leq t \leq T_{\varepsilon}$ we have $$\begin{aligned} \partial_t u(n,t)&=\frac{1}{2d}\sum_{\\|m-n\\|_1=1} \big[u(m,t)- u(n,t)\big]\\ &\leq\frac{1}{2d}\sum_{\\|m-n\\|_1=1}\big[f(m)+\varepsilon-u(n,t)\big]\\ &=f(n)+\varepsilon-u(n,t)+\frac{1}{2d}\sum_{\\|m-n\\|_1=1}\big[f(m)-f(n)\big]\\ &\leq f(n)+\varepsilon-u(n,t).\end{aligned}$$ Define $y(t)=u(n,t)-(f(n)+\varepsilon)$. From the inequality above we have $y'(t)\leq-y(t)$ and thus, for any $t\in [0,T_\varepsilon]$, $$y(t)\leq e^{-t}y(0)\leq -\varepsilon e^{-t}\leq-\varepsilon e^{-T_\varepsilon}.$$ Hence $$\label{lem_Sec2.1.2} u(n,T_\varepsilon)\leq (f(n)+\varepsilon)-\frac{\varepsilon}{\exp(T_\varepsilon)}$$ for any $n\in I^c$. Combining with we conclude that $u(n,t)\leq f(n) + \varepsilon$ for any $t$ with $$T_\varepsilon\leq t\leq T_\varepsilon+\frac{\varepsilon}{2\\|u_0\\|_\infty \exp(T_\varepsilon)},$$ which is in contradiction with the assumption that $T_\varepsilon < \infty$. Hence $T_\varepsilon=\infty$ and $$u(n,t)\leq f(n)+\varepsilon$$ for all $t\geq 0$ and all $n \in I^c$. To conclude the proof, take the limit $\varepsilon\to0^{+}$. Proof of Theorem \[thm\_disc\] ------------------------------ ### Step 1: Zorn’s lemma {#step-1-zorns-lemma .unnumbered} Recall that we are working here in the two cases: (i) $d=1$ and $1\leq p \leq \infty$ or (ii) $d>1$ and $p=2$. Consider the following family of functions: $${\mathcal}{S} = \left\{ \begin{array}{cc} f:{\mathbb{Z}}^d \to {\mathbb{R}};\\ \\ u_0(n) \leq f(n) \leq u^(n);& \ \ \forall n \in {\mathbb{Z}}^d;\\ \\ \\|\nabla f\\|_p \leq \\|\nabla u_0\\|_p\,.& \end{array} \right.$$ Note that ${\mathcal}{S}$ is non-empty since $u_0 \in {\mathcal}{S}$. We want to show ultimately that $u^* \in {\mathcal}{S}$. We put a partial order $\preceq$ in ${\mathcal}{S}$ by considering the pointwise order (i.e. $f \preceq g$ in ${\mathcal}{S}$ if and only if $f(n) \leq g(n)$ for all $n \in {\mathbb{Z}}^d$). Let us prove that $({\mathcal}{S,\preceq)}$ is inductive, i.e. every totally ordered subset has an upper bound in ${\mathcal}{S}$. Let $\{f_{\alpha}\}_{\alpha \in \Lambda}$ be a totally ordered subset and define $$\overline{f}(n) = \sup_{\alpha \in \Lambda}f_{\alpha}(n).$$ We claim that $\overline{f} \in {\mathcal}{S}$. It is clear that $u_0(n) \leq \overline{f}(n) \leq u^(n)$ for all $n \in {\mathbb{Z}}^d$. Let $J \subset {\mathbb{Z}}^d$ be a [finite]{} set and define $\widetilde{J} = \{n \in {\mathbb{Z}}^d;\ {\rm dist}(n,J) \leq 1\}$ (this distance is taken with respect to the $\\|\cdot\\|_1$-norm). There exists a sequence $\{f_k\}_{k \in {\mathbb{N}}} \subset \{f_{\alpha}\}_{\alpha \in \Lambda}$ such that $$\lim_{k \to \infty} f_k(n) = \overline{f}(n)$$ for all $n \in \widetilde{J}$. Thus, $$\sum_{n \in J} \big\|\nabla \overline{f}(n)\big\|^p = \lim_{k \to \infty} \sum_{n \in J} \|\nabla f_k(n)\|^p \leq \limsup_{k \to \infty} \\|\nabla f_k\\|_p^p \leq \\|\nabla u_0\\|_p^p.$$ Since this holds for any finite set $J$, we must have $$\big\\|\nabla \overline{f}\big\\|_p \leq \\|\nabla u_0\\|_p,$$ and thus $\overline{f} \in {\mathcal}{S}$. By Zorn’s lemma $({\mathcal}{S}, \preceq)$ has (at least) one maximal element, which we call $g$. ### Step 2: Conclusion {#step-2-conclusion .unnumbered} We now claim that $g = u^$. Suppose this is not true and let $I = \{n \in {\mathbb{Z}}^d;\ g(n) = u^(n)\}$. By Lemma \[lem3\], we know that $g$ cannot be superharmonic on $I^c$, and thus there is a point $n \in I^c$ such that $\Delta g(n) >0$. We first deal with the case $d \geq 1$ and $p=2$. Consider the function $$q_n(x) = \sum_{\\|m-n\\|_1=1} (g(m) - x)^2.$$ It is easy to see that $x \mapsto q_n(x)$ is a strictly convex function with its unique minimizer $x = x_n$ given by $$x_n = \frac{1}{2d} \sum_{\\|m-n\\|_1=1} g(m) = g(n) + \Delta g (n).$$ Therefore, if $\Delta g(n) >0$, we can consider the function $$\widetilde{g}(m) = \left\{ \begin{array}{lr} g(m),& \ \ {\rm if} \ \ m \neq n;\\ \\ \min\big\{u^(n), \,g(n) + \Delta g(n)\big\},& \ \ {\rm if}\ m =n. \end{array} \right.$$ Then $g \leq \widetilde{g} \leq u^$ pointwise and, since $g(n) < \widetilde{g}(n) \leq x_n$, the strict convexity of $q_n$ gives us $$q_n(g(n)) > q_n\big(\widetilde{g}(n)\big) \geq q_n(x_n).$$ This plainly implies that $$\\|\nabla g\\|_2^2 = \big\\|\nabla \widetilde{g}\big\\|_2^2 + q_n(g(n)) - q_n\big(\widetilde{g}(n)\big) > \big\\|\nabla \widetilde{g}\big\\|_2^2.$$ Thus $\widetilde{g} \in {\mathcal}{S}$ and this contradicts the maximality of $g$. Now we deal with the case $d=1$ and $1\leq p \leq \infty$. The idea is the same as above. If $1\leq p < \infty$ we simply observe that the function (for fixed $a,b \in {\mathbb{R}}$) $$q(x) = \|a-x\|^p + \|b-x\|^p$$ is convex (strictly convex if $1<p$) with minimizer $x = (a+b)/2$. If $p=\infty$ we note that $$q(x) = \max\{\|a-x\|, \|b-x\|\}$$ is also convex with minimizer $x = (a+b)/2$. This concludes the proof. Proof of Theorem \[thm\_disc\_Poisson\] - Discrete Poisson kernel ================================================================= In this section we keep, without loss of generality, the assumption $u_0 \geq0$ (since $\\|\nabla \|u_0\|\\|_p \leq \\|\nabla u_0\\|_p$ for any $1\leq p \leq \infty$). Preliminaries ------------- We start by proving the analogous statement to Lemma \[lem3\] for the discrete Poisson maximal function. \[lem7\_Poisson\_b\] Let $u_0 \in l^{\infty}({\mathbb{Z}}^d)$ and $u^{\star}$ be its discrete Poisson maximal function. Let $f \in l^{\infty}({\mathbb{Z}}^d)$ be such that $f \geq u_0$. Let $I = \{n \in {\mathbb{Z}}^d;\, u^{\star}(n)\leq f(n)\}$ and suppose that $\Delta f(n) \leq 0$ for all $n \in I^c$. Then $I = {\mathbb{Z}}^d$. For $y\in{\mathbb{Z}}^+$ we define $$\beta (y)=\sup_{n\in{\mathbb{Z}}^d} (u(n,y)-f(n))_+ \,,$$ where $t_+ := \max\{t,0\}$. Observe that $\beta(0)=0$ and that, for any $y$, we have $$0\leq\beta(y)\leq\\|u^{\star}\\|_\infty+\\|f\\|_\infty <\infty$$ and $$\label{pf_lem6_new_eq1} u(\cdot,y)\leq f+\beta(y).$$ Suppose that $I^c\neq\emptyset$. In this case, $$\beta(y)=\sup_{n\in I^c} \;( u(n,y)-f(n))_+.$$ Take $n\in I^c$ and $y>0$. Using the fact that $u$ is harmonic in the discrete upper half-space, together with and the hypothesis that $\Delta f(n) \leq 0$ for $n \in I^c$, we find $$\begin{aligned} 2(d+1)u(n,y)&=u(n,y-1)+u(n,y+1)+\sum_{\|\|m-n\|\|_1=1}u(m,y)\\ &\leq 2 f(n)+\beta(y-1) + \beta(y+1)+\sum_{\|\|m-n\|\|_1=1}\big(f(m)+\beta(y)\big)\\ &\leq 2(d+1)f(n)+\beta(y-1)+\beta(y+1)+2d \,\beta(y). \end{aligned}$$ Therefore $$\label{pf_lem6_new_eq2} 2(d+1)\big(u(n,y)-f(n)\big)\leq\beta(y-1)+\beta(y+1)+2d\, \beta(y).$$ Taking the supremum over $n\in I^c$ on the left hand side of we find $$\beta(y)\leq \dfrac{\beta(y-1)+\beta(y+1)}{2}$$ for any $y >0$. Since the sequence $y \mapsto \beta(y)$ is bounded and $\beta(0)=0$, we must have $\beta(y)=0$ for all $y>0$. Therefore $I^c$ would be empty, in contradiction with the original assumption $I^c\neq\emptyset$. Proof of Theorem \[thm\_disc\_Poisson\] --------------------------------------- Once we have established Lemma \[lem7\_Poisson\_b\], the proof of Theorem \[thm\_disc\_Poisson\] plainly follows by the argument based on Zorn’s lemma used in the proof of Theorem \[thm\_disc\] for the discrete heat flow maximal function. We will omit the details. Proof of Theorem \[thm\_cont\] - Continuous heat kernel ======================================================= Preliminaries ------------- We begin this section with a selection of lemmas that will be helpful as we move on to the proof. Throughout this section we assume without loss of generality that $u_0 \geq0$, for if $u_0 \in W^{1,p}({\mathbb{R}}^d)$ we have $\|u_0\| \in W^{1,p}({\mathbb{R}}^d)$ and $\|\nabla \|u_0\|\| = \|\nabla u_0\|$ a.e. if $u_0$ is real-valued (in the general case $u_0$ complex-valued we have $\|\nabla \|u_0\|\| \leq \|\nabla u_0\|$ a.e), and if $u_0$ is of bounded variation on ${\mathbb{R}}$ we have $V(\|u_0\|) \leq V(u_0)$. In what follows we write $${\rm Lip}(u) = \sup_{\stackrel{x,y \in {\mathbb{R}}^d}{x\neq y}} \frac{\|u(x) - u(y)\|}{\|x-y\|}$$ for the Lipschitz constant of a function $u:{\mathbb{R}}^d \to {\mathbb{R}}$. \[lem4\]. - If $u_0 \in C({\mathbb{R}}^d) \cap L^p({\mathbb{R}}^d)$, for some $1\leq p < \infty$, then $u^ \in C({\mathbb{R}}^d)$. - If $u_0$ is bounded and Lipschitz continuous then $u^$ is bounded and Lipschitz continuous with ${\rm Lip}(u^) \leq {\rm Lip}(u_0)$. \(i) Recall that $$u^(x) = \sup_{t >0} (u_0K_t)(x)$$ with the heat kernel $K_t$ defined in . Let us denote here $\tau_hu_0(x):= u_0(x- h)$. Given $\varepsilon >0$, there is a time $t_{\varepsilon} < \infty$ such that $$\begin{aligned} \|\tau_h u_0 - u_0\|K_t(x)& \leq \left(\|\tau_h u_0 - u_0\|^pK_t(x)\right)^{1/p}\\ & \leq \left(\\|\tau_h u_0 - u_0\\|_p^p \ \\|K_t\\|_{\infty}\right)^{1/p} = \frac{\\|\tau_h u_0 - u_0\\|_p}{(4\pi t)^{d/2p}}\\ & \leq \frac{2\\|u_0\\|_p}{(4\pi t)^{d/2p}} < \varepsilon\end{aligned}$$ whenever $t > t_{\varepsilon}$, for all $x, h \in {\mathbb{R}}^d$. Note that we used Jensen’s inequality in the first line above and Young’s inequality on the second line. On the other hand, given $x \in {\mathbb{R}}^d$ and if $0 < t \leq t_{\varepsilon}$, we can choose $\delta >0$ such that $$\begin{aligned} & \|\tau_h u_0 - u_0\|K_t(x) \\ &= \int_{\|y\| < \sqrt{ t_{\varepsilon}}} \|\tau_h u_0 - u_0\|(x-y)\,K_t(y)\,{\text{\rm d}y}+ \int_{\|y\| \geq \sqrt{t_{\varepsilon}}} \|\tau_h u_0 - u_0\|(x-y)\,K_t(y)\,{\text{\rm d}y}\\ & \leq \sup_{w \in B_{\sqrt{t_{\varepsilon}}}(x)}\|\tau_h u_0 - u_0\|(w) + \\|\tau_h u_0 - u_0\\|_p\,\\|\chi_{\{\|y\|\geq \sqrt{t_{\varepsilon}}\}} \,K_t\\|_{p'}< \varepsilon\end{aligned}$$ whenever $\|h\| < \delta$, where we have used the fact that $\\|\chi_{\{\|y\|\geq \sqrt{t_{\varepsilon}}\}}\, K_t\\|_{p'}$ is bounded for $0 < t \leq t_{\varepsilon}$. Using the sublinearity, we then arrive at $$\big\|\tau_h u^(x) - u^(x)\big\| \leq (\tau_h u_0 - u_0)^(x) \leq \varepsilon$$ for $\|h\| < \delta$, which proves that $u^$ is continuous at $x$. \(ii) It is easy to check that if $u_0$ is bounded by $M$ and has Lipschitz constant $L$, then for each time $t>0$ the function $u_0 K_t$ is also bounded by $M$ and admits the same Lipschitz constant $L$. In this case, the pointwise supremum of uniformly Lipschitz functions is still Lipschitz with (at most) the same constant. We will say here that a continuous function $f$ is [subharmonic]{} in an open set $A$ if, for every $x \in A$, and every ball $\overline{B_r(x)} \subset A$ we have $$f(x) \leq \frac{1}{\sigma_{d-1}}\int_{S^{d-1}} f(x +r\xi)\,{\rm d}\sigma(\xi),$$ where $\sigma_{d-1}$ denotes the surface area of the unit sphere $S^{d-1}$, and ${\rm d}\sigma$ is its surface measure. Here $B_r(x)$ denotes the open ball of radius $r$ and center $x$, and $\overline{B_r(x)}$ denotes the corresponding closed ball. \[lem5\] Let $u_0 \in C({\mathbb{R}}^d) \cap L^p({\mathbb{R}}^d)$ for some $1\leq p < \infty$ or $u_0$ be bounded and Lipschitz continuous. Then $u^$ is subharmonic in the open set $A = \{x \in {\mathbb{R}}^d; \,u^(x) > u_0(x)\}$. Note that by Lemma \[lem4\] we have $u^$ continuous and thus the set $A = \{x \in {\mathbb{R}}^d; \,u^(x) > u_0(x)\}$ is in fact open. Take $x_0 \in A$ and a radius $r>0$ such that $\overline{B_r(x_0)} \subset A$. Let $h:\overline{B_r(x_0)} \to {\mathbb{R}}$ be the solution of the Dirichlet boundary value problem $$\left\{ \begin{array}{rl} \Delta h = 0& \ {\rm in} \ B_r(x_0);\\ h = u^& \ {\rm in} \ \partial B_r(x_0). \end{array} \right.$$ Since $u^$ is a continuous function, this problem does admit a unique solution $h \in C^{2}(B_r(x_0)) \cap C\big(\overline{B_r(x_0)}\big)$. Now let $T>0$ and $\Omega = B_r(x_0) \times(0,T)$. Observe that $v(x,t) := u(x,t) - h(x) \in C^2(\Omega)\cap C\big(\overline{\Omega}\big)$ and solves the heat equation in $\Omega$. By the maximum principle for the heat equation, the maximum of $v$ in $\overline{\Omega}$ must be attained either in $\partial B_r(x_0) \times [0,T]$ or in $\overline{B_r(x_0)} \times \{t=0\}$. By construction, note that $$\max_{\partial B_r(x_0) \times [0,T]} v(x,t) = \max_{\partial B_r(x_0) \times [0,T]} u(x,t) - u^(x) \leq 0.$$ Let $y_0$ be such that $$\max_{\overline{B_r(x_0)}} v(x,0) = v(y_0,0).$$ We claim that $v(y_0,0) \leq 0$. In fact, let us suppose that $v(y_0,0) > 0$. Then, by the maximum principle, $v(y_0,t) \leq v(y_0,0)$ for any $0\leq t \leq T$, which in turn implies that $u(y_0,t) \leq u_0(y_0)$ for any $0\leq t \leq T$. Since $T$ is arbitrary, we would have $u^(y_0) = u_0(y_0)$ and thus $y_0 \notin A$, contradiction. Therefore $$\max_{\overline{B_r(x_0)}} v(x,0)\leq 0,$$ which plainly gives $$u(x_0, t) \leq h(x_0)$$ for any $0\leq t \leq T$. As $T$ is arbitrary we conclude that $$u^(x_0) \leq h(x_0),$$ which is the desired result since $h$ is harmonic and thus equal to its average over the sphere $\partial B_r(x_0)$, where $h = u^$ by construction. The next lemma will be important in the proof of the theorem for $p=2$ and $d\geq 1$. \[lem6\] Let $f, g \in C({\mathbb{R}}^d) \cap W^{1,2}({\mathbb{R}}^d)$ with $g$ Lipschitz. Suppose that $g \geq 0$ and that $f$ is subharmonic in the open set $J = \{x\in {\mathbb{R}}^d; \ g(x) >0\}$. Then $$\int_{{\mathbb{R}}^d} \nabla f\,.\,\nabla g\,\,{\text{\rm d}x}\leq 0.$$ Formally, the identity $$\int_{{\mathbb{R}}^d} \nabla f\,.\,\nabla g\,\,{\text{\rm d}x}= \int_{{\mathbb{R}}^d} (-\Delta f)\,g\,{\text{\rm d}x}$$ would imply the result, since $f$ subharmonic in the set $\{g>0\}$ would mean $-\Delta f \leq 0$ in this set. This justifies the intuition for the result. Our work here is to make this argument rigorous. Our first claim is that we can assume without loss of generality that $g$ has compact support. To see this let $\Psi \in C^{\infty}_c({\mathbb{R}}^d)$ be a non-negative function such that $\Psi(x) \equiv 1$ in $B_1$ and $\operatorname{\mathrm{supp}}(\Psi) \subset B_2$. For $N \in {\mathbb{N}}$, let $\Psi_N(x) := \Psi(x/N)$ and consider $g_N(x) := g(x)\,\Psi_N(x)$. If we could prove the result for each $g_N$ (note that $f$ is subharmonic in the set $\{g_N>0\}$ and each $g_N$ is Lipschitz since $g$ is bounded), since $g_N \to g$ in $W^{1,2}({\mathbb{R}}^d)$ we would obtain $$\int_{{\mathbb{R}}^d} \nabla f\,.\,\nabla g\,\,{\text{\rm d}x}= \lim_{N\to \infty} \int_{{\mathbb{R}}^d} \nabla f\,.\,\nabla g_N\,\,{\text{\rm d}x}\leq 0.$$ From now on we assume that $\operatorname{\mathrm{supp}}(g) \subset B_R$ for some $R>0$. Consider a non-negative function $\varphi \in C^{\infty}_c({\mathbb{R}}^d)$ with support on the unit ball $B_1$ and integral $1$. For $\varepsilon >0$ put $\varphi_{\varepsilon}(x) = \varepsilon^{-d}\varphi(x/\varepsilon)$ and write $$f_{\varepsilon} = f \varphi_{\varepsilon}.$$ We see that $f_{\varepsilon} \in C^{\infty}({\mathbb{R}}^d)$ and it is not hard to check that $$\partial_{x_i} f_{\varepsilon} = (\partial_{x_i} f) \varphi_{\varepsilon} = f(\partial_{x_i}\varphi_{\varepsilon}),$$ and $$\label{lem5_laplace} \partial_{x_ix_i} f_{\varepsilon} = (\partial_{x_i} f) (\partial_{x_i} \varphi_{\varepsilon}) = f(\partial_{x_ix_i}\varphi_{\varepsilon}).$$ Let us define the set $J_{\varepsilon} = \{x \in J;\ {\rm dist}(x,\partial J) >\varepsilon\}$. A simple computation shows that $f_{\varepsilon}$ is subharmonic on $J_{\varepsilon}$. In fact, if $x \in J_{\varepsilon}$ and $\overline{B_r(x)}\subset J_{\varepsilon}$, we have $$\begin{aligned} f_{\varepsilon}(x) &= \int_{B_{\varepsilon}} f(x - y) \, \varphi_{\varepsilon}(y)\,{\text{\rm d}y}\\ & \leq \int_{B_{\varepsilon}} \frac{1}{\sigma_{d-1}}\int_{S^{d-1}} f(x -y +r\xi)\,{\rm d}\sigma(\xi) \,\varphi_{\varepsilon}(y)\,{\text{\rm d}y}\\ & = \frac{1}{\sigma_{d-1}}\int_{S^{d-1}} f_{\varepsilon} (x +r\xi)\,{\rm d}\sigma(\xi).\end{aligned}$$ Since $f_{\varepsilon} \in C^{\infty}({\mathbb{R}}^d)$, this implies that $(-\Delta f_{\varepsilon})(x) \leq 0$ for $x \in J_{\varepsilon}$. For $\varepsilon >0$ and $\psi \in C^{\infty}_c({\mathbb{R}}^d)$ we can apply integration by parts to get $$\int_{{\mathbb{R}}^d} \nabla f_{\varepsilon}\,.\,\nabla \psi\,\,{\text{\rm d}x}= \int_{{\mathbb{R}}^d} (-\Delta f_{\varepsilon})\,\psi\,{\text{\rm d}x}.$$ Now since $\|\nabla f_{\varepsilon}\| \in L^2({\mathbb{R}}^d)$ and $\Delta f_{\varepsilon} \in L^2({\mathbb{R}}^d)$, we might approximate our function $g \in W^{1,2}({\mathbb{R}}^d)$ by such $\psi \in C^{\infty}_c({\mathbb{R}}^d)$ (in the $W^{1,2}$-norm) to obtain $$\begin{aligned} \label{lem5_limit} \begin{split} \int_{{\mathbb{R}}^d} \nabla f_{\varepsilon}\,.\,\nabla g\,\,{\text{\rm d}x}& = \int_{{\mathbb{R}}^d} (-\Delta f_{\varepsilon})\,g\,{\text{\rm d}x}\\ & = \int_{J \setminus J_{\varepsilon}} (-\Delta f_{\varepsilon})\,g\,{\text{\rm d}x}+ \int_{J_{\varepsilon}} (-\Delta f_{\varepsilon})\,g\,{\text{\rm d}x}\\ & \leq \int_{J\setminus J_{\varepsilon}} (-\Delta f_{\varepsilon})\,g\,{\text{\rm d}x}. \end{split}\end{aligned}$$ Let $C$ be the Lipschitz constant of $g$. Then, for any $x \in J\setminus J_{\varepsilon}$ we have $$\label{lem5_eq2} \|g(x)\| \leq C \varepsilon.$$ From we observe that $$\label{lem5_eq3} \partial_{x_ix_i} f_{\varepsilon} = (\partial_{x_i} f) \left(\varepsilon^{-1} (\partial_{x_i} \varphi)_{\varepsilon}\right),$$ where $(\partial_{x_i} \varphi)_{\varepsilon}(x) = \varepsilon^{-d} (\partial_{x_i} \varphi)(x/\varepsilon)$. From and , using Hölder’s inequality and Young’s inequality we get $$\begin{aligned} \label{lem5_limit2} \begin{split} \int_{J \setminus J_{\varepsilon}} &\big\|(-\Delta f_{\varepsilon})\,g\big\|\,{\text{\rm d}x}\leq C \int_{J \setminus J_{\varepsilon}} \sum_{i=1}^d \big\|(\partial_{x_i} f) (\partial_{x_i}\varphi)_{\varepsilon} \big\|\,{\text{\rm d}x}\\ & \leq C\,d \int_{J \setminus J_{\varepsilon}} \|\nabla f\|(\|\nabla\varphi\|)_{\varepsilon} \,\,{\text{\rm d}x}\\ & \leq C\,d \left(\int_{J \setminus J_{\varepsilon}} \big\| \|\nabla f\|(\|\nabla\varphi\|)_{\varepsilon}\big\|^2 \,\,{\text{\rm d}x}\right)^{1/2} \, m(J \setminus J_{\varepsilon})^{1/2}\\ & \leq C\,d \ \\|\nabla f\\|_2 \ \\|\nabla\varphi\\|_1 \ m(J \setminus J_{\varepsilon})^{1/2}. \end{split}\end{aligned}$$ Since $\operatorname{\mathrm{supp}}(g) \subset B_R$, we know that $m(J \setminus J_{\varepsilon}) \to 0$ as $\epsilon \to 0$. Finally, since $f_{\varepsilon} \to f$ in $W^{1,2}({\mathbb{R}}^d)$ we use and to get $$\begin{aligned} \int_{{\mathbb{R}}^d} \nabla f\,.\,\nabla g\,\,{\text{\rm d}x}& = \lim_{\varepsilon \to 0} \int_{{\mathbb{R}}^d} \nabla f_{\varepsilon}\,.\,\nabla g\,\,{\text{\rm d}x}\\ & \leq \lim_{\varepsilon \to 0} \int_{J \setminus J_{\varepsilon}} (-\Delta f_{\varepsilon})\,g\,{\text{\rm d}x}= 0.\end{aligned}$$ \[lem7\] In order to establish Theorem \[thm\_cont\] - parts (i) and (iv) - it suffices to consider the initial datum $u_0$ Lipschitz. If $p=\infty$, recall that a function $u_0 \in W^{1,\infty}({\mathbb{R}}^d)$ can be modified on a set of measure zero to become bounded and Lipschitz continuous. If $1 < p <\infty$, we take $\varepsilon >0$ and consider $$u_{\varepsilon}(x) = u_0K_{\varepsilon}(x).$$ It is clear that $u_{\varepsilon}$ is Lipschitz continuous. Suppose that the result is true for $u_{\varepsilon}$, i.e. that $u_{\varepsilon}^\in W^{1,p}({\mathbb{R}}^d)$ and $$\label{lem7_eq1} \\|\nabla u_{\varepsilon}^\\|_p \leq \\|\nabla u_{\varepsilon}\\|_p\,,$$ where $$u_{\varepsilon}^(x) = \sup_{\tau > 0} u_{\varepsilon}K_{\tau}(x) = \sup_{t > \varepsilon} u_0K_t(x).$$ From Young’s inequality we have $$\label{lem7_eq2} \\|u_{\varepsilon}\\|_p \leq \\|u_0\\|_p$$ and, together with Minkowski’s inequality, we also have $$\label{lem7_eq3} \\|\nabla u_{\varepsilon}\\|_p \leq \\|\nabla u_0\\|_p$$ for any $\varepsilon >0$. From , and we find that $u_{\varepsilon}^$ is uniformly bounded in $W^{1,p}({\mathbb{R}}^d)$. Note that $u_{\varepsilon}^$ converges pointwise to $u^$ as $\epsilon \to 0$. From the weak compactness of $W^{1,p}({\mathbb{R}}^d)$ we then conclude that $u^* \in W^{1,p}({\mathbb{R}}^d)$ and $u_{\varepsilon}^* \rightharpoonup u^$ as $\varepsilon \to 0$. Therefore $$\\|\nabla u^\\|_p \leq \liminf_{\varepsilon \to 0} \\|\nabla u_{\varepsilon}^\\|_p \leq \liminf_{\varepsilon \to 0} \\|\nabla u_{\varepsilon}\\|_p \leq \\|\nabla u_0\\|_p\,,$$ which completes the proof. Proof of part (iv) ------------------ In the case $p=\infty$, we know that $u_0 \in W^{1,\infty}({\mathbb{R}}^d)$ can be modified on a set of measure zero to become Lipschitz continuous with ${\rm Lip}(u_0) \leq \\|\nabla u_0\\|_{\infty}$. From Lemma \[lem4\], $u^$ will also be bounded and Lipschitz continuous, with ${\rm Lip}(u^) \leq {\rm Lip}(u_0)$, and the result follows, since in this case $u^ \in W^{1,\infty}({\mathbb{R}}^d)$ with $\\|\nabla u^\\|_{\infty} \leq {\rm Lip}(u^)$. If $p=2$, we are essentially done as well. In fact, from Lemma \[lem7\] it suffices to consider the case $u_0$ Lipschitz continuous in $W^{1,2}({\mathbb{R}}^d)$. In this case, from Lemma \[lem4\] we know that $u^$ is also Lipschitz continuous, and from Lemma \[lem5\] we have that $u^$ is subharmonic in the open set $A = \{x \in {\mathbb{R}}^d; \,u^(x) > u_0(x)\}$. Recall from our discussion in the introduction of the paper that we already have $u^ \in W^{1,2}({\mathbb{R}}^d)$, and thus the hypotheses of Lemma \[lem6\] apply to $f = u^$ and $g = (u^ - u_0)$. Therefore, $$\begin{aligned} \\|\nabla u_0\\|_2^2 &= \int_{{\mathbb{R}}^d} \|\nabla u_0\|^2\,{\text{\rm d}x}= \int_{{\mathbb{R}}^d} \|\nabla u^* - \nabla (u^* - u_0)\|^2\,{\text{\rm d}x}\\ & = \int_{{\mathbb{R}}^d} \|\nabla (u^-u_0)\|^2\,{\text{\rm d}x}- 2 \int_{{\mathbb{R}}^d} \nabla u^\,.\,\nabla (u^-u_0)\,{\text{\rm d}x}+ \int_{{\mathbb{R}}^d} \|\nabla u^\|^2\,{\text{\rm d}x}\\ & \geq \int_{{\mathbb{R}}^d} \|\nabla u^\|^2\,{\text{\rm d}x}= \\|\nabla u^\\|_2^2\,,\end{aligned}$$ which concludes the proof. Proof of part (i) - case $1< p < \infty$ ---------------------------------------- ### Step 1: Set up {#step-1-set-up .unnumbered} The initial considerations are the same as before. From Lemma \[lem7\] it suffices to consider the case $u_0$ Lipschitz continuous in $W^{1,p}({\mathbb{R}})$. From Lemma \[lem4\] we know that $u^$ is also Lipschitz, and from Lemma \[lem5\] we have that $u^$ is subharmonic in the open set $A = \{x \in {\mathbb{R}}; \,u^(x) > u_0(x)\}$. Let us explore the structure of ${\mathbb{R}}$ to write the open set $A$ as a countable union of disjoint open intervals $$A = \bigcup_{j} I_j = \bigcup_{j} \, (\alpha_j, \beta_j).$$ Note also that subharmonicity is equivalent to convexity in each $(\alpha_j, \beta_j)$, when we deal with continuous functions. ### Step 2: Zorn’s lemma {#step-2-zorns-lemma .unnumbered} As we did in the discrete case, we use here an argument based on Zorn’s lemma. Define the family of Lipschitz continuous functions $${\mathcal}{S} = \left\{ \begin{array}{cc} f:{\mathbb{R}}\to {\mathbb{R}};\\ \\ u_0(x) \leq f(x) \leq u^(x);& \ \ \forall x \in {\mathbb{R}};\\ \\ {\rm Lip}(f) \leq {\rm Lip}(u_0);&\\ \\ \\|f'\\|_p \leq \\|u_0'\\|_p;& \end{array} \right.$$ The family ${\mathcal}{S}$ is non-empty since $u_0 \in {\mathcal}{S}$. We put a partial order $\preceq$ in ${\mathcal}{S}$ by considering the pointwise order (i.e. $f \preceq g$ in ${\mathcal}{S}$ if and only if $f(x) \leq g(x)$ for all $x \in {\mathbb{R}}$). Let us prove that $({\mathcal}{S},\preceq)$ is inductive. Let $\{f_{\alpha}\}_{\alpha \in \Lambda}$ be a totally ordered subset and define $$\overline{f}(x) = \sup_{\alpha \in \Lambda}f_{\alpha}(x).$$ We claim that $\overline{f} \in {\mathcal}{S}$. Being a pointwise supremum of uniformly Lipschitz functions, we have ${\rm Lip}(\overline{f})\leq {\rm Lip}(u_0)$. Let us write $L = {\rm Lip}(u_0)$. For each $N \in {\mathbb{N}}$ consider the $2N^2 +1$ points $\{j/N\}$ with $-N^2 \leq j \leq N^2$, $j \in {\mathbb{Z}}$. For each of these $j's$ choose a function $f_{j,N} \in \Lambda$ such that $\overline{f}(j/N) - f_{j,N}(j/N) < \frac{1}{N}$. Then choose $$f_N = \sup_{-N^2 \leq j \leq N^2} f_{j,N}.$$ It is clear that for each $x \in [-N,N]$, since $\|x-j/N\| \leq 1/2N$ for some $j$, we have $$\overline{f}(x) - f_N(x) \leq \frac{1}{N} + \frac{L}{N}.$$ Therefore $f_N \to \overline{f}$ pointwise as $N\to \infty$. From the conditions on the family ${\mathcal}{S}$ we know that $\\|f_N\\|_{W^{1,p}}$ is uniformly bounded, and from the weak compactness of $W^{1,p}({\mathbb{R}})$ we must have $\overline{f} \in W^{1,p}({\mathbb{R}})$ and $f_N \rightharpoonup \overline{f}$. We then arrive at the bound $$\big\\|\big(\overline{f}\big)'\big\\|_p \leq \liminf_{N\to \infty} \\|f_N'\\|_p \leq \\|u_0'\\|_p\,,$$ which shows that $\overline{f} \in {\mathcal}{S}$. From Zorn’s lemma we guarantee the existence of (at least) one maximal element in $({\mathcal}{S}, \preceq)$, which we call $g$. ### Step 3: Finding an appropriate segment to cut {#step-3-finding-an-appropriate-segment-to-cut .unnumbered} We want to show that $g = u^$. Suppose this is not the case, i.e. that the open set $B = \{x \in {\mathbb{R}}; \,u^(x) > g(x)\} \subset A$ is non-empty. Let us write $B$ as a countable union $$B= \bigcup_{l} Q_l = \bigcup_{l} \, (\gamma_l, \delta_l).$$ We claim that $g$ cannot be superharmonic on $B$. In fact, if one of the intervals $(\gamma_l, \delta_l)$ is finite, since $u^(\gamma_l) = g(\gamma_l)$ and $u^(\delta_l) = g(\delta_l)$, the maximum principle would give us $u^* \equiv g$ in $[\gamma_l, \delta_l]$, a contradiction. If the interval is of type $(\gamma_l, \infty)$ (resp. $(-\infty, \delta_l)$), we would have $(u^* - g)$ strictly positive and convex in $(\gamma_l, \infty)$, with $(u^* - g)(\gamma_l)=0$. This is a contradiction since $(u^* - g)$ is Lipschitz and belongs to $L^p({\mathbb{R}})$, and thus must tend to zero at infinity. To conclude the proof of the claim, note that we cannot have $B = (-\infty,\infty)$, since $u_0(x_0) = g(x_0) = u^(x_0)$ at the global maximum $x_0$ of $u_0$. Therefore, there exists an interval $[a,b] \subset B$ such that $$\label{pf_thm1_eq1} g\left(\frac{a+b}{2}\right) < \frac{g(a) + g(b)}{2}.$$ Let $\ell(x)$ be the equation of the line connecting the points $(a, g(a))$ and $(b,g(b))$, i.e. $$l(x) = \frac{g(b) - g(a)}{b-a}(x-a) + g(a).$$ Let us consider the functions $\widetilde{u^}(x) := u^(x) - \ell(x)$ and $\widetilde{g}(x) := g(x) - \ell(x)$. Let $y_0$ be the point of minimum of $\widetilde{g}$ when restricted to $[a,b]$. From we have $\widetilde{g}(y_0) \leq \widetilde{g}((a+b)/2) < 0$. We claim that there exists a line $\widetilde{\ell}$ parallel to the $x$-axis such that the graph of $\widetilde{u^}$ is above $\widetilde{\ell}$ and the graph of $\widetilde{g}$ is below $\widetilde{\ell}$ in a neighborhood of $y_0$. To see this start by noting that $\widetilde{u^}(y_0) - \widetilde{g}(y_0) = C >0$. For each $- \widetilde{g}(y_0) > \varepsilon >0$ let $$a_{\varepsilon} = \max\{a \leq x \leq y_0;\ \widetilde{g}(x) \geq \widetilde{g}(y_0) + \varepsilon\}$$ and $$b_{\varepsilon} = \min\{y_0 \leq x \leq b;\ \widetilde{g}(x) \geq \widetilde{g}(y_0) + \varepsilon\}.$$ From this we have $\widetilde{g}(x) \leq \widetilde{g}(y_0) + \varepsilon$, for each $x \in [a_{\varepsilon}, b_{\varepsilon}]$, with equality on the endpoints. Suppose that for each $\varepsilon >0$ there exists a point $z_{\varepsilon} \in [a_{\varepsilon}, b_{\varepsilon}]$ such that $\widetilde{u^}(z_{\varepsilon}) < \widetilde{g}(y_0) + \varepsilon$. There will be a subsequence of $\{z_{\varepsilon}\}_{\varepsilon>0}$ that accumulates around a certain $z_0 \in [a,b]$ thus giving $$\widetilde{u^}(z_0) \leq \widetilde{g}(y_0) \leq \widetilde{g}(z_0) < \widetilde{u^}(z_0),$$ a contradiction. Therefore, we can find an $\varepsilon>0$ such that $$\widetilde{u^}(x) \geq \widetilde{g}(y_0) + \varepsilon$$ for each $x \in [a_{\varepsilon}, b_{\varepsilon}]$. If we undo the $\thicksim$ operation and return to the original picture, we have found a finite interval $[a_{\varepsilon}, b_{\varepsilon}]$ such that $g$ is below the line connecting $(a_{\varepsilon}, g(a_{\varepsilon}))$ to $(b_{\varepsilon}, g(b_{\varepsilon}))$ in $[a_{\varepsilon}, b_{\varepsilon}]$ (being strictly below in $(a_{\varepsilon}, b_{\varepsilon})$) and $u^$ is above this line. ### Step 4: Conclusion {#step-4-conclusion .unnumbered} Let us define a new function $$h(x) = \left\{ \begin{array}{lr} g(x)\,,&{\rm if}\ x \notin [a_{\varepsilon}, b_{\varepsilon}]\\ \\ \displaystyle\frac{g(b_{\varepsilon}) - g(a_{\varepsilon})}{b_{\varepsilon}-a_{\varepsilon}}(x-a_{\varepsilon}) + g(a_{\varepsilon})\,,&{\rm if}\ x \in [a_{\varepsilon}, b_{\varepsilon}]. \end{array} \right.$$ From the previous step we know that $u_0 \leq h \leq u^$ and it is also clear that ${\rm Lip}(h) \leq {\rm Lip}(g) \leq {\rm Lip}(u_0)$. Finally, by Jensen’s inequality we obtain $$\begin{aligned} \\|g'\\|_p^p & = \int_{[a_{\varepsilon}, b_{\varepsilon}]^c} \|g'(x)\|^p\,{\text{\rm d}x}+ \int_{[a_{\varepsilon}, b_{\varepsilon}]} \|g'(x)\|^p\,{\text{\rm d}x}\\ & \geq \int_{[a_{\varepsilon}, b_{\varepsilon}]^c} \|g'(x)\|^p\,{\text{\rm d}x}+ (b_{\varepsilon}-a_{\varepsilon}) \left(\int_{a_{\varepsilon}}^{b_{\varepsilon}} \|g'(x)\| \frac{{\text{\rm d}x}}{(b_{\varepsilon}-a_{\varepsilon})}\right)^p\\ & \geq \int_{[a_{\varepsilon}, b_{\varepsilon}]^c} \|g'(x)\|^p\,{\text{\rm d}x}+ (b_{\varepsilon}-a_{\varepsilon}) \left\|\int_{a_{\varepsilon}}^{b_{\varepsilon}} g'(x) \frac{{\text{\rm d}x}}{(b_{\varepsilon}-a_{\varepsilon})}\right\|^p\\ & = \int_{[a_{\varepsilon}, b_{\varepsilon}]^c} \|g'(x)\|^p\,{\text{\rm d}x}+ (b_{\varepsilon}-a_{\varepsilon}) \left\| \frac{g(b_{\varepsilon}) - g(a_{\varepsilon})}{(b_{\varepsilon}-a_{\varepsilon})}\right\|^p\\ & = \int_{{\mathbb{R}}} \|h'(x)\|^p\,{\text{\rm d}x}= \\|h'\\|_p^p.\end{aligned}$$ Therefore we get that $h \in {\mathcal}{S}$ but this is a contradiction since $h$ is strictly bigger than the maximal element $g$ in $(a_{\varepsilon}, b_{\varepsilon})$. This shows that $g = u^$ and the proof is concluded. Proof of part (ii) ------------------ The argument we shall use for this part is inspired in Tanaka’s [@Ta]. Recall that when $u_0 \in W^{1,1}({\mathbb{R}})$, after adjusting on a set of measure zero, $u_0$ may be taken to be absolutely continuous. From Lemma \[lem4\] we see that $u^$ is also continuous and the set $A = \{x \in {\mathbb{R}}; \ u^(x) > u_0(x)\}$ is open. Let us again write $A$ as a countable union of disjoint open intervals $$A = \bigcup_{j} I_j = \bigcup_{j} \, (\alpha_j, \beta_j).$$ From Lemma \[lem5\] we know that $u^$ is subharmonic (thus convex) in each subinterval $I_j = (\alpha_j, \beta_j)$. Therefore, $u^$ must be locally Lipschitz on each $I_j$ and, in particular, it is absolutely continuous on each compact subinterval of $I_j$. From this we conclude that $u^$ is differentiable a.e. on each $I_j$, with derivative that we will denote by $v$. We observe now that in each subinterval $I_j$ the variation of $u^$ is smaller than the variation of $u_0$. In fact, since $u^$ is convex on $I_j$, let $\gamma_j \in [\alpha_j, \beta_j]$ be a minimum of $u^$ on $[\alpha_j, \beta_j]$ (note that we might have $\gamma_j = \alpha_j$ or $\gamma_j = \beta_j$). The crucial observation is that $u^$ is monotone in $[\alpha_j, \gamma_j]$ and in $[\gamma_j, \beta_j]$, thus leading to (using the continuity of $u^$ and approaching by compacts from inside $I_j$) $$\begin{aligned} \label{part(ii)_eq3} \begin{split} \int_{I_j} \|v(x)\|\,{\text{\rm d}x}& = \big[u^(\alpha_j) - u^(\gamma_j)\big] + \big[u^(\beta_j) - u^(\gamma_j)\big]\\ & \leq \big[u_0(\alpha_j) - u_0(\gamma_j)\big] + \big[u_0(\beta_j) - u_0(\gamma_j)\big]\\ & \leq \int_{\alpha_j}^{\gamma_j} \|u_0'(x)\|\,{\text{\rm d}x}+ \int_{\gamma_j}^{\beta_j} \|u_0'(x)\|\,{\text{\rm d}x}\\ & = \int_{I_j} \|u_0'(x)\|\,{\text{\rm d}x}. \end{split}\end{aligned}$$ Note that in case $\alpha_j = -\infty$ (resp. $\beta_j = \infty$) we have $u_0(\alpha_j) = 0$ and $u^(\alpha_j) =0$ (resp. $u_0(\beta_j) = 0$ and $u^(\beta_j) = 0$) due to the fact that $u_0 \in W^{1,1}({\mathbb{R}})$ and $u^\in L^1_{weak}({\mathbb{R}})$ and is convex on $I_j$. In particular, since $u_0 \in W^{1,1}({\mathbb{R}})$, we have that $v \in L^1(A)$. We shall prove now that $u^$ is weakly differentiable with $$\label{part(ii)_eq2} (u^)' = \chi_{A^c} \,u_0' + \chi_{A} \,v,$$ where $\chi_{A}$ and $\chi_{A^c}$ denote the indicator functions of the sets $A$ and $A^c$. In fact, let $\varphi \in C^{\infty}_c({\mathbb{R}})$. Observe first that $$\label{part(ii)_eq1} \int_{I_j} u^(x)\,\varphi'(x)\,{\text{\rm d}x}= \big[u_0(\beta_j)\varphi(\beta_j)- u_0(\alpha_j)\varphi(\alpha_j)\big] - \int_{I_j} v(x)\, \varphi(x)\,{\text{\rm d}x},$$ obtained again by the continuity of $u^$ and a limiting argument approaching by compacts from inside $I_j$. From we have $$\begin{aligned} \begin{split} &\int_{{\mathbb{R}}} u^(x)\,\varphi'(x)\,{\text{\rm d}x}= \int_{A^c} u^(x)\,\varphi'(x)\,{\text{\rm d}x}+ \int_{A} u^(x)\,\varphi'(x)\,{\text{\rm d}x}\\ & = \int_{A^c} u_0(x)\,\varphi'(x)\,{\text{\rm d}x}+ \sum_j \big[u_0(\beta_j)\varphi(\beta_j)- u_0(\alpha_j)\varphi(\alpha_j)\big] - \int_{A} v(x)\, \varphi(x)\,{\text{\rm d}x}\\ & = \int_{A^c} u_0(x)\,\varphi'(x)\,{\text{\rm d}x}+ \left[\int_{A} u_0(x)\,\varphi'(x)\,{\text{\rm d}x}+ \int_{A} u_0'(x)\,\varphi(x)\,{\text{\rm d}x}\right] - \int_{A} v(x)\, \varphi(x)\,{\text{\rm d}x}\\ & = -\int_{{\mathbb{R}}} u_0'(x)\,\varphi(x)\,{\text{\rm d}x}+ \int_{A} u_0'(x)\,\varphi(x)\,{\text{\rm d}x}- \int_{A} v(x)\, \varphi(x)\,{\text{\rm d}x}\\ & = - \int_{{\mathbb{R}}} (\chi_{A^c} \,u_0' + \chi_{A} \,v)(x)\,\varphi(x)\,{\text{\rm d}x}, \end{split}\end{aligned}$$ as we wanted to show. We are now in position to conclude. From and we have $$\begin{aligned} \\|(u^)'\\|_1 & = \int_{{\mathbb{R}}} \|(u^)'(x)\|\,{\text{\rm d}x}\\ & = \int_{A^c} \|(u_0)'(x)\|\,{\text{\rm d}x}+ \int_{A} \|v(x)\|\,{\text{\rm d}x}\\ & \leq \int_{A^c} \|(u_0)'(x)\|\,{\text{\rm d}x}+ \int_{A} \|(u_0)'(x)\|\,{\text{\rm d}x}= \\|(u_0)'\\|_1.\end{aligned}$$ Proof of part (iii) ------------------- If $u_0$ has bounded variation, then $u_0$ is bounded. The distributional derivative $Du_0$ is a Radon measure with $\|Du_0\| \leq V(u_0)$, where $\|Du_0\|$ denotes the total variation of $Du_0$. For $\varepsilon>0$ we consider again $$u_{\varepsilon}(x) = u_0K_{\varepsilon}(x).$$ Note that $u_{\varepsilon} \in C^{\infty}({\mathbb{R}})$ is bounded and Lipschitz. We let $$u_{\varepsilon}^(x) = \sup_{\tau > 0} u_{\varepsilon}K_{\tau}(x) = \sup_{t > \varepsilon} u_0K_t(x).$$ From Lemma \[lem5\] we know that $u_{\varepsilon}^$ is subharmonic in the open set A = $\{x \in {\mathbb{R}};\ u_{\varepsilon}^(x) > u_{\varepsilon}(x)\}$. Let us write again $$A = \bigcup_{j} I_j = \bigcup_{j} \, (\alpha_j, \beta_j),$$ and thus we will have $u_{\varepsilon}^$ convex in each $I_j = (\alpha_j, \beta_j)$. Now consider a partition ${\mathcal}{P} = \{x_1,x_2,...,x_N\}$. Refine this partition by including the endpoints $\alpha_j$ and $\beta_j$ for which $x_k \in [\alpha_j, \beta_j]$ for $k=1,2,...,N$. Thus we obtain a new partition ${\mathcal}{P'} = \{y_1,...,y_M\} \supset {\mathcal}{P},$ where $M\geq N$. We now estimate the variation $V_{{\mathcal}{P}}(u_{\varepsilon}^)$ of the function $u_{\varepsilon}^$ with respect to the partition ${\mathcal}{P}$ by observing that $$\begin{aligned} \label{part(iii)_eq1} \begin{split} V_{{\mathcal}{P}}(u_{\varepsilon}^) & = \sum_{i=1}^{N-1} \big\| u_{\varepsilon}^(x_{i+1}) - u_{\varepsilon}^(x_{i})\big\| \\ & \leq V_{{\mathcal}{P'}}(u_{\varepsilon}^) = \sum_{j=1}^{M-1} \big\| u_{\varepsilon}^(y_{j+1}) - u_{\varepsilon}^(y_{j})\big\|\\ & \leq V(u_{\varepsilon})\,, \end{split}\end{aligned}$$ where in the last inequality we used the convexity of $u_{\varepsilon}^$ in each $I_j = (\alpha_j, \beta_j)$ and the fact that $u_{\varepsilon}^$ and $u_{\varepsilon}$ agree at the endpoints $\alpha_j$ and $\beta_j$ (minor modifications are needed for the cases $\alpha_j = -\infty$ or $\beta_j = \infty$). Since $$Du_{\varepsilon}(x) = Du_0K_{\varepsilon}(x),$$ from Young’s inequality we have $$\label{part(iii)_eq2} V(u_{\varepsilon}) = \|Du_{\varepsilon}\| \leq \|Du_0\|\,\\|K_{\varepsilon}\\|_1 = \|Du_0\| \leq V(u_0).$$ We now observe that as $\varepsilon \to 0$, we have $u_{\varepsilon}^* \to u^$ pointwise. From and we find that $$\begin{aligned} V_{{\mathcal}{P}}(u^) & = \sum_{i=1}^{N-1} \big\| u^(x_{i+1}) - u^(x_{i})\big\|\\ & = \lim_{\varepsilon \to 0} \sum_{i=1}^{N-1} \big\| u_{\varepsilon}^(x_{i+1}) - u_{\varepsilon}^(x_{i})\big\| \leq V(u_0).\end{aligned}$$ Since this holds for any partition ${\mathcal}{P}$ we obtain $$V(u^) \leq V(u_0),$$ and the proof is concluded. Proof of Theorem \[thm\_Poisson\] - Continuous Poisson kernel ============================================================= Throughout this proof we will again assume without loss of generality that $u_0 \geq 0$. Preliminaries ------------- The first result of this section is analogous to Lemma \[lem4\]. \[lem9\]. - If $u_0 \in C({\mathbb{R}}) \cap L^p({\mathbb{R}}^d)$, for some $1\leq p < \infty$, then $u^{\star} \in C({\mathbb{R}}^d)$. - If $u_0$ is bounded and Lipschitz continuous then $u^{\star}$ is bounded and Lipschitz continuous with ${\rm Lip}(u^{\star}) \leq {\rm Lip}(u_0)$. Just follow the proof of Lemma \[lem4\]. We now investigate the set where $u^{\star}$ disconnects from $u_0$. As in the heat kernel case, we will show that $u^{\star}$ is subharmonic in this set. The main tool for this will be the structure of the underlying Laplace equation (namely, the mean value property). \[lem10\] Let $u_0 \in C({\mathbb{R}}) \cap L^p({\mathbb{R}}^d)$ for some $1\leq p < \infty$ or $u_0$ be bounded and Lipschitz continuous. Then $u^{\star}$ is subharmonic in the open set $A = \{x \in {\mathbb{R}}^d; \,u^{\star}(x) > u_0(x)\}$. From Lemma \[lem9\] we see that $u^{\star}$ is continuous and $A$ is in fact an open set. In what follows we keep denoting by $B_r(x)$ the open $d$-dimensional ball centered in $x$ with radius $r$, and now we introduce $B_r(x,y)$ to denote the open $(d+1)$-dimensional ball centered in $(x,y)$ with radius $r$. Let $x_0 \in A$. Since $u_0(x_0) < u^{\star}(x_0)$ and $$\lim_{y \to 0^+} u(x_0,y) = u_0(x_0)\,,$$ there exists $\delta = \delta(x_0) >0$ such that if $y <\delta$ then $$\label{Lem10_eq0} u(x_0,y) < u^{\star}(x_0) - \tfrac{1}{2} (u^{\star}(x_0) - u_0(x_0)).$$ Now let $y_0 \geq \delta(x_0)$. Choose a radius $0< r_0 < \delta $ such that $\overline{B_{r_0}(x_0)} \subset A$. For any $r < r_0$ we have $\overline{B_{r}(x_0,y_0)} \subset A \times (0,\infty)$. Recall that the function $u(x,y)$ is harmonic in ${\mathbb{R}}^d \times (0,\infty)$ and therefore, by the mean value property, we have $$\label{Lem10_eq1} u(x_0,y_0) = \frac{1}{r^{d+1}\omega_{d+1}} \int_{B_{r}(x_0,y_0)} u(x,y)\,{\text{\rm d}x}\,{\text{\rm d}y}\,,$$ where $\omega_{d+1}$ denotes the volume of the $(d+1)$-dimensional unit ball. From we arrive at $$\begin{aligned} \label{Lem10_eq2} \begin{split} u(x_0,y_0) & \leq \frac{1}{r^{d+1}\omega_{d+1}} \int_{B_{r}(x_0,y_0)} u^{\star}(x)\,{\text{\rm d}x}\,{\text{\rm d}y}\\ & = \frac{1}{r^{d+1}\omega_{d+1}} \int_{B_{r}(x_0)} 2\, \sqrt{r^2 - \|x-x_0\|^2}\,\, u^{\star}(x)\,{\text{\rm d}x}. \end{split}\end{aligned}$$ By we know that $u^{\star}(x_0) = \sup_{y \geq \delta} u(x_0,y)$ and since holds for any $y_0 \geq \delta$, we have $$\label{Lem10_eq3} u^{\star}(x_0) \leq \frac{1}{r^{d+1}\omega_{d+1}} \int_{B_{r}(x_0)} 2\, \sqrt{r^2 - \|x-x_0\|^2}\,\, u^{\star}(x)\,{\text{\rm d}x}$$ for any $r < r_0 = r_0(x_0)$. Condition can be viewed as a “weighted" subharmonicity. We shall prove that it actually implies subharmonicity in the usual sense. First observe that is sufficient to establish the maximum principle for $u^{\star}$ in the domain $A$ (in each connected component to be precise), i.e. if $\Omega$ is an open connected set such that $\overline{\Omega} \subset A$, and $u^{\star}$ has an interior maximum in $\overline{\Omega}$ then $u^{\star}$ must be constant in $\overline{\Omega}$. With this in hand, consider any ball $\overline{B_s(x_0)} \subset A$ and let $h:\overline{B_s(x_0)} \to {\mathbb{R}}$ be the solution of the Dirichlet boundary value problem $$\left\{ \begin{array}{rl} \Delta h = 0& \ {\rm in} \ B_s(x_0);\\ h = u^{\star}& \ {\rm in} \ \partial B_s(x_0). \end{array} \right.$$ Consider the function $g= u^{\star} - h$. Let us prove that $g$ satisfies the same local “weighted" subharmonicity in $B_s(x_0)$. In fact, for a given $x_1 \in B_s(x_0)$, we know that holds in a neighborhood of $x_1$. Therefore, we can find a radius $r_1$ such that $\overline {B_{r_1}(x_1)} \subset B_s(x_0)$ and holds for $r < r_1$. Using the fact that $h$ is harmonic and that our weight is a radial function we arrive at $$\begin{aligned} g(x_1) & = u^{\star}(x_1) - h(x_1)\\ & \leq \left(\frac{1}{r^{d+1}\omega_{d+1}} \int_{B_{r}(x_1)} 2\, \sqrt{r^2 - \|x-x_1\|^2}\,\, u^{\star}(x)\,{\text{\rm d}x}\right) - h(x_1)\\ & = \left(\frac{1}{r^{d+1}\omega_{d+1}} \int_{B_{r}(x_1)} 2\, \sqrt{r^2 - \|x-x_1\|^2}\,\, \big\{ u^{\star}(x) - h(x)\big\} \,{\text{\rm d}x}\right) \\ & = \left(\frac{1}{r^{d+1}\omega_{d+1}} \int_{B_{r}(x_1)} 2\, \sqrt{r^2 - \|x-x_1\|^2}\,\, g(x) \,{\text{\rm d}x}\right).\end{aligned}$$ By the maximum principle, since $g$ is continuous in $\overline{B_s(x_0)}$, the maximum of $g$ in $\overline{B_s(x_0)}$ must be attained on the boundary. However, $g = 0$ in $\partial B_s(x_0)$, and therefore $$u^{\star}(x_0) \leq h(x_0),$$ which shows that $u^{\star}$ is subharmonic (in the usual sense, defined after Lemma \[lem4\]) since $h$ is harmonic and thus equal to its average over the sphere $\partial B_s(x_0)$, where $h = u^{\star}$ by construction. Proof of Theorem \[thm\_Poisson\] --------------------------------- Having established the subharmonicity of $u^{\star}$ in the disconnecting set (Lemma \[lem10\]), the proof of Theorem \[thm\_Poisson\] follows essentially in the same way as the proof of Theorem \[thm\_cont\], by observing that the Poisson kernel also satisfies the semigroup property $P_{y_1} P_{y_2} = P_{y_1 + y_2}$ to reduce to the Lipschitz case as in Lemma \[lem7\]. We shall omit the details. Acknowledgements ================ We would like to thank the following colleagues for helpful remarks during the preparation of this manuscript: William Beckner, Luis Caffarelli, Kevin Hughes, Milton Jara, Diego Moreira, Roberto Oliveira, Lillian Pierce, Luis Silvestre and Ralph Teixeira. E. C. acknowledges support from CNPq-Brazil grants $473152/2011-8$ and $302809/2011-2$, and FAPERJ grant $E-26/103.010/2012$. B. S. acknowledges support from CNPq-Brazil grants $302962/2011-5$, $474944/2010-7$, FAPERJ grant $E-26/102.940/2011$ and PRONEX Optimization. [99]{} J. M. Aldaz and J. Pérez-Lázaro, Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2443–2461. J. M. Aldaz. L Colzani and J. Pérez-Lázaro, Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function, J. Geom. Anal. 22 (2012), 132–167. J. Bober, E. Carneiro, K. Hughes and L. B. 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--- abstract: 'The semantic segmentation of 3D shapes with a high-density of vertices could be impractical due to large memory requirements. To make this problem computationally tractable, we propose a neural-network based approach that produces 3D augmented views of the 3D shape to solve the whole segmentation as sub-segmentation problems. 3D augmented views are obtained by projecting vertices and normals of a 3D shape onto 2D regular grids taken from different viewpoints around the shape. These 3D views are then processed by a Convolutional Neural Network to produce a probability distribution function (pdf) over the set of the semantic classes for each vertex. These pdfs are then re-projected on the original 3D shape and postprocessed using contextual information through Conditional Random Fields. We validate our approach using 3D shapes of publicly available datasets and of real objects that are reconstructed using photogrammetry techniques. We compare our approach against state-of-the-art alternatives.' author: - Davide Boscaini - Fabio Poiesi bibliography: - 'refs.bib' title: \| 3D Shape Segmentation\ with Geometric Deep Learning --- Introduction ============ Traditional Convolutional Neural Networks (CNNs) use a cascade of learned convolution filters, pooling operations and activation functions to transform image data into feature embeddings processable by fully connected layers that classify the image content [@He2016]. Typically, 3D deep-learning approaches extend traditional 2D methods to non-Euclidean domains as the convolution operation is not well defined in 3D [@Monti2017]. One of the most challenging researched topic related to 3D deep learning is the semantic segmentation of 3D shapes as it is key to support computer graphics applications such as shape editing [@Yu2004] and modelling [@Chen2015]. Challenges to segment 3D shapes include dealing with different topologies, handling noisy geometries and different resolutions, and modeling semantic representations for different segments. 3D segmentation can be performed through multi-view [@Kalogerakis2017; @Su2015], volumetric [@Wu2015] or intrinsic [@Qi2017; @Monti2017] deep learning-based approaches. Multi-view and volumetric approaches use Euclidean structures, such as 2D or 3D grids, respectively, to process 3D shapes with 2D CNNs [@Kalogerakis2017; @Su2015; @Wu2015]. In particular, multi-view approaches simplify the representation of a 3D model using a set of rendered depth images taken from different viewpoints around the model, thus making the segmentation independent of the 3D-model polygon density [@Kalogerakis2017; @Su2015]. Multi-view approaches cannot fully exploit the geometric properties of the 3D shape (e.g. face normals) because geometric information can be lost when data are projected in 2D. Volumetric approaches approximate the 3D shape using voxels which could overshadow geometric details of the object [@Wu2015]. Intrinsic approaches can be further divided into point-based and convolution-based approaches. Point-based approaches define feature extractors directly on the shape vertices [@Qi2017], whereas convolution-based approaches extend the traditional convolution operations from grid-like structures to triangular meshes [@Monti2017]. Point-based approaches mostly process each vertex of the shape independently and loosely exploit local information [@Qi2017]. The additional structures used by conventional convolution-based approaches increase the shape representation complexity hence prohibiting the processing of high-density polygon models [@Monti2017]. Typically, 3D segmentation approaches validate their performance on datasets collected in controlled scenarios, and they mostly lack of an evaluation carried out on 3D models reconstructed using photogrammetric techniques [@Nocerino2017]. In this paper we propose a novel 3D segmentation approach that retains both the advantages of view-based [@Kalogerakis2017] and intrinsic approaches [@Monti2017] by building 3D augmented views from multiple viewpoints around a 3D shape. 3D augmented views are a projection of 3D shape portions on 2D regular grids, where each cell of the grid encodes the information about depth and normal of the corresponding projected portion. This allows us to significantly reduce the number of parameters to learn and to perform 3D segmentation of shapes with diverse mesh topology (e.g. polygon structure and/or density). We evaluate our approach on synthetic 3D shapes from publicly available datasets, and on 3D shapes of objects we captured with a smartphone and reconstructed using photogrammetry techniques. Results show that the proposed approach can achieve state-of-the-art accuracy by using only $1\%$ of the parameters used by the alternative approaches. Our approach ============ Problem formulation {#sec:prob_formulation} ------------------- Given a 3D shape $\mathcal{X} \subset \mathbb{R}^3$ composed of vertices $x \in \mathcal{X}$, we design a neural-network based approach $\boldsymbol{p}(x) = \boldsymbol{\Gamma}_{\boldsymbol{\Theta}}(x)$ that outputs a probability distribution $\boldsymbol{p}(x)$ over the label space $\mathcal{L} = \{1, \dots, L\}$, where $L$ is the number of segmentation labels. The output segmentation of $\mathcal{X}$ is computed as $$h(x) = \operatorname{argmax}_{\ell = 1, \dots, L} \boldsymbol{p}(x),$$ where $h(x)$ is a label defining the segment class of the vertex $x$. The neural network $\boldsymbol{\Gamma_{\boldsymbol{\Theta}}}$ can be defined as a a parametric function in the set of learnable parameters (i.e. weights) $\boldsymbol{\Theta}$. $\boldsymbol{\Gamma_{\boldsymbol{\Theta}}}$ is composed of four modules, namely shape decomposition, feature extraction and classification, feature aggregation and prediction refinement. Shape decomposition* transforms the input 3D shapes into 3D augmented views, or 3D views. Each 3D view is processed by a feature extraction and classification network, namely ViewNet, that predicts the class of each vertex. Prediction aggregation re-projects the predictions of ViewNet of each 3D view onto the original 3D shape. Prediction refinement improves class prediction using contextual information on the original shape. Fig. \[fig:arch\] depicts the block diagram. [drawing\_arch.png]{} (1.5, 0) (12, 10) (12, 7) (12, 4) (33, 10) (33, 7) (33, 4) (54, 10) (54, 7) (54, 4) (75, 10) (75, 7) (75, 4) (90.2, 0) Shape decomposition ------------------- We simplify the 3D shape representation (e.g. triangular meshes, quad meshes, CAD models) by decomposing the input shape into several components. Shape decomposition can be performed by clustering shape vertices [@Hua2015], by using geometrical primitives [@Kaiser2019], or by generating range scans from different viewpoints [@Kalogerakis2017]. We use a similar approach to the latter in order to process the 3D shape regardless its 3D representation, resolution and vertex topology. Given $\mathcal{X}$ in the form of a triangular mesh with vertices $\boldsymbol{X} = (\boldsymbol{x}_1, \dots, \boldsymbol{x}_N)$, $\boldsymbol{x}_n \in \mathbb{R}^3$, $n=1, \dots, N$, we simplify $\mathcal{X}$ by building 3D views from $M$ different viewpoints. Let $\mathcal{I}(u, v; \boldsymbol{w}_m) = (u, v, d(u, v))$ be a range scan that is captured from the $m$th viewpoint $\boldsymbol{w}_m$, where $(u, v)$ is the coordinate of a pixel, $d(u, v)$ is the depth value of the 3D shape, and $m = 1, \dots, M$. Let $\mathcal{V}^{(m)}$ be the $m$th 3D view whose vertices $\boldsymbol{V}^{(m)} = (\boldsymbol{v}_1, \dots, \boldsymbol{v}_{N^{(m)}})$ are obtained by registering the coordinates $(u, v, d(u, v))$ of the range scan to the coordinates of the vertices $\boldsymbol{X}$. The faces of the 3D view are obtained by connecting depth values using the typical regular grid pattern of 2D images. For each vertex $\boldsymbol{v}_n \in \boldsymbol{V}^{(m)}$ we compute the surface normal $\boldsymbol{n}(\boldsymbol{v}_n) \in \mathbb{R}^3$ to define the signal on the 3D view as $\boldsymbol{f}(\boldsymbol{v}_n) = \left( \boldsymbol{v}_n, \boldsymbol{n}(\boldsymbol{v}_n) \right)$. The relation between a 3D view and the input 3D shape is defined by the correspondence function $t^{(m)} \colon \mathcal{V}^{(m)} \to \mathcal{X}$ that assigns the vertices of the $m$th 3D view to the corresponding vertices of the 3D shape. [drawing\_3Dviews.png]{} (18, -2) ------------------------------------------------------------------------ Feature extraction and classification ------------------------------------- The feature extraction and classification module processes the $M$ 3D views in parallel to learn features through a set of deep neural networks, namely ViewNets, with shared weights. Formally, each ViewNet is a non-linear parametric function $\boldsymbol{g}(\boldsymbol{v}_n) = \boldsymbol{\Phi}_{\boldsymbol{\Theta}_{\text{cla}}} \left( \boldsymbol{f}(\boldsymbol{v}_n) \right)$ that takes vertex-wise features $\boldsymbol{f}(\boldsymbol{v}_n)$ as input and produces the probability distribution $\boldsymbol{g}(\boldsymbol{v}_n) = (g_1(\boldsymbol{v}_n), \dots, g_L(\boldsymbol{v}_n))$ as output, where $L$ is the number of segmentation classes and $\boldsymbol{\Theta}_{\text{cla}} \subset \boldsymbol{\Theta}$ is the set of ViewNet learnable weights. Let $\boldsymbol{g}_{\mathcal{V}^{(m)}} \in {[0, 1]}^{N^{(m)} \times L}$ be the matrix containing the pdfs of all vertices of $\mathcal{V}^{(m)}$. A ViewNet module is defined as the composition of Intrinsic Convolutional (IC), Fully Connected (FC) and Softmax layers. FC and Softmax are standard layers, whereas the IC layer replaces the convolutional layer used in traditional Euclidean CNNs to perform convolution operations on 3D views [@Monti2017]. The convolution at $x \in \mathcal{X}$ using IC layers requires additional information, in the form of a local coordinate frame and a set of weighting functions that maps the signal of the local neighbourhood of $x$ to a fixed grid. Prediction aggregation ---------------------- Predictions inferred from each 3D view are re-projected and aggregated on the 3D shape $\mathcal{X}$ in order to transfer the segmentation result on the original input. We name this operation ProjNet. ProjNet employs a pooling operation that takes the ViewNet predictions $\boldsymbol{g}_{\mathcal{V}^{(m)}}$ on $\mathcal{V}^{(m)}$ as input and the correspondence function $t^{(m)} \colon \mathcal{V}^{(m)} \to \mathcal{X}$ for any $m$, to produce a single confidence map $\boldsymbol{g}_{\mathcal{X}}$ defined on $\mathcal{X}$. The pooling operation is defined as $$\boldsymbol{g}_{\mathcal{X}}(\boldsymbol{x}_n) = \frac{1}{\lvert \Omega(n) \rvert} \sum_{\tilde{m} \in \Omega(n)} \boldsymbol{g}_{\mathcal{V}^{(\tilde{m})}}(\boldsymbol{v}_{\tilde{n}}),$$ where $\Omega(n) = \{m\,:\,t^{(m)}(\tilde{n})=n\}$ is the set of 3D view indices relative to the vertex $\boldsymbol{x}_n \in \mathcal{X}$, and $\boldsymbol{g}_{\mathcal{V}^{(\tilde{m})}}(\boldsymbol{v}_{\tilde{n}})$ is the probability distribution over the segmentation classes associated to vertex $\boldsymbol{v}_{\tilde{n}}$ of the $\tilde{m}$th 3D view. Prediction refinement --------------------- The output of ProjNet is a point-wise prediction, i.e. the label prediction of each vertex is estimated independently from its neighbors, thus leading to likely local label inconsistencies. Moreover, some vertices of the input 3D shape may not have been projected on any of the 3D views, thus leading to vertices with undefined label predictions on $\mathcal{X}$. Therefore, we impose local label consistency by using a surface-based Conditional Random Field (CRF) approach [@Kalogerakis2017; @Zheng2015] that exploits contextual information to produce structured and dense predictions. For each vertex $\boldsymbol{x}_n \in \boldsymbol{X}$, let $y_n \colon \boldsymbol{x}_n \to \mathcal{L}$ be a random variable that assigns a label $\ell \in \mathcal{L}$ to it, and let $\boldsymbol{y} = (y_1, \dots, y_N)$ be the set of the random variables associated to the $N$ vertices of $\boldsymbol{X}$. The CRF energy associated to $\boldsymbol{y}$ is defined as: $$\label{eq:crf_energy} E(\boldsymbol{y}) = \sum_{n=1}^N \psi_\textrm{unary}(y_n) + \sum_{n=1}^N \sum_{\tilde{n}=n+1}^N \psi_\textrm{pairwise}(y_n, y_{\tilde{n}}),$$ where the unary term $\psi_{\textrm{unary}}(y_n)$ quantifies the assignment cost of $y_n$ to vertex $\boldsymbol{x}_n$ and the pairwise term $\psi_{\textrm{pairwise}}(y_n, y_{\tilde{n}})$ quantifies the joint assignment cost of $y_n, y_{\tilde{n}}$ to vertices $\boldsymbol{x}_n, \boldsymbol{x}_{\tilde{n}}$ [@Krahenbuhl2013]. Because $\boldsymbol{g}_\mathcal{X}(\boldsymbol{x}_n)$ measures the cost of assigning the vertex $\boldsymbol{x}_n$ to $\mathcal{L}$, we define the unary term as $\psi_{\textrm{unary}}(y_n) = - \log \left( \boldsymbol{g}_\mathcal{X}(\boldsymbol{x}_n) \right)$. The pairwise potential is instead defined as the weighted sum of three Gaussian kernels: $$\begin{aligned} \psi_\textrm{pairwise}(y_n, y_{\tilde{n}}) &= \mu(y_n, y_{\tilde{n}}) \bigg( w_{\textrm{near}}\; k_{\textrm{near}}(y_n, y_{\tilde{n}}) - w_{\textrm{far}}\; k_{\textrm{far}}(y_n, y_{\tilde{n}})\\ &+ w_{\textrm{feat}}\; k_{\textrm{feat}}(y_n, y_{\tilde{n}}) \bigg), \end{aligned}$$ where $$\begin{aligned} &k_{\textrm{near}}(y_n, y_{\tilde{n}}) = \exp \left( -\frac{ d_{\mathcal{X}}(\boldsymbol{x}_n, \boldsymbol{x}_{\tilde{n}}) }{\sigma_{\textrm{near}}} \right), & \nonumber\\ &k_{\textrm{far}}(y_n, y_{\tilde{n}}) = 1_{\mathcal{X}} - \exp \left( -\frac{ d_{\mathcal{X}}(\boldsymbol{x}_n, \boldsymbol{x}_{\tilde{n}}) }{ \sigma_{\textrm{far}} } \right), & \nonumber\\ &k_{\textrm{feat}}(y_n, y_{\tilde{n}}) = \exp \left( -\frac{\lVert \boldsymbol{f}(\boldsymbol{x}_n) - \boldsymbol{f}(\boldsymbol{x}_{\tilde{n}}) \rVert_2}{\sigma_{\textrm{feat}}} \right), &\end{aligned}$$ $d_{\mathcal{X}}(x, \tilde{x})$ is the geodesic distance between the vertices $x, \tilde{x} \in \mathcal{X}$, $1_{\mathcal{X}}$ is the identity function on $\mathcal{X}$, and $\mu(y_n, y_{\tilde{n}})$ is a label compatibility term. Similarly to [@Kalogerakis2017; @Zheng2015], $k_{\textrm{near}}$ favors local spatial consistency, while $k_{\textrm{feat}}$ promotes the assignment of similar labels to vertices with similar properties. The third kernel $k_{\textrm{far}}$ is novel and is introduced to disambiguate symmetries. Because symmetric parts are likely to be located far from each other (e.g. arms and legs in a human shape) we designed $k_{\textrm{far}}$ to avoid distant points to have similar labels. The set of CRF learnable parameters is defined as $\boldsymbol{\Theta}_{\text{CRF}} = \{ \mu, w_{\textrm{near}}, w_{\textrm{far}}, w_{\textrm{feat}}\}$, $\boldsymbol{\Theta}_{\text{CRF}} \subset \boldsymbol{\Theta}$. Fig. \[fig:crf\_kernels\] shows how CRF learns the relationships among segments through an example of learned parameters (i.e. $w_{\textrm{near}}$, $w_{\textrm{far}}$ and $\mu$) on human 3D shapes. In $w_{\textrm{near}}$ we can observe that the head weights suggest that there is a strong relationship between head and torso rather than between head and right foot/right arm. Similarly, the torso weights suggest that there is a strong relationship between torso and arms/legs rather than between torso and feet/hands. The most probable pdf configuration of $\boldsymbol{y}$ for $\mathcal{X}$ is obtained by minimizing the energy $E(\boldsymbol{y})$ defined in Eq. \[eq:crf\_energy\]. The exact inference of the CRF distribution is intractable, thus we use a mean-field approximation [@Kalogerakis2017; @Krahenbuhl2013]. The iterative algorithm for approximate mean-field inference can be implemented as a Recurrent Neural Network (RNN) by rephrasing each step of the algorithm as a CNN layer [@Zheng2015]. [crf.png]{} (14.5, 31.5)[$w_{\textrm{near}}$]{} (48.5, 31.5)[$w_{\textrm{far}}$]{} (83.5, 31.5)[$\mu$]{} Results ======= Experimental setup ------------------ We evaluate our 3D segmentation approach through two different experiments. Firstly, we use data from the publicly available Princeton Shape Benchmark (PSB) dataset [@Shilane2004] that contains synthetic shapes of several objects and animals; in particular, the rigid shapes of the Airplane class, and the non-rigid shapes of the Ant, Four Leg and Teddy classes. The segmentation labels of each object are defined as in [@Shilane2004]. Secondly, we use data of non-rigid human shapes; in particular, (i) synthetic people with different poses (FAUST dataset [@Bogo2014]), (ii) real people acquired with depth sensors (SCAPE dataset [@Anguelov2005]) and with structured light 3D body scanners (SHREC14 dataset [@Pickup2014]), and (iii) real people that we acquired with a smartphone and reconstructed using the photogrammetry pipeline COLMAP [@Schonberger2016]. We manually labelled the ground truth for FAUST and SCAPE datasets and used their training data to learn the neural network model for the human shapes. We have used this model to test our approach on all the other human shapes of FAUST, SCAPE, SHREC14 and COLMAP datasets. The segmentation labels for the non-rigid human shapes are: $\mathcal{L} = \{$head, torso, right arm, right hand, right leg, right foot, left arm, left hand, left leg, left foot$\}$. Training -------- Given a labelled training set, where each vertex $\boldsymbol{x}_n \in \boldsymbol{X}$ is associated to a ground-truth label $h(\boldsymbol{x}_n)$, the optimal parameters are obtained by minimizing the categorical cross-entropy loss, $$c(\delta_{h(\boldsymbol{x}_n)}, \boldsymbol{\Gamma}_{\boldsymbol{\Theta}}(x)) = -\sum_{n=1}^N \delta_{h(\boldsymbol{x}_n)} \log( \boldsymbol{\Gamma}_{\boldsymbol{\Theta}}(\boldsymbol{x}_n) ),$$ where $\delta_{h(\boldsymbol{x}_n)}$ is the Kronecker delta defined for the ground-truth label $h(\boldsymbol{x}_n)$. Our approach is trained end-to-end and from scratch. We use $M=10$ 3D views (Sec. 2.2, Fig. \[fig:3dviews\]) taken from equi-spaced viewpoints around the shape. For training we use the Adam optimizer [@Kingma2015] with a learning rate of $0.001$. The CRF weights are initialized with identity matrices, i.e. each segment class is only in relationship with itself. Evaluation ---------- PSB dataset: Table \[tab:PSB\_mAcc\] shows the quantitative results of our approach on a subset of PSB’s 3D shapes. We compare the accuracy of our approach with ShapeBoost [@Kalogerakis2010], Guo et al. [@Guo2015] and ShapePFCN [@Kalogerakis2017]. The first two approaches use classifiers that are learned from hand-crafted features, whereas the latter is an end-to-end deep learning approach similar to ours (i.e. features are also learned). We can observe that the accuracy of our approach is similar to that of state-of-the-art methods. However, compared to ShapePFCN [@Kalogerakis2017] that is based on the VGG16 architecture [@Simonyan2015], which uses $134$M parameters, our neural network uses $14$K parameters, i.e. $1\%$ of ShapePFCN’s parameters [@Kalogerakis2017]. Fig. \[fig:results\_PSB\] shows examples of segmentation results that are obtained on the Airplane category. The uncertainty map next to each segmentation result showed that the highest level of uncertainty is located where different segments intersect. Qualitatively, the results are very accurate and show only minor errors on the rudder region. ---------- ------------------------------- ----------------------- ------------------------------ ------ ShapeBoost [@Kalogerakis2010] Guo et al. [@Guo2015] ShapePFCN [@Kalogerakis2017] Ours Airplane 96.1 91.6 93.0 94.1 Ant 98.7 97.6 98.6 94.8 Four Leg 83.3 82.4 85.0 94.5 Teddy 98.7 97.3 97.7 92.8 ---------- ------------------------------- ----------------------- ------------------------------ ------ [psbAirplane.png]{} (100, 3.5) (100, 10.5) Non-rigid human shapes: Fig. \[fig:results\_FAUST\] to \[fig:results\_REPLICATE\] show examples of segmentation results that are obtained on the non-rigid human shapes. Beside each segmented shape we can observe their associated entropy map. The smaller the entropy the higher the uncertainty. As expected, the largest level of uncertainty is located at the joints between two segments, that is where transition is not well defined. Because we have annotations for FAUST and SCAPE, we quantified the accuracy [@Kalogerakis2017] and Intersection over Union (IoU) [@Qi2017] of the segmentation results. In FAUST we achieved an accuracy of 93.8% and IoU of 88.5% while in SCAPE we achieved an accuracy of 72.1% and IoU of 58.7%. This accuracy and IoU differences are due to the unbalanced number of training samples of the two datasets. FAUST annotations are much more numerous than those of SCAPE. A few of the poses of FAUST’s training shapes are also present in the test set. This does not occur in the case of SCAPE, where poses are only present once. Fig. \[fig:results\_SCAPE\] shows examples of the segmentation errors occurred in SCAPE test, e.g. on the right-hand block we can see that the legs of the shape in the middle have been segmented with inverted labels. [drawing2\_FAUST.jpg]{} (15.5, -3) ------------------------------------------------------------------------ (5, -2.5) (20, -2.5) (27, 2) (98.25, 4) (98.25, 25) [drawing\_SCAPE.jpg]{} (10, -3) ------------------------------------------------------------------------ (55.5, -3) ------------------------------------------------------------------------ (2.5, -2.5) (11.5, -2.5) (18, 1) (99.25, 3) (99.25, 17.5) Results in Fig. \[fig:results\_SHREC14\] and \[fig:results\_REPLICATE\] show that the method can generalize also to 3D shapes that have not been used for training. Interestingly, our approach can effectively generalize the mesh representation through the 3D augmented views and produce a reliable segmentation in the case of COLMAP’s shapes. Note that the mesh topology of COLMAP’s shapes is different from those used in training. This is because the meshing operation based on Poisson reconstruction of COLMAP produces highly irregular polygons [@Kazhdan2006]. However, it is also clear that COLMAP’s shapes are more challenging than SHREC14’s ones by looking at the respective confidence maps. Overall, results show that our approach can effectively segment 3D shapes of different subjects, despite their different pose. [drawing\_SHREC14.jpg]{} (2, -2) (9, 2.5) (97.5, 5) (97.5, 26) [drawing\_Replicate.jpg]{} (3, -2.5) (12, 2.5) (98, 4.5) (98, 28) Conclusions =========== We presented an approach to segment 3D shapes efficiently regardless their mesh topology. To achieve this we decomposed the segmentation problem into sub-segmentation problems by using 3D augmented views generated from the underlying 3D shape. This enabled us to train a neural network with 1% of the parameters used by alternative state-of-the-art solutions, while maintaining similar accuracy performance. We showed that our approach is generic and can be used to segment 3D shapes with arbitrary mesh topologies, like those computed with photogrammetry reconstruction techniques (e.g. Poisson reconstruction [@Kazhdan2006]) that have a high density of polygons and that are distributed irregularly. Moreover, our approach also showed evidence of being flexible to segment other categories of 3D shapes (e.g. airplanes) other than human ones. Future research directions include an extensive analysis of the results, evaluating the impact of a multi-scale approach applied on the 3D augmented views and exploring next-best-view approaches [@Potthast2014] to select the most suitable 3D views of the object of interest. We will also exploit the structured output of the CRF to build models for surface matching between 3D shapes [@Boscaini2016] and explore attention mechanisms to make the prediction of our approach robust to the clutter present on the 3D shape (i.e. untrained segmentation classes).
--- abstract: 'Factorizing low-rank matrices is a problem with many applications in machine learning and statistics, ranging from sparse PCA to community detection and sub-matrix localization. For probabilistic models in the Bayes optimal setting, general expressions for the mutual information have been proposed using powerful heuristic statistical physics computations via the replica and cavity methods, and proven in few specific cases by a variety of methods. Here, we use the spatial coupling methodology developed in the framework of error correcting codes, to rigorously derive the mutual information for the symmetric rank-one case. We characterize the detectability phase transitions in a large set of estimation problems, where we show that there exists a gap between what currently known polynomial algorithms (in particular spectral methods and approximate message-passing) can do and what is expected information theoretically. Moreover, we show that the computational gap vanishes for the proposed spatially coupled model, a promising feature with many possible applications. Our proof technique has an interest on its own and exploits three essential ingredients: the interpolation method first introduced in statistical physics, the analysis of approximate message-passing algorithms first introduced in compressive sensing, and the theory of threshold saturation for spatially coupled systems first developed in coding theory. Our approach is very generic and can be applied to many other open problems in statistical estimation where heuristic statistical physics predictions are available.' author: - \| Jean Barbier$^{\dagger}$, Mohamad Dia$^{\dagger}$ and Nicolas Macris$^{\dagger}$ firstname.lastname@epfl.ch\ $\dagger$ Laboratoire de Théorie des Communications, Faculté Informatique et Communications,\ Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.\ $$ International Center for Theoretical Physics, 11 Strada Costiera, I - 34151, Trieste, Italy. Florent Krzakala florent.krzakala@ens.fr\ Laboratoire de Physique Statistique, CNRS, PSL Universités et Ecole Normale Supérieure,\ Sorbonne Universités et Université Pierre & Marie Curie, 75005, Paris, France. Lenka Zdeborová lenka.zdeborova@gmail.com\ Institut de Physique Thé́orique, CNRS, CEA, Université Paris-Saclay,\ F-91191, Gif-sur-Yvette, France. bibliography: - 'sample.bib' title: 'Rank-one matrix estimation: analysis of algorithmic and information theoretic limits by the spatial coupling method' --- Sparse PCA, Wigner spike model, community detection, low-rank matrix estimation, spatial coupling, replica and cavity methods, interpolation method, approximate message-passing 0.2in
19truemm \ \[8truemm\] <span style="font-variant:small-caps;">Robin Horan[^1], Martin Lavelle[^2], and David McMullan</span>[^3]\ \[5truemm\] School of Mathematics and Statistics\ The University of Plymouth\ Plymouth, PL4 8AA\ UK > Abstract: In this article we investigate charged particles in gauge theories. After reviewing the physical and theoretical problems, a method to construct charged particles is presented. Explicit solutions are found in the Abelian theory and a physical interpretation is given. These solutions and our interpretation of these variables as the true degrees of freedom for charged particles, are then tested in the perturbative domain and are demonstrated to yield infra-red finite, on-shell Green’s functions at all orders of perturbation theory. The extension to collinear divergences is studied and it is shown that this method applies to the case of massless charged particles. The application of these constructions to the charged sectors of the standard model is reviewed and we conclude with a discussion of the successes achieved so far in this programme and a list of open questions. > Keywords: Gauge theories, infra-red, charged particles, confinement > PACS No.’s: 11.15.-q 12.20.-m 12.38.Aw 12.39.Hg Introduction ============ One of the most significant advances in particle physics was the definition of what a particle should be. Wigner [@Wigner:1939cj] identified a particle with an irreducible representation of the Poincaré group or its covering. Such representations have a well defined mass and spin. Building upon free creation and annihilation operators, and invoking [@weinberg:1995] the cluster decomposition theorem, one can construct the paradigm quantum field theoretic description of particles scattering into other particles. Physical particles generally carry additional quantum numbers such as isospin, electric and colour charge. These particles are described by the gauge theories of the standard model. Due to the masslessness of the gauge bosons, these field theories are plagued by infra-red divergences which dramatically alter the singularities of the Green’s functions of the matter fields. The states no longer form irreducible representations of the Poincaré group [@Buchholz:1986uj] and it is therefore widely believed [@kulish:1970] that there is no particle description of charges such as the electron. This review will describe the physics underlying this problem, the manifold consequences of this breakdown and how a particle description of charged fields may be recovered. The standard picture of a particle’s journey to a detector is as follows. When it is shot out of a scattering event it is, after enough time, a long way away from any other particle and the interaction between it and the other particles may be neglected. The free Hamiltonian then describes its dynamics and a particle description holds. This, the cornerstone of the interaction picture, breaks down for charged particles interacting via the gauge theories of the standard model. The importance of the large distance interactions, which mean that the residual interactions between particles at large separations cannot be neglected, is most obvious in Quantum Chromodynamics (QCD). Experimentalists observe hadrons and not quarks which are professed to be eternally confined. Although hadrons are colourless (chargeless as far as the strong nuclear force is concerned) their masses are rather well described in terms of building blocks which are the original Gell-Mann quarks. However, the current quarks of the QCD Lagrangian are not the constituent quarks of hadronic spectroscopy. This dichotomy is an unsolved puzzle and shows itself in many ways. For the light quark flavours the masses of the current and constituent quarks are very different (roughly two orders of magnitude for the $u$ and $d$ flavours). The division of the spin of the proton amongst its constituents has led to the proton spin crisis, which name shows the depth of our difficulties in understanding the experimental data. Finally, since constituent quarks are presumably constructs made from surrounding a Lagrangian matter field with a cloud of coloured glue, it is initially at least highly unclear how the constituents obtain a well-defined colour. A major aim of the programme of research [@Lavelle:1997ty] described here is to understand how constituent quarks arise in QCD, their mutual interactions and finally how it is that these effective degrees of freedom are confined. Electrons are not confined and so it may seem initially less obvious that the interaction picture paradigm breaks down here too. Yet the S-matrix elements of Quantum Electrodynamics (QED) are afflicted by infra-red divergences, as are the on-shell Green’s functions of the theory with external legs corresponding to charged particles (which we will sometimes generically refer to as electrons). The infra-red problems change the form of the Green’s functions such that we cannot associate a pole to the external legs. There are two different sorts of divergence here: soft divergences, which show up in Green’s functions and S-matrix elements, and phase divergences which occur in the phase. These latter divergences are often ignored in QED, but are of importance in QCD (see Sect. 3.4 of Ref. [@Ciafaloni:1989vs] and also [@DelDuca:1989jt]). The masslessness of the photon is the underlying cause of the infra-red problem[^4]. This vanishing of the mass means that photons can travel over a large distance and indeed that an infinite number of soft photons can be created for any finite amount of energy. Recognition of this led to the Bloch-Nordsieck answer to the infra-red problem in QED: since any finite experimental resolution does not restrict the number of photons which may accompany any charged particle, an experimental cross-section must come from summing over all these possibilities. In this sense QED is taken to be a theory defined only at the level of (measurable) cross-sections and not in terms of (unobservable) S-matrix elements. Although this is in no way incompatible with experiment, it is a radical conclusion. The survival of large distance interactions is responsible for the claim [@kulish:1970] that charged particles cannot be incorporated into relativistic quantum field theory. This conclusion followed from noting that as the usual free Hamiltonian does not determine the asymptotic dynamics, it must be modified. This leads to a description in terms of coherent states and there is no pole structure. Is particle physics then a misnomer? In fact we will see below that these problems can be resolved by realising that, as the asymptotic interaction Hamiltonian is not zero, electric charges are surrounded by an electromagnetic cloud, just as quarks are by glue. It will then be seen that such systems of dressed charges must be gauge invariant. We will demonstrate that the Green’s functions of such dressed charges are infra-red finite and have a good pole structure. ### General Properties of Charges The gauge dependence of the Lagrangian degrees of freedom means that it is hard to associate any physical meaning to them. This hinders any attempt to understand how phenomenological models and concepts can arise from the underlying theory. As an example of this we recall that the highly non-trivial gauge dependence of the Lagrangian vector potential in non-abelian gauge theories has put difficulties in the way of extracting such, phenomenologically useful, ideas as effective gluon masses from lattice calculations [@Henty:1996kv]. Constrained dynamics is the mathematical tool appropriate to finding gauge invariant degrees of freedom. The true degrees of freedom then correspond to locally gauge invariant constructs (which obey Gauss’ law). There have been many attempts to obtain such variables in gauge theories (see, e.g., [@Gogilidze:1997qq] and references therein). We now want to sketch some general properties that any description of charges must fulfill. Local gauge transformations in QED have the form $$\label{gauge} A_\mu(x)\to A_\mu(x) + \partial_\mu\theta(x)\,,\qquad \mathrm{and} \qquad\psi(x)\to {\mathrm{e}}^{ie\theta(x)}\psi(x) \,,$$ so if the coupling ${\mathrm{e}}$ could be switched off, then the Lagrangian fermion would be locally gauge invariant. Similarly Gauss’ law $$\label{gausslaw} \partial^i F_{i0}=-{\mathrm{e}}J_0 \,,$$ where $J_0$ is the charge density, would, in the $e=0$ limit, imply that only the transverse components of the field strength were physical. However, as we have noted the coupling does not vanish and so the matter fields cannot be identified with physical particles. Similarly Gauss’ law shows that there is an intimate link between the matter fields and the electromagnetic cloud which surrounds them. An immediate consequence of this is that any description of a charged particle cannot be local since the total charge can be written as a surface integral at infinity. We also see that objects which are invariant under global gauge transformations are chargeless, as the charge density is the generator of such global transformations. The Gauss law constraint generates local gauge transformations, so we demand invariance under local but not global gauge transformations of any description of a charged particle. The form of the cloud around a charge determines the electric and magnetic fields surrounding the charge. This implies a fundamental non-covariance: the velocity of any charged particle will determine the nature of the cloud. The implications of these inevitable properties of non-locality [@Maison:1975ex] and non-covariance [@Frohlich:1979uu] for any description of charged particles for the general properties of gauge theories have been investigated both for scattering theory [@Zwanziger:1975ka; @Zwanziger:1975jz] and in axiomatic approaches [@Morchio:1983ym; @Buchholz:1996]. ### Dirac’s Dressings To the best of our knowledge the interplay between a charged matter field and the electromagnetic cloud which inevitably surrounds it was first used by Dirac [@Dirac:1955ca] to construct a specific description of the electron. He suggested that one should use $$\label{dirac} {\psi_D}(x)\equiv \exp\left( -i{\mathrm{e}}\frac{\partial_i A_i}{\nabla^2}\right)\psi(x) \,.$$ This he motivated in the following way: it is locally gauge invariant[^5]. Using the fundamental equal-time commutator, $[E_i(x),A_j(y)]=i\delta_{ij}\delta(\boldsymbol{x}- \boldsymbol{y})$, and the representation $$\label{nabladef} \frac{\partial_iA_i}{\nabla^2}(x)= -\frac1{4\pi}\int\!d^3\boldsymbol{y} \frac{\partial_iA_i(x_0,\boldsymbol{y})}{{\vert {\boldsymbol{x}- \boldsymbol{y}}\vert}}$$ then the electric field of the state ${\psi_D}(x){\vert 0 \rangle}$ is found to be $$\label{Efield} E^i(x_0,{\boldsymbol{y}}){\psi_D}(x){\vert 0 \rangle}= -\frac {{\mathrm{e}}}{4\pi} \frac{{\boldsymbol{x}_i-\boldsymbol{y}_i}}{\vert{\boldsymbol{x}- \boldsymbol{y}}\vert^3}{\psi_D}(x){\vert 0 \rangle}\,,$$ which is what one would expect for a static charge. He further pointed out that this is actually a member of an entire class of composite fields $$\label{fdress} {\psi_f}(x)\equiv \exp\left( -i{\mathrm{e}}\int\!d^4zf^\mu(x-z)A_\mu(z) \right)\psi(x)\,,$$ which, he argued, are gauge invariant for all $f^\mu$ if it is demanded that ${\partial}_\mu f^\mu(w)=\delta^{(4)}(w)$ holds. These constructs are all evidently non-local. We refer to the cloud around the matter field as a dressing. It is clear that the dressing suggested by Dirac (\[dirac\]) is also non-covariant. This review will be concerned with generalising and refining such descriptions of charges. Such pictures have been rediscovered by various authors since Dirac and there have been many attempts to use certain examples of this wide class of dressings over the years [@dEmilio:1984; @dEmilio:1984a; @Steinmann:1984; @Prokhorov:1992; @Prokhorov:1993; @Kawai:1995zx; @Kawai:1995zv; @Kawai:1995zu; @Lusanna:1996ut; @Lusanna:1996us; @Kashiwa:1997; @Kashiwa:1996; @Chechelashvili:1997; @Haagensen:1997pi]. A natural extension of this is to generalise Eq. \[fdress\] to systems involving more than one matter field. Two opposite charges at different points can be made gauge invariant by including a dressing which keeps the entire system gauge invariant. Such a description could correspond to a positronium state, a hydrogen atom or a meson. If the cloud factorises so that each of the matter fields together with its part of the dressing is gauge invariant, then we can clearly speak of constituents. A dressing which does not factorise at all would mean that we had an effective meson field but could not really speak of constituent particles. The success of the constituent quark picture of hadrons, taken together with the rough equality of the constituent masses in mesons and baryons indicates that there must be at least a rough factorisation of the dressings in some dynamical domains of QCD. Once one has an ansatz for the dressing, it may be used to study the interaction between charges. The easiest manner to produce a gauge invariant description is to link the two matter fields by a path ordered exponential along some line. The potential is then obtained[^6] by taking the commutator of the Hamiltonian with this description: that part of the energy which depends on the separation is the potential. The string ansatz leads already in QED to a linear, confining potential. Furthermore the overall coefficient is, as a result of the infinite thinness of the string, divergent. This string description has no physical relevance, even in QCD the potential between two heavy quarks should have a Coulombic form at short distances [@Cahill:1979dq] and the finite string tension implies a cigar shaped dressing. Since the string model corresponds to an (infinitely) excited state, it is unstable [@Prokhorov:1993]. If we consider two extremely heavy, fixed charges and neglect pair creation, we may use the free Hamiltonian and thus solve the time development of the system starting from the string ansatz initial state. The electric and magnetic fields immediately broaden out and, of course, lead to the usual Coulombic field. An animation and detailed discussion of this can be found at the web site: `http://www.ifae.es/~roy/qed.html` After this fly-by tour of the subject, it is time to get to grips with the physics of charges. The structure of this article is as follows. In Sect. 2 the form of the interaction Hamiltonian at large times is investigated. It is shown not to vanish for the matter fields but we demonstrate that, for suitably dressed fields, the asymptotic interaction does indeed vanish. This result is then used to construct explicit charged fields. In Sect. 3 an alternative account of the charged fields in the heavy scalar theory is presented. These results and our interpretation of them are then tested in perturbation theory in Sect. 4. It is shown that the soft and phase divergences cancel and a pole structure is obtained. The massless electron limit is considered in Sect. 5 as a testing ground for the study of collinear divergences. The form of the asymptotic interaction Hamiltonian is shown to be such that the solutions of the dressing equation will remove these mass singularities as well. Subtleties of the dressing equation in this limit are investigated in this section. Finally Sect. 6 reviews what we have learned and presents a list of outstanding problems. Charged Particles ================= As we have seen, the masslessness of the photon implies that the interactions between charges cannot be switched off in the remote past or future. It is this long range nature of electromagnetic interactions that lies at the heart of the infra-red problem in QED. In this section we will review how the asymptotic dynamics found in such a gauge theory deviates from that of a free theory. As a consequence of this we will see that the correct identification of asymptotic, charged particle states can be made through a process of dressing the matter of the theory. Asymptotic Dynamics ------------------- The S-matrix codifies the intuitive picture of a scattering experiment whereby in-coming particles get sufficiently close to interact, resulting (after the dust settles) in some set of out-going particles. The in-coming and out-going particle regimes are identified with elements of the Fock space constructed out of the creation and annihilation operators found in the free theory. As discussed in the introduction, the reason for this is that it is these states that can be identified with (tensor products) of irreducible representations of the Poincaré group. As such, they are particles! The key assumption, then, in the S-matrix description of particle scattering is that in- and out-regimes where the dynamics is that of the free system exist, i.e., the interacting Hamiltonian must tend to zero in the remote past and future. It is this assumption that fails in a gauge theory [@dollard:1964; @kulish:1970]. To see how this arises in QED, we start with the gauge fixed Lagrangian density: $$\label{qed_lag} {\cal L}_{\mathrm{QED}}=-{{\textstyle\frac{1}{4}}}F_{\mu\nu}F^{\mu\nu}+i{\bar\psi}\gamma^\mu( {\partial}_\mu-i{\mathrm{e}}A_\mu)\psi-m{\bar\psi}\psi+{{\textstyle\frac{1}{2}}}\xi B^2+{\partial}_\mu A^\mu B\,.$$ In this we are working in the Lorentz class of gauges (Feynman gauge corresponding to the choice $\xi=1$). The gauge invariance of the physical states is encoded in the condition that $B^{+}\|\mathrm{phys}\bigr>=0$. For the present, we do not allow the mass of the electron to be zero ($m\ne0$). The interaction Hamiltonian is given by $$\label{intham} {{\cal H}_{\mathrm{Int}}}(t)=-e{\int\!d^3{{\boldsymbol x}}\;}A_\mu(t,{{\boldsymbol x}})J^\mu(t,{{\boldsymbol x}})\,,$$ where the conserved matter current is $J^\mu(t,{{\boldsymbol x}})={\bar\psi}(t,{{\boldsymbol x}})\gamma^\mu\psi(t,{{\boldsymbol x}})$. In order to construct the S-matrix, the fields that enter this part of the Hamiltonian are taken to be in the interaction picture. We recall that this means that the time evolution of the states is described by (\[intham\]) while the fields evolve under the free Hamiltonian. Thus in (\[intham\]) we should insert the free field expansions for the matter and gauge fields, this will then allow us to study in detail the large $t$ behaviour of the interaction. We take as our free field expansions $$\label{psi_free} \psi(x)={\int\!\frac{d^3{{\boldsymbol p}}}{(2\pi)^3}\;}\frac1{\sqrt{2E_{\smash{p}}}}\left\{ b({{\boldsymbol p}},s)u^s(p)e^{-ip{{\cdot}}x}+ {{d^{\dag}}}({{\boldsymbol p}},s)v^s(p)e^{ip{{\cdot}}x} \right\}$$ and $$\label{a_free} A_\mu(x)={\int\!\frac{d^3{{\boldsymbol k}}}{(2\pi)^3}\;}\frac1{2\omega_{\smash{k}}}\left\{ a_\mu({{\boldsymbol k}})e^{-ik{{\cdot}}x}+ {{a^{\dag}}}_\mu({{\boldsymbol k}})e^{ik{{\cdot}}x} \right\} \,,$$ where $E_p=\sqrt{\|{{\boldsymbol p}}\|^2+m^2}$ and $\omega_k=\|{{\boldsymbol k}}\|$. Inserting these into (\[intham\]) results in eight terms which we group according to the positive and negative frequency components of the fields. Each of these pieces will have a time dependence of the form $e^{i{\alpha}t}$ where ${\alpha}$ involves sums and differences of energy terms. As $t$ tends to plus or minus infinity, only terms with ${\alpha}$ tending to zero can survive and thus contribute to the asymptotic Hamiltonian. After performing the spatial integration, and using the resulting momentum delta function, only terms of the form $e^{\pm it(E_{p+k}- E_p\pm\omega_k)}$ have a large $t$-limit; there are four of them. The requirement that $E_{p+k}-E_p\pm\omega_k\approx0$ can only be met in QED because the photon is massless, in which case it implies that $\omega_k\approx0$, i.e., only the infra-red regime contributes to the asymptotic dynamics. From this observation it is straightforward to see that the full interacting Hamiltonian (\[intham\]) has the same asymptotic limit[^7] as $$\label{hint_as} {{\cal H}_{\mathrm{Int}}}^{\mathrm{as}}(t)=-\mathrm{e}{\int\!d^3{{\boldsymbol x}}\;}A_\mu(t,{{\boldsymbol x}})J^\mu_{\mathrm{as}}(t,{{\boldsymbol x}})$$ with $$\label{j_as} J^\mu_{\mathrm{as}}(t,{{\boldsymbol x}})={\int\!\frac{d^3{{\boldsymbol p}}}{(2\pi)^3}\;}\frac{p^\mu}{E_p}\rho(p)\delta^3 \biggl({{\boldsymbol x}}-\frac{{{\boldsymbol p}}}{E_p} t\biggr)\,.$$ The operator content of this current is only contained in the charge density $$\label{chargedensity} \rho(p)=\sum_{s}\Bigl({{b^{\dag}}}(p,s)b(p,s)-{{d^{\dag}}}(p,s)d(p,s)\Bigr)$$ which implied that the asymptotic current satisfies the trivial space-time commutator relation $$\label{jcomm} [J^\mu_{\mathrm{as}}(x),J^\nu_{\mathrm{as}}(y)]=0\,.$$ As such, this current can be interpreted as effectively the integral over all momenta of the current associated with a charged particle moving with velocity $p^\mu/E_p$. Such a current does not vanish as $t\to\infty$. We thus see that the asymptotic dynamics of QED is not that of a free theory. The non-triviality of the asymptotic dynamics dramatically alters the form of the in- and out- matter states. In particular, their propagator no longer has a pole-like structure, but instead behaves near its mass-shell like $(p^2-m^2)^{{\beta}-1}$ where the exponent ${\beta}$ is gauge dependent; in the Lorentz class of gauges it is given by [@Bogolyubov:1980nc] $$\label{yennie} {\beta}=\frac{{\mathrm{e}}^2}{8\pi^2}(\xi-3)\,.$$ It is this observation that lies at the heart of the statement [@kulish:1970] that there is no relativistic concept of a charged particle. Charges as Dressed Matter ------------------------- The persistence of the asymptotic dynamics in QED means that we cannot set the electromagnetic coupling to zero for the in-coming and out-going particle. As an immediate consequence of this we see that the matter field, $\psi(x)$, cannot be viewed as the field which creates or annihilates charges since it is not gauge invariant in the remote past or future. An equivalent statement of this fact is that the matter field, $\psi(x)$, does not satisfy Gauss’ law at any time. To construct a charged field we need to be able to find a functional of the fields, $h^{-1}(x)$, such that, under a gauge transformation described by the group element $U(x)=e^{i\mathrm{e}\theta(x)}$, we have $$\label{h_trans} h^{-1}(x)\to h^{-1}(x)U(x)\,.$$ Then, from (\[gauge\]) the charged field will be given by the gauge invariant product $$\label{cmatter} \Psi(x)= h^{-1}(x)\psi(x)\,.$$ It is this product which makes precise how we associate charges with dressed matter: a (chargeless) functional of the fields which transforms as in (\[h\_trans\]) is what we mean by a dressing; the product (\[cmatter\]) we then identify as charged matter. We continue to refer to the matter terms that enter directly into the Lagrangian simply as matter. There are many distinct ways to construct dressings, reflecting the fact that our identification (\[cmatter\]) of charged matter should be viewed as a minimal requirement. In specific applications the form of the dressing must be tailored to the physics at hand: charges that enter into bound states will have a very different structure to those that describe particle. In our programme to construct charges explicitly we have so far dealt exclusively with the construction of charged particles[^8]: it is an open and immensely interesting problem to extend our approach to bound states. In order for the charged matter to describe a particle, we demand that the dressing is such that the charged field creates a state which is an eigenstate of the energy-momentum tensor, i.e., that it has a sharp momentum. For the matter field, $\psi$, this was not possible due to the infra-red problem [@Buchholz:1986uj]. For our charged field, though, we shall see that the form of the dressing can be chosen so that there are no infra-red divergences. The interacting Hamiltonian (\[intham\]) is derived from the coupling term in the matter part of the Lagrangian density for QED: $$\label{matter_lag} i{\bar\psi}(x)\gamma^\mu({\partial}_\mu-i\mathrm{e}A_\mu(x))\psi(x)\,.$$ We can rewrite this in terms of the physical charged fields $\Psi(x)$ as $$\label{charged_lag} i\overline{\Psi}(x)\gamma^\mu({\partial}_\mu-i\mathrm{e}A^h_\mu(x))\Psi(x)\,$$ where $$\label{ah} A^h_\mu=h^{-1}A_\mu h+\frac1{i\mathrm{e}}{\partial}_\mu(h^{-1})h\,,$$ which we recognise as a (field dependent) gauge transformation[^9] of the vector potential. Thus, written in terms of the charged fields, the interacting Hamiltonian is $$\label{chint_ham} H^{\mathrm{c}}_{\mathrm{int}}(t)=-{\mathrm{e}}{\int\!d^3{{\boldsymbol x}}\;}A^h_\mu(t,{{\boldsymbol x}})J^\mu(t,{{\boldsymbol x}})\,,$$ which asymptotically becomes $$\begin{aligned} \label{chasy_ham} H^{\mathrm{c}}_{\mathrm{int}}(t)&\to& -{\mathrm{e}}{\int\!d^3{{\boldsymbol x}}\;}A^h_\mu(t,{{\boldsymbol x}})J^\mu_{\mathrm{as}}(t,{{\boldsymbol x}})\\ &=&-{\mathrm{e}}{\int\!d^3{{\boldsymbol x}}\;}{\int\!\frac{d^3{{\boldsymbol p}}}{(2\pi)^3}\;}\frac{A^h_\mu(t,{{\boldsymbol x}})p^\mu}{E_p}\rho(p)\delta^3 \biggl({{\boldsymbol x}}-\frac{{{\boldsymbol p}}}{E_p} t\biggr)\,.\end{aligned}$$ This would vanish if we could construct the dressing such that $A^h_\mu(t,{{\boldsymbol x}})p^\mu=0$. To investigate the extent to which this asymptotic interaction can vanish, we note that the momentum, $p^\mu$, is an arbitrary on-shell, four vector. Now through a gauge transformation, such an algebraic condition can be imposed on the vector potential at one point in the mass shell, but not for the whole mass shell. So we cannot expect to be able to construct a dressing such that the asymptotic interaction Hamiltonian (\[chasy\_ham\]) vanishes for the whole mass-shell. However, we can insist that our charged field creates an in-coming or out-going particle with a definite momentum. From this discussion we see that if we want the charged field (\[cmatter\]) to asymptotically create or annihilate a charged particle moving with four-velocity $u^\mu$, then the dressing must satisfy the additional kinematical condition that $$\label{dress1} u^\mu A^h_\mu(t,{{\boldsymbol x}})=0\,.$$ This will then ensure that, at the point in the mass-shell where $p^\mu=mu^\mu$, the asymptotic interaction Hamiltonian (\[chasy\_ham\]) vanishes and thus the state created by the field will have the appropriate sharp momentum. What this means in practical terms is that each charged particle must be constructed out of the matter fields with a different dressing — reflecting the velocity of the particle concerned. At first sight this looks very peculiar: we are used to particles being put on-shell as the mass of a particle is a well defined quantum number, but a particle’s velocity is not usually thought of as a quantum number of the system. For charged particles, though, velocity is a well defined quantum number. In order not to interrupt our account of how to construct charges we shall postpone a discussion of this interesting point until Sect. 3 where the connection with the heavy charge sector will also be discussed. Construction of Charges ----------------------- In order to construct an asymptotic charged particle moving with four-velocity $u^\mu$, we need to be able to find a dressing field $h^{-1}$ such that (\[h\_trans\]) and (\[dress1\]) hold. The dressing equation (\[dress1\]) is, for this purpose, best written as $$\label{dresseqtn} (\eta+v)^\mu{\partial}_\mu(h^{-1})=-i{\mathrm{e}}h^{-1}(\eta+v)^\mu A_\mu\,,$$ where we have written the four-velocity $u^\mu$ as $\gamma(\eta+v)^\mu$, where $\eta^\mu$ is the time like vector $(1,\boldsymbol{0})$, $v^\mu$ is the space-like vector $(0,\boldsymbol v)$ with $\boldsymbol v$ the three-velocity of the charged particle we wish to construct and $\gamma$ is just the standard relativistic factor $(1-\boldsymbol v^2)^{-\frac12}$. To see how to construct the dressing, we first restrict ourselves to lowest order in the coupling. That is, we take $h^{-1}(x)=1-i{\mathrm{e}}R(x)$ and, to this order, the dressing equation is $$\label{reqtn1} (\eta+v)^\mu{\partial}_\mu R(x)=(\eta+v)^\mu A_\mu(x)\,.$$ This equation can now be easily solved in terms of an integral along the world line of a particle moving with the given velocity: $$\label{rsol1} R(x)=\int_a^{x^0}(\eta+v)^\mu A_\mu(x_t)dt+\chi(x_a)\,.$$ In this expression we have, for convenience, parametrized the world line not by the proper time but by $x^\mu_t=x^\mu+(t-x^0)(\eta+v)^\mu$ and we have introduced an arbitrary reference time $a$. The term $\chi(x_a)$ is, for the moment, an unspecified field configuration in the kernel of the differential operator $(\eta+v)^\mu{\partial}_\mu$. It is useful here to group the $a$-dependent terms together under the integral, and write $R(x)$ as $$\begin{aligned} \label{rsol2} R(x)&=&\int_a^{x^0}\biggl((\eta+v)^\mu A_\mu(x_t)-\frac{d\chi}{dt}(x_t)\biggr)dt+\chi(x)\nonumber\\ &=& \int_a^{x^0}(\eta+v)^\mu\biggl( A_\mu(x_t)-\frac{{\partial}\chi}{{\partial}x_t^\mu}(x_t)\biggr)dt+\chi(x)\,.\end{aligned}$$ In addition to the dressing equation, we also need to ensure the correct gauge transformation properties for the dressing. To this order in the coupling, this means that under the gauge transformation $A_\mu\to A_\mu+{\partial}_\mu\theta$, we must have $$\label{rtrans} R(x)\to R(x)+\theta(x)\,.$$ This, in turn, implies that $$\label{chitrans} \chi(x)\to \chi(x)+\theta(x)\,.$$ This is precisely the type of transformation that Dirac investigated in (\[fdress\]) and we would be tempted to write $$\label{dansatz} \chi(x)=\int\!d^4z\,f^\mu(x-z) A_\mu(z)\,,$$ where $f^\mu(x-z)$ satisfies ${\partial}_\mu f^\mu(x-z)={\delta}^4(x-z)$. But this only implies (\[chitrans\]) if no surface terms arise when we integrate by parts. The restriction on the local gauge transformations to those that vanish at spatial infinity is quite natural as finite energy restrictions impose a $1/r$ fall-off on the potential. However, no such restriction applies to the fields at temporal infinity. Thus the form of $f^\mu(x-z)$ must be such that no such surface terms arise. As it stands, we can only infer from this that $f^0(x-z)$ should be zero outside of some bounded region in the $z^0$-direction. To get more from this, we make the ansatz that in fact $$\label{ansatz} \chi(x)=\int\!d^4z\,G(x-z){{\cal G}}^\mu A_\mu(z)\,,$$ where ${{\cal G}}^\mu$ is a first order differential operator and ${{\cal G}}{\cdot}{\partial}\, G(x-z)={\delta}(x-z)$. In order to avoid the surface terms that would obstruct the gauge transformation properties of the dressing, we must have that the operator ${{\cal G}}{\cdot}{\partial}$ cannot involve any time derivatives. Given this restriction, we see that $G(x-z)={\delta}(x^0-z^0)G({{\boldsymbol x}}-{{\boldsymbol z}})$. We shall, for convenience, write $\chi$ as $$\label{chinotation} \chi(x)=\frac{{{\cal G}}{\cdot}A}{{{\cal G}}{\cdot}{\partial}}(x)\,.$$ The temporal parameter $a$ which enters into the form of the dressing (\[rsol1\]) has been introduced by hand and thus should not directly affect any physical results. Taking the derivative of (\[rsol2\]) with respect to $a$ yields the gauge invariant term $$\label{aderiv1} -(\eta+v)^\mu(A_\mu(x_a)-{\partial}_\mu \chi(x_a))\equiv -(\eta+v)^\mu\biggl(A_\mu(x_a)-{\partial}_\mu \frac{{{\cal G}}^\nu A_\nu}{{{\cal G}}{{\cdot}}{\partial}}(x_a)\biggr)\,.$$ This we can write in a form where the gauge invariance is manifest as $$\label{aderiv2} -(\eta+v)^\mu\biggl(\frac{{{\cal G}}^\nu {\partial}_\nu A_\mu}{{{\cal G}}{{\cdot}}{\partial}}(x_a)- \frac{{{\cal G}}^\nu{\partial}_\mu A_\nu}{{{\cal G}}{{\cdot}}{\partial}}(x_a)\biggr) =-(\eta+v)^\mu\frac{{{\cal G}}^\nu F_{\nu\mu}}{{{\cal G}}{{\cdot}}{\partial}}(x_a)\,.$$ The condition selecting physical states is that they are annihilated by the $B$-field, thus any physical observable must commute with $B(x)$. The relevant equation of motion that follows from (\[qed\_lag\]) is, to this order, ${\partial}^\nu F_{\nu\mu}={\partial}_\mu B$. This tells us that, in order for (\[aderiv2\]) to act trivially on physical states, it must be equal to $$\label{aderiv3} (\eta+v)^\mu\frac{{\partial}^\nu F_{\nu\mu}}{{{\cal G}}{{\cdot}}{\partial}}(x_a)\,.$$ This now allows us to find the form for ${{\cal G}}^\mu$ and hence the dressing to this order. The first order operator ${{\cal G}}^\mu$ is constructed out of the vectors ${\partial}^\mu$, $n^\mu$ and $v^\mu$ that characterise the theory. The anti-symmetry of the field strength $F_{\nu\mu}$ in (\[aderiv2\]) implies that (\[aderiv3\]) will follow if $$\label{g1} {{\cal G}}^\nu=-{\partial}^\nu+(\eta+v)^\nu\bigl({\alpha}(n{{{\cdot}}}{\partial})+{\beta}(v{{{\cdot}}}{\partial})\bigr)\,.$$ In which case the second order operator ${{\cal G}}{\cdot}{\partial}$ is $$\label{gpa} {{\cal G}}{\cdot}{\partial}=\nabla^2+{\beta}(v{{\cdot}}{\partial})^2+({\alpha}-1)(\eta{{\cdot}}{\partial})^2+({\alpha}+{\beta})(v{{\cdot}}{\partial}) (\eta{{\cdot}}{\partial})\,,$$ where $\nabla={\partial}_i{\partial}_i$. We have seen that it is essential for the gauge invariance of the charges that this operator has no time derivatives in it. This follows only if ${\alpha}=1$ and ${\beta}=-1$. Hence we see that $$\label{g2} {{\cal G}}^\mu=(\eta+v)^\mu(n-v){{\cdot}}{\partial}-{\partial}^\mu\,.$$ Thus, to lowest order in the coupling, we have seen that $h^{-1}(x)=1- i\mathrm{e}R(x)$ where $R(x)$ is the sum of two terms: $$\label{horder1} R(x)=-\int_a^{x^0}\!ds\,(\eta+v)^\nu\frac{{\partial}^\mu F_{\mu\nu}}{{{\cal G}}{\cdot}{\partial}}(x_t) +\frac{{{\cal G}}^\mu A_\mu}{{{\cal G}}{\cdot}{\partial}}(x)\,.$$ As will become apparent when we perform perturbative test of our construction, this decomposition of the dressing into two terms reflects the two manifestations of the infra-red in QED that were discussed in the introduction: the soft divergence and the phase divergence. Indeed, the first (gauge invariant) part of $R(x)$ will be shown to be responsible for controlling the phase structure of charged matter; while the second term will be responsible for the soft dynamics of the charged particle. It is, perhaps, helpful to specialise to the static situation in order to get a better feel for the form of the dressing we have been constructing. In that case $v=0$ and we get $$\label{rstatic} R(x)=-\int_a^{x^0}\!ds\,\frac{{\partial}^i F_{i0}}{\nabla^2}(s,{{\boldsymbol x}}) +\frac{{\partial}_i A_i}{\nabla^2}(x)\,,$$ where $1/\nabla^2$ has been defined in (\[nabladef\]). The expected spatial non-locality of the charge is now manifest. More generally, the inverse to ${{\cal G}}{\cdot}{\partial}$ is given by $$\label{nonlocalv} \frac1{{{\cal G}}{\cdot}{\partial}}f(t,{{\boldsymbol x}})=-\frac{1}{4\pi}{\int\!d^3{{\boldsymbol y}}\;}\frac{f(t,{{\boldsymbol y}})}{\\|{{\boldsymbol x}}-{{\boldsymbol y}}\\|_v}\,,$$ where $$\label{vdef} \frac1{\\|{{\boldsymbol z}}\\|_v}:= \frac1{2\pi^2}\int\! d^3{{\boldsymbol k}}\,\frac{e^{i {{\boldsymbol k}}{{\cdot}}{{\boldsymbol z}}}} {V^v{\cdot}k}\,,$$ and $V^v_\mu=(\eta+v)_\mu(\eta-v){\cdot}k-k_\mu$. In order to go beyond this first order result we must take into account the fact that the fields which enter into the dressing are operators, and thus we need to know their commutation properties. Equal time commutators between the fields can be directly read off from the Lagrangian (\[qed\_lag\]). However, the non-locality displayed in (\[horder1\]) means that we need general space-time commutators between the fields. For the fully interacting theory there is no general approach to finding these commutators without first solving the whole theory. Charges as particles, though, only make physical sense in the asymptotic regime governed by the interacting Hamiltonian (\[hint\_as\]). In this regime, the relevant dynamics of the gauge field is described by the simpler set of equations: $$\begin{aligned} \label{eqofmotion} {\partial}^\nu F_{\nu\mu}&=& {\partial}_\mu B-{\mathrm{e}}J_\mu^{\mathrm{as}}\\ {\partial}^\mu A_\mu&=&-\xi B\,,\end{aligned}$$ where the matter coupling is now through the asymptotic current (\[j\_as\]). For this theory the space-time commutators can be explicitly constructed, as we will now show. In Feynman gauge ($\xi=1$) the potential satisfies $$\label{boxaj} \Box A_\mu=-{\mathrm{e}}J^{\mathrm{as}}_\mu\,.$$ The general solution to this is $$\label{sola} A_\mu(x)=A_\mu^{\mathrm{free}}(x)-{\mathrm{e}}\int\!d^4yD_{R}(x-y)J^{\mathrm{as}}_\mu(y)\,,$$ where $D_{R}(x-y)$ is the retarded Green’s function for the $\Box$ operator and $A_\mu^{\mathrm{free}}$ is a solution of the homogeneous equation $\Box A_\mu^{\mathrm{free}}=0$. From the form of this solution for the potential, and given the triviality of the asymptotic current commutations (\[jcomm\]), the space-time commutator for the potential in this interacting theory are seen to be the same as that for the free theory. The free field equation $\Box A_\mu^{\mathrm{free}}=0$ implies that $$\label{atrick} A_\mu^{\mathrm{free}}(x)=\int\!d^3z\bigl({\partial}_0^zD(x-z) A_\mu^{\mathrm{free}}(z)-D(x-z){\partial}_0A_\mu^{\mathrm{free}}(z)\bigr)$$ where $D(x-z)$ is the commutator function for free fields: $$\label{commfn} D(x-y) =-{\int\!\frac{d^3{{\boldsymbol k}}}{(2\pi)^3}\;}\frac1{\omega_k}e^{i{{\boldsymbol k}}{{\cdot}}({{\boldsymbol x}}-{{\boldsymbol y}})}\sin \Bigl(\omega_k(x^0-y^0)\Bigr)\,.$$ The identification in (\[atrick\]) is made by first observing that the right hand side is independent of $z^0$: setting $z^0=x^0$ then implies the result. Exploiting this $z^0$-independence, the commutator $[A_\mu^{\mathrm{free}}(x),A_\nu^{\mathrm{free}}(y)]$ is simply calculated by using (\[atrick\]) with $z^0=y^0$. Then the equal time commutation relations $[A_\mu(y),\dot{A}_\nu(z)]_{\mathrm{et}}=-ig_{\mu\nu}{\delta}({{\boldsymbol y}}-{{\boldsymbol z}})$ can be used. This, in conjunction with our observation that the free and asymptotic interacting fields have the same commutators, results in the space time commutators in Feynman gauge being: $$\label{acomm} [A_\mu(x),A_\nu(y)]=[A_\mu^{\mathrm{free}}(x),A_\nu^{\mathrm{free}}(y)]=-ig_{\mu\nu}D(x-y)\,.$$ Using this, the first order result (\[horder1\]) can be exponentiated (while still preserving the $a$-independence of the construction on physical states) to give $$\label{hexpo} h^{-1}(x)=e^{-i{\mathrm{e}}K(x)}e^{-i{\mathrm{e}}\chi(x)}\,,$$ where $$\label{softbitofh} \chi(x)=\frac{{{\cal G}}^\mu A_\mu}{{{\cal G}}{\cdot}{\partial}}(x)\,,$$ and $$\begin{aligned} \label{phasebitofh} K(x)&=&-\int_a^{x^0}ds\,(\eta+v)^\mu\frac{{\partial}^\nu F_{\nu\mu}}{{{\cal G}}{\cdot}{\partial}}(x_s)+ {\mathrm{e}}\int_{-\infty}^a ds (\eta+v)_\mu \frac{J^{\mathrm{as}\mu}}{{{\cal G}}{\cdot}{\partial}}(x_s)\\ &&\qquad\qquad\qquad-{{\textstyle\frac{1}{2}}}{\mathrm{e}}\gamma^{-1}u{{\cdot}}x{\int\!\frac{d^3{{\boldsymbol k}}}{(2\pi)^3}\;}\frac1{{{\boldsymbol k}}^2-({{\boldsymbol k}}{{\cdot}}{{\boldsymbol v}})^2}\end{aligned}$$ These results are, by construction, independent of $a$. Setting $a=-\infty$ gives (modulo the field independent, tadpole term) $$\label{phaseinh} K(x)=-\int_{-\infty}^{x^0}ds\,(\eta+v)^\mu\frac{{\partial}^\nu F_{\nu\mu}}{{{\cal G}}{\cdot}{\partial}}(x_s)\,,$$ which is a form adapted to performing perturbative calculations. The full derivation of these results will be presented elsewhere. In summary, we have seen that charged fields corresponding to charged particles moving with momentum, $p^\mu=m u^\mu$ may be described by $$\label{hexponentiated} \Psi_p(x)=e^{-i{\mathrm{e}}K_p(x)}e^{-i{\mathrm{e}}\chi_p(x)}\psi(x)\,,$$ where the dressing satisfies both the gauge transformation property (\[h\_trans\]) and the dressing equation (\[dresseqtn\]) with the four velocity appropriate for the particular momentum $p$. Our construction has shown that the on-shell Green’s functions of these dressed fields, taken at the correct points on their mass shells, should not suffer from any infra-red divergences. Such formal arguments must be tested: in Sect. 4 we will show that this prediction [@Lavelle:1997ty] is true to all orders in perturbation theory [@Bagan:1998kg]. We now conclude this construction of charges in QED with an observation and a comment. Note that in the static limit the charge field is given by $$\label{diracdressing} \exp\left(i{\mathrm{e}}\int_{-\infty}^{x^0}\!ds \frac{{\partial}^iF_{i0}}{\nabla^2}(s,{{\boldsymbol x}})\right) \exp\left(-i{\mathrm{e}}\frac{{\partial}_i A_i}{\nabla^2}\right)\psi(x)= \exp\left(i{\mathrm{e}}\int_{-\infty}^{x^0}\!ds \frac{{\partial}^iF_{i0}}{\nabla^2}(s,{{\boldsymbol x}})\right){\psi_D}(x)$$ where ${\psi_D}$ was Dirac’s proposal (\[dirac\]) for a static charged field based on the form of the electric field produced by such a charge. The additional term in (\[diracdressing\]), that follows from the more fundamental dressing equation approach presented here, does not alter the form of the electromagnetic field and was thus missed by Dirac. Its role in the cancellation of the phase divergence will be presented in Sect. 4. The construction of the charge above has been done in Feynman gauge. It is essential to verify that the results hold for the full Lorentz class of gauges. While it is true that the very construction of the charges (\[cmatter\]) ensures gauge invariance, it is not immediately clear that this is equivalent to the independence of the construction from the gauge fixing parameter $\xi$. In particular, changing $\xi$ from one will modify the form of the space-time commutators, (\[acomm\]), and thus could have a potentially non-trivial impact on the construction of the dressing. A full proof of the gauge invariance of these dressed fields will be presented elsewhere, however, in Sect 4, we shall demonstrate perturbatively that the $n$-point Green’s functions constructed out of the charged fields are manifestly $\xi$-independent. Before making these tests, we will now present an alternative approach to the physical restrictions on dressings. Heavy Charges ============= We have seen in the last section that the construction of the dressing for a charged particle depends on the velocity of the charge. In this section we wish to discuss, in more general terms, the role the velocity of a charged particle plays in the structure of the states of QED . We will then look at heavy charges, where velocity is also singled out, and see how the asymptotic regime, governed by the asymptotic Hamiltonian (\[hint\_as\]), can be characterised in terms of the mass of the charge. In addition, we will also see a further derivation for the dressing equation (\[dresseqtn\]). Moving Charges -------------- It is not possible [@Haag:1992hx] to talk about the space of states of QED without first identifying the observables of the theory and their algebra. The dual requirements of gauge invariance and locality on any physical observable have immediate and quite striking implications for the structure of QED. A familiar example of this is the fact that charge is superselected: the state space is a direct sum of different charge sectors and the action of any physical observable cannot change the charge. In order to derive this physically important result we recall that Gauss’ law (\[gausslaw\]) implies that on physical (gauge invariant) states there is a precise relation between the electric field and the charge density. In particular, the total charge $Q$ is given on such states by the spatial integral $$\label{chargeq} Q={\int\!d^3{{\boldsymbol x}}\;}{\partial}_iE_i(t,{{\boldsymbol x}})\,.$$ Using Gauss’ theorem we can relate this expression for the charge to a surface integral and, in particular, to one at spatial infinity. That is, if we define the electric flux in the direction $\hat{{{\boldsymbol x}}}$ by $$\label{flux} \mathcal{E}_i( \hat{{{\boldsymbol x}}}):=\lim_{R\to\infty} R^2E_i({{\boldsymbol x}}+R\hat{{{\boldsymbol x}}})\,,$$ then the total charge is given by the flux at infinity through $$\label{qinfinity} Q=\int_{S^2_\infty} d\boldsymbol{s}\cdot\boldsymbol{\mathcal{E}}\,.$$ The superselection properties of the charge then follow immediately from the obvious fact that any local observable will always commute with the flux at spatial infinity. This simple argument can be extended to show why velocity plays such a central role in the identification of charged particles. Recall that the the electric field of a charge whose present position is $y$ and moving with velocity ${{\boldsymbol v}}$ is $$\label{movinge} E_i({{\boldsymbol x}})=-\frac{{\mathrm{e}}\gamma}{4\pi} \frac{(x-y)_i}{\left(\gamma^2\Bigl(({{\boldsymbol v}}\cdot({{\boldsymbol x}}-{{\boldsymbol y}})\Bigr)^2 +\|{{\boldsymbol x}}-{{\boldsymbol y}}\|^2\right)^{3/2}}\,.$$ The electric flux at spatial infinity is then $$\label{fluxv} \mathcal{E}_i(\hat{{{\boldsymbol x}}})=-\frac{{\mathrm{e}}\gamma}{4\pi} \frac{\hat{x}_i}{\left(\gamma^2({{\boldsymbol v}}\cdot\hat{{{\boldsymbol x}}})^2 +1\right)^{3/2}}$$ Just as above for the total charge, this flux will commute with all local observables and thus can be used to characterise distinct sectors of the theory. As this flux only depends on the velocity of the charge we see that, in the asymptotic regime where charged particles exist, their velocity is a well defined quantum number. Within the class of local observables, this means that velocity is superselected. However, the very non-locality of the construction of the charges implies that we have to extend the algebra of observables to include non-local constructions such as dressings. As we will see in the next section, the perturbative construction of Green’s functions for charged matter will involve non-local interactions between different charges and thus allow charges with different velocities to interact. This does not mean, though, that the total charge $Q$ is no-longer superselected: the dressings are all chargeless and hence the interactions they induce commute with the charge. Heavy Matter ------------ In our discussion of the asymptotic interactions found in QED, we have simply taken the large $t$-limit of the interaction Hamiltonian in the interaction picture. A disadvantage of studying this limit, though, is that it does not supply a description of the scale for the onset of the asymptotic dynamics and thus of the domain within which velocity is a valid quantum number. More properly, then, we should investigate the limit as some dimensionless parameter gets large. In QED with massive matter, the natural parameter is the product of $t$ and the mass scale of the theory set by $m$, the lightest mass of the system. In this sense, the dynamics at large time is equivalent to that of a theory with heavy charges. Given the central role of the asymptotic dynamics in our programme, it is important to understand precisely how the asymptotic interactions emerge in the heavy sector, and to understand the significance of the dressing equation for heavy matter. It is well known [@weinberg:1995] that the infra-red structures found in QED are independent of the spin of the charged particles, thus it is instructive to see how the interaction (\[hint\_as\]) also emerges from the heavy sector of scalar QED. The matter part of the QED Lagrangian is now $$\label{scalarmatter} (D_\mu\phi)^\dagger(D^\mu\phi)-m^2\phi^\dagger\phi\,.$$ The heavy limit [@Georgi:1990um] can only be taken at specific points on the mass-shell of the particle. To this end, one introduces the rescaled fields $$\label{phitilde} {\tilde{\phi}}(x):=\sqrt{2m}e^{imu{{\cdot}}x}\phi(x)\,,$$ where we have chosen a four-velocity $u^\mu$ ($u{\cdot}u=1$) that will ultimately describe the velocity of the heavy charge. In terms of these new fields, the matter part of the Lagrangian becomes $$\label{scalarmatter2} i{\tilde{\phi}}^\dagger u^\mu D_\mu{\tilde{\phi}}+\frac1{2m}(D_\mu{\tilde{\phi}})^\dagger(D^\mu{\tilde{\phi}})\,.$$ In the large $m$-limit only the first term survives and the equations of motion for the heavy matter become $$\label{hmeqm} u^\mu D_\mu{\tilde{\phi}}=0\,.$$ The interaction Hamiltonian is easily identified in this limit and is constructed out of the current $$\label{heavyj} J^\mu_{\mathrm{heavy}}(x)=u^\mu{\tilde{\phi}}^\dagger(x){\tilde{\phi}}(x)\,,$$ which has, as expected, precisely the same form as the asymptotic current (\[j\_as\]) for the specific point on the mass-shell described by $u^\mu$. Heavy charged matter also needs to be constructed out of the heavy matter through the process of dressing. Thus the heavy scalar charge is given by the gauge invariant field $$\label{hcharge} \tilde\Phi(x)=h^{-1}(x){\tilde{\phi}}(x)\,,$$ where the dressing is that appropriate to a charge moving with the given four velocity $u^\mu$: $$\label{heavyde} u^\mu{\partial}_\mu(h^{-1})=-i{\mathrm{e}}h^{-1} u^\mu A_\mu\,.$$ This dressing equation, in conjunction with the equation of motion for the heavy matter (\[hmeqm\]) implies that $$\label{hceqm} u^\mu {\partial}_\mu\tilde\Phi=0\,.$$ That is, the state created by the field $\varphi=e^{-imu{{\cdot}}x}\tilde\Phi$ is an eigenstate of the momentum operator: $$\label{eigenstate} P_\mu\varphi\|0\rangle=mu_\mu\varphi\|0\rangle\,,$$ i.e., as claimed, it is a particle with momentum $mu^\mu$. Having seen that the heavy sector characterises the regime described by the asymptotic Hamiltonian (\[hint\_as\]), we can now identify the onset of the asymptotic dynamics, and thus the emergence of a charged particle interpretation, with the kinematical regime where the lightest mass $m$ of the matter dominates the momentum flow in any process. This is precisely the domain of the eikonal approximation in soft photon amplitudes [@sterman:1993] whereby the photon has a spin-independent coupling to the matter, which acts as an external point-like source. It is also the domain where scattering theory applies and we will now test our construction. Perturbation Theory =================== Our aim now is to show how we may apply the physical, dressed fields we derived in the last section to remove infra-red divergences already at the level of Green’s functions. We will start with a one-loop example, where we will classify the various sorts of mass singularities present in an unbroken Abelian gauge theory. Then we will demonstrate [@Bagan:1998kg] their explicit cancellation when physical, dressed fields are used. Finally we will show that this cancellation is in fact general and occurs at all orders. In this review we will solely concern ourselves with the IR-structure and not with the renormalisation of UV-singularities — full calculations of the renormalisation constants associated with the physical propagators of spinor and scalar QED were, however, given in Ref.’s [@Bagan:1997su] and [@Bagan:1997dh] respectively. Ref.’s [@weinberg:1995; @muta:1987; @sterman:1993] offer introductions to infra-red divergences in perturbation theory. One-Loop Calculations --------------------- The full spectrum of infra-red divergences that characterise gauge theories where the charged particles are massive all occur in pair creation. We thus start by briefly reviewing this process for undressed matter fields. We will use the theory of scalar QED since these divergences are insensitive to the spin of the particles and we may so avoid the technical complications of Diracology. The Feynman rules are [fig41]{} $$\parbox{20mm}{\begin{fmfgraph}(20,10) \fmfleft{in} \fmfright{out} \fmf{fermion,lab=$p$}{in,out} \end{fmfgraph}}\; = {\displaystyle \frac{i}{p^2-m^2}} \,,\quad \parbox{18mm}{\begin{fmfgraph}(18,10) \fmfleft{in} \fmfright{out} \fmf{photon,lab=$k$}{in,out} \fmfv{lab=$\mu$,lab.angle=90}{in} \fmfv{lab=$\nu$,lab.angle=90}{out} \end{fmfgraph}} \; ={\displaystyle \frac{-ig_{\mu\nu}}{k^2}}\,,\quad \parbox{18mm}{\begin{fmfgraph}(18,20) \fmfleft{upp} \fmfright{in,out} \fmf{fermion,lab=p}{in,vtx} \fmf{fermion,lab=q}{vtx,out} \fmf{photon}{vtx,upp} \fmfv{lab=$\mu$,lab.angle=90}{upp} \end{fmfgraph}}\; =i{\mathrm{e}}(p+q)^\mu$$ Fig. 4.1: The Feynman rules. We have chosen to work in Feynman gauge, but the physical results will be shown to be gauge invariant below. The seagull ($\phi^{}\phi AA$) vertex does not alter the spin independent IR-behaviour and is therefore irrelevant to our purpose. ### One-Loop Pair Creation Consider now pair creation from a classical source[^10] as shown in Fig. 4.2. [fig42]{} $${{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes,tension=1.5}{source,ver} \fmf{vanilla}{ver,v1} \fmfv{label=$q$,label.angle=-80,label.dist=.1w}{v1} \fmf{vanilla}{ver,v2} \fmfv{label=$p$,label.angle=80,label.dist=.1w}{v2} \fmf{fermion}{v1,pone} \fmf{fermion}{v2,ptwo} \fmffreeze \fmf{photon,tension=1}{v1,v2} \end{fmfgraph}}}$$ Fig. 4.2:* Covariant pair production diagram. The vertex function corresponding to Fig. 4.2 is given in Feynman gauge by $$\label{gamone} \Gamma= \frac{-i{\mathrm{e}}^2}{[p^2-m^2][{q}^2-m^2]} {\int\!\frac{d^4k}{(2\pi)^4}}\frac{(2q+k)^\mu(2p-k)^\nu}{[(q+k)^2-m^2+i{\epsilon}][ (p-k)^2-m^2+i{\epsilon}]}\frac{g_{\mu\nu}}{k^2+i{\epsilon}} \,.$$ Naive power counting tells us that this does not have an infra-red divergence when the outgoing particles are off-shell. However, if we extract a simple pole for each of our outgoing particles, then the on-shell residue may be seen to be IR-divergent $$\label{resone} \Gamma_{\hbox{\scriptsize IR}}= \frac{i{\mathrm{e}}^2}{[p^2-m^2][{q}^2-m^2]} {\int\!\frac{d^4k}{(2\pi)^4}}\frac{{q}^\mu p^\nu}{(q\cdot k+i{\epsilon}) (p\cdot k-i{\epsilon})}\frac{g_{\mu\nu}}{k^2+i{\epsilon}} \,,$$ where we have dropped higher powers of $k$ as they do not lead to IR-divergences (this means that we are henceforth restricting ourselves to loop momenta smaller than some cutoff). The easiest way to calculate these divergences is to perform the $k_0$ integral using Cauchy’s theorem. The four poles are: $$k_0=\pm{\vert {\boldsymbol{k}}\vert} \mp i{\epsilon}\,,\qquad k_0={\boldsymbol{q}}\cdot {\boldsymbol{k}}/q_0 -i{\epsilon}\,,\qquad k_0={\boldsymbol{p}}\cdot {\boldsymbol{k}}/p_0 +i{\epsilon}\,.$$ The results of the two different sets of poles have different physical interpretations. To see this distinction, let us take them one at a time. The contribution from the $k^2$ poles to (\[resone\]) is easily found to be $$\label{restwo} \frac{{\mathrm{e}}^2}{(p^2-m^2)({q}^2-m^2)}\frac1{8\pi^2}\frac1{{\vert { \boldsymbol{v}_{\hbox{\scriptsize rel}}}\vert}}\log\left(\frac{1+{\vert { \boldsymbol{v}_{\hbox{\scriptsize rel}}}\vert}} {1-{\vert {\boldsymbol{v}_{\hbox{\scriptsize rel}}}\vert}}\right) \int \frac{d{\vert {\boldsymbol{k}}\vert}}{{\vert { \boldsymbol{k}}\vert}} \,,$$ where $\boldsymbol{v}_{\hbox{\scriptsize rel}}$ is the relative velocity between the two charged particles [@weinberg:1995]. In (\[restwo\]) there is a logarithmic divergence in the small ${\vert {\boldsymbol{k}}\vert}$ limit, i.e., there is a divergent contribution from soft virtual photons. This is called a soft divergence. The other poles are often neglected as they correspond to divergences in the (unobservable) phase of the Green’s function, which cancel in cross-sections. These structures yield in (\[resone\]) $$\frac{-{\mathrm{e}}^2}{(p^2-m^2)({q}^2-m^2)} \frac1{4\pi^2} \int^1_{-1} \frac{du}{{\vert {\boldsymbol{v}_{\hbox{\scriptsize rel}}}\vert}u-i{\epsilon}} \int \frac{d{\vert {\boldsymbol{k}}\vert}}{{\vert {\boldsymbol{k}}\vert}} \,.$$ An IR-divergence is again visible. To perform the $u$-integral we can now employ the relation $$\label{dlt} \frac1{u-i{\epsilon}}=\mbox{\textsf{PV}}\frac1u +i\pi \delta(u)\,,$$ where denotes the principle value. Only the last term in (\[dlt\]) contributes as the other leads to an odd integral in $u$. We so obtain $$\label{resthree} \frac{-i{\mathrm{e}}^2}{(p^2-m^2)({q}^2-m^2)}\frac1{4\pi}\frac1{{\vert { \boldsymbol{v}_{\hbox{\scriptsize rel}}}\vert}} \int \frac{d{\vert {\boldsymbol{k}}\vert}}{{\vert {\boldsymbol{k}}\vert}} \,.$$ The extra factor of $i$ here, compared to (\[restwo\]), betrays the fact that this divergence occurs in the (unobservable) phase. Such singularities are called phase divergences. (Note that this is why we chose a pair creation process: if we had taken a scattering process then both of the poles in ${\boldsymbol{p}}_i \cdot {\boldsymbol{k}}$ would be in the same half-plane and could be neglected.) Now we want to demonstrate that the use of the correct, gauge invariant fields removes both of these divergences. ### Perturbation Theory with Dressed Fields Using dressed fields means calculating Green’s functions of the fields given in (\[hexponentiated\]). Since the dressings explicitly depend on the coupling, ${\mathrm{e}}$, we must also expand the dressings as well as including the usual interaction vertices. Thus we introduce new vertices and hence new diagrams. The diagrammatic rules for dressed Green’s functions are then just the standard ones augmented by the extra vertices which come from expanding the dressings in powers of the coupling. The two factors in the dressing each yield a different vertex structure. The Feynman rules corresponding to the dressings are given in Fig. 4.3. [fig43]{} $$\parbox{20mm}{\begin{fmfgraph}(20,15) \fmfleft{i}\fmfright{o}\fmftop{g}\fmf{fermion,label=$p- k$}{i,o} \fmf{photon,tension=0,label=$\searrow^{{\displaystyle k}}$,label.side=left}{g,o} \fmflabel{$\mu$}{g} \fmfv{decor.shape=circle,decor.filled=empty,decor.size=2mm}{o} \end{fmfgraph}} \quad ={\displaystyle \frac{{\mathrm{e}}V_p^\mu}{V_p\cdot k}}\,,\qquad\quad \parbox{20mm}{\begin{fmfgraph}(20,20) \fmfleft{i}\fmfright{o}\fmftop{g}\fmf{fermion,label=$p- k$}{i,o} \fmf{photon,tension=0,label=$ \searrow^{{\displaystyle k}}$,label.side=left}{g,o} \fmflabel{$\mu$}{g} \fmfv{decor.shape=cross,decor.size=3mm}{o} \end{fmfgraph}}\quad ={\displaystyle \frac{{\mathrm{e}}W^\mu_p}{V_p\cdot k}} \,.$$ Fig. 4.3: The Feynman rules from expanding the dressings. The first vertex comes from the soft ($\chi$) part of the dressing, and the latter corresponds to the phase ($K$) term. The dressings are, of course, dependent upon the momentum of the particle being dressed and so here we have defined $$\label{vwdef} V^\mu_p:= (\eta+v)^\mu(\eta-v){\cdot}k - k^\mu \,,\quad W_p^\mu:= \frac{(\eta+v){\cdot}k k^\mu - (\eta+v)^\mu k^2}{k{\cdot}(\eta+v)}$$ where $v=(0,\boldsymbol{v})$ is the velocity of a particle with momentum $p=m\gamma(1, \boldsymbol{v})$. Since the dressed fields are, by construction, gauge invariant, so are their (connected) Green’s functions. In the connected vertex function, there are two further diagrams with extra interaction vertices, Fig.’s 4.4.b-c. The remaining diagrams of Fig. 4.4 come from expanding both parts of the dressing. [fig44]{} $$\begin{array}{ccc} {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes,tension=1.5}{source,ver} \fmf{vanilla}{ver,v1} \fmfv{label=$ q$,label.angle=-80,label.dist=.1w}{v1} \fmf{vanilla}{ver,v2} \fmfv{label=$ p$,label.angle=80,label.dist=.1w}{v2} \fmf{fermion}{v1,pone} \fmf{fermion}{v2,ptwo} \fmffreeze \fmf{photon,tension=1}{v1,v2} \end{fmfgraph}}\atop{(a)}} & {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,pone} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,ptwo} \fmffreeze \fmf{photon,left}{v1,v2} \end{fmfgraph}}\atop{(b)}} & {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,ptwo} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,pone} \fmffreeze \fmf{photon,left}{v1,v2} \end{fmfgraph}}\atop{(c)}} \\ [5mm] {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,u3} \fmf{vanilla,tension=3}{u3,ptwo} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,v3} \fmf{vanilla,tension=3}{v3,pone} \fmffreeze \fmf{photon,right=0.15}{pone,ptwo} \fmfv{decor.shape=circle,decor.filled=empty,decor.size=2mm}{pone} \fmfv{decor.shape=circle,decor.filled=empty,decor.size=2mm}{ptwo} \end{fmfgraph}}\atop{(d)}} & {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,u3} \fmf{vanilla,tension=3}{u3,ptwo} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,v3} \fmf{vanilla,tension=3}{v3,pone} \fmffreeze \fmf{photon,right}{u2,ptwo} \fmfv{decor.shape=circle,decor.filled=empty,decor.size=2mm}{ptwo} \end{fmfgraph}}\atop{{(e)}}} & {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,u3} \fmf{vanilla,tension=3}{u3,ptwo} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,v3} \fmf{vanilla,tension=3}{v3,pone} \fmffreeze \fmf{photon,left}{v2,pone} \fmfv{decor.shape=circle,decor.filled=empty,decor.size=2mm}{pone} \end{fmfgraph}}\atop{(f)}} \\ [5mm] {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,u3} \fmf{vanilla,tension=3}{u3,ptwo} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,v3} \fmf{vanilla,tension=3}{v3,pone} \fmffreeze \fmf{photon,right=0.15}{pone,ptwo} \fmfv{decor.shape=cross,decor.size=3mm}{pone} \fmfv{decor.shape=cross,decor.size=3mm}{ptwo} \end{fmfgraph}}\atop{(g)}} & {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,u3} \fmf{vanilla,tension=3}{u3,ptwo} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,v3} \fmf{vanilla,tension=3}{v3,pone} \fmffreeze \fmf{photon,right}{u2,ptwo} \fmfv{decor.shape=cross,decor.size=3mm}{ptwo} \end{fmfgraph}}\atop{{(h)}}} & {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,u3} \fmf{vanilla,tension=3}{u3,ptwo} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,v3} \fmf{vanilla,tension=3}{v3,pone} \fmffreeze \fmf{photon,left}{v2,pone} \fmfv{decor.shape=cross,decor.size=3mm}{pone} \end{fmfgraph}}\atop{(i)}} \\ [5mm] {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,u3} \fmf{vanilla,tension=3}{u3,ptwo} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,v3} \fmf{vanilla,tension=3}{v3,pone} \fmffreeze \fmf{photon,right=0.15}{pone,ptwo} \fmfv{decor.shape=circle,decor.filled=empty,decor.size=2mm}{pone} \fmfv{decor.shape=cross,decor.size=3mm}{ptwo} \end{fmfgraph}}\atop{(j)}} & {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,u1,u2} \fmf{vanilla}{u2,u3} \fmf{vanilla,tension=3}{u3,ptwo} \fmf{vanilla}{ver,v1,v2} \fmf{vanilla}{v2,v3} \fmf{vanilla,tension=3}{v3,pone} \fmffreeze \fmf{photon,right=0.15}{pone,ptwo} \fmfv{decor.shape=cross,decor.size=3mm}{pone} \fmfv{decor.shape=circle,decor.filled=empty,decor.size=2mm}{ptwo} \end{fmfgraph}}\atop{(k)}} & \\ \end{array}$$ Fig. 4.4: All one-loop Feynman diagrams in the pair creation process which contain IR-divergences. Diagrams $(a)-(c)$ are covariant; $(d)-(f)$ involve the perturbative expansion of the $\chi$ term in the dressing; $(g)-(i)$ comes from expanding the $K$ term; finally the diagrams $(j)$ and $(k)$ are cross terms from expanding both dressing structures. Our procedure is to extract the IR-divergences in the on-shell residue after we have taken out a pole for each external leg. For simplicity we work in Feynman gauge, however, the gauge invariance of our dressed Green’s functions will be fully apparent in our final results. We will employ dimensional regularisation since it preserves gauge invariance. We may neglect the sorts of diagram shown in Fig. 4.5: those of type (a) are massless tadpoles (and so vanish in dimensional regularisation), type (b) do not yield IR-divergences when we extract the poles, which reflects the spin independence of the infra-red structure, and diagrams of type (c) are also found not to yield IR-divergences in the residue. [fig45]{} $${{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes,tension=1.5}{source,ver} \fmf{fermion}{ver,pone} \fmf{fermion}{ver,v3} \fmf{phantom, tension=20}{v3,ptwo} \fmffreeze \fmf{photon,tension=0.9,right=90}{v3,v3} \fmfv{decor.shape=square,decor.filled=empty,decor.size=2mm}{v3} \end{fmfgraph}}\atop{(a)}} \qquad {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,v2} \fmf{fermion}{v2,pone} \fmf{vanilla}{ver,v1} \fmf{fermion}{v1,ptwo} \fmffreeze \fmf{photon,right=90,tension=0.9}{v1,v1} \end{fmfgraph}}\atop{(b)}} \qquad {{\begin{fmfgraph}(30,25) \fmfleft{source} \fmfright{ptwo,pone} \fmf{dashes}{source,ver} \fmf{vanilla}{ver,v2} \fmf{fermion}{v2,pone} \fmf{vanilla}{ver,v1} \fmf{fermion}{v1,ptwo} \fmffreeze \fmf{photon}{pone,v1} \fmfv{decor.shape=square,decor.filled=empty,decor.size=2mm}{pone} \end{fmfgraph}}\atop{(c)}}$$ Fig. 4.5: Classes of diagrams which do not contribute IR-divergences to the residue. The square vertex here and below signifies that the generic contributions of both parts of the dressing are meant. Let us now consider the contribution from the $\chi$ part of the dressing. Since the other term in the dressing is itself gauge invariant, this part, taken together with the covariant diagrams, must be gauge invariant. The diagrams 4.4.e-f have two poles already, but Fig. 4.4.d appears not to have any poles: it is, however, already IR-divergent even off-shell. This has nothing to do with the usual IR-divergences which, we recall, arise when we go on-shell. We can, however, extract poles from such diagrams as we now describe. ### Factorisation This sort of off-shell IR-divergence appears when one or more photons are exchanged between two dressings. We will call such diagrams rainbow diagrams. When we extract poles associated with the external legs, we find the residue has an on-shell IR-divergence. Together with the other diagrams of Fig. 4.4, the divergences from the rainbow diagrams will then (if our predictions of the IR-finiteness of the physical Green’s functions is correct) cancel the soft and phase divergences of the covariant (dressing independent) diagrams. This procedure makes much use of the following algebraic identity: $$\label{ident} \frac1{(p-k)^2-m^2}=\frac1{p^2-m^2}\left[1+ \frac{2p\cdot k-k^2}{(p-k)^2-m^2}\right] \,.$$ The integrand from the Feynman rules for Fig. 4.4.d may be thus rewritten as (we do not write superfluous factors) $$\begin{aligned} \label{rnone} & & \!\!\!\!\!\! \!\!\!\!\!\! {\int\!\frac{d^4k}{(2\pi)^4}}\frac{V_p\cdot V_{q}}{V_p\cdot k\,V_q\cdot k\, k^2} \frac1{[{({p}+k)^2-m^2+i{\epsilon}}][{({q}-k)^2-m^2+i{\epsilon}}]} \nonumber \\ & =& \!\!\!\! {\int\!\frac{d^4k}{(2\pi)^4}}\frac{V_p\cdot V_{q}}{V_p\cdot k\,V_q\cdot k\, k^2} \frac1{p^2-m^2}\frac1{{({q}-k)^2-m^2+i{\epsilon}}}\left[1-\frac{2p\cdot k+k^2}{{({p}+k)^2-m^2+i{\epsilon}}} \right]\!.\end{aligned}$$ The first term in the square bracket here is the only relevant one. Before we consider it, let us show that the second term does not contribute to the residue: extracting the pole in $1/(q^2-m^2)$ we obtain from this term $$\frac{1}{[p^2-m^2][q^2-m^2]} {\int\!\frac{d^4k}{(2\pi)^4}}\frac{V_p\cdot V_{q}}{V_p\cdot k\,V_q\cdot k \,k^2} \frac{k^2-2p\cdot k}{{({p}+k)^2-m^2+i{\epsilon}}} \left[1+\frac{2q\cdot k-k^2}{{({q}-k)^2-m^2+i{\epsilon}}} \right] \,.$$ We have extracted the two poles and may now go on-shell: but the square bracket here, and hence the contribution of this second term, vanishes on-shell. Returning now to (\[rnone\]) the contribution of the first term is easily found to be $$\label{fall} \frac1{[p^2-m^2][q^2-m^2]} {\int\!\frac{d^4k}{(2\pi)^4}}\frac{V_p\cdot V_{q}}{V_p\cdot k\,V_q\cdot k\, k^2} \left[1+\frac{2q\cdot k-k^2}{{({q}-k)^2-m^2+i{\epsilon}}} \right] \,.$$ However, the one in the square bracket here is just a massless tadpole and thus vanishes. On-shell the second term becomes $-1$ and we obtain the final result that, as far as IR-divergent terms in the residue are concerned, we have $$\begin{aligned} {\int\!\frac{d^4k}{(2\pi)^4}}&&\!\!\!\!\!\! \!\!\!\!\!\! \frac{V_p\cdot V_{q}}{V_p\cdot k\,V_q\cdot k\, k^2} \frac1{[{({p}+k)^2-m^2+i{\epsilon}}][{({q}-k)^2-m^2+i{\epsilon}}]} \nonumber \\ & \to& \frac1{[p^2-m^2][q^2-m^2]}\times (-1) {\int\!\frac{d^4k}{(2\pi)^4}}\frac{V_p\cdot V_{q}}{V_p\cdot k\,V_q\cdot k\, k^2} \,,\end{aligned}$$ i.e., the rainbow line has been stripped off the original integral to yield a factor of $$\label{Cdefn} C_{pq}=-{\int\!\frac{d^4k}{(2\pi)^4}}\frac{V_p\cdot V_{q}}{V_p\cdot k\,V_q\cdot k\, k^2} \,,$$ times the Feynman rules for the diagram without this line. It may be demonstrated that this property of the factorisation of rainbow lines is completely general. For a diagram with $n$ rainbow lines, as far as the soft divergences in the on-shell residue of the poles are concerned, the rainbow lines may be stripped off and replaced by $(C_{pq})^n$. A diagrammatic proof of the factorisation property will be published elsewhere. We will make extensive use of this property in the all orders proof below. ### Tadpoles With the factorisation property we are able to calculate all of the one-loop diagrams. However, it seems worthwhile to first remark on the identification of IR- and UV-singularities since, as is well known, in dimensional regularisation we can exchange IR- and UV-divergences because massless tadpoles vanish in this scheme. We have used this cancellation above in (\[fall\]). Using this identity the badly defined, soft, off-shell divergences were removed and replaced by terms which yield IR divergences only when we went on shell. These on-shell singularities, as we will soon show, help to cancel the soft on-shell divergences of the other diagrams. This replacement of ill-defined, off-shell singularities in rainbow diagrams is the only place in our calculations where we drop tadpoles. At no stage do we alter structures which first develop soft or phase divergences only on-shell. However, there are other massless tadpole diagrams associated with dressed Green’s functions, as well as the tadpoles which we dropped above in Eq. \[fall\]. Let us now consider their interplay — firstly for the dressed propagator. Fig. 4.6 shows the generic diagrams which enter here: [fig46]{} $${{\begin{fmfgraph}(30,25) \fmfleft{in} \fmfright{out} \fmf{vanilla}{in,v1} \fmf{vanilla}{v1,v2} \fmf{vanilla}{v2,out} \fmffreeze \fmf{photon,left}{v1,v2} \end{fmfgraph}}\atop{(a)}} \qquad {{\begin{fmfgraph}(30,25) \fmfleft{in} \fmfright{out} \fmf{vanilla}{in,v1} \fmf{vanilla}{v1,out} \fmffreeze \fmfv{decoration.shape=square,decoration.filled=empty,decor.size=2mm}{in} \fmf{photon,left=.75}{in,v1} \end{fmfgraph}}\atop{(b)}} \qquad {{\begin{fmfgraph}(30,25) \fmfleft{in} \fmfright{out} \fmf{vanilla}{in,out} \fmffreeze \fmfv{decoration.shape=square,decoration.filled=empty,decor.size=2mm}{in} \fmfv{decoration.shape=square,decoration.filled=empty,decor.size=2mm}{out} \fmf{photon,left=.5}{in,out} \end{fmfgraph}}\atop{(c)}} \qquad {{\begin{fmfgraph}(30,25) \fmfleft{in} \fmfright{out} \fmf{phantom,tension=20}{in,v1} \fmf{vanilla}{v1,out} \fmffreeze \fmf{photon,tension=1,right=90}{v1,v1} \fmfv{decoration.shape=square,decoration.filled=empty,decor.size=2mm}{v1} \end{fmfgraph}}\atop{(d)}}$$ Fig. 4.6: The classes of diagrams in the one-loop dressed propagator. Here there are massless tadpole diagrams, Fig. 4.6.d, and rainbow diagrams, Fig. 4.6.c. The propagator has, of course, no phase divergence, so let us solely consider the contribution of the soft part of the dressing here[^11]. Using (\[ident\]) to strip off the rainbow line we obtain from this diagram (for the soft part of the dressing) the massless tadpole integral $$\label{klapsch} \frac{i{\mathrm{e}}^2}{p^2-m^2}{\int\!\frac{d^4k}{(2\pi)^4}}\frac{V_p^2}{(V_p\cdot k)^2}\frac{-i}{k^2} \,,$$ which we can drop in dimensional regularisation. However, the contribution of the two massless tadpole diagrams, of the type shown in Fig. 4.6.d, is rapidly found from the Feynman rules to be equal to (\[klapsch\]) but with the opposite sign and so cancels it exactly[^12]. The remaining integrals conspire to yield an infra-red finite propagator as has been shown elsewhere in detail [@Bagan:1997su; @Bagan:1997dh]. This exact cancellation does not carry through to higher Green’s functions. In general a gauge invariant subset of tadpoles survives. This appears to be a subtlety concerned with taking products of distributions. It has been suggested that such problems may perhaps be removed by using the Hertz formulation of QED [@dEmilio:1984]. The removal of these tadpoles in a general fashion is a worthy object for further study, here we content ourselves with noting that they vanish in dimensional regularisation and that we do not drop tadpoles connected with on-shell infra-red divergences. ### Cancellation of the Soft and Phase Divergences Putting together the one-loop diagrams relevant to the soft structure, i.e., , extracting a pole for each of the external legs and going on-shell, we obtain the following terms in the residue which are by power counting IR-divergent: $$\begin{aligned} \label{indone} I_{\hbox{\scriptsize IR}}= {\int\!\frac{d^4k}{(2\pi)^4}}\Bigg\{\Bigg. && \!\!\!\!\!\! \!\!\!\! \left(\frac{{p}^\mu}{p\cdot k}-\frac{{V_q}^\mu}{{V_q}\cdot k}\right) \frac{g_{\mu\nu}}{k^2} \left(\frac{{V_p}^\nu}{V_p\cdot k}-\frac{{q}^\nu}{{q}\cdot k}\right) \nonumber \\ && \quad - \left(\frac{{p}^\mu}{p\cdot k}-\frac{{q}^\mu}{{q}\cdot k}\right) \frac{g_{\mu\nu}}{k^2} \left(\frac{{p}^\nu}{p\cdot k}-\frac{{q}^\nu}{q\cdot k}\right) \Bigg.\Bigg\} \,,\end{aligned}$$ where we have removed a factor of $i{\mathrm{e}}^2/[(p^2-m^2)(q^2-m^2)]$. This is a gauge invariant result: replacing the Feynman gauge propagator by any more general form necessarily brings in a factor of $k_\mu$ or $k_\nu$ which vanishes on multiplication into the brackets in (\[indone\]). Similar gauge invariant structures are obtained in the study of the dressed propagator and in other physical vertex functions. We now have to show that the soft divergences seemingly apparent in (\[indone\]) in fact cancel. The cancellation may be demonstrated by performing the integrals explicitly, using the methods outlined above. However, it is simpler to realise that $$\frac{V_p^\mu}{V_p\cdot k}= \frac{(\eta+v)^\mu(\eta-v)\cdot k -k^\mu}{{\boldsymbol{k}}^2 -({\boldsymbol{k}}\cdot{\boldsymbol{v}})^2} \,,$$ which, at the soft pole, $k^2=0$, we may write as $$\frac{V_p^\mu}{V_p\cdot k}= \frac{(\eta+v)^\mu(\eta-v)\cdot k -k^\mu}{(k\cdot \eta)^2 -({\boldsymbol{k}}\cdot{\boldsymbol{v}})^2} \,.$$ We may drop the $k^\mu$ term in the numerator here (cf, the argument for the gauge invariance of Eq. \[indone\]). So effectively we have $$\frac{V_p^\mu}{V_p\cdot k}= \frac{(\eta+v)^\mu(\eta-v)\cdot k}{(k\cdot \eta)^2 -({\boldsymbol{k}}\cdot{\boldsymbol{v}})^2} =\frac{(\eta+v)^\mu}{(\eta+v)\cdot k} =\frac{p^\mu}{p\cdot k} \,,$$ where it is important to note that we have taken $p$ to be at the correct point on the mass shell, $p=m\gamma(\eta+ v)$, which defines the appropriate dressing. (If we were to go on-shell at a different point, this equality would not hold and the soft divergences do not cancel.) In all cases of gauge invariant Green’s functions, we are thus able to replace $$\frac{V_p^\mu}{V_p\cdot k}= \frac{p^\mu}{p\cdot k}\,,\qquad \hbox{\rm and}\qquad \frac{V_q^\mu}{V_q\cdot k}= \frac{q^\mu}{q\cdot k} \,,$$ as far as the soft divergences are concerned. This equality makes the cancellation of the soft divergences in (\[indone\]) and other dressed Green’s functions immediately apparent. Up till now we have not included the other gauge invariant part of the dressing. It is straightforward to check that the contribution of this structure to the pair creation process is gauge invariant. It may also be demonstrated that it does not spoil the above cancellation of soft divergences. (Some of the individual diagrams which involve this part of the dressing, Fig.’s 4.4.g-k, contain soft divergences, but overall they cancel.) The role of this structure is, we expect, to cancel the phase divergence of Fig. 4.4.a. The only phase divergent contribution from the entire dressing comes from a single diagram involving only the $K$-structure: Fig. 4.4.g. From the Feynman rules we obtain from this figure after factorisation $${\int\!\frac{d^4k}{(2\pi)^4}}\frac{W_p^\mu}{V_p{\cdot}k} \frac{W_q^\nu}{V_q{\cdot}k} \frac{g_{\mu\nu}}{k^2} = {\int\!\frac{d^4k}{(2\pi)^4}}\frac{(\eta+v){\cdot}(\eta+v')k^2}{V_p{\cdot}k V_q{\cdot}k [(\eta+v){\cdot}k-i\epsilon][(\eta+v'){\cdot}k +i\epsilon]} \,,$$ where we have not written the two poles $-i{\mathrm{e}}^2/([p^2-m^2][q^2-m^2])$ and have thrown away terms which do not yield phase divergences. This result may be readily seen to cancel exactly with (\[resthree\]) when we go on-shell at the right point. To summarise the one-loop results: the dressed Green’s functions are, by construction, gauge invariant. If we go on-shell at the points on the mass shells such that the momenta of the external lines and the velocities that define the various dressings agree, then the soft and phase divergences cancel. We now want to show that this holds at all orders. All Orders ---------- The physical dressed propagator was shown elsewhere [@Bagan:1997su; @Bagan:1997dh] to be IR-finite at one loop. However, we can show that this holds at all orders through a slight extension of the work of Jackiw and Soloviev [@soloviev:1968]. These authors used low energy theorems and spectral representations to study the behaviour of the propagator around the pole, $p^2=m^2$, in various gauges. In covariant gauges they regained the usual result (\[yennie\]), while in non-covariant gauges there is a pole if the following integral vanishes (this is their Eq. 3.41) $$\label{jacsol} F_R=\frac{{\mathrm{e}}^2}{(2\pi)^3}\int^4k e^{ik{{\cdot}}k}\theta(k_0) \delta(k^2) \frac{r_\mu r_\nu}{(r{{\cdot}}k)^2} \Pi^{\mu\nu}_R\,,$$ here $r$ corresponds to the mometum of the matter field and $\Pi$ is the photon propagator up to a factor of $i/k^2$. Now our dressed propagator, for a charge with momentum $r=m\gamma(\eta,\boldsymbol{v})$ corresponds to the usual matter propagator in the gauge where $$\Pi^{\mu\nu}_R = -g^{\mu\nu} +\frac{k^\mu (\eta+v)^\nu + (\eta+v)^\mu k^\nu }{k{{\cdot}}(\eta+v)} -\frac{k^\mu k^\nu}{(k{{\cdot}}(\eta+v))^2} \left[ V_r^2+2V_r{\cdot}k\right]\,.$$ taking the $\delta(k^2)$ factor of the integrand into account, it is easy to see that at the right point on the mass shell (\[jacsol\]) vanishes and the propagator has a pole. This was pointed out explicitly for the static case (Coulomb gauge) in Ref. [@soloviev:1968]. ### The Dressed Propagator Since we now know that the dressed propagator has a pole, we see that the wave function renormalisation constants are IR-finite. We can therefore multiply the dressed Green’s functions by them without introducing new IR-divergences. We will use this freedom to show the cancellation of soft and phase singularities in dressed Green’s functions at all orders. Our approach is diagrammatic. First we shall consider the types of diagrams which can introduce IR-divergences in the propagator, in this way we can describe the structure of the wave function renormalisation constants. The non-dressed propagator, i.e., just the covariant diagrams, can be represented at all orders in the following way: [ufig41]{} $$ \parbox{40mm}{\begin{fmfgraph}(40,15) \fmfleft{in}\fmfright{out} \fmf{vanilla,tension=2}{in,j} \fmf{vanilla}{j,jj} \fmf{vanilla,tension=3}{jj,v1} \fmf{vanilla}{v1,u} \fmf{vanilla,tension=2}{u,v2} \fmf{dashes}{v2,v3} \fmf{vanilla,tension=2}{v3,v4} \fmf{vanilla}{v4,out} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v1,u,v4} \end{fmfgraph}} \quad =:\frac i{p^2-m^2-\Sigma}\,.$$ When we take dressings into account, these chains must be supplemented by various possible structures which we now enumerate. The first is [ufig42]{} $$2\,\parbox{40mm}{\begin{fmfgraph}(40,15) \fmfleft{in}\fmfright{out} \fmf{vanilla,tension=2}{in,j} \fmf{vanilla}{j,jj} \fmf{vanilla,tension=3}{jj,v1} \fmf{vanilla}{v1,u} \fmf{vanilla,tension=2}{u,v2} \fmf{dashes}{v2,v3} \fmf{vanilla,tension=2}{v3,v4} \fmf{vanilla}{v4,v5} \fmf{brown_muck}{v5,out} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v1,u,v4} \end{fmfgraph}} \quad =:\frac{2i\tilde\Sigma}{p^2-m^2-\Sigma}\,.$$ where the 2 accounts for the possibility that dressing corrections, $\tilde\Sigma$, can be indistinguishably attached at either end in the scalar theory. There may of course also be such dressing corrections at both ends. This is diagrammatically [ufig43]{} $$\parbox{41mm}{ {\begin{fmfgraph}(42,10) \fmfleft{in}\fmfright{out} \fmf{brown_muck}{in,j} \fmf{vanilla}{j,jj} \fmf{vanilla,tension=3}{jj,v1} \fmf{vanilla}{v1,u} \fmf{vanilla,tension=2}{u,v2} \fmf{dashes}{v2,v3} \fmf{vanilla,tension=2}{v3,v4} \fmf{vanilla}{v4,v5} \fmf{brown_muck}{v5,out} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v1,u,v4} \end{fmfgraph}}} \quad =:\frac{i{\tilde\Sigma}^2}{p^2-m^2-\Sigma}\,.$$ Taking all of these possibilities together we see that these contribution to the propagator have the form $$=\frac{i(1+\tilde\Sigma)^2}{p^2-m^2-\Sigma}\,.$$ Of course we can also exchange one or more photons from one dressing to the next. This can be done for each and every one of the above sets of diagrams. We so obtain the following rainbow diagrams [ufig44]{} $$\parbox{42mm}{\begin{fmfgraph}(40,32) \fmfleft{in}\fmfright{out} \fmf{vanilla,tension=2}{in,j} \fmf{vanilla}{j,jj} \fmf{vanilla,tension=3}{jj,v1} \fmf{vanilla}{v1,u} \fmf{vanilla,tension=2}{u,v2} \fmf{dashes}{v2,v3} \fmf{vanilla,tension=2}{v3,v4} \fmf{vanilla}{v4,out} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v1,u,v4} \fmffreeze \fmf{photon,left=.82, lab=${ { \cdot \atop n }\atop \cdot}$, lab.sid=right}{in,out} \fmf{photon,left=.40}{in,out} \end{fmfgraph}} + 2\;\,\parbox{42mm}{\begin{fmfgraph}(40,15) \fmfleft{in}\fmfright{out} \fmf{vanilla,tension=2}{in,j} \fmf{vanilla}{j,jj} \fmf{vanilla,tension=3}{jj,v1} \fmf{vanilla}{v1,u} \fmf{vanilla,tension=2}{u,v2} \fmf{dashes}{v2,v3} \fmf{vanilla,tension=2}{v3,v4} \fmf{vanilla}{v4,v5} \fmf{brown_muck}{v5,v6} \fmf{phantom,tension=9}{v6,out} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v1,u,v4} \fmffreeze \fmf{photon,left=.82, lab=${ { \cdot \atop n }\atop\cdot}$, lab.sid=right}{in,out} \fmf{photon,left=.40}{in,out} \end{fmfgraph}} +\; \parbox{42mm}{{\begin{fmfgraph}(42,15) \fmfleft{in}\fmfright{out} \fmf{phantom,tension=9}{in,jjj} \fmf{brown_muck}{jjj,j} \fmf{vanilla}{j,jj} \fmf{vanilla,tension=3}{jj,v1} \fmf{vanilla}{v1,u} \fmf{vanilla,tension=2}{u,v2} \fmf{dashes}{v2,v3} \fmf{vanilla,tension=2}{v3,v4} \fmf{vanilla}{v4,v5} \fmf{brown_muck}{v5,v6} \fmf{phantom,tension=9}{v6,out} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v1,u,v4} \fmffreeze \fmf{photon,left=.82, lab=${ { \cdot \atop n }\atop\cdot}$, lab.sid=right}{in,out} \fmf{photon,left=.40}{in,out} \end{fmfgraph}}}$$ Using the factorisation property, summing over all possible numbers of rainbow lines, and taking into account the $1/n!$ symmetry factor which accompanies a diagram with $n$ rainbow lines, we can re-express this algebraically as $$\label{aop} \frac{i(1+\tilde\Sigma)^2}{p^2-m^2-\Sigma} \exp{\left(-C_{pp}\right)} \,,$$ where $-C_{pq}$ was defined in Eq. \[Cdefn\]. Of course there can be other diagrams such as [ufig45]{} $${\begin{fmfgraph}(35,13) \fmfleft{in}\fmfright{out} \fmf{vanilla}{in,j} \fmf{vanilla}{j,jj} \fmf{vanilla}{jj,out} \fmffreeze \fmf{photon,right=.7}{in,jj} \fmf{photon,left=.7}{j,out} \fmfv{decor.shape=square,decor.filled=empty,decor.size=2mm}{in} \fmfv{decor.shape=square,decor.filled=empty,decor.size=2mm}{out} \end{fmfgraph}}$$ where the dressing corrections overlap in a non-rainbow form, i.e., diagrams which are not one-particle irreducible even after rainbow lines are stripped off. Such diagrams do not yield poles and so we can drop them here. We neglect loops of matter fields since these also remove the soft divergences. Eq. \[aop\] is therefore our final result for the infra-red divergences in and the pole structure of the physical propagator. We see that the dressings factorise into a covariant and a non-covariant part. For the full non-covariant wave function renormalisation constants, we thus have the attractive result that $$\label{aoztwo} Z_2^p=Z_2^{\mathrm{cov}}(1+\tilde\Sigma)^2\exp(-C_{pp}) \,.$$ Although only a partial exponentiation of the dressing contribution is apparent, we know that it is in fact IR-finite at all orders. However, (\[aoztwo\]) will suffice to show that the pair creation vertex is finite to all orders. ### All Orders Pair Creation The general class of diagrams with double poles and possible IR-divergences in the pair creation process is [ufig46]{} $$\parbox{50pt}{\begin{fmfgraph}(49,27) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes,tension=0.7}{source,ver} \fmf{vanilla}{ver,v,v1} \fmf{dots,tension=0.8}{v1,v2,v3} \fmf{vanilla}{v3,v8} \fmf{fermion}{v8,uv} \fmf{vanilla,tension=5}{uv,vv} \fmf{brown_muck,tension=0.5}{vv,pone} \fmf{vanilla}{ver,u,v4} \fmf{dots,tension=0.8}{v4,v5,v6} \fmf{vanilla}{v6,v7} \fmf{fermion}{v7,vu} \fmf{vanilla,tension=5}{vu,v9} \fmf{brown_muck,tension=0.5}{v9,ptwo} \fmffreeze \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=5mm}{ver} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v1} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v4} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v3} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v6} \fmfdot{pone,ptwo} \fmf{photon, right=0.95}{pone,ptwo} \fmf{photon, right=0.18}{pone,ptwo} \end{fmfgraph}}\qquad \qquad\qquad\qquad\!\!\!{\scriptstyle\cdots n\cdots}\qquad \times \frac1{\sqrt{Z_2^p Z_2^{q}}}$$ where we have chosen to multiply the diagrams by an (IR-finite) factor: the inverse square root of the appropriate wave function renormalisation constant (as given by Eq. \[aoztwo\]) for each of the legs. This is useful as it makes the exponentiation of the infra-red divergences apparent. The diagrams we have retained are those which can yield IR-divergences, i.e., where there are possible covariant vertex corrections from one leg to another, covariant corrections on the external legs, dressing corrections at the end of the external legs and possible rainbow corrections from one dressing to the other. The use of a black blob, $\bullet$, here denotes that there may or may not be a dressing correction ($\tilde\Sigma$) at the ends of one or both of the lines. Other diagrams (e.g, where a line from a dressing connects to a covariant interaction vertex on the other line) will not give two poles or will be infra-red finite. We now factorise the rainbow dressings, this means (summing over all possible rainbow lines and including the $1/n!$ symmetry factor for a diagram with $n$ such lines) that we may write [ufig47]{} $$\parbox{44pt}{\begin{fmfgraph}(43,27) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes,tension=0.7}{source,ver} \fmf{vanilla}{ver,v,v1} \fmf{dots,tension=0.8}{v1,v2,v3} \fmf{vanilla}{v3,v8} \fmf{fermion}{v8,uv} \fmf{vanilla,tension=5}{uv,vv} \fmf{brown_muck,tension=0.5}{vv,pone} \fmf{vanilla}{ver,u,v4} \fmf{dots,tension=0.8}{v4,v5,v6} \fmf{vanilla}{v6,v7} \fmf{fermion}{v7,vu} \fmf{vanilla,tension=5}{vu,v9} \fmf{brown_muck,tension=0.5}{v9,ptwo} \fmffreeze \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=5mm}{ver} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v1} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v4} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v3} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v6} \fmfdot{pone,ptwo} \end{fmfgraph}}\qquad \qquad \times \exp\left(-C_{pq}\right) \frac1{\sqrt{Z_2^pZ_2^{q}}}$$ Since there may or may not be a dressing correction at the end of these lines, we may write these end factors as $(1+\tilde\Sigma)$. They clearly then cancel the $\tilde\Sigma$ dependence from each of the external leg wave function renormalisation factors as given by (\[aoztwo\]). Diagrammatically we thus have [ufig48]{} $$\parbox{40pt}{\begin{fmfgraph}(38,25) \fmfleft{source} \fmfright{pone,ptwo} \fmf{dashes,tension=0.8}{source,ver} \fmf{vanilla}{ver,v,v1} \fmf{dots}{v1,v2,v3} \fmf{vanilla}{v3,v8} \fmf{fermion}{v8,vv} \fmf{vanilla,tension=1.5}{vv,pone} \fmf{vanilla}{ver,u,v4} \fmf{dots}{v4,v5,v6} \fmf{vanilla}{v6,v7} \fmf{fermion}{v7,v9} \fmf{vanilla,tension=1.5}{v9,ptwo} \fmffreeze \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=5mm}{ver} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v1} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v4} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v3} \fmfv{decor.shape=circle,decor.filled=shaded,decor.size=4mm}{v6} \end{fmfgraph}}\qquad \qquad \times \exp\left( {-C_{pq}+\frac12 C_{pp}+ \frac12 C_{qq}}\right) \frac1{Z_2^{\mathrm{cov}}}$$ and we see that the dressing effects have exponentiated. Since the covariant diagrams which are left over are well known to exponentiate, we see that all the soft and phase effects exponentiate in the residue of the double poles. We have seen that they cancel at one loop, this now trivially implies that the dressed process is IR-finite at all orders at the level of Green’s functions. We stress that this holds for both soft and phase divergences. The extension of this approach to scattering and to higher vertices is straightforward. We have thus seen that the dressings we have introduced in Sect. 2 remove soft and phase divergences in massive Electrodynamics already at the level of Green’s functions and permit a particle interpretation. The immediate question now is whether these perturbative successes can be extended to QCD. In particular, we need to clarify the generalisation to a different class of mass singularities, collinear divergences, which arise in QCD. Since collinear divergences also characterise QED with massless charged particles, the next section is dedicated to a study of that theory. Massless Charges ================ Massless charges are an essential ingredient of the standard model. In QCD the colour charge of the massless gluon results in its self-interaction which in turn leads to asymptotic freedom. In addition, gluonic bremsstrahlung plays a dominant role in jet formation [@Dokshitser:1991wu]. However, such massless charges are poorly understood, even at the level of cross-sections, since the Block-Nordsieck procedure of summing over degenerate final states fails to yield finite results [@Doria:1980ak; @Di'Lieto:1981dt]. Massless charges can be modelled in QED by taking the limit of vanishing electron mass. As one would expect, QED then has many new and unusual features. In particular, massless electron pairs can now be created by expending arbitrarily small amounts of energy. This freedom leads to a new type of singularity, the collinear divergences. In perturbation theory these express themselves through the divergence of on-shell massless propagators. The inverse propagator, $(p-k)^2-m^2$, vanishes in the $m\to 0$ limit if $p$ is on-shell and $k$ is parallel to $p$. (Note that ${{\boldsymbol k}}$ is not necessarily small.) For further details see [@weinberg:1995]. The matter pairs, which can now be easily created, screen the initial charge and the effective coupling vanishes [@Espriu:1996sk]. Here we will see that the argument of Sect. 2 for the vanishing of the asymptotic interaction Hamiltonian, if correctly dressed matter is used, also applies in the massless limit. Finally we discuss some aspects of the construction of dressings in this limit. Collinear Asymptotic Dynamics ----------------------------- The asymptotic behaviour of the interaction Hamiltonian for massive QED was discussed in Sect. 2. In the massless case we will show that the interaction Hamiltonian in the distant past or future has a far richer structure than in the massive case. The interaction Hamiltonian, ${{\cal H}_{\mathrm{Int}}}(t)$, is given by (\[intham\]), where the conserved matter current and the free field expansions for matter and gauge fields are still those of (\[psi\_free\]) and (\[a\_free\]). The all-important difference is that, since $m=0$, the energy in the massless case is given by $E_p=\|{{\boldsymbol p}}\|$. When the expansions (\[psi\_free\]) and (\[a\_free\]) are substituted into (\[intham\]), then, of the eight possible terms, the two involving $a_\mu \overline{v} \gamma ^\mu u$ and $a^\dagger_\mu \overline{u} \gamma^\mu v$ will have a time dependence of the form $e^{i\alpha t}$ with $\alpha$ being either positive or negative. These cannot survive for large values of $t$ and so do not appear in the asymptotic limit. This leaves six terms which are potential survivors, of which two vanish in the massive case. A closer look at some of these structures will show how to evaluate all the terms. The first term to be considered in detail is one of the off-diagonal (in the spinors) terms: $$\label{hterm1} -{\mathrm{e}}\int \frac{d^3x\,d^3k\,d^3p\,d^3q}{(2\pi)^9} \frac{a_\mu(k)}{2\omega_k \sqrt{4E_pE_{q}}} b^\dagger (q,s)d^\dagger(p,r)\overline{u}^s(q)\gamma^\mu v^r(p)e^{-ik\cdot x} e^{iq\cdot x}e^{ip\cdot x}\,.$$ Integrating out the $\boldsymbol{x}$-integral will give a term involving $\delta({{\boldsymbol q}}+{{\boldsymbol p}}-{{\boldsymbol k}})$. Following this by integrating out the ${{\boldsymbol p}}$ integral, the expression (\[hterm1\]) becomes $$\label{hterm2} -{\mathrm{e}}\int \frac{d^3k\,d^3q}{(2\pi)^6}\frac{a_\mu(k)}{2\omega_k \sqrt{4E_qE_{k-q}}}b^\dagger (q,s)d^\dagger (k-q,r) \overline{u}^s(q)\gamma^\mu v^r(k-q)e^{it(E_q+E_{k-q}-\omega_k)}\,.$$ If this is to survive as $t\rightarrow \infty$ then the coefficient in the exponent must vanish. This is equivalent to demanding that $\omega_k=E_q+E_{k-q}$, or $\|{{\boldsymbol k}}\|=\|{{\boldsymbol q}}\|+\|\boldsymbol{ k}-{{\boldsymbol q}}\|$. Since these three vectors represent the three sides of a triangle, plainly a solution to this equation will exist if and only if the vectors ${{\boldsymbol k}}$ and ${{\boldsymbol q}}$ are parallel and $\|{{\boldsymbol k}}\|\geq \|{{\boldsymbol q}}\|$. In this region, where the photon is collinear with the matter field, this contribution to the asymptotic interaction Hamiltonian does not vanish. The momentum, ${{\boldsymbol k}}$, is no longer restricted to the soft region, ${{\boldsymbol k}}=0$, but must be larger than the matter momenta, since this term corresponds to pair creation. The second term to be examined is one of the diagonal terms and has the form $$\label{hterm3} -{\mathrm{e}}\int \frac{d^3x\,d^3k\,d^3p\,d^3q}{(2\pi)^9} \frac{a_\mu(k)}{2\omega_k \sqrt{4E_pE_{q}}} d(q,s)d^\dagger(p,r)\overline{v}^s(q)\gamma^\mu v^r(p)e^{-ik\cdot x} e^{-iq\cdot x}e^{ip\cdot x}\,.$$ Integrating out the $\boldsymbol{x}$ again gives a delta function, this time $\delta({{\boldsymbol p}}-{{\boldsymbol k}}-{{\boldsymbol q}})$. Integrating out the ${{\boldsymbol q}}$ integral now yields $$\label{hterm4} -{\mathrm{e}}\int \frac{d^3k\,d^3p}{(2\pi)^6}\frac{a_\mu(k)}{2\omega_k \sqrt{4E_pE_{p-k}}}d(p-k,s) d^\dagger (p,r) \overline{v}^s(p-k)\gamma^\mu v^r(p)e^{it(E_p-E_{p-k}-\omega_k)}\,.$$ The exponent in (\[hterm4\]) must vanish for large $t$ and this implies that $E_p-E_{p-k}-\omega_k$ must vanish. This is similar to the previous case except that now we must have $\|{{\boldsymbol p}}\|=\|{{\boldsymbol k}}\|+\|{{\boldsymbol p}}-{{\boldsymbol k}}\|$. Reasoning as before, we find that ${{\boldsymbol p}}$ must be parallel to ${{\boldsymbol k}}$ but now $\|{{\boldsymbol p}}\| \geq \|{{\boldsymbol k}}\|$, i.e., $E_p \geq \omega _k$. This clearly corresponds to photon production. If the six terms are evaluated using (\[terms2\]) then the final form of the asymptotic interaction Hamiltonian in the massless case is [@havemann:1985] $$\label{hasymp} {{\cal H}_{\mathrm{Int}}}^{\mathrm{as}}(t)=-{\mathrm{e}}\int\frac{d^3k}{2\omega_k}[a_\mu(k)J^\mu_{as}(k,t) +\mathrm{h.c.}]\,,$$ where h.c. denotes the hermitian conjugate of the first term in brackets, and $$\label{current} J^\mu_\mathrm{as}(k,t)=\int\frac{d^3p}{(2\pi)^3}\frac{p^\mu}{E_p} \Bigl(\rho_\mathrm{scatt}(p,k) +\rho_\mathrm{prod}(p,k)\Bigr)\,.$$ The two structures in the asymptotic current are respectively $$\label{scatt} \rho_\mathrm{scatt}(p,k)=\sum_s[b^\dagger(p,s)b(p-k,s)-d^\dagger(p,s)d(p-k,s)] e^{it(E_p-E_{p-k}-\omega_k)}$$ in the region $\omega_k\leq E_p$, and $$\label{prod} \rho_\mathrm{prod}(p,k)=\sum_sb^\dagger (p,s)d^\dagger(k-p,r)\xi^{s\dagger}\boldsymbol{\sigma} \cdot\hat{\boldsymbol{ q}}\xi^re^{it(E_p+E_{p+k}-\omega_k)}$$ in the region $\omega_k\geq E_p$. The two terms in (\[hasymp\]) have different physical interpretations: the $\rho_\mathrm{scatt}$ part corresponds to photon radiation and includes the soft region which was responsible for the non-vanishing of the usual soft asymptotic interaction Hamiltonian, (\[hint\_as\]). However, the photon momenta is only required to be collinear and not extremely soft. The other term, $\rho_\mathrm{prod}$, is completely new and corresponds to the production of massless matter pairs with momenta less than that of the initial photon. Even though photons cannot radiate other photons, these two structures in this model theory already show the basic processes underlying the collinear production of gluons and quark-antiquark pairs in jet creation. We now note that from the form of (\[hasymp\]), and in particular the explicit $p^\mu$ factor in the asymptotic current, the solution to the dressing equation, (\[dress1\]) or (\[dresseqtn\]), will, also here in the massless theory, correspond to a particle since the asymptotic interaction Hamiltonian vanishes at the correct point on the mass shell. We thus predict that the Green’s functions of the dressed fields will also be free of collinear divergences. In the context of this discussion of the asymptotic interaction Hamiltonian, it should be noted that, for massive electrons, if we allow the photon a small mass there is no momentum configuration such that the exponential survives at asymptotic times. In perturbation theory it may be easily seen that a small photon mass regulates the infra-red singularities. However, if the mass of the photon becomes large enough to open decay channels into matter pairs, then once again the interaction picture breaks down and this results in apparently gauge dependent S-matrix elements (see, e.g., [@Nowakowski:1993iu]). We now turn to the construction of the solutions of the dressing equation in the massless limit. Collinear Dressings ------------------- In the massless theory, the dressing must still fulfill the requirements of gauge invariance (\[h\_trans\]) and the dressing equation (\[dresseqtn\]), where now $(\eta+v)^2=0$. Following the procedure outlined in Sect. 2 for solving these equations, we will require the commutators of the potential in the theory described by the asymptotic current, (\[current\]). As we saw for the massive case, the potential can now be written as in (\[sola\]). However, in this massless theory, the asymptotic currents no longer commute and the commutators of the theory are no longer those of the free theory. This makes the construction of the dressing of massless charges somewhat more difficult. Rather than discuss the details of how to extend the analysis of Sect. 2, we will now examine the $v\to 1$ limit of the massive case and then study, in this limit, the route taken by Dirac to the dressings. The naive massless limit of a dressed charge (\[hexponentiated\]) moving in the $x^1$-direction results in integrals of the form $$\label{naive} -\frac{1}{4\pi}\int d^3z \frac{(\partial _2 A_2 +\partial _3A_3-E_1)(x^0,\boldsymbol{z})}{\|x^1-z^1\|}\,.$$ Clearly this is ill-defined, and although $1/\|x^1-z^1\|$ can be defined in terms of generalised functions [@jones:1966], the serious problems with this naive limit cannot be circumvented in this fashion. Further evidence that a naive approach to the massless theory will not suffice, comes from the detailed perturbative calculation of the one-loop propagator [@Bagan:1997dh; @Bagan:1997su]. The wave-function renormalisation constants found in these papers diverge as $v\to1$. To obtain a deeper insight into the form of the dressings, we can follow Dirac’s lead and construct a dressing that reproduces the electric and magnetic fields for a massless charge. The electric field of an ultra-relativistic charge contracts in the direction of motion. For a massless charge this contraction leads to a singular field configuration. It has been argued that, for motion in the $x^1$-direction, the electromagnetic fields are [@kabat:1992; @robinson:1984] $$\begin{aligned} \label{field} E_1=0,& & E_i=\frac{-ex^i\delta(x^0-x^1)}{2\pi r^2}\nonumber \\ & & \\ B_1=0,& &B_i=\frac{\varepsilon^{ij}x^j\delta(x^0-x^1)}{2\pi r^2} \nonumber\end{aligned}$$ where $i,j=2,3$ and $r^2=(x^2)^2+(x^3)^2 = \|x_\perp \|^2$. This result should follow from the soft dressing term. We argue that this part of the dressing should have the form $$\label{green2} \chi _1(x^0,\boldsymbol{x}) =\frac{1}{2\pi}\int d^2z^\perp (\partial _2 A_2 + \partial _3A_3-E_1)(x^0,x^1,z^\perp)\log \|x^\perp -z^\perp \| + g(x)$$ where $g(x)$ is a function which is harmonic in $x^\perp$, i.e., $g(x)$ is in the kernel of the two dimensional Laplacian. Although this is not the naive limit (\[naive\]), it can be understood as arising from that part of the dressing equation which describes the soft dynamics, ${{\cal G}}{\cdot}{\partial}( \chi) = {{\cal G}}{\cdot}A$. For this configuration, the equation becomes $$\label{green1} ((1-v^2)\partial _1^2 +\partial _2^2 +\partial _3^2)\chi _v(x^0,\boldsymbol{x})=((1-v^2)\partial _1A_1+\partial _2 A_2 +\partial _3A_3-vE_1)(x^0,\boldsymbol{x})\,.$$ Now in the $v\to1$ limit, one sees that the limiting value of $\chi_v$, which we write $\chi_1$, must satisfy $$\label{light} (\partial _2^2 +\partial _3^2)\chi _1(x^0,\boldsymbol{x})= (\partial _2 A_2 +\partial _3A_3-E_1)(x^0,\boldsymbol{x})\,,$$ with solution (\[green2\]). It may be checked that the dressing constructed out of this soft term, with $g(x)$ set to zero, indeed yields the electric and magnetic fields, (\[field\]). This result strengthens our claim that a careful analysis of the dressing equation will allow us to construct massless charges. Discussion ========== In this review we have demonstrated that it is indeed possible to construct relativistic charged particles. Let us now recall what we have seen. Our starting point was the non-vanishing of the asymptotic interaction Hamiltonian which characterises gauge theories. The misidentification of the asymptotic interactions of the free Hamiltonian with the true asymptotic dynamics of gauge theories causes the infra-red problem. Since the interaction Hamiltonian does not vanish at large times, Gauss’ law tells us that charged particles are not just the matter fields of the Lagrangian: physical particles like the electron are always accompanied by an electromagnetic dressing. In practical calculations this expresses itself in the lack of a pole structure in the on-shell Green’s functions of the Lagrangian matter fields and in divergences in S-matrix elements. Any physical degree of freedom must be gauge invariant. For charged particles this means that the dressing together with the matter field at its core must be locally gauge invariant — this translates into an equation for the dressing, (\[h\_trans\]). This requirement, though, does not suffice to construct charged particles: any gauge invariant solution is in principle a physical degree of freedom, but it is not necessarily one that physics chooses to use. To restrict ourselves to the solutions which are physically relevant, we require a second equation, (\[dresseqtn\]). This latter relation was deduced from demanding that the velocity of an incoming or outgoing charged particle is well defined and the dressing must take this into account. The form of the asymptotic interaction Hamiltonian is such that it vanishes in the propagator of a correctly dressed particle, i.e., one which satisfies (\[h\_trans\]) and (\[dresseqtn\]). It follows that dressed charges are then described by the free Hamiltonian and a relativistic particle description is indeed possible. In Sect. 2 dressings which solve these equations were explicitly constructed. These are physical degrees of freedom with a specific physical interpretation. The dressings factorise into two parts, each of which was interpreted as playing a different role in the infra-red physics of QED. Sect. 3 then illuminated the role of velocity in any description of charged particles and showed how the form of the dressing could be derived from the theory of heavy charges. These arguments have been tested by explicit perturbative calculations described in Sect. 4. The interpretations of the two different terms in the dressing were such that they were expected to each cancel a different type of infra-red divergence. The constructions passed these one-loop tests with flying colours and a gauge invariant pole structure was obtained. This was then generalised to an all orders proof of the cancellation of infra-red divergences in the on-shell Green’s functions of QED with dressed fields. These results give us great confidence in the requirements (\[h\_trans\]) and (\[dresseqtn\]), in the solutions (\[hexponentiated\]) and also in the interpretation we associate with these physical degrees of freedom. Finally in Sect. 5 collinear divergences were studied in the framework of QED with massless fermions. Here the structure of the interaction Hamiltonian which survives at large times is much richer than in QED with massive matter. However, it still vanishes if correctly dressed matter, satisfying (\[dresseqtn\]), is used. This implies that the on-shell Green’s functions of the solutions to (\[h\_trans\]) and (\[dresseqtn\]) in the massless theory will also be free of all infra-red singularities, including collinear divergences. The construction of these dressings was considered. Charges in the Standard Model ----------------------------- Most of this paper has been given over to QED. The other interactions in the standard model of particle physics are also described by gauge theories, however, the charged particles in these theories are very different to those of the unbroken abelian theory. In this subsection we will sketch some of the most important differences which characterise the charged particles of the weak and strong nuclear forces. The asymptotic Hamiltonian in QCD also does not reduce to the free one. However, there is a new problem with defining charges in non-abelian gauge theories such as QCD. The colour charge $$Q^a=\int d^3x(J^a_0(x)-f^a_{bc}E^b_i(x)A^c_i(x)) \,,$$ in sharp contrast to that of QED, is not invariant under gauge transformations. It is therefore natural to wonder how we may speak of colour charged particles. But on physical states, where Gauss’ law holds, the charge may be written as $$\label{Qwho} Q^a=\frac1{g}\int d^3x\partial_i E_i^a(x) \,.$$ Using Gauss’ theorem it follows that the colour charge expressed in this way is in fact invariant under gauge transformations which reduce, in a directionally independent manner, to elements of the centre of the group at spatial infinity. Thus the concept of colour charge is only meaningful for locally gauge invariant fields and then only if this restriction on the allowed gauge transformations is imposed. Since the Lagrangian matter fields are not gauge invariant, coloured quarks in QCD are necessarily dressed by glue [@Lavelle:1996tz; @Lavelle:1997ty]. The dressing equation (\[dresseqtn\]) of QED may be directly generalised [@Bagan:1998kg] to QCD. The non-abelian solutions will be asymptotically described by a free Hamiltonian provided that the field transformations required are, in fact, admissible. To the extent that they may be constructed, it follows that the various mass singularities would cancel at the correct points in the mass shell of the dressed quark fields. However, there is a fundamental obstruction to the construction of dressed quarks (and gluons) which follows from the nature of non-abelian gauge transformations. Any gauge invariant description of a quark could be used to construct a gauge fixing. However, the boundary conditions which must be imposed on the allowed gauge transformations in order for colour to be associated with the physical degrees of freedom, are such that the Gribov ambiguity [@Gribov:1978wm; @Singer:1978dk] holds. There is then a fundamental, non-perturbative limit on the construction of gauge-invariant, coloured charges. Thus the true degrees of freedom in QCD outside of the perturbative domain do not include quarks and gluons. Colourless gauge-invariant fields can of course be constructed, but quarks and gluons are confined. The picture of confinement which emerges from studying the construction of coloured charges is as follows. When a $Q\bar Q$-system is separating, but the matter fields are still at a short distance from each other, the interaction potential is essentially Coulombic and the dynamics is described by perturbative dynamics. The short-distance, Coulombic inter-quark potential may be described using low-order perturbative solutions to the dressing equations for each of the individual quarks. Deviations from this such that a confining potential arises in a $Q\bar Q$-system are expected to come from a mesonic dressingwhich does not factorise into two parts. Finally, it might be objected that the non-abelian theories which underly the weak interaction do not lead to confinement of weakly charged particles such as the $W$ or indeed the electron. This is easily understood: in spontaneously broken gauge theories we are entitled to use the Higgs sector to dress charged particles [@Lavelle:1995rh]. (The ability to choose the unitary gauge circumvents the gauge fixing ambiguity.) In this way gauge invariant solutions corresponding to weakly charged particles may be constructed. Open Questions -------------- This review has sketched out a systematic approach to the construction of charged particles. The qualitative results to date of the program are fully in accord with phenomenology and the calculational tests to which the charged particles have been subjected have all supported the methods. But there are still a large number of unanswered questions ranging from calculational procedures to the extension of the applicability of this programme to physics at finite temperature and density. We will now conclude by listing some of the most pressing tasks. a) The above proof of the infra-red finiteness of the dressed on-shell Green’s functions needs to be extended to full calculations of these Green’s functions, in particular the UV renormalisation of the $n$-point functions of these composite operators must be performed. The dressed propagators of both fermionic and scalar QED have been carried out at one-loop. This now needs to be extended to higher loops and to vertex functions. b) The perturbative tests need to be extended to collinear divergences. Massless QED is the natural testing ground here. First the dressing equation needs to be solved within the framework of the remarks of Sect. 5. Then the solutions of the equivalent dressing equation for QCD should be constructed and tested. c) Although a brute force, direct perturbative solution of the dressing equations for the non-abelian theory is feasible at low orders, a systematic and practical approach to the construction of a gauge invariant dressed quark fulfilling the dressing equation of QCD for an arbitrary velocity is needed. In the appendix to Ref. [@Lavelle:1997ty] an algorithm was presented with which the calculation of a particular gauge invariant dressed quark solution to any order in perturbation theory became rather simple and some of the first terms in this perturbative construction were given. In this context we also refer to Ref. [@haller:1997] where all orders expressions apparently corresponding to the gauge invariant extension of this term were presented. This corresponded to the gauge invariant extension of the soft term in the QED dressing of a static charge. Such work needs to be generalised to both terms and indeed to arbitrary velocities. Furthermore the success of phenomenological constituent quark models in describing hadronic structure strongly indicates that, within the overall constraint of confinement, some non-perturbative input into the construction of dressed quarks should be possible. Here we are thinking especially of the role of instantons and condensates in chiral symmetry breaking. We stress that it will not be possible to incorporate all the non-perturbative, topological aspects of QCD into dressings of individual quarks or gluons. This, we have argued above, is how confinement makes itself manifest. d) The importance of the perturbative chromo-electric and chromo-magnetic dressings is that they determine the short distance interactions between quarks and have implications for jet physics. We would urge, e.g., a comparative study of the distribution of glue in the dressings around quarks and gluons which could shed light on the different development of such jets. The fruitful concept of parton-hadron duality [@Khoze:1997dn] could we suggest be potentially replaced by a refined version of a duality between (perturbatively) gauge invariant, dressed colour charges and the resulting physical hadrons. e) The qualitative proof of quark confinement which was described above needs to be quantified. How can the hadronic scale be found in this way? That quarks are not part of the physical degrees of freedom of QCD will only become apparent at larger distances where that part of the non-perturbative dynamics which is sensitive to the Gribov ambiguity becomes first significant and finally dominant. This is, of course, a tough non-perturbative calculation. One natural technique is the lattice, where there has been some work on studying the gauge fixing problem. It will be important here to distinguish between lattice artifacts and the true, physical limitation on gauge fixing [@Parrinello:1991fp; @deForcrand:1995mz]. We also suggest here that the construction of dressed charges in monopole backgrounds be studied. f) In view of the difficulties inherent in non-perturbative calculations, phenomenological modelling of dressings is desirable. Thus we feel that the construction of dressed quarks with phenomenologically desirable properties, e.g., running masses should be investigated. A corollary of this last point is the construction of mesonic dressings which lead to phenomenological inter-quark potentials. The stability of such model dressings should then be tested using variational methods. We recall from the introduction the instability of the (confined) $e^+e^-$-system where the two matter fields are linked by a string [@Prokhorov:1993; @Haagensen:1997pi]. g) One of the most important questions in particle physics is how mass is generated. The construction of charges in theories with spontaneous symmetry breaking deserves further, quantitative investigation. Both perturbative and non-perturbative effects should be studied. A major question closely related to this is how one should describe unstable charged particles, such as those occurring in the weak interaction. It is well known that there are problems with obtaining gauge invariant results in the presence of such fields [@Nowakowski:1993iu; @Stuart:1995ba; @Papavassiliou:1995fq]. We have seen, at the end of Sect. 5.1, the non-vanishing of the asymptotic interaction Hamiltonian in such theories. We urge the development of, in some sense dressed, admixtures of fields which would fulfill the following properties: i) if the coupling is set by hand to zero it should reduce to the Lagrangian field, ii) they must be gauge invariant and iii) the asymptotic interaction Hamiltonian should vanish for the propagator of these constructs. h) To what extent can we talk about charged particles beyond the standard model? The construction of descriptions of charged particles in theories such as technicolour and with (broken) supersymmetry could cast light on their experimental signatures. Here it is important to see if the gauge invariant, dressed charged particles are still related by supersymmetries, or if this just holds for the Lagrangian fields. Furthermore we strongly suggest a study of the construction and dynamics of charged particles in unbroken supersymmetric models so as to clarify the role of (supersymmetrically) charged particles in the Seiberg-Witten description of confinement in such theories [@Seiberg:1994rs]. i) Constrained dynamics is the mathematical framework for extracting physical degrees of freedom in theories such as QED and QCD. In modern formulations BRST symmetry is used to single out gauge invariant, local fields. However, as will have become apparent above, gauge invariance is not enough: one must still make a clear identification between the true degrees of freedom and the observed particles. Quark confinement in QCD is merely the most obvious example; even in Quantum Electrodynamics the infrared problem, and the associated superselection structures labelling charges with different velocities, show that this identification is not direct. The interplay between the asymptotic interaction Hamiltonian and the physical observables needs further study. It should be noted that the BRST method is a local construction and, as we have seen, charged fields are necessarily non-local. In this context we point out a further symmetry of QED which was noted in Ref. [@Lavelle:1993xf] and which singled out a subset of gauge invariant fields. We believe that this is only one representative of a class of symmetries with whose aid the true degrees of freedom may be interpreted. Finally, the incorporation of nonperturbative effects is the outstanding question in mathematical physics today. In constrained dynamics this entails extending the usual reduction procedure to theories with superselection sectors characterised by the existence of nontrivial surface terms and other global structures. A proper understanding of these effects is essential if constrained dynamics is to be able to discuss nonperturbative physics whether it be the quark structure of low energy QCD or string phenomenology. Acknowledgements: First and foremost we wish to thank Emili Bagan in collaboration with whom the work of Sect. 4 was carried out. Between us we also thank the following for discussions and correspondence: David Broadhurst, Emilio d’Emilio, Tomeu Fiol, O.W. Greenberg, Kurt Haller, Marek Nowakowski, John Ralston, Nicolas Roy, Dieter Schütte, Sergei Shabanov, Manfred Stingl, Shogo Tanimura, Rolf Tarrach, John C. Taylor and Izumi Tsutsui. Appendix ======== To obtain the asymptotic Hamiltonians of Sect. 2 and Sect. 5 we must evaluate the terms $\overline{u}^r(p)\gamma^\mu u^s(q)\,$, $\overline{v}^r(p)\gamma^\mu v^s(q)\,$, $\overline{v}^r(p)\gamma^\mu u^s(q)$ and $ \overline{u}^r(p)\gamma^\mu v^s(q)$. The results we will obtain, and indeed some generalisations, are quoted in Appendix J of Ref. [@Bialynicki-Birula:1975yp]. Given an on-shell $4$-vector, $p$, so that $p^2=m^2$, then $E_p$ is given by $E_p=\sqrt{\|{{\boldsymbol p}}\|^2+m^2}$. We introduce the notation $$\label{not1} N_p=\frac{1}{E_p+m} \qquad \mbox{and} \qquad \hat{{{\boldsymbol p}}}=N_p{{\boldsymbol p}}\,.$$ Let $\xi^{1\dagger}=(1,0)$ and $\xi^{2\dagger}=(0,1)$. The Dirac spinors for $r=1,2$ are taken as $$u^r(p)= \frac{1}{\sqrt{N_p}} \left( \begin{array}{c} \xi^r \\ \boldsymbol{\sigma }\cdot \hat{{{\boldsymbol p}}} \xi^r \end{array} \right) \, \qquad v^r(p)= \frac{1}{\sqrt{N_p}} \left( \begin{array}{c} \boldsymbol{\sigma}\cdot \hat{{{\boldsymbol p}}} \xi^r\\ \xi^r \end{array} \right)$$ The $u^r(p)$ and $v^r(p)$ are, respectively, positive and negative energy solutions to the Dirac equation. Let us define $$\Lambda=\left( \begin{array}{cc} 0&1 \\ 1&0 \end{array} \right)\,.$$ Then we have $$u^r(p)= \Lambda v^r(p)\,,$$ and $$\overline{u}^r(p)= \overline{v}^r(p)\gamma ^0 \Lambda \gamma ^0\,.$$ Our gamma matrix convention is $$\gamma^0= \left( \begin{array}{cc} 1&0 \\ 0&-1 \end{array} \right) \,, \qquad \gamma ^i= \left( \begin{array}{cc} 0&\sigma^i \\ -\sigma^i&0 \end{array} \right)\,,$$ where $\sigma^i$ are the Pauli matrices. It is now straightforward to show that $$\begin{aligned} \overline{u}^r(p)\gamma^\mu u^s(q)=&\overline{v}^r(p)\gamma^\mu v^s(q)\,,\\ \overline{u}^r(p)\gamma^\mu v^s(q)=&\overline{v}^r(p)\gamma^\mu u^s(q)\,.\end{aligned}$$ This observation is based upon the easily verifiable identity $$\gamma ^0 \Lambda\gamma^0\gamma^\mu \Lambda =\gamma^\mu\,.$$ Two other useful observations for the evaluation of the above expressions are the following. The first is $$\overline{u}^r(p)\gamma^\mu u^s(q)=\frac{1}{2m}\overline{u}^r(p) (\not{\!p }\gamma^\mu+\gamma^\mu\!\! \not{\! q})u^s(q)$$ which uses the fact that $u^r(p)$ is a positive frequency solution to the Dirac equation; a similar identity can be obtained for $\overline{v}^r(p)\gamma^\mu u^s(q)$. The second is the observation that $$\not{\!p }\gamma^\mu+\gamma^\mu\!\! \not{\! q }=(p+q)^\mu+i(p-q)_\nu\sigma^{\mu\nu}\,,$$ where $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu ,\gamma^\nu ]$. Routine calculations now lead to the following results. $$\label{terms1} \begin{array}{ll} \overline{v}^r(p)\gamma^0 u^s(q)=&\frac{1}{\sqrt{N_pN_q}}\xi^{r\dagger} \boldsymbol{ \sigma}\cdot(\hat{{{\boldsymbol p}}}+\hat{{{\boldsymbol q}}}) \xi^s\,,\\ \overline{v}^r(p)\gamma^i u^s(q)=&\frac{1}{\sqrt{N_pN_q}}\xi^{r\dagger} ([(\boldsymbol{\sigma}\cdot\hat{{{\boldsymbol p}}})\hat{{{\boldsymbol q}}}+ (\boldsymbol{\sigma}\cdot\hat{\boldsymbol{ q}})\hat{{{\boldsymbol p}}}-i{{\boldsymbol p}}\times \boldsymbol{ q}]^i+(1-\hat{{{\boldsymbol p}}}\cdot\hat{{{\boldsymbol q}}})\sigma ^i)\xi^s\,,\\ \overline{u}^r(p)\gamma^0 u^s(q)=&\frac{1}{\sqrt{N_pN_q}}\xi^{r\dagger} (1+\hat{{{\boldsymbol p}}}\cdot\hat{{{\boldsymbol q}}}+i\boldsymbol{ \sigma}\cdot\hat{{{\boldsymbol p}}}\times\hat{{{\boldsymbol q}}})\xi^s\,, \\ \overline{u}^r(p)\gamma^i u^s(q)=&\frac{1}{\sqrt{N_pN_q}}\xi^{r\dagger} ({{\boldsymbol p}}+{{\boldsymbol q}}+i\boldsymbol{ \sigma}\times (\hat{\boldsymbol{ p}}-\hat{{{\boldsymbol q}}}))^i\xi^s\,. \end{array}$$ The latter pair of results is sufficient to calculate the asymptotic interaction Hamiltonian (\[hint\_as\]) of the massive theory. In the $m=0$ limit, these results simplify in an attractive way. Recall from the discussion of Sect. 5 that the asymptotic interaction Hamiltonian only receives a contribution from the region where the momenta are collinear. As an illustration, we shall examine the first of the terms in (\[terms1\]), in the collinear region where we have $\|{{\boldsymbol k}}\|=\|{{\boldsymbol q}}\|+\|{{\boldsymbol k}}-{{\boldsymbol q}}\|$ (c.f. the paragraph after Eq. \[hterm2\]). In the massless case we have $N_p=1/E_p$, so that $\hat{{{\boldsymbol p}}}={{\boldsymbol p}}/E_p$. It is also clear that $\hat{{{\boldsymbol p}}}^2=1$ and $\hat{{{\boldsymbol p}}}\cdot{{\boldsymbol p}}=E_p$. We thus obtain for massless charges for this momentum configuration $$\begin{aligned} \label{apex} \overline{v}^r(q)\gamma^0 u^s(k-q)&=&\frac{1}{\sqrt{N_{k-q}N_q}}\xi^{r\dagger} \boldsymbol{\sigma}\cdot(\hat{{{\boldsymbol q}}}+\widehat{({{\boldsymbol k}}- {{\boldsymbol q}})})\xi^s\,, \nonumber\\ &=& \sqrt{E_qE_{k-q}}\,\xi^{r\dagger}\boldsymbol{\sigma} \cdot\left(\frac{{{\boldsymbol q}}}{E_q}+ \frac{{{\boldsymbol k}}-{{\boldsymbol q}}}{E_{k-q}}\right)\xi^s\,.\end{aligned}$$ Now in the asymptotic interaction Hamiltonian this spinor combination only occurs if the momenta are such that $k$ and $q$ are parallel light-like vectors, and using this in (\[apex\]), we see that $$\begin{aligned} \overline{v}^r(q)\gamma^0u^s(k-q) &=&\sqrt{E_qE_{k-q}}\,\xi^{r\dagger} {{{\boldsymbol q}}}\cdot\boldsymbol{\sigma} \xi^s\left(\frac{1}{E_q}+\frac{1}{E_q}\right)\\ &=&2\sqrt{E_qE_{k-q}}\, \xi^{r{\dagger}}\hat{{{\boldsymbol q}}}\cdot\boldsymbol{\sigma} \xi^s\,.\end{aligned}$$ The other terms in (\[terms1\]) can be found in a similar manner. 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[^8]: In non-abelian gauge theories the gauge field also needs to be dressed in order for it to carry colour [@Lavelle:1996tz; @Lavelle:1994xa; @Lavelle:1997ty]. [^9]: The non-abelian appearance of this gauge transformation in QED arises from the fact that the dressing is field dependent and thus might not commute with the potential. [^10]: The Feynman rule for the vertex with the source is here taken to be unity for simplicity, it could also be given the renormalisation group invariant form, $m^2\phi^{*}\phi$. [^11]: It is easy to check that the phase part of the dressing does not bring in any soft divergences [^12]: Note that the massless tadpole diagrams of Fig. 4.5.b do not cancel, but they, of course, do not alter the on-shell dependence of the IR-structure
--- abstract: 'We report the specific heat, magnetic, magnetotransport and galvanomagnetic properties of polycrystalline GdNiC$_2$. In the intermediate temperature region above $T_N$ = 20 K, we observe large negative magnetoresistance due to Zeeman splitting of the electronic bands and partial destruction of a charge density wave ground state. Our magnetoresistance and Hall measurements show that at low temperatures a magnetic field induced transformation from antiferromagnetic order to a metamagnetic phase results in the partial suppression of the CDW.' author: - 'Kamil K. Kolincio, Karolina Górnicka, Michał J. Winiarski, Judyta Strychalska - Nowak, Tomasz Klimczuk' title: 'Field induced suppression of charge density wave in GdNiC$_2$' --- Introduction ============ The interest in quasi-low dimensional materials lies in their unconventional physical properties. Low dimensionality often results in anisotropy of thermoelectric and transport properties or electronic instabilities such as charge or spin density waves (CDW and SDW, respectively) [@gruner_density_1994; @Monceau2012; @Gruner1988]. The coupling between CDW, magnetic field and magnetic order is a long standing area of interest. In particular, the application of external magnetic field leads to a rich variety of phenomena such as: suppression of CDW due to the Zeeman splitting of the electronic bands[@tiedje_magnetoresistance_1975], enhancement of the CDW [@Balseiro1985] or field induced CDW condensation [@brooks_magnetic_2006; @zanchi_phase_1996; @graf2004]. Since the discovery of the coexistence of CDW and antiferromagnetic order in metallic Cr [@Young1974; @Fawcett1988], the extensive efforts have been issued for understanding of the coupling between CDW and magnetism, however the number of compounds exhibiting both Peierls instability and magnetic ordering is limited. In fact, the case of chromium still engages great interest of researchers [@jacques2014; @nicholson2016; @jacques2016]. Recently, much attention has been devoted to the two families of intermetallic materials: M$_5$Ir$_4$Si$_{10}$, where M = (Er, Yb, Dy, Ho, Y and Tm) [@Yang1991; @Ghosh1993; @galli_charge-density-wave_2000; @galli_coexistence_2002; @Hossain2005; @vansmaalen2004] and RNiC$_2$, where R = (Sm, Tb, Nd and Gd)[@hanasaki_magnetic_2012; @Yamamoto_2013; @Shimomura2016], in which the emergence of CDW and magnetic ordering has been observed. The study of physical properties of these compounds opens a wide road to explore the interplay between both phenomena. GdNiC$_2$ belongs to the group of ternary RNiC$_2$ compounds, forming in the orthorhombic CeNiC$_2$-type structure with a space group of Amm2 [@bodak_o._i.__1979; @jeitschko_ternary_1986; @matsuo_antiferromagnetism_1996]. In this system, magnetic order originates entirely from the 4$f$ electrons of rare-earth elements, while Ni atoms have been found to carry no magnetic moments [@onodera_magnetic_1998; @KOTSANIDIS_1989]. For R = Sm, the magnetic ground state is ferromagnetic (FM), while compounds with R = (Gd, Tb and Nd) show antiferromagnetic (AF) character[@onodera_magnetic_1998; @ONODERA_1995; @Schafer_1997]. The anomalous temperature dependence of electrical resistivity and lattice constants of RNiC$_2$ [@murase] have been identified as genuine Peierls transitions associated with partial nesting of the Fermi surface (FS) built of warped sheets perpendicular to $a$ axis[@laverock_electronic_2009; @kim_chemical_2013]. This charge density wave instability, associated with distortion of Ni atoms [@wolfel_commensurate_2010], is accompanied with opening of an electronic gap and condensation of a certain portion of electronic carriers [@ahmad_evidence_2015]. GdNiC$_2$ shows a CDW transition with Peierls temperature $T_P$ = 205 K. Initially, the FS nesting occurs with slightly incommensurate wavevector $q = \left[ \frac{1}{2}; \eta; 0\right]$, which evolves into a doubly commensurate value of $q =\left[ \frac{1}{2}; \frac{1}{2}; 0\right] $ [@Shimomura2016]. A recent X-Ray diffuse scattering study of the satellite reflections [@Shimomura2016] has shown that, in contrast to SmNiC$_2$, the CDW in GdNiC$_2$ survives the transition to a magnetically ordered (AF) state at $T_N$ = 20 K. Owing to the rich phase diagram of GdNiC$_2$ [@Hanasaki2011], it becomes interesting to study the CDW response to the application of external magnetic field and the evolution of magnetic order. Here we report the magnetoresistance, Hall effect, magnetization and specific heat measurements and discuss the destructive influence of magnetic field and magnetic transitions on the CDW in GdNiC$_2$. Experimental ============ A polycrystalline sample of GdNiC$_2$ was synthesized by arc-melting stoichiometric amounts of elemental precursors (from Alfa Aesar): gadolinium (99.9%), nickel (99.999%) and carbon (99.997%). Melting took place in a water-cooled copper hearth, under an ultra-high purity argon atmosphere. A zirconium button was used as an oxygen getter. To homogenize the specimen, the obtained sample was remelted four times. The arc-melted button was wrapped in tantalum foil, placed in an evacuated quartz tube and annealed at 900[$^\circ$]{}C for 10 days and quickly cooled down by water-ice quenching. Magnetization measurements were carried out using the AC Susceptibility Option (ACMS) of a Quantum Design Physical Properties Measurement System (PPMS). A piece of the sample was fixed in a standard polyethylene straw holder. Thin slices of the sample for transport and Hall effect measurements were cut with wire saw and polished. Platinum wires serving as electrical leads were spark-welded to the sample surface. The experiments were performed using the PPMS. Resistivity was measured employing the standard four probe technique. The Hall voltage was measured in the direction perpendicular to the electrical current in a presence of magnetic field perpendicular to the sample surface. The data were collected reversing the orientation of applied magnetic field, in order to subtract the parasitic longitudinal magnetoresistance voltage component due to small misalignment of contacts. Specific heat measurements were done by means of the standard $2\tau$ relaxation method of the PPMS system on a flat polished sample (approx. 4.5 mg). Results and discussion ====================== ![image](CP5.eps) ![\[moment\](a) Magnetisation of GdNiC$_2$ measured at various temperatures as a function of $H$. Panels (b) and (c) show the expanded views of hysteresis observed at high and low fields, respectively.](MmiBvsH.eps) The purity and crystallographic structure of the sample was tested by powder X-ray diffraction - see the Supplemental Material. The results of specific heat measurements in the vicinity of the antiferromagnetic transition are shown in fig. \[CP\] compared with the magnetic phase diagram proposed by Hanasaki et al. [@Hanasaki2011] based on magnetizatin measurements. The temperature of a transition from paramagnetic (PM 1) to antiferromagnetic (AF 1) phase is almost unaffected by applied magnetic field up to ca. 4.5 T, above which the specific heat peak at the Néel temperature ($T_N$ = 20 - 23 K) is suppressed and two new peaks appear (see Fig. \[CP\](a)). Between 5 and 7 T two peaks are seen gradually shifting towards higher temperatures with increasing external magnetic field. While the small change of $T_N$ with applied field is consistent with the phase diagram of Hanasaki et al. [@Hanasaki2011], the transition between the different AF and PM phases (AF 1 - AF 2 and PM 1 - PM 2, see Fig. \[CP\]) could not be observed within the available measurement accuracy. The phase boundaries between paramagnetic PM 2 and ’intermediate’ phases IM 1 and IM 2 seem to be in qualitative agreement with peaks positions shifted towards higher temperatures. A large $C_p$ peak arising at 9 T between 25 and 27 K can likely be attributed to the field-induced ferromagnetic transition observed previously in magnetization measurements [@Hanasaki2011] at slightly lower fields and temperatures. These differences may be caused by the effects of crystal structure disorder that are much larger in a polycrystalline sample than in a single-crystal. Detailed analysis of heating-cooling curves recorded by the PPMS calorimeter have not revealed any discernible distortions that should be seen in case of first order phase transitions [@PPMS_HC_manual]. No significant peak of specific heat was observed at $T_P$, which is typical of CDW transition with small lattice deformation. Results of magnetization measurements vs. applied field ($M$ vs. $\mu _0 H$) are presented in Fig. \[moment\]. At temperatures down to 60 K the sample magnetization shows a linear behavior without hysteresis. No features are observed in $M$ vs. $T$ (not shown) at $T_P$, which is not surprising, since the change of Pauli - Landau magnetic components are expected to be significantly weaker than the Curie - Weiss term from the local strong magnetic moments. Between 60 K and the Néel temperature $T_N$, which is slightly above 20 K, curves start to depart from linearity and a small hysteresis loop is formed between approximately 4 and 8 T. Below the $T_N$ two hysteresis loops are observed - larger, between 3 and 6 T (see Fig. \[moment\](b)) and smaller at fields up to ca. 0.2 T (see Fig. \[moment\](c)). The low-field hysteresis can either be attributed to a previously overlooked phase transition or to a trace amount of ferromagnetic impurity; however lack of an additional specific heat anomaly seems to support the latter. It is interesting that the high-field hysteresis starts to develop already between 40 and 30 K, well above the $T_N$. At 10 K the curve does not saturate even at 9 T, in agreement with previous reports [@Onodera_1997], where the saturation field at 4.2 K was found to be 9.7 or 12 T, depending on the crystallographic axis. ![\[RT\] Thermal dependence of resistivity of GdNiC$_2$ at various magnetic fields. Inset: expanded view of the range corresponding to low temperature phase transitions.](rhoTB.eps) ![\[MR\] Magnetoresistance of GdNiC$_2$, as a function of magnetic field. Solid lines correspond to $-H^2$ fits to the experimental data. Inset: scaling of MR with eq. \[Zeeman\] for 30 K $ \leq T \leq $ 180 K.](MR2in1.eps) We have used the transport measurements to explore the influence of magnetic field and magnetic transitions to CDW. The main panel of figure \[RT\] shows the thermal dependence of electrical resistivity of GdNiC$_2$ measured at various magnetic fields. At high temperatures, the zero field resistivity ($\rho_0$) shows typical metallic behavior. At $T_P$ = 196 K, a clear anomaly is pronounced as a metal - metal transition, which is characteristic for the CDW transition in a quasi-2D material, in which the nesting of the Fermi surface is imperfect with fragments of FS still remaining below $T_P$ [@kolincio]. At $T$ = 20 K, which corresponds to $T_N$, one observes an abrupt decrease of resistivity. This crossover is an universal feature of the RNiC$_2$ family and can be attributed both to the reconstruction of the conduction bands driven by magnetoelastic modification of the crystallographic structure [@laverock_electronic_2009; @murase] upon a transition to the magnetically ordered state and to the consequent destruction of CDW state resulting in release of condensed carriers. Despite the $\simeq 40\%$ resistivity drop in GdNiC$_2$, its magnitude is notably smaller than in SmNiC$_2$ [@prathiba_tuning_2016]. These observations are in agreement with the X-Ray data recently collected on single crystals by Shimomura et al. [@Shimomura2016], who have shown that, in contrast to complete suppression of CDW in the FM state of SmNiC$_2$ [@hanasaki_magnetic_2012; @Shimomura_2009], the Peierls instability, although weakened, survives in the magnetically ordered state of GdNiC$_2$. Note that the values of $\rho_0$ found by us are an order of magnitude larger than in the single crystals studied by Shimomura et al. Also, in our polycrystalline sample, $\rho_0(T)$ does not show any influence of the lock-in transition occurring at $T_{lock-in}\approx90$ K as seen in resistivity measured along $c$ axis of the single crystal. The polycrystalline nature of our sample is also responsible for a lower, in comparison to single crystal, value of $T_P$ (although the value of $T_P$ found by us converges with data reported by Murase et al. [@murase]). In the metallic regime above $T_P$, the magnetoresistance (MR = $\frac{\rho(H)-\rho_0}{\rho_0}$) is negligibly small. Below $T_P$, one observes a significant decrease of resistivity in the presence of magnetic fields and the MR remains large and negative even at lowest temperatures, softening the drop of $\rho(H)$ in proximity of $T_N$. An interesting observation is the occurrence of a small minimum followed by a hump at temperatures slightly below $T_N$ in the presence of fields ranging from 5 to 7 T (see inset of figure \[RT\]). Considering the phase diagram of GdNiC$_2$ (see fig \[CP\]), one can attribute this effect to the transition towards an intermediate magnetic phase. This effect will be more extensively discussed in the next paragraphs. At temperatures $T_N<T<T_P$, the MR follows an $\sim-H^2$ dependence which is depicted in figure \[MR\]. This behavior suggests that the main source of magnetoresistance in this temperature interval is the Zeeman splitting of the electronic bands at the Fermi level towards spin-up and spin-down ones separated by $\Delta E = 2 \mu_B H$. Theoretical work of Dieterich and Fulde [@Dieterich1973] predicted that a sufficiently strong magnetic field reduces the pairing interaction between electrons, lowers the CDW electronic gap ($\Delta_{CDW}$) and suppresses the CDW ground state. As a consequence, the Peierls temperature follows the BCS relation with magnetic field; $$\label{MReq2} \frac{T_P(H)-T_P(0)}{T_{P}(0)}=\frac{\gamma}{4}\left( \frac{\mu_BH}{k_BT_P(0)}\right)^2$$ where $\gamma$ is a constant of the order of unity. When $\Delta_{CDW}$ is much larger than $\mu_B H$, the negative magnetoresistance due to an increase of free electronic carriers can be expressed by the formula [@tiedje_magnetoresistance_1975]: $$\label{Zeeman} MR = \frac{\rho (H)-\rho_{0}}{\rho_0}= -\frac{1}{2} \left( \frac{\mu_BH}{k_BT}\right)^2+0 \left( \frac{\mu_BH}{k_BT}\right)^4$$ ![image](MRlt.eps) As depicted in the inset of figure \[MR\], the magnetoresistance of GdNiC$_2$ scales with $ -\left( \frac{\mu_BH}{k_BT}\right)^2$ in the wide temperature interval of $30 $ K $ \leq T \leq 180 $ K. This evidences that Zeeman suppression of the CDW is a driving force of large negative MR in this temperature range. An interesting observation, is that the description of MR with eq. \[Zeeman\] requires the introduction of a prefactor of $\simeq$ 30. Typically, in CDW materials this coefficient or equivalently the $\gamma$ factor in eq. \[MReq2\] is smaller than unity. Good examples are the magnetoresistance scaled by 0.25 in Li$_{0.9}$Mo$_6$O$_{17}$ [@xu_directional_2009] or a number of organic compounds with $\gamma<1$ [@brooks_magnetic_2008; @graf_suppression_2004; @Graf2005; @monchi_international_1999; @matos_modification_1996]. Matos et al.[@matos_modification_1996] shown that the presence of weakly magnetic chains in (Per)$_2$Pt(mnt)$_2$ leading to the local increase of internal magnetic field, enhances the CDW suppression with $H$. As a result one observes the $\gamma$ parameter larger in comparison with a similar compound (Per)$_2$Au(mnt)$_2$ in which the magnetic chains are absent. This effect is in some extent similar to the case of GdNiC$_2$. However, in contrast to the aforementioned systems, where the local magnetic moments are insignificant, in GdNiC$_2$, the Gd$^{3+}$ ions carry a large moment of ca. $8 \mu_B$ [@kittel_2004]. The presence of such local moments produces a strong internal magnetic field acting on neighboring Ni atoms, and that results in the large value of magnetoresistance prefactor. Such ’transferred’ magnetic fields have been observed through Mössbauer spectroscopy (MS) measurements in hyperfine structures of various nonmagnetic atoms embedded in magnetic systems [@Nowik_2012]. The MS measurements on GdNiC$_2$ at 4.2 K have revealed a hyperfine magnetic field of 34 T acting at Gd nuclei [@matsuo_antiferromagnetism_1996; @Onodera_1997], but no reports on Ni hyperfine fields exist to the author’s knowledge. We also suggest, that due to the presence of such strong magnetic moments in GdNiC$_2$ one can safely assume, that the enhancement of the CDW suppression originates mostly from the spin magnetic moments from Gd$^{3+}$ and the contribution to $\gamma$ from the orbital effects, observed in (Per)$_2$Au(mnt)$_2$ [@graf_suppression_2004], is insignificant in comparison to the influence of the spin mechanism. Although the magnetoresistance stands in agreement with theory, we have found no visible modification of the Peierls temperature in magnetic field. The shift of $T_P$ due to Zeeman suppression of the CDW gap has been observed only in several materials with the $T_P$ as low as 8 K or 12 K[@bonfait_research_1995-1]. Since the Peierls temperature in GdNiC$_2$ is relatively high, $k_BT_P$ is two orders of magnitude larger than $\mu_BH$ at our maximum field of 9 T. Then, even considering the factor $\gamma$ = 30 in eq. \[MReq2\], the expected $T_P$ shift is only $\simeq$ 1.3 K at 9 T. Such a small difference is difficult to observe within experimental resolution for the polycrystalline sample. Furthermore, lack of visible deviation from the mean-field (eq. \[Zeeman\]) scaling, including signs of saturation, which was seen in Li$_{0.9}$Mo$_6$O$_{17}$ [@xu_directional_2009] due to complete destruction of CDW at high magnetic fields suggests, that the suppression of the charge density wave observed in GdNiC$_2$ is not complete for $T_N<T<T_P$, and the Peierls instability (at least partly) survives in the presence of the external magnetic field of 9 T. Figure \[MRLT\] shows the magnetoresistance of GdNiC$_2$ measured at temperatures in the vicinity and below $T_N$. The curve measured at $T$ = 30 K shows $-H^2$ behavior described above. As temperature is lowered to 24 K, one can observe two kinks in MR$(H)$. First, one appears at, roughly, 4.5 T and is accompanied by a small hysteresis; a second one is found at $\simeq$ 6.5 T and shows no hysteretic behavior. Upon further decreasing the temperature, the former term becomes sharper and dominates over the latter one. Eventually, at 10 K, MR$(H)$ shows an abrupt drop at fields between 4 and 6 T. At higher fields, MR shows a further, yet much slower, decrease. Owing to the phase diagram of GdNiC$_2$ and our magnetization data, we find that both anomalies correspond to the phase transitions towards a metamagnetic phase (MM in fig \[CP\]b). Above 10 K this process occurs via the intermediate magnetic phase, which explains the existence of two kinks in MR($H$). In contrast to that, at T $\leq$ 10 K, the increase of magnetic field transforms the system directly from AF to metamagnetic phase without any intermediate stage, which is pronounced by a single anomaly. This steep decrease of resistivity is reminiscent with the behavior of magnetoresistance in NdNiC$_2$ [@Yamamoto_2013], where the CDW state surviving the transition to AF state was definitely suppressed by a magnetic field induced spin-flop transition. A similar effect was observed slightly above the Curie temperature in SmNiC$_2$ [@hanasaki_magnetic_2012], where application of magnetic field causes the transition into a ferromagnetic state, which results in a destruction of CDW. This suggests, that the effect observed in GdNiC$_2$ is of the same origin as in the compounds recalled above. The wavevector $q$ of the CDW modulation appears to play a key role in the interplay between CDW and AF. Agreement between $q$ and the magnetic propagation vector leads to a sort of resonance, which prevents the local magnetic moments from breaking the pairing interaction and, in this scenario CDW and AF orders coexist [@Yamamoto_2013]. Deviation from this condition leads then to breaking of the singlet electron - hole pairs forming the CDW. Although the magnetic structure of GdNiC$_2$ has not been precisely defined yet, Matsuo et al. [@matsuo_antiferromagnetism_1996] proposed the propagation vector of $\left[ \frac{1}{2}; \frac{1}{2}; 0\right] $. This value corresponds with the CDW modulation vector, which becomes doubly commensurate at $T_{lock-in}$ [@Shimomura2016]. When a magnetic field induces a change of magnetic structure this resonance becomes disturbed, which leads to suppression of the CDW and release of condensed carriers. One shall also consider the possibility of the electronic bands structure modification upon the AF - MM transition. This leads to the change of the nesting conditions and may eventually act as another mechanism suppressing the CDW. To support this scenario, we have followed an analysis proposed by Yamamoto et al. [@Yamamoto_2013] and compared the ratio of resistivity in GdNiC$_2$ measured at 2 K (thus in presence of magnetic order), with and without magnetic field, respectively and at a temperature slightly above the AF transition, with corresponding values obtained for and NdNiC$_2$. In GdNiC$_2$, $\frac{\rho_{0T,2K}}{\rho_{0T,23K}}$ = 0.51 and $\frac{\rho_{9T,2K}}{\rho_{0T,23K}}$= 0.18 [@Yamamoto_2013]. These quantities parallel relevant ratios for NdNiC$_2$: in the presence of AF, partially suppressing CDW: $\frac{R_{0T,5K}}{R_{0T,20K}} \simeq$ 0.4 and $\frac{R_{9T,5K}}{R_{0T,20K}}\simeq$ 0.16 at the same temperature, albeit in the presence of a magnetic field in which the CDW is completely suppressed. To compare, in SmNiC$_2$[@Shimomura_2009; @prathiba_tuning_2016], where the FM order entirely destroys the CDW, the ratio $\frac{R_{5K}}{R_{20K}}\simeq$ 0.1. These results are consistent with the scenario of partial destruction of the CDW in the AF state of GdNiC$_2$ and the further suppression of the Peierls instability with increasing magnetic field, which drives the metamagnetic crossover. We emphasize that, although this comparison suggests a strong suppression of the Peierls instability in the MM state, X-Ray diffuse scattering experiment showing the absence of satellite reflections is required to deliver unambiguous evidence of complete CDW destruction. Due to polycrystalline nature of our samples, we were unable to explore the thermal and magnetic field evolution of CDW satellite peaks via X-Ray diffuse scattering. Instead, we have conducted a study of the Hall effect to complement the magnetoresistance data. Figure \[Hall1\] shows the thermal dependence of the Hall resistivity normalized by magnetic field ($\frac{\rho_{xy}}{\mu_0H}$) measured at various magnetic fields. Above $T_P$, $\frac{\rho_{xy}}{\mu_0H}$ is almost temperature independent. At the Peierls temperature one observes a decrease of Hall resistivity due to condensation of part of the electronic carriers into the CDW state. The lock - in transition is pronounced as an inflection in the $\frac{\rho_{xy}}{\mu_0H}(T)$ curve. Note that, as $T$ approaches $T_N$, one observes an increase of $\frac{\rho_{xy}}{\mu_0H}$, which is significantly enhanced at strong magnetic fields. Similar behaviors have been reported at $T_P$ and $T_C$ of SmNiC$_2$ [@Kim_2012], corresponding to formation and suppression of the CDW, respectively. Relating the increase of Hall resistivity at $T \rightarrow T_P$ exclusively to the release of free carriers due to destruction of the CDW by AF order would be too far a simplification. In materials exhibiting significant magnetic ordering, the Hall resistance consists of two components related to external magnetic field and magnetization respectively[@karplus_hall_1954]: $$\label{Halleq} \rho_{xy}=R_0\mu_0H+4\pi R_SM$$ $R_0$ is the ordinary Hall coefficient, which for a single band system, is a direct measure of electronic concentration $n$ ($R_0=\frac{1}{en}$). $R_s$ represents the anomalous Hall effect associated with skew and side jump scattering. The separation of those parameters is not straightforward and usually requires measurements in magnetic fields strong enough to observe saturation of $M(H)$ [@Berger1980; @shiomi_extrinsic_2009; @Granovskii2012], which in an antiferromagnetic metal can be as large as tens of Teslas. Considering the $M(H)$ dependence, we can propose at least a qualitative discussion of the evolution of $R_0$ (thus of $n$) as a function of $H$. ![\[Hall1\]Hall coefficient of GdNiC$_2$ vs temperature at various fields. Arrows indicate characteristic phase transitions temperatures.](Hall1.eps) ![\[Hall2\]Hall resistance of GdNiC$_2$ as a function of H for various temperatures above 60 K. Solid lines show the extended fits to the linear parts of the curves. For clarity, the curves have been vertically shifted.](Rxynonlin.eps) Firstly, we have followed the Hall resistance as a function of $H$ for $T\geq60 $ K. The idea is that in this temperature range the magnetization is a linear function of magnetic field. Hence, the anomalous Hall resistance term is proportional to $H$ as well, and any departure from linearity of $\rho_{xy}(H)$ is a fingerprint of some modification of free carrier concentration. Figure \[Hall2\] shows the Hall resistivity as a function of magnetic field for $T\geq60 $ K. At 200 K, thus above $T_P$ where GdNiC$_2$ exhibits ordinary metallic character, the Hall resistivity shows a classical linear dependence of $H$. At temperatures below $T_P$, one observes a clear deviation from this linear scaling. This effect becomes more pronounced as $T$ decreases and $H$ increases. This indicates the increase of free electronic concentration and is consistent with the negative Zeeman magnetoresistance due to partial release of CDW condensed electrons, observed in this temperature range. ![\[por20\] Comparison of magnetoresistance, Hall resistivity and magnetization of GdNiC$_2$ at $T$ = 20 K. Arrows in the upper and lower panels show the direction of the magnetic field sweep. ](por20.eps) ![\[por10\] Comparison of magnetoresistance, Hall resistivity and magnetization of GdNiC$_2$ at $T$ = 10 K. Arrows in the upper and lower panels show the direction of the magnetic field sweep. ](por10.eps) ![\[rhest\] Hall resistivity of GdNiC$_2$ measured at 10 K (black squares) and $\rho_{xy}$ after subtraction of the estimated anomalous component (red circles) - see text for details](rhest.eps) Figures \[por20\] and \[por10\] compare of magnetoresistance, Hall resistivity and magnetization of GdNiC$_2$ measured at 20 and 10 K, respectively. In both cases, at fields up to 3.5 T, the Hall resistance follows a linear dependence on $H$. At 4 T, there is an upturn of $\rho_{xy}(H)$ concomitant with a decrease of resistance and increase of magnetization discussed in previous paragraphs. At 20 K, the second kink observed in MR and $M$ is also reflected in $\rho_{xy}$ and is pronounced as an upturn of Hall resistivity. The departure of $\rho_{xy}$ from its linear field dependence show large similarities with the data collected at $T\geq$ 60 K (see figure \[Hall2\]), where the carrier concentration is increased due to partial suppression of the CDW. Nevertheless, to avoid overinterpretation of this result one has to analyze the data also in respect to the anomalous part of the Hall effect. Due to the complicated shapes of $\rho_{xy}(H)$ and $M(H)$, the separation of ordinary and anomalous components of the Hall resistance requires several crude assumptions, which cause the approximate nature of the following analysis. Firstly, we assume that $R_S$ does not change appreciably across the observed magnetic phase transitions. A second assumption is that at high fields $R_0\mu_0H\ll4\pi R_SM$. Then, equation \[Halleq\] reduces to $\rho_{xy}=4\pi R_SM$. The anomalous Hall coefficient $R_S$ was evaluated from a linear fit (not shown here) to $\rho_{xy}(M)$ measured at 10 K. Then, the anomalous component was subtracted from the measured $\rho_{xy}(H)$. The result is shown in fig. \[rhest\]. A clear deviation of this curve from a linear function of H suggests the increase of the carrier concentration at field stronger than 4 T. This supports the scenario of partial CDW destruction due to a magnetic transition. conclusions =========== We have studied the specific heat, magnetic, magnetotransport and galvanomagnetic properties of polycrystalline GdNiC$_2$. In the absence of antiferromagnetic order, above $T_N$, we observe a strong negative magnetoresistance due to Zeeman splitting of conduction bands and partial suppression of the CDW. This result is confirmed by the increase of electronic carrier concentration revealed by Hall measurements. The presence of large local magnetic moments of Gd$^{3+}$ ions is presumably responsible for the anomalously strong magnetic field dependence of magnetoresistance. In order to investigate this problem more deeply, a measurement of local magnetic field, especially acting on Ni atoms, should be performed using a local-probe technique, like Mössbauer spectroscopy. At temperatures below $T_N$, we observe a significant decrease of resistance as magnetic field drives the crossover from aniferromagnetic to metamagnetic order. We suggest that the evolution of the magnetic propagation vector upon the MM transition distorts the resonance between AF magnetic order and doubly commensurate CDW. As a result the electron - hole pairing interaction is substantially weakened. This, together with possible modification of the Fermi surface nesting conditions, results in strong suppression of the charge density wave instability in GdNiC$_2$ and a prominent release of electronic carriers, which we confirm by magnetoresistance and Hall effect measurements. We also suggest, that an X-Ray study of structural modulation response to the application of magnetic fields performed on a single crystal is necessary to investigate further the behavior of the CDW ground state in this strongly magnetic system.\ Acknowledgments =============== We would like to thank to Joe D. Thompson (LANL) for helpful discussions. Authors gratefully acknowledge the financial support from National Science Centre (Poland), grant number: UMO-2015/19/B/ST3/03127. [55]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop []{}, Vol. (, , ) [, ()](\doibase 10.1080/00018732.2012.719674) [, ()](\doibase 10.1103/RevModPhys.60.1129) [, ()](\doibase 10.1139/p75-202) [, ()](\doibase 10.1103/PhysRevLett.55.2336) [, ()](\doibase 10.1016/j.cap.2005.01.039) [, ()](\doibase 10.1103/PhysRevB.53.1240) [, ()](\doibase 10.1103/PhysRevLett.93.076406) [, ()](http://stacks.iop.org/0305-4608/4/i=8/a=023) [, ()](\doibase 10.1103/RevModPhys.60.209) [, ()](\doibase 10.1103/PhysRevB.89.245127) [, ()](\doibase 10.1103/PhysRevLett.117.136801) [, ()](\doibase 10.1103/PhysRevLett.117.156401) [, ()](\doibase 10.1103/PhysRevB.43.7688) [, ()](\doibase 10.1103/PhysRevB.48.4152) [, ()](\doibase 10.1103/PhysRevLett.85.158) [, ()](\doibase 10.1088/0953-8984/14/20/302) [, ()](\doibase 10.1103/PhysRevB.71.060406) [, ()](\doibase 10.1103/PhysRevB.69.014103) [, ()](\doibase 10.1103/PhysRevB.85.092402) [, ()](\doibase 10.7566/JPSJ.82.123701) [, ()](\doibase 10.1103/PhysRevB.93.165108) @noop [, ()]{} [, ()](\doibase 10.1016/0022-5088(86)90225-0) [, ()](\doibase 10.1016/S0304-8853(96)01282-6) [, ()](\doibase 10.1016/S0304-8853(97)01011-1) [, ()](\doibase http://dx.doi.org/10.1016/0022-5088(89)90096-9) [, ()](\doibase http://dx.doi.org/10.1016/0304-8853(95)00077-1) [, ()](\doibase http://dx.doi.org/10.1016/S0925-8388(96)02564-9) [, ()](\doibase 10.1143/JPSJ.73.2790) [, ()](\doibase 10.1103/PhysRevB.80.125111) [, ()](\doibase 10.1088/1367-2630/15/12/123018) [, ()](\doibase 10.1103/PhysRevB.82.054120) [, ()](\doibase 10.1002/pssb.201552005) [, ()](\doibase 10.1088/1742-6596/320/1/012072) @noop []{}, () @noop [**, ()]{} [, ()](\doibase 10.1103/PhysRevB.93.235126) [, ()](\doibase 10.1038/srep26530) [, ()](\doibase 10.1103/PhysRevLett.102.076404) [, ()](\doibase 10.1007/BF01397717) [, ()](\doibase 10.1103/PhysRevLett.102.206602) [, ()](\doibase 10.1088/0034-4885/71/12/126501) [, ()](\doibase 10.1103/PhysRevB.69.125113) [, ()](\doibase http://dx.doi.org/10.1016/j.synthmet.2005.07.306) [, ()](\doibase 10.1016/S0379-6779(98)00319-1) [, ()](\doibase 10.1103/PhysRevB.54.15307) @noop []{}, ed. 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--- abstract: \| By applying a singular perturbation approach, canard limit cycles exhibited by a general family of singularly perturbed planar piecewise linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both a supercritical and a subcritical Hopf bifurcation. The obtained results are completely comparable with those obtained for smooth vector fields. In some sense, the manuscript can be understood as an extension towards the PWL framework of the results obtained for smooth systems by Krupa and Szmolyan [@KS01]. In addition, some novel slow-fast behaviors are obtained. In particular, in the supercritical case, and under suitable conditions, it is proved that the limit cycles are organized along a curve exhibiting two folds. Each of these folds corresponds to a saddle-node bifurcation of canard limit cycles, one involving headless canard cycles, whereas the other involving canard cycles with head. This configuration allows the coexistence of three canard limit cycles. author: - 'V. Carmona' - 'S. Fernández-García' - 'A. E. Teruel' title: \| Saddle-node canard cycles in\ planar piecewise linear differential systems --- Introduction ============ Planar slow-fast systems are differential systems involving two variables which evolve with different velocities. The ratio of the velocities of both variables is the so-called singular parameter. From a qualitative point of view, slow-fast systems exhibit neither remarkable properties, nor special bifurcations. Phase portraits of slow-fast systems are topologically equivalent to phase portraits of differential systems without different time scales, and canard limit cycles are topologically equivalent to usual limit cycles. The main difference between slow-fast systems and those which are not, is the rate of variation of interesting quantitative information versus the variation of some parameters, specially when the singular parameter is small. This is the case of the phenomenon called canard explosion, where the amplitude of a limit cycle, born at a supercritical Hopf bifurcation, increases very rapidly while a system parameter varies in an exponentially small range, see [@dumortier; @eckhaus; @KS01]. The canard explosion phenomenon was discovered and analyzed by Benoit et al. in 1981 [@benoit] in the Van der Pol oscillator and explains the fast transition, by varying a parameter, from a small amplitude limit cycle to the relaxation oscillation appearing in this system. Relaxation oscillations are oscillatory behavior characterized by long periods of quasi-static behavior interspersed with short periods of rapid transition. Since this behavior is usual in real-life phenomena, see [@KS01] and references therein, models based in slow-fast differential equations are ubiquitous in many applications, such as chemical and biological ones, and in particular in neuroscience [@FN1; @izhikevich; @FN2]. The main tools for the analysis of the slow-fast dynamics are provided by Geometric Singular Perturbation Theory, and rely on the ability of reconstructing the global dynamics by splitting and then joining, in a suitable way, the fast and slow behaviors. Under hyperbolicity conditions, Fenichel Theorem describes the existence of invariant slow manifolds close to compact parts of the fast nullcline and also describes the stability properties of these slow manifolds [@fenichel]. When the fast nullcline folds, the flow of big parts of the phase space evolves in a narrow neighborhood of the stable branch of the slow manifold, and close to the fold. This fact has dynamical consequences since the existence of limit cycles can be deduced from the behavior of the slow manifold around the fold. In some sense, global dynamical behaviors, like limit cycles, can be deduced from local aspects as the local transition next to the fold. A usual technique to analyze this local transition is the blow up of the fold [@dumortier; @KS00; @KS01]. Some other authors have analyzed this transition by reproducing it in different contexts, which result more friendly for analysis. Piecewise smooth systems, both continuous and discontinuous, have received the main attention of the authors, and many progress have been made. For instance, in [@R16; @glend] the authors analyze the canard explosion in a continuous piecewise smooth context. Nevertheless, the simplest scenario where this phenomenon can be reproduced is the piecewise linear (PWL) framework. Even when some dynamical aspects of the slow-fast behavior had been observed in PWL systems, see [@DFK16; @DGPPRT16] and references therein, it has taken some time to understand the way of reproducing the slow-fast dynamics properly, see [@chapter; @DFHPT13; @DGPPRT16]. In [@FDKT] the authors reproduce part of the canard explosion phenomenon in the PWL context, in particular the one involving hyperbolic headless canards. In this work, we consider an extension of the system analyzed in [@FDKT], that allows for the existence of both canards with and without head and both, hyperbolic and non-hyperbolic canard cycles. In particular, the system is able to reproduce saddle-node bifurcations of canard limit cycles. The obtained results are completely comparable with those obtained, for smooth vector fields, by Krupa and Szmolyan in [@KS01]. Moreover, we find new scenarios that, as far as we are concerned, have not been previously reported in the smooth framework. In particular, we find situations where two saddle-node bifurcations of canard cycles take place, one of headless canards and another one of canards with head. In such a case, we show the coexistence of three canard limit cycles. The article is organized as follows. In Section \[sec:background\], we provide a brief overview of canard explosion and saddle-node canard cycles in the smooth case. After that, in Section \[sec:pwlsystem\], we introduce the class of systems we aim to study. In Section \[sec\_mainresults\], we present the Main Results of the article. Section \[sec\_proofs\] is devoted to the proofs of the Main Results. Finally, Section \[sec\_conclusions\] is devoted to conclusions and possible extensions of the present work. The technical issues of the proofs have been left to the Appendix. Background on canard cycles: canard explosion {#sec:background} ============================================= In this section, based on [@KS00; @KS01], we briefly review the basic ingredients of the canard oscillatory behavior appearing in the smooth framework. Typically, canard solutions take place in planar differential systems of the form $$\label{smooth1} \left\{\begin{array}{l} \varepsilon\dot x=f(x,y,a,\varepsilon), \\ \noalign{\medskip} ~~\dot y= g(x,y,a,\varepsilon), \end{array} \right.$$ where $f,g\in \mathcal{C}^r, r\geq 3, a\in\mathbb{R}, 0<\varepsilon\ll 1$ and the dot denotes the derivative with respect to the temporal variable $\tau.$ Since the velocity of the solutions of system is very different depending on the regions of the phase plane they are crossing through (far from the $x$-nullcline the velocity is fast, and close to the $x$-nullcline it is slow), system is often called [slow-fast system]{}. After the rescaling in time $t=\tau/\varepsilon$, system writes as $$\label{smooth2} \left\{\begin{array}{l} x'=f(x,y,a,\varepsilon), \\ \noalign{\medskip} y'= \varepsilon g(x,y,a,\varepsilon), \end{array} \right.$$ where the prime denotes the derivative with respect to the fast time $t.$ Systems (\[smooth1\]) and (\[smooth2\]) are equivalent through the identity when $\varepsilon>0$, but they have not the same limit for $\varepsilon=0$. In fact, the limit of system (\[smooth1\]), called [slow subsystem]{}, is a semi-explicit differential algebraic equation (DAE), where the relation between the variables is given by $$S=\{(x,y):f(x,y,a,0)=0\}.$$ Assuming that $f_y(x,y,a,0)\neq 0$ it follows that $S$ is the graph of a differentiable function $y=\varphi_a(x)$, and the DAE reduces to the differential equation $$\begin{aligned} \label{eq:reduced} f_x(x,\varphi_a(x),a,0)\dot{x}=-f_y(x,\varphi_a(x),a,0)g(x,\varphi_a(x),a,0),\end{aligned}$$ which is called the [reduced equation]{}. On the other hand, the limit for $\varepsilon=0$ of system (\[smooth2\]), called [fast subsystem]{}, is a differential equation having $S$ as the locus of every equilibrium point. From here $S$ is called the [critical manifold]{}. From Fenichel’s Theorems, we obtain an approximation of the overall slow dynamics through the slow subsystem, and an approximation of the overall fast dynamics through the fast subsystem. In brief, Fenichel’s theory asserts that compact subsets $S_0 \subset S$ formed by normally hyperbolic equilibrium points of the critical manifold, persist as locally invariant slow manifolds $S_{\varepsilon}$ for $\varepsilon>0,$ which can be extended by the flow. Moreover, $S_{\varepsilon}$ is an attracting or repelling manifold depending on the same character, under the flow defined by the fast subsystem, of $S_0$. Furthermore, the flow over $S_{\varepsilon}$ is a regular perturbation of the flow defined by the slow subsystem, or equivalently the reduced equation. Typically, the breakdown of the normal hyperbolicity takes place at the points $(x_0,y_0)\in S$ at which the manifold folds, i.e., $f_x(x_0,y_0,a,0)=0$ and $f_{xx}(x_0,y_0,a,0)\neq0$. It is possible to assume, without loss of generality, that the fold point is at the origin when $a= 0$, in that case we avoid the subindex at function $\varphi_a$, and therefore $\varphi(x)=\eta x^2+O(x^3)$. Hence, in a neighborhood of the origin two different branches of the critical manifold coexist, the attracting one, $ S^a_0=\{(x,y): f_x(x,y,0,0)<0\}$, and the repelling one, $S^r_0=\{(x,y):f_x(x,y,0,0)>0\}$. Over each of these two branches the reduced equation can be written as the EDO $$\dot{x} = -\frac{f_y(x,\varphi(x),0,0)}{f_x(x,\varphi(x),0,0)}g(x,\varphi(x),0,0)= \frac {g(x,\varphi(x),0,0)}{\varphi'(x)},$$ and, assuming that $g(0,0,0,0)\neq 0$ , the flow defined by it has opposite orientation over each of these branches. In this case the fold point is called [jump point]{}. A special situation occurs when the fold point at the origin satisfies that $g(0,0,0,0)=0$ together with the non-degeneracy condition $g_x(0,0,0,0)\neq 0$. The reduced equation can be then regularized, defining a solution of the desingularized system which passes from one branch, $S_0^a$, to the other, $S_0^r$, through the fold point. In this case the fold point is called a [canard point]{}. After perturbation, i.e., for $\varepsilon>0$, Fenichel slow manifolds $S^a_{\varepsilon}$ and $S^r_{\varepsilon}$ behave in a different way near a jump fold point and near a canard point. Around a jump point, an attracting Fenichel slow manifold $S^a_\varepsilon$ may follow closely the attracting branch $S_0^a,$ pass in the vicinity of the fold point, and continue following approximately the fast dynamics, giving rise to the possibility of relaxation oscillations. However, around a canard point, Fenichel slow manifold $S^a_\varepsilon$ may follow closely the attracting branch $S_0^a,$ pass in the vicinity of the fold point and then, surprisingly, continue following closely the repelling branch $S_0^r$, see Figure \[fig:intro\]. From this behavior it can be concluded the existence of solutions of system with $0<\varepsilon\ll 1$ containing canard segments. ![Representation of the critical manifold $S$ in the neighborhood of a canard point. Attracting branch, $S_{\varepsilon}^a$, and repelling branch, $S_{\varepsilon}^r$, of the slow manifold obtained after singular perturbation are also represented. The canard cycles with and without head are obtained provided $S_{\varepsilon}^a$ flows along one side or the other of the repelling manifold $S_{\varepsilon}^r.$ []{data-label="fig:intro"}](dib_intro.eps){width="6.5cm"} (0,0) (-10,30)[$S_{\varepsilon}^a$]{} (-180,100)[$S_{\varepsilon}^a$]{} (-20,90)[$S$]{} (-120,95)[$S_{\varepsilon}^r$]{} Under the existence of another attracting Fenichel slow manifold, see Figure \[fig:intro\], when $S^a_\varepsilon$ flows along one side or the other of the $S^r_\varepsilon$, they can exist canards without head or [canards with head]{}, respectively. Since slow manifolds are exponentially close to one another, the presence of exponentially small terms in the expansions in power series of $\varepsilon$ of the slow manifolds implies that their respective position can change upon an exponentially small parameter variation. This phenomenon is known as the [canard explosion]{}. Moreover, the transition from canards without head to canards with head occurs typically when $S^a_\varepsilon$ connects to $S^r_\varepsilon$. This connection takes place along a curve ${a}_c(\varepsilon)$ in the parameter plane $(\varepsilon,a)$ and the associated canard solution is said to be a maximal canard. Canard cycles develop along a branch born at a Hopf bifurcation, at $a=a_H$, and the canard explosion takes place at a value which is at a distance of $O(\varepsilon)$ from the $a_H$. This means that very close to the bifurcation point $a_H$, before the explosion, the cycles have the characteristics of typical Hopf cycles. This Hopf bifurcation arises only for $\varepsilon>0$ and is usually known as a singular Hopf bifurcation [@braaksma; @guckenheimer]. The existence of saddle-node bifurcation of canard cycles in the smooth framework has been analyzed in [@KS01]. There, the authors consider two different cases, depending whether the Hopf bifurcation where the cycle is born is supercritical or subcritical. Thus, after proving the existence of the maximal canard, they distinguish two different scenarios: - Supercritical case: In Theorem 3.3, authors state the existence of a family of periodic orbits. These periodic orbits can be stable Hopf-type limit cycles, canard limit cycles or relaxation oscillations. To analyze the stability of the canard limit cycles, they use the way in-way out function $R(s)$, which is the limit of the integral of the divergence along the slow manifolds when $\varepsilon\to 0$. In Theorem 3.4, assuming that this function is negative, the authors state that the canard limit cycles of the family are stable. - Subcritical case: In Theorem 3.5, authors state the existence of other family of periodic orbits. The orbits of that family can be unstable Hopf-type limit cycles, canard limit cycles or relaxation oscillations. Again, to analyze the stability of canard cycles, they use the way in-way out function $R(s)$. In Theorem 3.6, assuming that this function has exactly one simple zero at $s=s_{lp,0}$, the authors state that there exists a function $s_{lp}(\sqrt{\varepsilon})$ having limiting point at $s_{lp,0}$ when $\varepsilon\to 0$, such that canard limit cycles are unstable for $s<s_{lp}(\sqrt{\varepsilon})$ and stable for $s>s_{lp}(\sqrt{\varepsilon}).$ Instead of the way in-way out function $R(s)$ used in [@KS01], other authors use a similar concept known as slow divergence integral, see [@D11]. Statement of the piecewise linear system. {#sec:pwlsystem} ========================================= In this section we introduce the family of PWL differential systems we are going to work with, together with some basic elements of their dynamics. We also define some functions and quantities which are needed for stating the main results in the next section. Let us consider the following family of planar differential systems depending on the four dimensional parameter $\bm{\eta}=(a,k,m,\varepsilon)$, $$\label{systempal} \left\{\begin{array}{l} x'=y-f(x,a,k,m,\varepsilon),\\ y'=\varepsilon(a-x), \end{array}\right.$$ where the prime denotes the derivative with respect to the time $t$, $(x,y)^T\in\mathbb{R}^2$, $0<\varepsilon\ll 1,$ and the $x$-nullcline is defined by the graph of the continuous PWL function with four segments given by $$\label{vectorfield4z} f(x,a,k,m,\varepsilon)= \left\{ \begin{array}{ll} x+1-k(\sqrt{\varepsilon}-1)-m(\sqrt{\varepsilon}+a),&\quad \mbox{if }x<-1\\ -k(x+\sqrt{\varepsilon})-m(\sqrt{\varepsilon}+a), &\quad \mbox{if }-1<x\leq-\sqrt\varepsilon,\\ m(x-a), &\quad \mbox{if } \|x\|\leq\sqrt\varepsilon,\\ x-\sqrt\varepsilon+m(\sqrt\varepsilon-a), &\quad \mbox{if } x>\sqrt\varepsilon, \end{array}\right.$$ with $k>0$ and $\|m\|<2\sqrt{\varepsilon}$. The phase space is splitted into four regions: the lateral half-planes $LL=\{(x,y): x\leq -1\}$ and $R=\{(x,y): x\geq \sqrt{\varepsilon}\}$, and the central bands $L=\{(x,y): -1 \leq x\leq -\sqrt{\varepsilon}\}$ and $C=\{(x,y): \|x\|\leq\sqrt{\varepsilon}\}$. Restricted to any of these regions, the vector field is linear and it can be expressed in a matrix way as ${F}_i(\mathbf{x}) =A_i\mathbf{x} +\mathbf{b}_i$ with $i\in\{LL,L,C,R\}$ and $$A_{LL}=\left( \begin{array}{cc} -1 & 1 \\ -\varepsilon & 0 \end{array} \right),\ A_L=\left( \begin{array}{cc} k & 1 \\ -\varepsilon & 0 \end{array} \right),\ A_C=\left( \begin{array}{cc} -m & 1 \\ -\varepsilon & 0 \end{array} \right),\ A_R=\left( \begin{array}{cc} -1 & 1 \\ -\varepsilon & 0 \end{array} \right),$$ $$\mathbf{b}_{LL}=\left( \begin{array}{c} (k+m)\sqrt{\varepsilon}+ma-(k+1) \\ \varepsilon a \end{array} \right),\ \mathbf{b}_L=\left( \begin{array}{c} (k+m)\sqrt{\varepsilon}+ma \\ \varepsilon a \end{array} \right),\ \mathbf{b}_C=\left( \begin{array}{c} ma \\ \varepsilon a \end{array} \right),$$ $$\mathbf{b}_R=\left( \begin{array}{c} \sqrt{\varepsilon}(1-m)+ma \\ \varepsilon a \end{array} \right).$$ The local behavior of the flow of system - at any of the regions $i\in\{LL,L,C,R\}$ is determined by the trace $t_i$, the determinant $d_i=\varepsilon$, the discriminant $\Delta_i=t_i^2-4\varepsilon$, the slow eigenvalue $\lambda_i^s$, the fast eigenvalue $\lambda_i^q$, the slow eigenvector $\mathbf{v}_i^s=(\lambda_i^s,-\varepsilon)^T$ and the fast eigenvector $\mathbf{v}_i^q=(\lambda_i^q,-\varepsilon)^T$ of the matrix $A_i$, and by the location of the point $\mathbf{e}_i=-A_i^{-1}\mathbf{b}_i$. We summarize all this information in Table \[tab\_1\] and Table \[tab\_2\]. $LL$ $R$ ---------------- ------------------------------------------------------------------------------------- -------------------------------------------------------------------------------- $t_i $ $-1$ $-1$ $\Delta_i$ $1-4\varepsilon$ $1-4\varepsilon$ $\lambda_i^s$ $\frac {-1+\sqrt{1-4\varepsilon}}{2}=-\varepsilon- \varepsilon^2 + \ldots$ $\frac {-1+\sqrt{1-4\varepsilon}}{2}=-{\varepsilon}- {\varepsilon^2} + \ldots$ $\lambda_i^q$ $-1-\lambda_i^s$ $-1-\lambda_i^s$ $\mathbf{e}_i$ $\left(\begin{array}{c} a \\ 1+k-\sqrt{\varepsilon}(m+k)-a(m-1) \end{array}\right)$ $\left(\begin{array}{c} a \\ (m-1)(\sqrt{\varepsilon}-a)\end{array} \right)$ : Significant quantities for the dynamics of system - in the lateral half-planes $LL$ and $R$. Point $\mathbf{e}_i=-A_i^{-1}\mathbf{b}_i$ with $i\in\{LL,R\}$ is an equilibrium point only when $\mathbf{e}_{LL}\in LL$ or $\mathbf{e}_{R}\in R$.[]{data-label="tab_1"} We remark that $\mathbf{e}_i$ are equilibrium points only when they are located in the region where the system - behaves as the linear system $\mathbf{F}_i(\mathbf{x})=A_i\mathbf{x}+\mathbf{b}_i$. Otherwise, these points are called [virtual equilibrium points]{}, and they also organise the dynamic behaviour in that region even when they are not equilibrium points. $L$ $C$ ---------------- -------------------------------------------------------------------------------------------------- --------------------------------------------------------- $t_i $ $k$ $-m$ $\Delta_i$ $k^2-4\varepsilon$ $m^2-4\varepsilon$ $\lambda_i^s$ $\frac {k-\sqrt{k^2-4\varepsilon}}{2}=\frac{\varepsilon}{k}+\frac {\varepsilon^2}{k^3} + \ldots$ $-\frac{m}{2} \pm \frac {\sqrt{4\varepsilon-m^2}}{2} i$ $\lambda_i^q$ $k-\lambda_i^s$ $\mathbf{e}_i$ $\left(\begin{array}{c} a \\ -(m+k)(\sqrt{\varepsilon}+a)\end{array}\right)$ $\left(\begin{array}{c} a \\ 0 \end{array}\right)$ : Significant quantities for the dynamics of system - in the central bands $L$ and $C$. Point $\mathbf{e}_i=-A_i^{-1}\mathbf{b}_i$ is an equilibrium point only when $\mathbf{e}_i$ is contained in its own region, that is, $\mathbf{e}_{L}\in L$ or $\mathbf{e}_{C}\in C$.[]{data-label="tab_2"} From Lemma 4 in [@PTV16], the slow manifold $S_{\varepsilon}$ of system -, with $0<\varepsilon\ll 1$, is locally formed by segments, each of them contained in a region $i\in \{LL,L,R\}$ and defined by the slow eigenvector $\mathbf{v}_i^s=(\lambda_i^s,-\varepsilon)^T$ associated to the slow eigenvalue $\lambda_i^s$. Hence $$\label{def:Se} S_{\varepsilon}=\left\{ \begin{array}{ll} \mu_{LL}=\mathbf{e}_{LL}+r\mathbf{v}_{LL}^s & r\in \left[-\frac {1+a}{\lambda_{LL}^s},+\infty \right),\\ \noalign{\smallskip} \mu_{L}=\mathbf{e}_{L}-r\mathbf{v}_{L}^s & r\in\left[\frac{\sqrt{\varepsilon}+a}{\lambda_L^s},\frac{1+a}{\lambda_L^s}\right],\\ \noalign{\smallskip} \mu_{R}=\mathbf{e}_{R}-r\mathbf{v}_{R}^s & r\in\left[\frac {a-\sqrt{\varepsilon}}{\lambda_R^s},+\infty\right).\\ \end{array} \right.$$ Since $\|m\|<2\sqrt{\varepsilon}$, the matrix $A_C$ has complex eigenvalues with modulus equal to $\sqrt{\varepsilon}$, see Table \[tab\_2\]. Therefore, there is not neither a fast eigenvector nor a slow eigenvector. Thus, the slow manifold in the central region is not a segment but it is formed by two pieces of curve, the repelling one connecting with $\mu_L$ and the attracting one connecting with $\mu_R$. In Figure \[dib:geommetry\] both pieces have been represented by a unique curve. Finally, since there is not a real splitting between fast and slow behavior at the level of the eigenvalues, it follows that neither the repelling effect nor the attracting effect of the slow manifold in the central region increases drastically as $\varepsilon$ tends to zero. This phenomena was introduced in [@DFHPT13] and limit cycles flowing close to this kind of slow manifold are called quasi-canards. For this reason, we do not consider this part of the slow manifold as a part neither of the attracting branch, nor the repelling branch. We conclude that $S_{\varepsilon}^a=\mu_{LL} \cup \mu_R$ and $S_{\varepsilon}^r=\mu_L$ are the attracting branch and the repelling branch, respectively, of the slow manifold $S_{\varepsilon}$. Moreover, the attracting branch $S_{\varepsilon}^a$ intersects with the switching lines $x=-1$ and $x=\sqrt{\varepsilon}$ at the points $$\label{q1R} \mathbf{q}_1^{LL}= \begin{pmatrix} -1\\-\lambda_{LL}^s(1+a)-k(\sqrt{\varepsilon}-1)-m(\sqrt{\varepsilon}+a) \end{pmatrix},\quad \mathbf{q}_1^R=\begin{pmatrix} \sqrt{\varepsilon}\\ (m+\lambda_R^s)(\sqrt{\varepsilon}-a) \end{pmatrix},$$ respectively, see Figure \[dib:geommetry\], whereas the repelling branch $S_{\varepsilon}^r$ intersects the switching lines $x=-1$ and $x=-\sqrt{\varepsilon}$ at the points $$\label{q0L} \mathbf{q}_0^{L}=\begin{pmatrix} -\sqrt{\varepsilon}\\-(m+\lambda_L^s)(\sqrt{\varepsilon}+a) \end{pmatrix},\quad \mathbf{q}_1^L=\begin{pmatrix} -1\\-(m+k)(\sqrt{\varepsilon}+a)+(1+a)\lambda_L^q \end{pmatrix},$$ respectively. We also highlight the intersection points of the $x$-nullcline with the switching lines $x=-1,x=-\sqrt{\varepsilon}$ and $x=\sqrt{\varepsilon}$, $$\label{pLpLL} \mathbf{p}_{LL}=\begin{pmatrix} -1\\k(1-\sqrt{\varepsilon})-m(\sqrt{\varepsilon}+a) \end{pmatrix},\quad \mathbf{p}_L=\begin{pmatrix} -\sqrt{\varepsilon}\\-m(\sqrt{\varepsilon}+a) \end{pmatrix},\quad \mathbf{p}_R=\begin{pmatrix} \sqrt{\varepsilon}\\m(\sqrt{\varepsilon}-a) \end{pmatrix},$$ respectively. Note that the flow at these points is tangent to the switching line. ![Representation of the dynamical objects of system -. Linearity regions $LL,L,C$ and $R$ and switching lines $x=-1, x=-\sqrt{\varepsilon}$ and $x=\sqrt{\varepsilon}$. The fast nullcline given by the graph of the function $y=f(x,a,k,m,\varepsilon)$ and the intersection points with the switching lines $\mathbf{p}_{LL},\mathbf{p}_L$ and $\mathbf{p}_R$. The slow nullcline $x=a$ and the equilibrium point at the intersection with the fast nullcline. Slow manifold $S_{\varepsilon}$ with the attracting branch $S_{\varepsilon}^a=\mu_{LL} \cup \mu_{R}$, the repelling branch $S_{\varepsilon}^r=\mu_{L}$, and the intersection points with the switching lines $\mathbf{q}_1^{LL}, \mathbf{q}_1^R$ and $\mathbf{q}_0^L, \mathbf{q}_1^L$, respectively. Limit cycles $\Gamma_{x_0}$ are also represented both cycles with head $x_0<-1$ and cycles without head $x_0\in(-1,-\sqrt{\varepsilon})$.[]{data-label="dib:geommetry"}](dibujo_nonhyper_canards.eps "fig:") (-210,35)[$\mathbf{q}_0^L$]{} (-195,63)[$\mathbf{p}_L$]{} (-240,140)[$(-\sqrt{\varepsilon},h)$]{} (-250,120)[$\tau_L$]{} (-240,105)[[$(x_0,f(x_0))$]{}]{} (-265,70)[$\tau_L$]{} (-315,95)[$\tau_{LL}$]{} (-120,45)[$\mathbf{q}_1^R$]{} (-137,75)[$\mathbf{p}_{R}$]{} (-117,142)[$(\sqrt{\varepsilon},h)$]{} (-110,110)[$\tau_R$]{} (-175,168)[$\Gamma_{x_0}$]{} (-175,142)[$\Gamma_{x_0}$]{} (-180,27)[$\tau_C$]{} (-170,5)[$x=a$]{} (-320,75)[$(-1,h)$]{} (-45,134)[$\tau_{RR}$]{} (-280,165)[$\mathbf{q}_1^{LL}$]{} (-280,147)[$\mathbf{p}_{LL}$]{} (-302,119)[$\mathbf{q}_1^{L}$]{} (-118,170)[$\mathbf{q}_0^{RR}$]{} (-375,95)[$\mu_{LL}$]{} (-275,95)[$\mu_{L}$]{} (-60,90)[$\mu_{R}$]{} (-325,10)[$x=-1$]{} (-245,10)[$x=-\sqrt{\varepsilon}$]{} (-120,10)[$x=\sqrt{\varepsilon }$]{} Regarding the invariant sets under the flow of system -, in the following result we show that all of them are included into the rhomboid $\mathcal{R}$ which is limited by the straight lines $\mathbf{e}_{LL} +r\mathbf{v}_{LL}^s$, $\mathbf{e}_{LL} +r\mathbf{e}_{1}$, $\mathbf{e}_R +r\mathbf{v}_R^s$, and $\mathbf{e}_R +r\mathbf{e}_1$, where $r\in \mathbb{R}$ and $\mathbf{e}_1^T=(1,0)$. The result follows straightforward by analysing the orientation of the flow over the boundary of $\mathcal{R}$. \[lem:inv\_reg\] - The rhomboid $\mathcal{R}$, previously defined, is positively invariant under the flow defined by the system -. - The rhomboid $\mathcal{R}$, previously defined, contains the equilibrium point and every periodic orbit of the system -. Every limit cycle $\Gamma$ of system - intersects the $x$-nullcline $(x,f(x,a,k,m,\varepsilon))$ at exactly one point $(x_{\Gamma},f(x_{\Gamma},a,k,m,\varepsilon))$ with $x_{\Gamma}<a$. From now on, we call width of the limit cycle $\Gamma$, to the first coordinate of this intersection point, that is $x_{\Gamma}$. One special limit cycle, assuming that it exists, is the one having width $x=-1$. Such a limit cycle is tangent to the switching line $\{x=-1\}$ at the point $\mathbf{p}_{LL}$, and therefore, it is the separation cycle between the limit cycles intersecting the lateral region $LL$ and those that do not intersect it. In a similar way, the limit cycle having width $x=-\sqrt{\varepsilon}$ is tangent at $\mathbf{p}_L$ to the switching line $\{x=-\sqrt{\varepsilon}\}$ and it is the separation cycle between the limit cycles intersecting the region $L$ and those that do not intersect it. When $\varepsilon$ is small enough, the limit cycles with width $-1<x<-\sqrt{\varepsilon}$ will be referred to as headless canard limit cycles whereas limit cycles with width $x<-1$ will be referred to as canard limit cycles with head. Therefore, the limit cycle with width $x=-1$ will be referred as [the transitory canard]{}, see [@DDR14], and it is the boundary between headless canard cycles and canard cycles with head. In addition, every limit cycle with head, $\Gamma_{x_0}$ where $x_0<-1$, intersects the separation line $\{x=-1\}$ at two points. Let $(-1,h)$ be the one below the point $\mathbf{p}_{LL}$, see Figure \[dib:geommetry\]. Moreover, every headless limit cycle, $\Gamma_{x_0}$ with $x_0\in(-1,-\sqrt{\varepsilon})$, intersects the separation line $\{x=-\sqrt{\varepsilon}\}$ also at two points. Let $(-\sqrt{\varepsilon},h)$ be the one over the point $\mathbf{p}_{L}$. We referred to $h$ as the [height]{} of the limit cycle $\Gamma_{x_0}$ both in the case with head and in the case without head. Therefore, any limit cycle can be labeled by the two different quantities that we have denoted by the width and by the height. Let $\Phi$ be the piecewise function which maps the width of a limit cycle into its height, i.e. $$\Phi(x)=\left\{ \begin{array}{ll} \Phi_{3z}(x) & \text{if } x\in[-1,-\sqrt{\varepsilon}), \\ \Phi_{4z}(x) & \text{if } x<-1, \end{array} \right.$$ where $\Phi_{3z}$ is defined by the flow of the linear system in the region $L$, and $\Phi_{4z}$ is defined by the flow, in backward time, of the linear system in the region $LL$. Therefore, through $h=\Phi(x)$ we can pass from the width $x$ of a limit cycle $\Gamma$ to its height $h$. Typically, the height $h$ is more convenient for computational purposes, whereas the width $x$ is more convenient for stating the results. To analyze the stability of the canard limit cycles in the PWL framework, it is not possible to use the same approach that it is used in the smooth context, since the singularity of the reduced flow can not be removed through the desingularization process, [@KS01]. In fact, the reduced equation associated to the system - writes as $$\dot{x}= \left\{ \begin{array}{ll} \frac{x}{k} & x < 0,\\ -x & x> 0, \end{array} \right.$$ whose desingularized flow is discontinuous at $x=0$. Therefore, the analysis of the stability of limit cycles through techniques based on the singular flow, such as the way in-way out function, [@KS00; @KS01], or the slow divergence integral, see [@DDR14] and references there in, can not be successfully applied in this context. Nevertheless, this analysis can be performed directly when $\varepsilon>0$ by explicitly computing the integral of the divergence as the sum of the products of the traces of the linear systems and the time of flight in each region of linearity [@freire1]. Statement of the Main Results {#sec_mainresults} ============================= In this section we present the main results in the paper. These results concern to the existence of a one parameter family of canard limit cycles in the PWL system -, and to the description about how this family organizes along a curve in the plane $(x,a)$, where $x$ is the width of the canard limit cycle and $a$ is the parameter value. The results also provide information about the stability of the limit cycles, paying special attention to semi-stable ones. In order to be fluid in the exposition, we left the proofs and their technical issues for next sections. In the first result we assure that, the starting point of the curve organizing the family of limit cycles exhibited by system - takes place at a Hopf-like bifurcation [@freire1]. At this bifurcation a limit cycle appears after the change of stability of the singular point, just like in the Hopf bifurcation. The difference between both kind of bifurcations is the relation between the amplitude of the limit cycle and the bifurcation value, this relation is linear in the Hopf-like bifurcation and a square root in the Hopf bifurcation. The proof of the following result is a straightforward conclusion of Theorem 5.1 and Theorem 5.2 in [@S19], see also [@freire1] and [@S18]. \[th:hopf\] System - has a unique singular point $\mathbf{e}=(a,f(a))$ which converges to the fold of the critical manifold at the origin as $(\varepsilon,a)$ tends to zero. Moreover, the function, $$a_H(\sqrt{\varepsilon})= \left\{ \begin{array}{ll} \sqrt{\varepsilon}& m=-\sqrt{\varepsilon},\\ -\sqrt{\varepsilon} & m=\sqrt{\varepsilon}, \end{array} \right.$$ satisfies that the equilibrium is stable for $a>a_H(\sqrt{\varepsilon})$ and looses stability through a Hopf-like bifurcation as $a$ passes through $a_H(\sqrt{\varepsilon}).$ In particular, if $m=-\sqrt{\varepsilon}$, a stable limit cycle appears when $a<a_H(\sqrt{\varepsilon})$ in a supercritical bifurcation, and if $m=\sqrt{\varepsilon}$, a unstable limit cycle appears when $a>a_H(\sqrt{\varepsilon})$ in a subcritical bifurcation. In both cases, the size of the limit cycle depends linearly on the distance $\|a_{H}(\sqrt{\varepsilon})-a\|$. Next theorem is devoted to the existence of a trajectory connecting the attracting branch and the repelling branch of the slow manifold. This connection is usually referred to as the [ maximal canard]{} trajectory. \[th:connection\] Set $m=\pm\sqrt{\varepsilon}.$ There exist a value $\varepsilon_0>0$ and a function $a=\tilde{a}(k,\varepsilon; m),$ analytic as a function of $(k,\sqrt{\varepsilon})$, defined in the open set $U=(0,+\infty)\times (0,\varepsilon_0)$ and such that, for $(k,\varepsilon)\in U$, a solution of system - starting in the attracting branch of the slow manifold, $\mu_R$, connects to the repelling branch of the slow manifold, $\mu_L$, if and only if $a=\tilde{a}(k,\varepsilon;m)$. In such case, the time of flight of the transition is $\tau_C(k,\varepsilon; m)>0$. First terms of the expansions of $\tilde{a}(k,\varepsilon; m)$ and $\tau_C(k,\varepsilon; m)$ are given as follows, $$\label{aepmenos} \tilde{a}(k,\varepsilon;m)= \left\{\begin{array}{l} \dfrac{ {{e}^{\frac{\pi }{\sqrt{3}}}}-1}{{{e}^{\frac{\pi }{\sqrt{3}}}}+1}\sqrt{\varepsilon} -\dfrac{{{e}^{\frac{\pi }{\sqrt{3}}}}}{{{\left( {{e}^{\frac{\pi }{\sqrt{3}}}}+1\right) }^{2}}} \left(\dfrac{1-k^2}{{k}^{2}}\right) {{\varepsilon}^{{3}/{2}}} +O(\varepsilon^2),\quad \mbox{if }m=-\sqrt{\varepsilon}, \\ -\dfrac {e^{\frac{\pi}{\sqrt{3}}}-1} {e^{\frac{\pi}{\sqrt{3}}}+1}\sqrt{\varepsilon} -\dfrac {e^{\frac{\pi}{\sqrt{3}}}} {\left(e^{\frac{\pi}{\sqrt{3}}}+1\right)^2} \left( \dfrac {1-k^2}{k^2} \right) \varepsilon^{3/2}+O(\varepsilon^2),\quad \mbox{if }m=\sqrt{\varepsilon}, \end{array}\right.$$ and $$\label{tauepmenos} \tau_C(k,\varepsilon; m)= \left\{\begin{array}{l} \dfrac{2 \pi }{\sqrt{3}}\dfrac 1{\sqrt{\varepsilon}} -\dfrac{1+k}{k} -\dfrac{1-k^2}{2 {{k}^{2}}} \sqrt{\varepsilon} +O(\varepsilon),\quad \mbox{if }m=-\sqrt{\varepsilon}, \\ \noalign{\smallskip} \dfrac{2\pi}{\sqrt{3}} \dfrac1{\sqrt{\varepsilon}} -\dfrac {1+k}{k} + \dfrac {1-k^2}{2k^2} \sqrt{\varepsilon} +O(\varepsilon) ,\quad \mbox{if }m=\sqrt{\varepsilon}. \end{array}\right.$$ The existence of the maximal canard trajectory, together with the divergence of the flow in a neighborhood of the slow manifold, provide the arguments we use in Section \[sec\_proofs\] to proof the following result about the existence of canard cycles of any suitable width. To state the result in a proper way we introduce the following values $$\label{def:xs} x_r= -(1+k)+k\sqrt{\varepsilon}-\lambda_L^s (\sqrt{\varepsilon}+a), \quad x_s= -\sqrt{\varepsilon}-\lambda_L^s (\sqrt{\varepsilon}+a).$$ This values correspond with the end points of the interval such that limit cycles having width contained in $(x_r,x_s)$ are canard limit cycles. In fact, limit cycles having width $x<x_r$ are relaxation oscillations whereas limit cycles having width $x>x_s$ are still under the effect of the Hopf-like bifurcation. \[th:exist\_Gamma\] Fix $\varepsilon_0$ sufficiently small and set $m=\pm\sqrt{\varepsilon}$. There exists a function $a=\hat{a}(k,\varepsilon,x_0; m),$ $C^\infty$ function of $(k,\sqrt{\varepsilon},x_0)$, defined in the open set $U=(0,+\infty)\times (0,\varepsilon_0)\times(x_r,x_s)$, fulfilling $$\begin{array}{ll} \| \hat{a}(k,\varepsilon,x_0; m)-\tilde{a}(k,\varepsilon;m)\| \approx \|x_0\| e^{-\frac {x_0}{\varepsilon^{3/2}}} & x_0\in[-1,x_s),\\ \|\hat{a}(k,\varepsilon,x_0; m)-\tilde{a}(k,\varepsilon;m)\| \approx \|x_0-x_r\| e^{-\frac {x_0-x_r}{\varepsilon}} & x_0\in(x_r,-1), \end{array}$$ with $\tilde{a}(k,\varepsilon;m)$ the function defined in Theorem \[th:connection\], and such that, for $(k,\varepsilon,x_0)\in U$ and $a=\hat{a}(k,\varepsilon,x_0; m)$ system - possesses a canard limit cycle, $ \Gamma_{x_0}$, passing through $(x_0,f(x_0))$. The canard limit cycle is headless if $x_0\in(-1,{x}_s)$ and with head if $x_0\in(x_r,-1).$ Previous result describe the canard explosion taking place in the PWL framework. There, it can be observed that the slope of the explosion is different before and after the maximal canard. In the next result we establish the stability of the canard limit cycles obtained in the previous theorem. To do this, we compute a piecewise smooth function $$R(x)=\left\{ \begin{array}{ll} R_{3z}(x) & x\in[-1,x_s),\\ R_{4z}(x) & x\in (x_r,x_u), \end{array} \right.$$ see and , approximating the integral of the divergence along the limit cycle, $\Gamma_{x}$, and use the sign of this function to conclude the stability of $\Gamma_{x}$. Nevertheless, this approach does not produce accurate results if the canard limit cycle is close to the transitory canard, the one having width $x=-1$. The interval where the sign of $R(x)$ does not provide the stability of the canard limit cycles is given by $(x_u,-1)$, where $$\label{def:xu} x_u=-1+\lambda_{LL}^s(1+a).$$ We organise the results into two theorems depending on whether the Hopf-like bifurcation is supercritical or subcritical. We illustrate the theorems with Figure \[figteorneg\] and Figure \[figteorpos\]. \[th:supercritico\] Set $\varepsilon>0$ small enough, $m=-\sqrt{\varepsilon}$, $x_0\in (x_r,x_u)\cup [-1,x_s)$ and $a=\hat{a}(k,\varepsilon,x_0;m)$. Let $\Gamma_{x_0}$ be the canard limit cycle of system - whose existence has been proved in Theorem \[th:exist\_Gamma\]. The following statements hold: - For $k\leq 1$, the canard limit cycle $\Gamma_{x_0}$ is hyperbolic and stable. - For $k>1,$ there exist exactly two values $x_1\in(-1,x_s)$ and $x_2\in(x_r,x_u)$ such that the canard limit cycle $\Gamma_{x_0}$ is hyperbolic and stable if $x_0\in(x_r,x_2)\cup(x_1,x_s),$ hyperbolic and unstable if $x_0\in(x_2,x_u)\cup(-1,x_1),$ and a saddle-node canard cycle if $x_0=x_1$ and $x_0=x_2.$ ![Representation of the width of limit cycles of system -, versus the parameter $a$, in the supercritical case $m=-\sqrt{\varepsilon}$. The dotted line in all the panels corresponds with the width of the transitory canard, i.e. the limit between headless canard cycles and canard cycles with head. Moreover, discoloured parts of the curves refers to the regions where the sign of the functions $R_{3z}$ and $R_{4z}$ do not guarantee the stability of the limit cycle. In panels (a) and (b) we represent the cases where $k<1$ and $k=1$, respectively. In these cases, the limit cycle appearing after the supercritical bifurcation at $a_{H}$ exhibits a canard explosion. Panel (c) corresponds with the case $k>1$. Here, two saddle nodes of width $x_1<x_2$, take place at the values $a^1_{sn}<a^2_{sn}$ after the Hopf bifurcation at $a_{H}$.[]{data-label="figteorneg"}](amp_vs_a_supcr_kmenor1.eps "fig:"){width="4.5cm"}![Representation of the width of limit cycles of system -, versus the parameter $a$, in the supercritical case $m=-\sqrt{\varepsilon}$. The dotted line in all the panels corresponds with the width of the transitory canard, i.e. the limit between headless canard cycles and canard cycles with head. Moreover, discoloured parts of the curves refers to the regions where the sign of the functions $R_{3z}$ and $R_{4z}$ do not guarantee the stability of the limit cycle. In panels (a) and (b) we represent the cases where $k<1$ and $k=1$, respectively. In these cases, the limit cycle appearing after the supercritical bifurcation at $a_{H}$ exhibits a canard explosion. Panel (c) corresponds with the case $k>1$. Here, two saddle nodes of width $x_1<x_2$, take place at the values $a^1_{sn}<a^2_{sn}$ after the Hopf bifurcation at $a_{H}$.[]{data-label="figteorneg"}](amp_vs_a_supcr_k=1.eps "fig:"){width="4.5cm"}![Representation of the width of limit cycles of system -, versus the parameter $a$, in the supercritical case $m=-\sqrt{\varepsilon}$. The dotted line in all the panels corresponds with the width of the transitory canard, i.e. the limit between headless canard cycles and canard cycles with head. Moreover, discoloured parts of the curves refers to the regions where the sign of the functions $R_{3z}$ and $R_{4z}$ do not guarantee the stability of the limit cycle. In panels (a) and (b) we represent the cases where $k<1$ and $k=1$, respectively. In these cases, the limit cycle appearing after the supercritical bifurcation at $a_{H}$ exhibits a canard explosion. Panel (c) corresponds with the case $k>1$. Here, two saddle nodes of width $x_1<x_2$, take place at the values $a^1_{sn}<a^2_{sn}$ after the Hopf bifurcation at $a_{H}$.[]{data-label="figteorneg"}](amp_vs_a_supcr_kmayor1.eps "fig:"){width="4.5cm"} (0,0) (-358,-8)[$0$]{} (-301,-8)[$a_{H}$]{} (-405,48)[with head]{} (-405,35)[headless]{} (-220,-8)[$0$]{} (-164,-8)[$a_H$]{} (-80,-8)[$0$]{} (-60,-8)[$a^1_{sn}$]{} (-40,-8)[$a^2_{sn}$]{} (-17,7)[$a_H$]{} (-415,65)[(a)]{} (-277,65)[(b)]{} (-140,65)[(c)]{} (-18,27)[$x_1$]{} (-18,46)[$-1$]{} (-18,70)[$x_2$]{} \[th:subcritico\] Set $\varepsilon>0$ small enough, $m=\sqrt{\varepsilon}$, $x_0\in (x_r,x_u)\cup [-1,x_s)$ and $a=\hat{a}(k,\varepsilon,x_0;m)$. Let $\Gamma_{x_0}$ be the canard limit cycle of system - whose existence has been proved in Theorem \[th:exist\_Gamma\]. The following statements hold: - For $k<1,$ there exists exactly one value $x_1\in(-1,x_s)$ such that $\Gamma_{x_0}$ is an hyperbolic limit cycle, if $x_0\in(x_r,x_u)\cup(-1,x_s)\setminus \{x_1\}$, and a saddle-node canard cycle, if $x_0=x_1$. Moreover, $\Gamma_{x_0}$ is stable if $x_0<x_1$ and unstable if $x_0>x_1.$ - For $k=1,$ the canard limit cycle $\Gamma_{x_0}$ is hyperbolic and stable if $x_0\in(x_r,x_u)$ and hyperbolic and unstable if $x_0\in(-1,x_s).$ - For $k>1,$ there exists exactly one value $x_2\in(x_r,x_u)$ such that $\Gamma_{x_0}$ is hyperbolic, if $x_0\in(x_r,x_u)\cup(-1,x_s)\setminus \{x_2\}$, and a saddle-node canard cycle, if $x_0=x_2$. Moreover, $\Gamma_{x_0}$ is stable if $x_0<x_2$ and unstable if $x_0>x_2.$ ![Representation of the width of limit cycles of system -, versus the parameter $a$, in the subcritical case $m=\sqrt{\varepsilon}$. The dotted line in all the panels corresponds with the transitory canard cycle with width $x=-1$. Moreover, discolored parts of the curves refers to the regions where the sign of the functions $R_{3z}$ and $R_{4z}$ do not guarantee the stability of the limit cycle. In panels (a) and (b) we represent the cases where $k<1$ and $k=1$, respectively, and in panel (c) we represent the case $k>1$.[]{data-label="figteorpos"}](amp_vs_a_subcr_kmenor1.eps "fig:"){width="4.5cm"}![Representation of the width of limit cycles of system -, versus the parameter $a$, in the subcritical case $m=\sqrt{\varepsilon}$. The dotted line in all the panels corresponds with the transitory canard cycle with width $x=-1$. Moreover, discolored parts of the curves refers to the regions where the sign of the functions $R_{3z}$ and $R_{4z}$ do not guarantee the stability of the limit cycle. In panels (a) and (b) we represent the cases where $k<1$ and $k=1$, respectively, and in panel (c) we represent the case $k>1$.[]{data-label="figteorpos"}](amp_vs_a_subcr_k=1.eps "fig:"){width="4.5cm"}![Representation of the width of limit cycles of system -, versus the parameter $a$, in the subcritical case $m=\sqrt{\varepsilon}$. The dotted line in all the panels corresponds with the transitory canard cycle with width $x=-1$. Moreover, discolored parts of the curves refers to the regions where the sign of the functions $R_{3z}$ and $R_{4z}$ do not guarantee the stability of the limit cycle. In panels (a) and (b) we represent the cases where $k<1$ and $k=1$, respectively, and in panel (c) we represent the case $k>1$.[]{data-label="figteorpos"}](amp_vs_a_subcr_kmayor1.eps "fig:"){width="4.5cm"} (0,0) (-338,-8)[$a_{H}$]{} (-303,-8)[$0$]{} (-405,48)[with head]{} (-405,35)[headless]{} (-200,-8)[$a_H$]{} (-164,-8)[$0$]{} (-27,-8)[$0$]{} (-60,-8)[$a_H$]{} (-415,65)[(a)]{} (-277,65)[(b)]{} (-139,65)[(c)]{} (-20,48)[-1]{} In the last main result we state that for every width between the smallest canard cycle and the relaxation oscillation cycle, that is for every $x_0\in(x_r,x_u)\cup [-1,x_s)$, there exist values of the parameters such that system - exhibits a saddle-node canard limit cycle $\Gamma_{x_0}$ of width $x_0$. \[th:supercritico\_LP\] Consider system - with $m=-\sqrt{\varepsilon}$ or $m=\sqrt{\varepsilon}.$ For each $x_0\in(x_r,x_u)\cup(-1,x_s),$ there exists a value $\varepsilon_0$ and a function $k_{x_0}(\varepsilon)$ defined for $\varepsilon\in(0,\varepsilon_0),$ such that system - with parameters $k=k_{x_{0}}(\varepsilon)$ and $a=\hat{a}(k_{x_{0}}(\varepsilon),\varepsilon,x_0;m)$ exhibits the saddle-node canard $\Gamma_{x_{0}}$ whose existence has been stated in Theorem \[th:supercritico\] for $m=-\sqrt{\varepsilon}$ and in Theorem \[th:subcritico\] for $m=\sqrt{\varepsilon},$ respectively. Proofs of the Main Results {#sec_proofs} ========================== Let us begin by introducing some notation. For chosen parameters $\bm{\eta}=(a,k,m,\varepsilon)$, and a point $\mathbf{p} \in \mathbb{R}^2$, we denote by $$\varphi(t;\mathbf{p},\bm{\eta})=\left( x(t;\mathbf{p},\bm{\eta}), y(t;\mathbf{p},\bm{\eta})\right)$$ the solution of system with initial condition $\varphi(0;\mathbf{p},\bm{\eta})=\mathbf{p}$. The coordinates of $\varphi(t;\mathbf{p},\bm{\eta})$ will be referred to as $x^i(t;\mathbf{p},\bm{\eta})$ and $y^i(t;\mathbf{p},\bm{\eta})$, with $i\in \{LL,L,C,R\}$, depending on the region where the solution belongs to, for that value of $t$. Proof of Theorem \[th:connection\] {#sec_theo1} ---------------------------------- The existence of the maximal canard solution reduces to the existence of an orbit connecting points $\mathbf{q}_1^R$ and $\mathbf{q}_0^L$, see Figure \[dib:geommetry\]. The set of conditions characterizing this connection is given by the existence of ${\tau}_C>0$ and parameters $\bm{\eta}=(a,k,m,\varepsilon)$, with $0<\varepsilon\ll 1,$ $k>0$ and $\|a\|<\sqrt\varepsilon$, and such that the following conditions hold: $$\begin{aligned} x^C(\tau_C; \mathbf{q}_1^R , \bm{\eta})&=&-\sqrt\varepsilon,\label{E1}\\ y^C(\tau_C; \mathbf{q}_1^R, {\bm{\eta}})&=& -(m+\lambda_L^s)(\sqrt{\varepsilon}+a),\label{E2}\\ x^C(s; \mathbf{q}_1^R, {\bm{\eta}})&\in&\left(-\sqrt\varepsilon,\sqrt\varepsilon\right)\; \mbox{ for all }\; s\in(0,{\tau}_C), \label{Ineq}\end{aligned}$$ where $m=\pm \sqrt{\varepsilon}$. Next we proof the existence of a solution for system - and inequality in the supercritical case $m=-\sqrt{\varepsilon}$. The existence of a solution in the subcritical case, $m=\sqrt{\varepsilon}$, follows by completely analogous arguments. Let us define the following functions, $$\label{defFG} \left\{\begin{array}{l} F(\tau,a,k,\varepsilon)=x^C(\tau; \mathbf{q}_1^R , \bm{\eta})+\sqrt\varepsilon,\\ \noalign{\medskip} G(\tau,a,k,\varepsilon)= y^C(\tau; \mathbf{q}_1^R, {\bm{\eta}})+(\lambda_L^s-\sqrt{\varepsilon})(\sqrt{\varepsilon}+a). \end{array}\right.$$ Thus, system (\[E1\])-(\[E2\]) is equivalent to system $$\label{sysFG} \left\{\begin{array}{l} F(\tau,a,k,\varepsilon)=0,\\ G(\tau,a,k,\varepsilon)=0, \end{array}\right.$$ and inequality (\[Ineq\]) is equivalent to $$F(s,a,k,\varepsilon)\in\left(0,2\sqrt\varepsilon\right)\; \mbox{ for all }\; s\in(0,{\tau_C}). \label{IneqF}$$ By integrating the linear system defined in the central band $C$ with initial condition ${\mathbf{q}}_1^R,$ we obtain the following explicit expression of the solution, $$\label{solxy} \left\{\begin{array}{ll} x^C(\tau;\mathbf{q}_1^R,\bm{\eta})&= \displaystyle \frac {e^{\frac{\sqrt{\varepsilon}}2\tau}(\sqrt{\varepsilon}-a)}{\sqrt{3}} \left( \left( \frac{2\lambda_R^s}{\sqrt{\varepsilon}}-1 \right) \sin\left(\frac{\sqrt{3\varepsilon}}{2} \tau\right) +\sqrt{3}\cos\left(\frac{\sqrt{3\varepsilon}}{2} \tau\right) \right)+a \\ \noalign{\medskip} y^C(\tau;\mathbf{q}_1^R,\bm{\eta})&= \displaystyle \frac {e^{\frac{\sqrt{\varepsilon}}2\tau}(\sqrt{\varepsilon}-a)}{\sqrt{3}} \left((\lambda_R^s-\sqrt{\varepsilon})\sqrt{3} \cos\left(\frac{\sqrt{3\varepsilon}}{2} \tau\right) -(\lambda_R^s+\sqrt{\varepsilon}) \sin\left(\frac{\sqrt{3\varepsilon}}{2} \tau\right) \right). \end{array}\right.$$ Consider the rescaling of parameters $\tau$ and $a,$ as follows : $$\label{rescaling} \quad \tau=\bar\tau/\sqrt\varepsilon,\quad a=\bar a\sqrt{\varepsilon}.$$ It follows from the definition of $F$ (see (\[defFG\])) and formula for $x^C$ (see (\[solxy\])) that $F=\sqrt\varepsilon\bar F$, where $\bar F$ is given by $$\label{eq-hatf} \bar F(\bar{\tau},\bar a,\varepsilon)=\sqrt{3} (\bar{a}-1) e^{\bar{\tau}/2} \left(\left(2 \sqrt{\varepsilon}+1\right) \sin \left(\frac{\sqrt{3} \bar{\tau}}{2}\right)-\sqrt{3} \cos \left(\frac{\sqrt{3} \bar{\tau}}{2}\right)\right)+3 \bar{a}+3+O(\varepsilon).$$ In the same way, from the definition of $G$ (see (\[defFG\])) and expressions for $x^C$ and $y^C$ (see (\[solxy\])), it holds that $G=\varepsilon\bar G$, where $\bar G$ is given by $$\bar G(\bar\tau,\bar a,k,\varepsilon)=\frac{(\bar{a}+1) \left(\sqrt{\varepsilon}-k\right)}{k}+\frac{(\bar{a}-1) e^{\bar{\tau}/2} \left(\sqrt{3} \left(\sqrt{\varepsilon}+1\right) \cos \left(\frac{\sqrt{3} \bar{\tau}}{2}\right)-\left(\sqrt{\varepsilon}-1\right) \sin \left(\frac{\sqrt{3} \bar{\tau}}{2}\right)\right)}{\sqrt{3}}+O(\varepsilon).$$ Hence, solving $F=G=0$ for $\varepsilon> 0$ is equivalent to solving $\bar F=\bar G=0$. For $\varepsilon=0,$ we have $$\begin{array}{ll}\label{computationFG} \bar F&=\sqrt{3} (\bar{a}-1) e^{\bar{\tau}/2} \left(\sin \left(\frac{\sqrt{3} \bar{\tau}}{2}\right)-\sqrt{3} \cos \left(\frac{\sqrt{3} \bar{\tau}}{2}\right)\right)+3 \bar{a}+3,\\ \noalign{\smallskip} \bar G&=-(\bar{a}+1)+\displaystyle\frac{(\bar{a}-1) e^{\bar{\tau}/2} \left(\sqrt{3} \cos \left(\frac{\sqrt{3} \bar{\tau}}{2}\right)+\sin \left(\frac{\sqrt{3} \bar{\tau}}{2}\right)\right)}{\sqrt{3}}. \end{array}$$ For $\bar a=(e^{\frac{\pi}{\sqrt{3}}}-1)/(e^{\frac{\pi}{\sqrt{3}}}+1), \bar\tau=2\pi/\sqrt{3}$ equations $\bar F=0,\bar G=0$ hold. To apply the implicit function theorem, it is necessary to prove that $\det(J(2\pi/\sqrt{3},(e^{\frac{\pi}{\sqrt{3}}}-1)/(e^{\frac{\pi}{\sqrt{3}}}+1),k,0))\neq 0,$ where $$J(\bar \tau,\bar a,k,\varepsilon)= \left( \begin{array}{cc} \displaystyle\frac{\partial \bar F}{\partial \bar \tau}(\bar \tau,\bar a,k,\varepsilon)& \displaystyle\frac{\partial \bar F}{\partial \bar a}(\bar \tau,\bar a,k,\varepsilon)\medskip\\ \displaystyle\frac{\partial \bar G}{\partial \bar \tau}(\bar \tau,\bar a,k,\varepsilon)& \displaystyle\frac{\partial \bar G}{\partial \bar a}(\bar \tau,\bar a,k,\varepsilon)\\ \end{array} \right).$$ It is easy to see that the partial derivatives take the following values at the point $(2\pi/\sqrt{3},(e^{\frac{\pi}{\sqrt{3}}}-1)/(e^{\frac{\pi}{\sqrt{3}}}+1),k,0),$ $$\frac{\partial \bar F}{\partial \bar \tau}=0,\quad \frac{\partial \bar F}{\partial \bar a}=3(1+e^{\frac{\pi}{\sqrt{3}}}),$$ $$\frac{\partial \bar G}{\partial \bar \tau}=\frac{2e^{\frac{\pi}{\sqrt{3}}}}{1+e^{\frac{\pi}{\sqrt{3}}}},\quad \frac{\partial \bar G}{\partial \bar a}=-1-e^{\frac{\pi}{\sqrt{3}}},$$ and then, $\det(J(2\pi/\sqrt{3},(e^{\frac{\pi}{\sqrt{3}}}-1)/(e^{\frac{\pi}{\sqrt{3}}}+1),k,0))=-6e^{2\frac{\pi}{\sqrt{3}}}\neq 0.$ Thus, from the implicit function theorem we conclude that there exist a value $\varepsilon_0>0$ and functions $\tilde{a}(k,\varepsilon;m)$ and $\tau_C(k,\varepsilon;m)$, analytic and smooth as a function of $(k,\sqrt{\varepsilon})$, respectively, defined in the open set $U=(0,+\infty)\times (0,\varepsilon_0)$ and such that equations (\[E1\])-(\[E2\]) has the solution $a=\tilde{a}(k,\varepsilon;m)$ and $\tau=\tau_C(k,\varepsilon;m)$. Moreover, the lower order terms in $\sqrt{\varepsilon}$ of such a solution coincides with those in the expression (\[aepmenos\])-(\[tauepmenos\]). The remainder terms in this approximated solution can be obtained by the method of the undetermined coeficients. Finally, since the angle travelled by the solution from $\mathbf{q}_1^R$ to $\mathbf{q}_0^L$ satisfies that $\beta \tau_C(k,\varepsilon;m)<\pi$ and $\|\tilde{a}(k,\varepsilon;m)\|<\sqrt{\varepsilon}$, it follows that inequality (\[Ineq\]) is fulfilled. Therefore, we conclude that for $a=\tilde{a}(k,\varepsilon;m)$ system - exhibits an orbit connecting the slow manifolds with time of flight equal to $\tau_C(k,\varepsilon;m)>0$. Proof of Theorem \[th:exist\_Gamma\] ------------------------------------ We perform the proof for the headless canard limit cycles. The proof for cycles with head follows in a similar way. Consider a point $(x_0,f(x_0))$ with $x_0\in(-1,{x}_s)$, and the orbit $\Gamma_{x_0}$ through this point, see Figure \[dib:geommetry\]. Since the point is contained in the positive invariant region $\mathcal{R}$, see Lemma \[lem:inv\_reg\], then by integrating both in forward and backward time, the orbit targets the switching line $\{x=-\sqrt{\varepsilon}\}$ at two points respectively $$\begin{pmatrix} -\sqrt{\varepsilon}\\h \end{pmatrix},\quad\quad \mathbf{p}_0 = \mathbf{q}_0^L+ \begin{pmatrix} 0 \\ h e^{-\frac {k h}{\varepsilon(\sqrt{\varepsilon}-a)}} \end{pmatrix}.$$ The expression of $\mathbf{p}_0$ follows from Lemma \[lem:Pmaps\], since $h=\Phi_{3z}(x_0) > (-m+\varepsilon^{-\nu}\lambda_L^s)(\sqrt{\varepsilon}+a)$ provided that $x_0\in (-1,x_s)$, see Lemma \[lem:dom\]. Moreover, $\Gamma_{x_0}$ also intersects in forward time the switching line $\{x=\sqrt{\varepsilon}\}$ at the two points $$\begin{pmatrix} \sqrt{\varepsilon}\\h \end{pmatrix},\quad\quad \mathbf{p}_1=\mathbf{q}_1^R+ \begin{pmatrix} 0 \\ h e^{-\frac {h}{\varepsilon(\sqrt{\varepsilon}-a)}} \end{pmatrix}.$$ The conditions on the points $\mathbf{p}_0$ and $\mathbf{p}_1$ in order to be $\Gamma_{x_0}$ a limit cycle is the existence of values for the parameters $\bm{\eta}=(a,k,m,\varepsilon)$ such that the solution of the linear system in the central band $C$ with initial condition at $\mathbf{p}_1$ targets first time the switching line $\{x=-\sqrt{\varepsilon}\}$ at the point $\mathbf{p}_0$, i.e., $e^{\tau A_c}(\mathbf{p}_1-\mathbf{e}_c) + \mathbf{e}_c-\mathbf{p}_0 = \mathbf{0}$ with $\beta \tau<\pi$. By substituting the values of $\mathbf{p}_0$ and $\mathbf{p}_1$, previous equation writes as $$\label{aux0:lem_43} e^{\tau A_c}(\mathbf{q}_1^R -\mathbf{e}_c) + \mathbf{e}_c-\mathbf{q}_0 + e^{\tau A_c} \begin{pmatrix} 0 \\ h e^{-\frac {h}{\varepsilon(\sqrt{\varepsilon}-a)}} \end{pmatrix} -\begin{pmatrix} 0 \\ h e^{-\frac {k h}{\varepsilon(\sqrt{\varepsilon}-a)}} \end{pmatrix} = \mathbf{0}.$$ Considering the change of variables $\tau=\frac{\tilde{\tau}}{\sqrt{\varepsilon}}$ and $a=\hat{a}\sqrt{\varepsilon}$ and multiplying previous equations by $\frac {1}{\sqrt{\varepsilon}}$ and $\frac 1{\varepsilon}$, respectively, we obtain that expression writes as $E_p(\tilde{\tau},\hat{a},k,m,\varepsilon)=\mathbf{0}$, where $$\label{aux:lem_43} E_p(\tilde{\tau},\hat{a},k,m,\varepsilon,h)= E_q(\tilde{\tau},\hat{a},k,m,\varepsilon) + \begin{pmatrix} \frac 1{\sqrt{\varepsilon}} & 0 \\ 0 & \frac 1{\varepsilon} \end{pmatrix} \left( e^{\tilde{\tau} A_c} \begin{pmatrix} 0 \\ h e^{-\frac {h}{\varepsilon(\sqrt{\varepsilon}-\hat{a})}} \end{pmatrix} -\begin{pmatrix} 0 \\ h e^{-\frac {k h}{\varepsilon(\sqrt{\varepsilon}-\hat{a})}} \end{pmatrix} \right),$$ and $E_q(\tilde{\tau},\hat{a},k,m,\varepsilon)=e^{\tilde{\tau} A_c}(\mathbf{q}_1^R -\mathbf{e}_c) + \mathbf{e}_c-\mathbf{q}_0$ are the equations given in (\[computationFG\]) and establishing the connection between $\mathbf{q}_1^R$ and $\mathbf{q}_0^L$. Then, the Jacobian matrix respect to the variables $\tau$ and $a$ satisfies $$\begin{aligned} \left. D_{\tilde{\tau},\hat{a}}E_p\right\|_{(\tilde{\tau},\hat{a},k,m,\varepsilon,h)}&= \left.D_{\tilde{\tau},\hat{a}}E_q\right\|_{(\tilde{\tau},\hat{a},k,m,\varepsilon)}\\ &+ \begin{pmatrix} \frac 1{\sqrt{\varepsilon}} & 0 \\ 0 & \frac 1{\varepsilon} \end{pmatrix} \left( e^{\tilde{\tau} A_c} \begin{pmatrix} h e^{-\frac {h}{\varepsilon(\sqrt{\varepsilon}-\hat{a})}} & 0 \\ 0 & -\frac{h^2 e^{-\frac {h}{\varepsilon(\sqrt{\varepsilon}-\hat{a})}}}{\varepsilon(\sqrt{\varepsilon}-\hat{a})^2} \end{pmatrix} - \begin{pmatrix} 0 \\ -\frac{kh^2 e^{-\frac {k h}{\varepsilon(\sqrt{\varepsilon}-\hat{a})}}} {\varepsilon(\sqrt{\varepsilon}-\hat{a})^2} \end{pmatrix} \right).\end{aligned}$$ The trace $t_c=-\sqrt{\varepsilon}$ and the determinant $d_c=\varepsilon$ of the matrix $A_C$ tend to zero when $\varepsilon$ tends to zero. Hence, for every $h$ with $$\label{aux2:lem43} h>O(\varepsilon^{3/2}),$$ we conclude that $E_p(\tilde{\tau},\hat{a},k,0,0,h)= E_q(\tilde{\tau},\hat{a},k,0,0)$ and $\left. D_{\tilde{\tau},\hat{a}}E_p\right\|_{(\tilde{\tau},\hat{a},k,0,0,h)}= \left.D_{\tilde{\tau},\hat{a}}E_q\right\|_{(\tilde{\tau},\hat{a},k,0,0)}$. From the proof of Theorem \[th:connection\], it follows that $$\begin{aligned} E_p(2\pi/\sqrt{3},(e^{\frac{\pi}{\sqrt{3}}}-1)/(e^{\frac{\pi}{\sqrt{3}}}+1),k,0,0,h)&=\mathbf{0},\\ \det\left(\left.D_{\tilde{\tau},\hat{a}}E_p\right\|_{(2\pi/\sqrt{3},(e^{\frac{\pi}{\sqrt{3}}}-1)/(e^{\frac{\pi}{\sqrt{3}}}+1),k,0,0,h)}\right)&= -6e^{\frac{2\pi}{\sqrt{3}}}.\end{aligned}$$ Thus, we can apply the Implicit Function Theorem to the set of equations $E_p(\tilde{\tau},\hat{a},k,m,\varepsilon,h)=\mathbf{0}$, which establishes the connection between $\mathbf{p}_{1}$ and $\mathbf{p}_{0}$, and conclude the existence of the functions $\tilde{\tau}=\tilde{\tau}_C(k,\varepsilon,h;m)$ and $\hat{a}=\hat{a}(k,\varepsilon,h;m)$ satisfying $$E_p(\tilde{\tau}_C(k,\varepsilon,h;m),\hat{a}(k,\varepsilon,h;m),k,m,\varepsilon,h)=\mathbf{0}.$$ Furthermore, it follows that $$\begin{aligned} E_p(\tilde{\tau}_C(k,\varepsilon,h;m),\hat{a}(k,\varepsilon,h;m),k,m,\varepsilon,h) & - E_p(\tau_C(k,\varepsilon;m),\tilde{a}(k,\varepsilon;m),k,m,\varepsilon,h) = \\ \tilde{E}_p(\tilde{\tau}_C,\hat{a},\tau_C,\tilde{a},k,m,\varepsilon,h) &\begin{pmatrix} \tilde{\tau}_C(k,\varepsilon,h;m) - \tau_C(k,\varepsilon;m)\\ \hat{a}(k,\varepsilon,h;m)-\tilde{a}(k,\varepsilon;m) \end{pmatrix},\end{aligned}$$ where $\tilde{E}_p(\tilde{\tau}_C,\hat{a},\tau_C,\tilde{a},k,m,\varepsilon,h)=\int_0^1 D_{\tilde{\tau},\hat{a}}E_p(s\tilde{\tau}_C+(1-s)\tau_C,s\hat{a}+(1-s)\tilde{a},k,m,\varepsilon,h)\, ds$. Applying now equation , we conclude that for $\varepsilon$ small enough $$\begin{pmatrix} \tilde{\tau}_C(k,\varepsilon,h;m) - \tau_C(k,\varepsilon;m)\\ \hat{a}(k,\varepsilon,h;m)-\tilde{a}(k,\varepsilon;m) \end{pmatrix} \approx \frac {6e^{\frac{2\pi}{\sqrt{3}}}}{ \varepsilon} \left( e^{\bar{\tau} Ac} \begin{pmatrix} 0 \\ he^{-\frac{h}{\varepsilon^{3/2}}} \end{pmatrix}- \begin{pmatrix} 0 \\ he^{-\frac{k h}{\varepsilon^{3/2}}} \end{pmatrix} \right),$$ where $\bar{\tau}=(e^{\frac{\pi}{\sqrt{3}}}-1)/(e^{\frac{\pi}{\sqrt{3}}}+1)$. From here we conclude the proof. Proof of theorems \[th:supercritico\] and \[th:subcritico\] {#sec:3z} ----------------------------------------------------------- This subsection is devoted to the proof of theorems \[th:supercritico\] and \[th:subcritico\]. The outline of the proof is: First, we study the hyperbolicity/non-hyperbolicity of headless canard cycles. Second, we analyze the hyperbolicity/non-hyperbolicity of canard cycles with head. Finally, we prove the correspondence of non-hyperbolic canard cycles to saddle-node bifurcations. ### Hyperbolicity/non-hyperbolicity of headless canard cycles Let us begin by defining the Poincaré map in neighborhood of periodic orbits visiting zones $L$, $C$ and $R$. Consider a point $\mathbf{p}_0=(\sqrt{\varepsilon},y_0)$ in the switching line $\{x=\sqrt{\varepsilon}\}$ and located between $\mathbf{p}_R$ and $\mathbf{q}_{R}^1$, see Figure \[dib:geommetry\]. From expressions and it follows that $(m+\lambda_R^s)(\sqrt{\varepsilon}-a)<y_0<m(\sqrt{\varepsilon}-a)$. Assume now that there exists a time of flight $\tau_{Cd}>0$ such that $x^C(\tau_{Cd};\mathbf{p}_0,\bm{\eta})=-\sqrt{\varepsilon}$ and $x^C(s;\mathbf{p}_0,\bm{\eta})\in(-\sqrt{\varepsilon},\sqrt{\varepsilon})$ for all $s\in(0,\tau_{Cd})$, where $x^C(s;\mathbf{p}_0,\bm{\eta})$ is the first coordinate of the solution through $\mathbf{p}_0$ reduced to the central region $C$, see . In such a case, we can define the Poincaré half-map between the switching lines $\{x=\sqrt{\varepsilon}\}$ and $\{x=-\sqrt{\varepsilon}\}$ at the point $y_0$ as $\Pi_{C_d}(y_{0},{\bm{\eta}})= {y^C(\tau_{Cd};\mathbf{p}_0,{\bm{\eta}})}.$ Similarly, we can define the Poincaré half-map between the switching lines $\{x=-\sqrt{\varepsilon}\}$ and $\{x=\sqrt\varepsilon\}$ at a point $y_2>-m(\sqrt{\varepsilon}+a)$ as $\Pi_{C_u}(y_2,{\bm{\eta}})=y^C\left(\tau_{C_u};\mathbf{p}_2,{\bm{\eta}}\right)$, where $\tau_{C_u}>0$ is the time of flight and $\mathbf{p}_2=(-\sqrt{\varepsilon},y_2).$ Consider now a point $\mathbf{p}_1=(-\sqrt{\varepsilon},y_1)$ in the switching line $\{x=-\sqrt{\varepsilon}\}$ and located between $\mathbf{q}_L^0$ and $\mathbf{p}_L$, that is $y_1\in\left(-(m+\lambda_L^s)(\sqrt{\varepsilon}+a),-m(\sqrt{\varepsilon}+a)\right)$. Assume that there exists a time of flight $\tau_{L}>0$ such that $x^L(\tau_{L};\mathbf{p}_1,\bm{\eta})=-\sqrt{\varepsilon}$ and $x^L(s;\mathbf{p}_1,\bm{\eta})\in(-1,-\sqrt{\varepsilon})$ for all $s\in(0,\tau_L).$ Here $x^L(\tau_{L};\mathbf{p}_1,\bm{\eta})$ is the first coordinate of the solution through $\mathbf{p}_1$ and reduced to the region $L$. In such a case, we can define the Poincaré half-map between the switching line $\{x=-\sqrt{\varepsilon}\}$ and itself at the point $y_1$ as $\Pi_L(y_1,{\bm{\eta}})=y^L\left(\tau_{L};\mathbf{p}_1,{\bm{\eta}}\right)$. Similarly, we can define the Poincaré half-map between the switching line $\{x=\sqrt{\varepsilon}\}$ and itself at the point $y_3>m(\sqrt{\varepsilon}-a)$ as $\Pi_R(y_3,{\bm{\eta}})=y^R\left(\tau_{R};\mathbf{p}_3,{\bm{\eta}}\right)$, where $\tau_{R}>0$ is the time of flight and $\mathbf{p}_3=(\sqrt{\varepsilon},y_3).$ Expressions for $\Pi_L^{-1}$ and $\Pi_R$ can be found in Lemma \[lem:Pmaps\]. At this point, the Poincaré map for an orbit of system visiting zones $L$, $C$ and $R$ can be defined. \[defipoincareydesplazamiento\] The Poincaré map $\Pi$ in the neighborhood of an orbit $\Gamma_{x_0}$ of system visiting zones $L$, $C$ and $R$ is defined as $$\Pi(y_0,\bm{\eta})=\Pi_R(\Pi_{C_u}(\Pi_L(\Pi_{C_d}(y_0,\bm{\eta}),\bm{\eta}),\bm{\eta}),\bm{\eta}),$$ provided the composition of Poincaré half-maps is possible, where $y_0=\Pi_{Cd}^{-1}(\Pi_{L}^{-1}(\Phi_{3z}(x_0),{\bm \eta}),{\bm \eta})$. For $\varepsilon$ fixed and small enough, suppose the existence of a headless canard limit cycle $\Gamma_{x_0}$, see Figure \[dib:geommetry\], obtained under the parameter relation $a=\hat{a}(k,\varepsilon,x_0; m)$ given in Theorem \[th:exist\_Gamma\]. The cycle $\Gamma_{x_0}$ corresponds to the fixed point of the Poincaré map $\Pi(y_0,{\bm \eta}),$ where $y_0=\Pi_{Cd}^{-1}(\Pi_{L}^{-1}(\Phi_{3z}(x_0),{\bm \eta}),{\bm \eta})$. To take into account the non-hyperbolicity of the canard cycle $\Gamma_{x_0}$, we consider the derivative of the Poincaré map, which corresponds to the exponential of the integral of the divergence of the system along $\Gamma_{x_0}$, see [@chicone]. In the particular case of PWL systems, the integral of the divergence can be explicitly computed as the sum of the products of the traces and the time of flight of $\Gamma_{x_0}$ in each region of linearity, see [@freire1]. Let $\tau_L$ and $\tau_R$ be the time of flight of $\Gamma_{x_0}$ along the regions $L$ and $R$, respectively, and let $\tau_C=\tau_C(k,\varepsilon;m)$ be the time of flight from $\mathbf{q}_1^R$ to $\mathbf{q}_0^L$ obtained in Theorem \[th:connection\]. From Lemma \[lem:dom\], when $x_0\in[-1,x_s)$ it follows that $\Gamma_{x_0}$ intersect the switching lines $\{x=-\sqrt{\varepsilon}\}$ and $\{x=\sqrt{\varepsilon}\}$ exponentially close to $\mathbf{q}_0^L$ and $\mathbf{q}_1^R$, respectively. Therefore, the values of $\tau_L$ and $\tau_R$ can be approximated by the time of flight of the orbit from $\mathbf{q}_0^L$ to $(-\sqrt{\varepsilon},h)$ and from $(\sqrt{\varepsilon},h)$ to $\mathbf{q}_1^R$, respectively. Hence, $\tau_L=\tau_L(h)$ and $\tau_R=\tau_R(h)$ are the ones computed in Lemma \[lem:times\]. Notice that when $x_s<x_0<-\sqrt{\varepsilon}$, we can not assure that $\Gamma_{x_0}$ intersect neither $\{x=-\sqrt{\varepsilon}\}$ nor $\{x=\sqrt{\varepsilon}\}$ exponentially close to $\mathbf{q}_0^L$ and $\mathbf{q}_1^R$, respectively, see Figure \[fig:Zoom1\]. ![Zoom of the flow in a neighbourhood of the contact point $\mathbf{p}_L$. Orbits having width in $x_s<x<-\sqrt{\varepsilon}$ do not pass exponentially close of $\mathbf{q}_0^L$. Therefore, the time of flight $\tau_L$ can not be computed as it is done in Lemma \[lem:times\].[]{data-label="fig:Zoom1"}](dibujo_ampliado1.eps) (0,0) (-80,4)[$x_s$]{} (-53,12)[$\mathbf{q}_0^L$]{} (-35,30)[$\mathbf{p}_L$]{} (-60,98)[$\{x=-\sqrt{\varepsilon}\}$]{} In such case, expressions in Lemma \[lem:times\] are not good approximations to $\tau_L$ and $\tau_R$, respectively. Therefore we have eliminated the interval $(x_s,-\sqrt{\varepsilon})$ from the scope of the Theorem \[th:supercritico\] and of the Theorem \[th:subcritico\]. Finally, the time of flight from $(-\sqrt{\varepsilon},h)$ to $(\sqrt{\varepsilon},h)$ (dashed arc on $\Gamma_{x_0}$ in Figure \[dib:geommetry\]) is order one in $\varepsilon$, so we can consider it as zero. Then, the derivative of the Poincaré map $\Pi(y_0,{\bm \eta})$, where $y_0=\Pi_{Cd}^{-1}(\Pi_{L}^{-1}(h,{\bm \eta}),{\bm \eta})$ and $h=\Phi_{3z}(x_0)$, can be approximated by $$\label{derprim} \frac{\partial \Pi}{\partial y}(y_0,\bar{\bm{\eta}})\approx e^{t_L \tau_L-m\tau_C+t_R\tau_R}.$$ A necessary condition on the limit cycle $\Gamma_{x_0}$ to be non-hyperbolic is that $\frac{\partial \Pi}{\partial y}(y_0,\bar{\bm{\eta}})=1$ with $y_0=\Pi_{Cd}^{-1}(\Pi_{L}^{-1}(\Phi_{3z}(x_0),{\bm \eta}),{\bm \eta})$. In terms of the right side of this condition writes as $R_{3z}(h,k,\varepsilon;m)=e^{t_L \tau_L+t_R\tau_R}-e^{m\tau_C}=0.$ By using the expression of $\tau_L$, $\tau_R$ and $\tau_C$ previously computed it follows that $$\label{cond1_3z} R_{3z}(h,k,\varepsilon;m)= \left(1 + \frac {h+(m+\lambda_L^s)(\sqrt{\varepsilon}+a)}{(\lambda_L^q-\lambda_L^s)(\sqrt{\varepsilon}+a)} \right)^{\frac k{\lambda_L^s}} \left(1 + \frac {(m+\lambda_R^s)(\sqrt{\varepsilon}-a)-h}{(\lambda_R^q-\lambda_R^s)(\sqrt{\varepsilon}-a)} \right)^{\frac 1{\lambda_R^s}}- e^{m\tau_C}.$$ Notice that the function $R_{3z}$ writes in terms of the height $h$ of the cycle $\Gamma_{x_0}$, with $h=\Phi_{3z}(x_0)$. In the next result we compute the stability of a canard cycle through the sign of the function $R_{3z}$. \[prop:R3z\] For $\varepsilon$ fixed and small enough, there exists $0<\delta\ll 1$ such that for $x_0\in[-1,x_s)$ and $h=\Phi_{3z}(x_0)$: - if $R_{3z}(h,k,\sqrt{\varepsilon}; m)<-\delta$, then $\Gamma_{x_0}$ is a hyperbolic headless stable canard cycle; - if $h$ is a simple root of $R_{3z}(h,k,\sqrt{\varepsilon}; m)$, then in a neighborhood of $\Gamma_{x_0}$ there is a nonhyperbolic headless canard cycle; - if $R_{3z}(h,k,\sqrt{\varepsilon}; m)>\delta$, then $\Gamma_{x_0}$ is a hyperbolic headless unstable canard cycle. The proposition is a straight forward consequence of the definition of $R_{3z}(h,k,\varepsilon;m)$ and the equation . By fixing parameters $k$ and $\varepsilon$ we next describe the qualitative behavior of $R_{3z}(h,k,\varepsilon;m)$ as a function of $h$, both in the supercritical case, $m=-\sqrt{\varepsilon}$, and in the subcritical case, $m=\sqrt{\varepsilon}$. Even when the domain of definition of $R_{3z}$, as a function of $h$, is greater, we only consider the reduction of $R_{3z}$ to the interval $(h_s,h_M]$ where $h_s=\Phi_{3z}(x_s)$ and $h_M=\Phi_{3z}(-1)$, see Lemma \[lem:dom\]. We pay special attention to the existence of simple zeros of $R_{3z}$. \[th:nhcond3z\] Fixed $\varepsilon$ small enough, we consider the function $R_{3z}(h,k,\varepsilon;m)$ defined in . - Under the supercritical condition $m=-\sqrt{\varepsilon}$ we obtain that: - for $k\leq 1$, then $R_{3z}(h,k,\varepsilon;m)<0$ when $h\in(h_s,h_M]$, and - for $k>1$, the function $R_{3z}(h,k,\varepsilon;m)$ behaves as it is represented in Figure \[dib:FunsSyh\](a). More specifically, - $\lim_{h\searrow h_s}R_{3z}(h,k,\varepsilon;m)<0$ and $ R_{3z}(h_M,k,\sqrt{\varepsilon};m)>0$, - let $h^\in(h_s,h_M]$ be a zero of $R_{3z}(h,k,\varepsilon;m)$, then $ \left. \frac {\partial R_{3z}}{\partial h} \right\|_{(h^,k,\sqrt{\varepsilon};m)} >0$, and - denoting by $h^(k,\varepsilon;m)$ the unique positive zero of $R_{3z}(h,k,\varepsilon;m)$ in $(h_s,h_M]$, then $$h^(k,\varepsilon;m)=\frac {2}{1+e^{-\frac {\pi}{\sqrt{3}}}} k^{\frac{k^2}{k^2-1}} e^{\frac{\pi}{\sqrt{3}}\frac{1-2\varepsilon}{k^ 2-1}}\sqrt{\varepsilon} +O(\varepsilon).$$ - Under the subcritical condition $m=\sqrt{\varepsilon}$ we obtain that: - for $k<1$, the function $R_{3z}(h,k,\sqrt{\varepsilon};m)$ behaves as it is represented in Figure \[dib:FunsSyh\](c). More specifically, - $\lim_{h\searrow h_s}R_{3z}(h,k,\varepsilon;m)>0$ and $R_{3z}(h_M,k,\sqrt{\varepsilon};m)=-e^{\frac {2\pi}{\sqrt{3}}}$, - let $h^>0$ be a zero of $R_{3z}(h,k,\varepsilon;m)=0$, then $ \left. \frac {\partial R_{3z}}{\partial h} \right\|_{(h^,k,\sqrt{\varepsilon}:m)} <0.$ - denoting by $h^(k,\varepsilon;m)$ the unique zero of $R_{3z}(h,k,\varepsilon;m)=0$ in $(h_s,h_M]$, then $$h^(k,\varepsilon;m)=\frac {2}{1+e^{\frac {\pi}{\sqrt{3}}}} k^{\frac{k^2}{k^2-1}} e^{\frac{\pi}{\sqrt{3}}\frac{1-2\varepsilon}{1-k^ 2}}\sqrt{\varepsilon} +O(\varepsilon),$$ - for $k\geq 1$, then $R_{3z}(h,k,\varepsilon;m)>0$ when $h\in(h_s,h_M]$. By straightforward computations we write $$\begin{aligned} R_{3z}(0,k,\sqrt{\varepsilon};m) &= \left(1 + \frac {m+\lambda_L^s}{\lambda_L^q-\lambda_L^s} \right)^{\frac k{\lambda_L^s}} \left(1 + \frac {m+\lambda_R^s}{\lambda_R^q-\lambda_R^s} \right)^{\frac 1{\lambda_R^s}}- e^{m \tau_C}\\ &= \left(1 + \frac 1 {\frac {\lambda_L^q-\lambda_L^s}{m+\lambda_L^s}} \right)^{\frac k{\lambda_L^s}} \left(1 + \frac 1 {\frac {\lambda_R^q+m}{-(\lambda_R^s+m)}} \right)^{-\frac 1{\lambda_R^s}}- e^{m \tau_C}\\ &=\left(1+\frac 1{z_1}\right)^{\frac {k}{\lambda_L^S}} \left(1+\frac 1{z_2}\right)^{-\frac {1}{\lambda_R^S}},\end{aligned}$$ where $$\begin{aligned} z_1&=\frac {\lambda_L^q-\lambda_L^s}{m+\lambda_L^s}=\frac{k}{m}-1+O(m),\\ z_2&=\frac {\lambda_R^q+m}{-(m+\lambda_R^s)} = \frac{1}{m}-O(m),\end{aligned}$$ tend to $\infty$ (resp. to $-\infty$) as $\varepsilon$ tends to zero when $m=\sqrt{\varepsilon}$ (resp. $m=-\sqrt{\varepsilon}$. Therefore $$\lim_{\varepsilon \searrow 0} R_{3z}(0,k,\sqrt{\varepsilon};m)= e^{\lim_{\varepsilon \searrow 0} \frac 1 {z_1} \frac k {\lambda_L^s}-\frac 1 {z_2} \frac 1 {\lambda_R^s} }-e^{\frac {2\pi}{\sqrt{3}}}=e^{\lim_{m \searrow 0} \frac{k+1}{m}+1+O(m)}-e^{\frac {2\pi}{\sqrt{3}}}.$$ Since $h_s$ tends to zero when $\varepsilon$ does, we conclude that $R_{3z}(h_s,k,\sqrt{\varepsilon};m)<0$ when $m=-\sqrt{\varepsilon}$ and $R_{3z}(h_s,k,\sqrt{\varepsilon};m)>0$ when $m=\sqrt{\varepsilon}$, provided $\varepsilon$ is small enough. Expanding the different operands in the expression in power series of $\varepsilon$, and keeping the lower order terms, we obtain the following approximation of $R_{3z}(h,k,\varepsilon;m)$ which is valid for $\varepsilon$ small enough $$\begin{aligned} R_{3z}(h,k,\varepsilon;m) &\approx \left(\frac{h\left(1+e^{s\frac{\pi}{\sqrt{3}}}\right)}{2k\sqrt{\varepsilon}} \right)^{\frac {k^2}{\varepsilon}} \left(\frac{2 e^{s\frac{\pi}{\sqrt{3}}} \sqrt{\varepsilon} }{h\left(1+e^{s\frac{\pi}{\sqrt{3}}}\right)} \right)^{\frac {1}{\varepsilon}}- e^{s\frac {2\pi}{\sqrt{3}}}\\ &\approx \left( \frac {2\sqrt{\varepsilon}}{h \left(1+e^{s\frac{\pi}{\sqrt{3}}}\right)} \right)^{\frac {1-k^2}{\varepsilon}} \left( \frac {e^{s\frac{\pi}{\sqrt{3}}}}{k^{k^2}} \right)^{\frac 1{\varepsilon}} -e^{s\frac {2\pi}{\sqrt{3}}},\end{aligned}$$ where $s$ is the $\rm{sign}(m)$. From here, we obtain the sign of the function $R_{3z}$ at $h_M=\Phi_{3z}(-1)$, depending on the parameters. That is $R_{3z}(h_M,k,\sqrt{\varepsilon};m)>0$ when $k>1$, $R_{3z}(h_M,k,\sqrt{\varepsilon};m)<0$ when $k<1$, and when $k=1$ the sign of the function $R_{3z}(h_M,k,\sqrt{\varepsilon};m)$ is equal to the sign of $m$. From the previous assertions, we conclude the existence of a zero $h^\in (h_s,h_M]$ of the function $R_{3z}(h,k,\sqrt{\varepsilon};m)$ for the parameters $m=-\sqrt{\varepsilon}$ and $k<1$, and for $m=\sqrt{\varepsilon}$ and $k>1$. The expression of $h^$ stated in the theorem follows by equalizing to zero the approximation of $R_{3z}(h,k,\sqrt{\varepsilon};m)$ given above. The partial derivative of $R_{3z}(h,k,\varepsilon;m)$ with respect to $h$ can be written as follows $$\begin{aligned} \left.\frac {\partial R_{3z}}{\partial h}\right\|_{(h,k,\sqrt{\varepsilon};m)}&= \left( R_{3z}(h,k,\varepsilon;m)+ e^{s\frac {2\pi}{\sqrt{3}}}\right)\\ &\hspace{0.4cm}\left( \frac{k} {\lambda_L^s((\lambda_L^q+m)(\sqrt{\varepsilon}+a)+h)} -\frac{1} {\lambda_R^s((\lambda_R^q+m)(\sqrt{\varepsilon}-a)-h)} \right)\\ &=\left( R_{3z}(h,k,\varepsilon;m)+ e^{s\frac {2\pi}{\sqrt{3}}}\right) \left( \frac{k^2-1}{h m^2}+O(\varepsilon^{-\frac 1 2}) \right).\end{aligned}$$ Therefore, assuming the existence of a zero $h^$ of $R_{3z}(h,k,\sqrt{\varepsilon};m)$, we obtain that $\left.\frac {\partial R_{3z}}{\partial h}\right\|_{(h^,k,\sqrt{\varepsilon})}<0$ if $k<1$, and $\left.\frac {\partial R_{3z}}{\partial h}\right\|_{(h^,k,\sqrt{\varepsilon})}>0$ if $k>1$, which implies the uniqueness of such zero. From here, we conclude that function $R_{3z}$ does not change sign in $(h_s,h_M]$ neither when $m=-\sqrt{\varepsilon}$ and $k\leq 1$ nor when $m=\sqrt{\varepsilon}$ and $k\geq 1$. ![Graphical representation of the functions $R_{3z}(h,k,\sqrt{\varepsilon};m)$ and $h^(k,{\varepsilon};m)$ analysed in Proposition \[th:nhcond3z\] for different values of $\varepsilon$. In particular, blue/dotted curves correspond with $\varepsilon=0.05$, green/dashed curves with $\varepsilon=0.01$ and red/solid curves with $\varepsilon=0.005$. First column contains the graphs of the function $R_{3z}(h,k,\varepsilon;m)$ as a function of $h$ for the previous values of $\varepsilon$: panel (a) when $k=2.5$ and $m=-\sqrt{\varepsilon}$ and panel (c) when $k=0.75$ and $m=\sqrt{\varepsilon}$. Second column contains the graphs of the function $h^(k,\varepsilon;m)$ as a function of $k$ for the previous values of $\varepsilon$: panel (b) when $m=-\sqrt{\varepsilon}$ and panel (c) when $m=\sqrt{\varepsilon}$.[]{data-label="dib:FunsSyh"}](S3z-n-kg1.eps "fig:")![Graphical representation of the functions $R_{3z}(h,k,\sqrt{\varepsilon};m)$ and $h^(k,{\varepsilon};m)$ analysed in Proposition \[th:nhcond3z\] for different values of $\varepsilon$. In particular, blue/dotted curves correspond with $\varepsilon=0.05$, green/dashed curves with $\varepsilon=0.01$ and red/solid curves with $\varepsilon=0.005$. First column contains the graphs of the function $R_{3z}(h,k,\varepsilon;m)$ as a function of $h$ for the previous values of $\varepsilon$: panel (a) when $k=2.5$ and $m=-\sqrt{\varepsilon}$ and panel (c) when $k=0.75$ and $m=\sqrt{\varepsilon}$. Second column contains the graphs of the function $h^(k,\varepsilon;m)$ as a function of $k$ for the previous values of $\varepsilon$: panel (b) when $m=-\sqrt{\varepsilon}$ and panel (c) when $m=\sqrt{\varepsilon}$.[]{data-label="dib:FunsSyh"}](h3z-n.eps "fig:")\ ![Graphical representation of the functions $R_{3z}(h,k,\sqrt{\varepsilon};m)$ and $h^(k,{\varepsilon};m)$ analysed in Proposition \[th:nhcond3z\] for different values of $\varepsilon$. In particular, blue/dotted curves correspond with $\varepsilon=0.05$, green/dashed curves with $\varepsilon=0.01$ and red/solid curves with $\varepsilon=0.005$. First column contains the graphs of the function $R_{3z}(h,k,\varepsilon;m)$ as a function of $h$ for the previous values of $\varepsilon$: panel (a) when $k=2.5$ and $m=-\sqrt{\varepsilon}$ and panel (c) when $k=0.75$ and $m=\sqrt{\varepsilon}$. Second column contains the graphs of the function $h^(k,\varepsilon;m)$ as a function of $k$ for the previous values of $\varepsilon$: panel (b) when $m=-\sqrt{\varepsilon}$ and panel (c) when $m=\sqrt{\varepsilon}$.[]{data-label="dib:FunsSyh"}](S3z-p-kl1.eps "fig:")![Graphical representation of the functions $R_{3z}(h,k,\sqrt{\varepsilon};m)$ and $h^(k,{\varepsilon};m)$ analysed in Proposition \[th:nhcond3z\] for different values of $\varepsilon$. In particular, blue/dotted curves correspond with $\varepsilon=0.05$, green/dashed curves with $\varepsilon=0.01$ and red/solid curves with $\varepsilon=0.005$. First column contains the graphs of the function $R_{3z}(h,k,\varepsilon;m)$ as a function of $h$ for the previous values of $\varepsilon$: panel (a) when $k=2.5$ and $m=-\sqrt{\varepsilon}$ and panel (c) when $k=0.75$ and $m=\sqrt{\varepsilon}$. Second column contains the graphs of the function $h^(k,\varepsilon;m)$ as a function of $k$ for the previous values of $\varepsilon$: panel (b) when $m=-\sqrt{\varepsilon}$ and panel (c) when $m=\sqrt{\varepsilon}$.[]{data-label="dib:FunsSyh"}](h3z-p.eps "fig:") (0,0) (-360,275)[(a)]{} (-236,254) (-280,250)[[$R_{3z}(h,k,\sqrt{\varepsilon};m)$]{}]{} (-220,147)[$h$]{} (-352,165)[$0$]{} (-248,165) (-302,165) (-314,165) (-255,170)[$h^$]{} (-170,275)[(b)]{} (-86,147)[$1$]{} (-30,147)[$k$]{} (-165,245)[$h^$]{} (-60,147)[$2.5$]{} (-55,160)[(0,1)[25]{}]{} (-55,162) (-55,166) (-55,178) (-65,204) (-69,200)[[$h^(k,\sqrt{\varepsilon};m)$]{}]{} (-360,130)[(c)]{} (-352,115)[$0$]{} (-278,85) (-290,82)[[$R_{3z}(h,k,\sqrt{\varepsilon};m)$]{}]{} (-220,4)[$h$]{} (-170,130)[(d)]{} (-86,4)[$1$]{} (-30,4)[$k$]{} (-165,100)[$h^$]{} (-104,64) (-145,61)[[$h^(k,\sqrt{\varepsilon};m)$]{}]{} ### Hyperbolicity/non-hyperbolicity of canard cycles with head As in the previous section, we start this one by defining the Poincaré map in neighborhood of orbits visiting the four regions $LL$, $L$, $C$ and $R$. Consider the Poincaré half-maps $\Pi_{Cd}$, $\Pi_{Cu}$, $\Pi_R$, and the time of flight $\tau_{Cd}$, $\tau_{Cu}$, and $\tau_{R}$ previously defined. Let $\mathbf{p}_1=(-\sqrt{\varepsilon},y_1)$ be a point in the switching line $\{x=-\sqrt{\varepsilon}\}$ and located below the point $\mathbf{p}_L$, and assume that there exists a time of flight $\tau_{Ld}>0$ such that $x^L(\tau_{Ld};\mathbf{p}_1,\bm{\eta})=-1$ and $-1<x^L(s;\mathbf{p}_1,\bm{\eta})<-\sqrt{\varepsilon}$ for all $s\in(0,{\tau}_{Ld}).$ In such a case, we define the Poincaré half-map between the switching lines $\{x=-\sqrt{\varepsilon}\}$ and $\{x=-1\}$ at the point $y_1$ as ${\Pi}_{Ld}(y_1,{\bm{\eta}})=y^L\left(\tau_{Ld};\mathbf{p}_1,{\bm{\eta}}\right)$. Moreover, consider a point $\mathbf{p}_2=(-1,y_2)$ located below the point $\mathbf{p}_{LL}$ and assume that there exists a time of flight $\tau_{LL}>0$ such that $x^{LL}({\tau}_{LL};\mathbf{p}_1,\bm{\eta})=-1$ and $x^{LL}(s;\mathbf{p}_1,\bm{\eta})<-1$ for all $s\in(0,{\tau}_{LL}).$ We define the Poincaré half-map between the switching line $\{x=-1\}$ and itself at the point $y_2$ as $\Pi_{LL}(y_2,{\bm{\eta}})=y^{LL}\left(\tau_{LL};\mathbf{p}_2,{\bm{\eta}}\right)$. Finally, let $\mathbf{p}_3=(-1,y_3)$ be a point in the switching line $\{x=-1\}$, located over the point $\mathbf{p}_{LL}$, and assume that there exists $\tau_{Lu}>0$ such that $x^L(\tau_{Lu};\mathbf{p}_3,\bm{\eta})=-\sqrt{\varepsilon}$ and $-1<x^L(s;\mathbf{p}_3,\bm{\eta})<-\sqrt{\varepsilon}$ for all $s\in(0,{\tau}_{Lu}).$ We define the Poincaré half-map between the switching lines $\{x=-1\}$ and $\{x=-\sqrt{\varepsilon}\}$ at the point $y_3$ as ${\Pi}_{Lu}(y_3,{\bm{\eta}})=y^L\left(\tau_{Lu};\mathbf{p}_3,{\bm{\eta}}\right)$. At this point, the Poincaré map for an orbit of system - visiting zones $LL$, $L$, $C$ and $R$ can be defined. \[defipoincareydesplazamiento\] The Poincaré map $\Pi$ in the neighborhood of an orbit $\Gamma_{x_0}$ of system - visiting zones $LL$, $L$, $C$ and $R$ is defined as $$\Pi(y_0,\bm{\eta})=\Pi_{R}(\Pi_{Cu}(\Pi_{Lu}(\Pi_{LL}(\Pi_{Ld}(\Pi_{Cd}(y_0,\bm{\eta}),\bm{\eta}),\bm{\eta}),\bm{\eta}),\bm{\eta}),\bm{\eta}),$$ provided the composition of Poincaré half-maps is possible, where $y_0=\Pi_{Cd}^{-1}(\Pi_{Ld}^{-1}(\Phi_{4z}(x_0),{\bm \eta}),{\bm \eta})$. For $\varepsilon$ fixed and small enough, suppose the existence of a canard limit cycle with head $\Gamma_{x_0}$, see Figure \[dib:geommetry\], obtained under the parameter relation $a=\hat{a}(k,\varepsilon,x_0; m)$ given in Theorem \[th:exist\_Gamma\]. The cycle $\Gamma_{x_0}$ corresponds to the fixed point of the Poincaré map $\Pi(y_0,{\bm \eta}),$ where $y_0=\Pi_{Cd}^{-1}(\Pi_{Ld}^{-1}(\Phi_{4z}(x_0),{\bm \eta}),{\bm \eta})$. The non-hyperbolicity of $\Gamma_{x_0}$ can be obtained, similarly as in the case of headless canard cycles, through the sum of the products of the traces of the matrices of the differential linear systems, and the corresponding time of flight, namely $\tau_{Cd},\tau_{Ld},\tau_{LL}$, $\tau_{Lu}$, $\tau_{Cu}$, and $\tau_{R}$. By analogous arguments than those for headless canard limit cycles, we conclude that $\tau_{Cd} =\tau_C(k,\varepsilon;m)$ obtained in Theorem \[th:connection\], and that the values of $\tau_{Cu}$ and $\tau_{Lu}$ are negligible. On the other hand, when $x_0\in (x_r,x_u)$ the canard cycle $\Gamma_{x_0}$ intersects the switching line $\{x=-1\}$ exponentially close to $\mathbf{q}_1^{LL}$, see Lemma \[lem:dom\] and Lemma \[lem:Pmaps\]. Therefore, the value of $\tau_{LL}$ can be approximated by the time of flight of $\Gamma_{x_0}$ from the point $(-1,h)$, where $h=\Phi_{4z}(x_0)$, to the point $\mathbf{q}_1^{LL}$. Then, from Lemma \[lem:times\], we obtain $\tau_{LL}=\tau_{LL}(h)$ and $$\label{eq:tau_RR} \tau_R=\tau_R(h_0)=- \frac1{\lambda_R^s} \ln\left( 1+ \frac{ \lambda_R^s(\sqrt{\varepsilon}-a)+\lambda_{LL}^s(1+a)+k(\sqrt{\varepsilon}-1)+2m\sqrt{\varepsilon} } { (\lambda_R^q-\lambda_R^s)(\sqrt{\varepsilon}-a)} \right),$$ where $h_0=-\lambda_{LL}^s(1+a)-k(\sqrt{\varepsilon}-1)-m(\sqrt{\varepsilon}+a)$ is the second coordinate of $\mathbf{q}_1^{LL}$. Notice that when $x_u<x_0<-1$, we can not assure that $\Gamma_{x_0}$ intersects $\{x=-1\}$ exponentially close to $\mathbf{q}_1^{LL}$, see Figure \[fig:Zoom2\]. ![Zoom of the flow in a neighborhood of the contact point $\mathbf{p}_{LL}$. Orbits having width in $x_u<x_0<-1$ do not pass exponentially close of $\mathbf{q}_1^{LL}$. Therefore, the time of flight $\tau_{LL}$ can not be computed as in Lemma \[lem:times\].[]{data-label="fig:Zoom2"}](dibujo_ampliado2.eps) (0,0) (-78,-7)[$x_u$]{} (-75,83)[$\mathbf{q}_1^{LL}$]{} (-50,60)[$\mathbf{p}_{LL}$]{} (-53,97)[$\{x=-1\}$]{} In this case, neither expression in Lemma \[lem:times\] nor expression are good approximations for $\tau_{LL}$ and $\tau_R$, respectively. Therefore, we have eliminated the interval $(x_u,-1)$ from the stated of the Theorem \[th:supercritico\] and of the Theorem \[th:subcritico\]. Following similar arguments that those applied in Section \[sec:3z\], a necessary condition on the canard cycle $\Gamma_{x_0}$ to be non-hyperbolic can be written as $R_{4z}(h,k,\varepsilon;m)=e^{t_L \tau_{Ld} + t_{LL} \tau_{LL} + t_R \tau_{R}}-e^{m \tau_C }=0$. By using the expressions of $\tau_{Ld}, \tau_{LL},\tau_{R}$ and $\tau_C$ we obtain $$\begin{aligned} \label{cond1_4z} R_{4z}(h,k,\varepsilon;m)= &\left( 1 + \frac {h+m(\sqrt{\varepsilon}+a)+\lambda_L^s(2\sqrt{\varepsilon}+a-1)} {(\lambda_L^q-\lambda_L^s)(\sqrt{\varepsilon}+a)} \right)^{\frac k{\lambda_L^s}}\nonumber\\ &\left( 1+ \frac{ h+m(\sqrt{\varepsilon}+a)+k(\sqrt{\varepsilon}-1)+\lambda_{LL}^s(1+a)} { (\lambda_{LL}^q-\lambda_{LL}^s)(1+a) } \right)^{\frac1{\lambda_{LL}^s}}\\ &\left( 1+ \frac{ \lambda_R^s(\sqrt{\varepsilon}-a)+\lambda_{LL}^s(1+a)+k(\sqrt{\varepsilon}-1)+2m\sqrt{\varepsilon} } { (\lambda_R^q-\lambda_R^s)(\sqrt{\varepsilon}-a)} \right)^{\frac1{\lambda_R^s}}-e^{m\tau_C}.\nonumber \end{aligned}$$ \[prop:R4z\] For $\varepsilon$ fixed and small enough, there exists $0<\delta\ll 1,$ such that, for $x_0\in(x_r,x_u)$ and $h=\Phi_{4z}(x_0)$: - if $R_{4z}(h,k,\sqrt{\varepsilon}; m)<-\delta$, then $\Gamma_{x_0}$ is a hyperbolic stable canard cycle with head; - if $h$ is a simple root of $R_{4z}(h,k,\sqrt{\varepsilon}; m)$, then in a neighborhood of $\Gamma_{x_0}$ there is a nonhyperbolic canard cycle with head; - if $R_{4z}(h,k,\sqrt{\varepsilon}; m)>\delta$, then $\Gamma_{x_0}$ is a hyperbolic unstable canard cycle with head. The proposition follows similarly to Proposition \[prop:R3z\]. Next, we describe the qualitative behavior of $R_{4z}(h,k,\varepsilon;m)$, as a function of $h$, for fixed values of the parameters $k$ and $\varepsilon$. Even when the domain of the function $R_{4z}$ is greater, we consider it reduced to $(h_r,h_u)$, where $h_r=\Phi_{4z}(x_r)$ and $h_u=\Phi_{4z}(h_u)$, see Lemma \[lem:dom\]. \[th:nhcond4z\] Fixed $\varepsilon$ small enough, we consider the function $R_{4z}(h,k,\varepsilon;m)$ defined in . - If $k<1$, or $k=1$ and $m=-\sqrt{\varepsilon}$, then $R_{4z}(h,k,\sqrt{\varepsilon}; m)<0$ in $(h_r,h_u)$. - If $k>1$, or $k=1$ and $m=\sqrt{\varepsilon}$, then $R_{4z}(h,k,\sqrt{\varepsilon};m)$ behaves as in Figure \[fig:FunS4z\](a) or in Figure \[fig:FunS4z\](c), depending on the supercritical case, $m=-\sqrt{\varepsilon}$, or the subcritical case, $m=\sqrt{\varepsilon}$, respectively. More specifically: - $R_{4z}(h_r,k,\sqrt{\varepsilon};m)<0$ and $R_{4z}(h_u,k,\sqrt{\varepsilon};m)>0$, see Figure \[fig:FunS4z\](a) and (c); - let $h^\in(h_r,h_u)$ be a zero of $R_{4z}(h,k,\varepsilon;m)$, then $ \left. \frac {\partial R_{4z}}{\partial h} \right\|_{(h^,k,\sqrt{\varepsilon};m)} >0$; - denoting by $h^(k,\sqrt{\varepsilon};m)$ the unique zero of $R_{4z}(h,k,\varepsilon;m)$ in $(h_r,h_u)$, it follows that, if $k>1$, then $$h^(k,\sqrt{\varepsilon};m)=(k+1)e^{\frac {2-k^2}{2}} (\sqrt{\varepsilon})^{\frac {k^2-1}{k^2}}+O(\sqrt{\varepsilon}),$$ see Figure \[fig:FunS4z\](b) and (d), and if $k=1$ then $$h^(1,\sqrt{\varepsilon};m)= \frac {2}{1+e^{\frac{\pi}{\sqrt{3}}}}+O(\sqrt{\varepsilon}),$$ see Figure \[fig:FunS4z\](d). Setting $m=\pm \sqrt{\varepsilon}$ and $s=\rm{sign}(m)$, and expanding in power series of $\varepsilon$ every term in the expression of $R_{4z}(h,k,\sqrt{\varepsilon}; m)$, we obtain that $$\begin{aligned} R_{4z}(h,k,\sqrt{\varepsilon}; m) =& \left( 1+k-h -\frac{2k+h(e^{s\frac{\pi}{\sqrt{3}}}-1)}{1+e^{s \frac{\pi}{\sqrt{3}}}} \sqrt{\varepsilon} +O(\varepsilon) \right)^{-\frac{1}{\varepsilon}+1+O(\varepsilon)}\\ &\left( \frac{k(1+e^{-s\frac{\pi}{\sqrt{3}}})}{2} \frac 1{\sqrt{\varepsilon}} -\frac{k(1+e^{-s\frac{\pi}{\sqrt{3}}})-2}{2} +O(\varepsilon^{\frac 1 2}) \right)^{-\frac{1}{\varepsilon}+1+O(\varepsilon)}\\ &\left( \frac{h(1+e^{s\frac{\pi}{\sqrt{3}}})}{2k} \frac 1{\sqrt{\varepsilon}} +1 -\left( \frac{e^{s\frac{\pi}{\sqrt{3}}}+1-2ks}{2k^2} +\frac{e^{s\frac{\pi}{\sqrt{3}}}(k^2-5)-4}{4k^3}h \right)\sqrt{\varepsilon} +O(\varepsilon) \right)^{\frac{k^2}{\varepsilon}-1+O(\varepsilon)}\\ &-e^{s\frac{2\pi}{\sqrt{3}}-s\frac{k+1}{k}\sqrt{\varepsilon}+O(\varepsilon)}. \end{aligned}$$ By changing the sign in the exponent of the first two terms, it follows that $$\begin{aligned} \label{ex:TayS} R_{4z}(h,k,\sqrt{\varepsilon}; m) =& \left( \frac{1}{1+k-h} +\frac{ 2k+h(e^{s\frac{\pi }{\sqrt{3}}}-1) }{\left( 1+ e^{s\frac{\pi }{\sqrt{3}}} \right) {{\left(1+k-h\right) }^{2}}}\sqrt{\varepsilon} +O(\varepsilon) \right)^{\frac{1}{\varepsilon}-1+O(\varepsilon)}\nonumber \\ &\left( \frac{2}{k(1+e^{-s\frac{\pi}{\sqrt{3}}})} {\sqrt{\varepsilon}} +O(\varepsilon) \right)^{\frac{1}{\varepsilon}-1+O(\varepsilon)}\\ &\left( \frac{h(1+e^{s\frac{\pi}{\sqrt{3}}})}{2k} \frac 1{\sqrt{\varepsilon}} +1 -\left( \frac{e^{s\frac{\pi}{\sqrt{3}}}+1-2ks}{2k^2} +\frac{e^{s\frac{\pi}{\sqrt{3}}}(k^2-5)-4}{4k^3}h \right)\sqrt{\varepsilon} +O(\varepsilon) \right)^{\frac{k^2}{\varepsilon}-1+O(\varepsilon)} \nonumber \\ &-e^{s\frac{2\pi}{\sqrt{3}}-s\frac{k+1}{k}\sqrt{\varepsilon}+O(\varepsilon)}. \nonumber \end{aligned}$$ Then $$\begin{aligned} R_{4z}(0,k,\sqrt{\varepsilon};m) =& \left( \frac{1}{1+k} +O(\sqrt{\varepsilon}) \right)^{\frac{1}{\varepsilon}-1+O(\varepsilon)} \left( \frac{2}{k(1+e^{-s\frac{\pi}{\sqrt{3}}})} \sqrt{\varepsilon} +O(\varepsilon) \right)^{\frac{1}{\varepsilon}-1+O(\varepsilon)}\\ &\left( 1 -\frac{e^{s\frac{\pi}{\sqrt{3}}}+1-2sk}{2k^2} \sqrt{\varepsilon} +O(\varepsilon) \right)^{\frac{k^2}{\varepsilon}-1+O(\varepsilon)}\\ &-e^{s\frac{2\pi}{\sqrt{3}}-s\frac{k+1}{k}\sqrt{\varepsilon}+O(\varepsilon)}, \end{aligned}$$ which can be approximated, for $\varepsilon$ small enough, as follows $$R_{4z}(0,k,\sqrt{\varepsilon};m) \approx \left( \frac{2\sqrt{\varepsilon}}{k(1+k)(1+e^{-s\frac{\pi}{\sqrt{3}}})} -\frac{(e^{s\frac{\pi}{\sqrt{3}}}+1-2sk)\varepsilon}{k(1+k)(1+e^{-s\frac{\pi}{\sqrt{3}}})} + O(\varepsilon^{\frac 3 2}) \right)^{\frac 1 {\varepsilon}} -e^{s\frac{2\pi}{\sqrt{3}}-s\frac{k+1}{k}\sqrt{\varepsilon}+O(\varepsilon)}.$$ First operand in the right side of previous equation tends to zero provided $\varepsilon$ does. Hence, for $\varepsilon$ small enough we obtain that $-e^{s\frac{2\pi}{\sqrt{3}}}< R_{4z}(0,k,\sqrt{\varepsilon}; m)<0$. Same expression is also satisfied by $R_{4z}(h_r,k,\sqrt{\varepsilon}; m)$ since $\lim_{\varepsilon \searrow 0} h_r=0$. Thus, we conclude that $-e^{s\frac{2\pi}{\sqrt{3}}}< R_{4z}(h_r,k,\sqrt{\varepsilon}; m)<0$. On the other hand, from expression it follows that $$\begin{aligned} R_{4z}(k,k,\sqrt{\varepsilon}; m) &=\left( 1+k\sqrt{\varepsilon}+O(\varepsilon) \right)^{\frac{1}{\varepsilon}-1+O(\varepsilon)} \left( \frac{2}{k(1+e^{-s\frac{\pi}{\sqrt{3}}})} \sqrt{\varepsilon} +O(\varepsilon) \right)^{\frac{1}{\varepsilon}-1+O(\varepsilon)}\\ &\phantom{=}\left( \frac{1 + e^{s\frac{\pi}{\sqrt{3}}}}{2} \frac 1{\sqrt{\varepsilon}} +1 +O(\sqrt{\varepsilon}) \right)^{\frac{k^2}{\varepsilon}-1+O(\varepsilon)}-e^{s\frac{2\pi}{\sqrt{3}}-s\frac{k+1}{k}\sqrt{\varepsilon}+O(\varepsilon)}\\ &=\left( \frac {2^{1-k^2}(1+e^{s\frac{\pi}{\sqrt{3}}})^{k^2-1} e^{s\frac{\pi}{\sqrt{3}}}}{k} \varepsilon^{\frac{1-k^2}{2}} + \ldots \right)^{\frac 1 {\varepsilon}}-e^{s\frac{2\pi}{\sqrt{3}}}. \end{aligned}$$ Hence, for $\varepsilon$ small enough we obtain that if $k>1$, then $R_{4z}(k,k,\sqrt{\varepsilon}; m)>0$; and if $k<1$, then $R_{4z}(k,k,\sqrt{\varepsilon}; m)<0$. Moreover, when $k=1$ it follows that $R_{4z}(k,k,\sqrt{\varepsilon}; m)$ has the same sign than $m$. Since $\lim_{\varepsilon \searrow 0} h_u=k$, previous inequalities are also satisfied by $R_{4z}(h_u,k,\sqrt{\varepsilon}; m)$. Let $h_r<h^<h_u$ be a zero of $R_{4z}(h,k,\sqrt{\varepsilon}; m)$. Straight forward computations shows that $$\begin{aligned} \left.\frac{\partial R_{4z}}{\partial h}\right\|_{(h^,k,\sqrt{\varepsilon};m)}&= e^{s\frac{2\pi}{\sqrt{3}}-s\frac{k+1}{k}\sqrt{\varepsilon}+O(\varepsilon)} \left( \frac{ k^3+k^2(1-h^)+h^} {h^(k+1-h^)\varepsilon} +O\left(\frac 1 {\sqrt{\varepsilon}}\right) \right)>0, \end{aligned}$$ which implies that, when exists, the zero $h^$ is unique. Moreover, by keeping the lower order terms in , we obtain the following implicit expression for an approximation to the solution $h^(k,\sqrt{\varepsilon})$ of $R_{4z}(h,k,\sqrt{\varepsilon}; m)=0$, that is $$\left( \frac 1{1+k-h^} \right) \left( \frac {2\sqrt{\varepsilon}}{k(1+e^{-s\frac {\pi}{\sqrt{3}}})} \right) \left( \frac {h^(1+e^{s\frac {\pi}{\sqrt{3}}})}{2k} \frac 1{\sqrt{\varepsilon}} \right)^{k^ 2} =e^{s \frac {2\pi}{\sqrt{3}} \varepsilon},$$ When $k=1$, the equation above can be solved and it follows that $$h^= \frac {2}{1+e^{s\frac{\pi}{\sqrt{3}}}}+O(\sqrt{\varepsilon}).$$ which implies that $h^<h_u$ only if $m=\sqrt{\varepsilon}$. When $k\neq 1$ the solution of the equation can be approximated by the undetermined coefficients method $$h^=(k+1)e^{\frac {2-k^2}{2}} (\sqrt{\varepsilon})^{\frac {k^2-1}{k^2}}+O(\sqrt{\varepsilon}).$$ In Figure \[fig:FunS4z\](a) we represent $R_{4z}(h,k,\varepsilon;m)$ as a function of $h$ for different values of $\varepsilon$ and $k>1$. We notice that function $R_{4z}(h,k,\varepsilon;m)$ is negative for $h$ close to $h_r$, and positive for values close to $h_{u}$. Moreover, as it follows from the proof of the previous theorem, $\lim_{\varepsilon \searrow 0} R_{4z}(0,k,\sqrt{\varepsilon};m)=-e^{\frac {2\pi}{\sqrt{3}}}$. Furthermore, the unique zero $h^(k,\sqrt{\varepsilon})$ tends to zero as $\varepsilon$ tends to zero. In Figure \[fig:FunS4z\](b) we represent function $h^(k,\sqrt{\varepsilon})$ as a function of $k$ for different values of $\varepsilon$. Notice that all the curves have a common point at $k=1$ and $h^=2{e}^{\frac 1 2}\approx 3.297442541400256\ldots>k,$ which lies out of the interval $(h_r,h_u)$ since $h_u\approx k$. Therefore this part of the curves do not correspond to non-hyperbolic canard cycles, see Figure \[dib:hk34\]. ![Graphical representation of the functions $R_{4z}(h,k,\sqrt{\varepsilon};m)$ and $h^(k,\varepsilon;m)$ analyzed in Theorem \[th:nhcond4z\] for different values of $\varepsilon$. In particular blue/dotted curves correspond with $\varepsilon=0.05$, green/dashed curves with $\varepsilon=0.01$ and red/solid curves with $\varepsilon=0.005$. First column contains the graphs of the function $R_{4z}(h,k,\varepsilon;m)$ as a function of $h$ for the previous values of $\varepsilon$: panel (a) when $k=1.3$ and $m=-\sqrt{\varepsilon}$ and panel (c) when $k=0.75$ and $m=\sqrt{\varepsilon}$. Second column contains the graphs of the function $h^(k,\varepsilon;m)$ as a function of $k$ for the previous values of $\varepsilon$: panel (b) when $m=-\sqrt{\varepsilon}$ and panel (c) when $m=\sqrt{\varepsilon}$.[]{data-label="fig:FunS4z"}](S4z-n-kg1.eps "fig:")![Graphical representation of the functions $R_{4z}(h,k,\sqrt{\varepsilon};m)$ and $h^(k,\varepsilon;m)$ analyzed in Theorem \[th:nhcond4z\] for different values of $\varepsilon$. In particular blue/dotted curves correspond with $\varepsilon=0.05$, green/dashed curves with $\varepsilon=0.01$ and red/solid curves with $\varepsilon=0.005$. First column contains the graphs of the function $R_{4z}(h,k,\varepsilon;m)$ as a function of $h$ for the previous values of $\varepsilon$: panel (a) when $k=1.3$ and $m=-\sqrt{\varepsilon}$ and panel (c) when $k=0.75$ and $m=\sqrt{\varepsilon}$. Second column contains the graphs of the function $h^(k,\varepsilon;m)$ as a function of $k$ for the previous values of $\varepsilon$: panel (b) when $m=-\sqrt{\varepsilon}$ and panel (c) when $m=\sqrt{\varepsilon}$.[]{data-label="fig:FunS4z"}](h4z-n.eps "fig:")\ ![Graphical representation of the functions $R_{4z}(h,k,\sqrt{\varepsilon};m)$ and $h^(k,\varepsilon;m)$ analyzed in Theorem \[th:nhcond4z\] for different values of $\varepsilon$. In particular blue/dotted curves correspond with $\varepsilon=0.05$, green/dashed curves with $\varepsilon=0.01$ and red/solid curves with $\varepsilon=0.005$. First column contains the graphs of the function $R_{4z}(h,k,\varepsilon;m)$ as a function of $h$ for the previous values of $\varepsilon$: panel (a) when $k=1.3$ and $m=-\sqrt{\varepsilon}$ and panel (c) when $k=0.75$ and $m=\sqrt{\varepsilon}$. Second column contains the graphs of the function $h^(k,\varepsilon;m)$ as a function of $k$ for the previous values of $\varepsilon$: panel (b) when $m=-\sqrt{\varepsilon}$ and panel (c) when $m=\sqrt{\varepsilon}$.[]{data-label="fig:FunS4z"}](S4z-p-kg1.eps "fig:")![Graphical representation of the functions $R_{4z}(h,k,\sqrt{\varepsilon};m)$ and $h^(k,\varepsilon;m)$ analyzed in Theorem \[th:nhcond4z\] for different values of $\varepsilon$. In particular blue/dotted curves correspond with $\varepsilon=0.05$, green/dashed curves with $\varepsilon=0.01$ and red/solid curves with $\varepsilon=0.005$. First column contains the graphs of the function $R_{4z}(h,k,\varepsilon;m)$ as a function of $h$ for the previous values of $\varepsilon$: panel (a) when $k=1.3$ and $m=-\sqrt{\varepsilon}$ and panel (c) when $k=0.75$ and $m=\sqrt{\varepsilon}$. Second column contains the graphs of the function $h^(k,\varepsilon;m)$ as a function of $k$ for the previous values of $\varepsilon$: panel (b) when $m=-\sqrt{\varepsilon}$ and panel (c) when $m=\sqrt{\varepsilon}$.[]{data-label="fig:FunS4z"}](h4z-p.eps "fig:") (0,0) (-360,275)[(a)]{} (-236,254) (-246,254) (-280,250)[[$R_{4z}(h,k,\sqrt{\varepsilon};m)$]{}]{} (-220,147)[$h$]{} (-352,167)[$0$]{} (-246,171) (-242,171) (-237,171) (-259,175)[$h^$]{} (-170,275)[(b)]{} (-86,147)[$1$]{} (-30,147)[$k$]{} (-165,245)[$h^$]{} (-60,147)[$1.3$]{} (-55,160)[(0,1)[25]{}]{} (-55,162) (-55,166) (-55,171) (-61,204) (-73,215)[[$h^(k,\sqrt{\varepsilon};m)$]{}]{} (-360,130)[(c)]{} (-352,65)[$0$]{} (-274,85) (-280,82)[[$R_{4z}(h,k,\sqrt{\varepsilon};m)$]{}]{} (-220,4)[$h$]{} (-170,130)[(d)]{} (-86,4)[$1$]{} (-30,4)[$k$]{} (-165,100)[$h^$]{} (-104,64) (-145,61)[$h^(k,\sqrt{\varepsilon};m)$]{} ### Correspondence to saddle-node bifurcations To finish with the proof of the theorems \[th:supercritico\] and \[th:subcritico\], in this subsection we are going to prove that the non-hyperbolic canard limit cycles whose existence has been proved in the previous sections correspond, indeed, to a saddle-node bifurcation. We focus the proof in the case of headless canards. The proof in the case of canards with head is analogous. We will prove that the non-degeneracy conditions on the Poincaré map hold, that is, the second derivative of the Poincaré map with respect to the initial condition and the derivative of the Poincaré map with respect to parameter $a$ are both different from zero, see [@K04; @Perko]. Consider a non-hyperbolic headless canard cycle $\Gamma_{x_0}$ corresponding to a fixed point of the Poincaré map $y_0$ for parameters $\bm{\eta}=\bar{\bm{\eta}},$ i.e. $y_0=\Pi_{Cd}^{-1}(\Pi_{L}^{-1}(\Phi(x_0),\bar{{\bm \eta}}),\bar{{\bm \eta}})$. First, we compute the second derivative of the Poincaré map with respect to the initial condition and we will see that, in a headless non-hyperbolic canard cycle this derivative is nonzero if and only function $R_{3z}$ is non zero at $\bar{h}=\Phi_{3z}(x_0)$. Second, we compute the derivative of the Poincaré map with respect to parameter $a$ and we test that this derivative does not vanish. From expression (\[derprim\]), the second derivative of Poincaré map with respect to $h$ in the non-hyperbolic fixed point $\bar{h}$ is given by $$\label{derseg} \frac{\partial^2 \Pi}{\partial y^2}(y_0,\bar{\bm{\eta}})=\frac{\partial G}{\partial h}(\bar{h},\bar{\bm{\eta}}), $$ where $$G({h},\bm{\eta})=e^{t_L \tau_L-m\tau_C+t_R\tau_R}.$$ It is easy to see that $$G({h},\bm{\eta})=(R_{3z}(h,k,\varepsilon;m)+e^{m\tau_C})e^{-m\tau_C}=R_{3z}(h,k,\varepsilon;m)e^{-m\tau_C}+1.$$ As in a non-hyperbolic headless canard cycle $R_{3z}(h,k,\varepsilon;m)=0,$ it is easy to see that, $$\frac{\partial G}{\partial h} (\bar{h},\bar{\bm{\eta}})=\frac{\partial R_{3z}}{\partial h} (\bar{h},\bar{\bm{\eta}})e^{-m\tau_C}.$$ Thus, from expression (\[derseg\]), in a non-hyperbolic headless canard cycle, $$\label{dersegfin} \frac{\partial^2 \Pi}{\partial y^2}(y_0,\bar{\bm{\eta}})=\frac{\partial R_{3z}}{\partial h} (\bar{h},\bar{\bm{\eta}})e^{-m\tau_C}, $$ and then the sign of $\frac{\partial^2 \Pi}{\partial y^2}(y_0,\bar{\bm{\eta}})$ is that of $\frac{\partial R_{3z}}{\partial h} (\bar{h},\bar{\bm{\eta}}),$ which is non zero from Theorem \[th:nhcond3z\] (a-2-2), (b-1-2) and Theorem \[th:nhcond4z\] (b-2). To compute the derivative of the Poincaré map with respect to parameter $a$ in the neighborhood of a non-hyperbolic headless canard cycle $\Gamma_{x_0}$, we will use the following reasoning. Taking a point $(\sqrt{\varepsilon},y_1)$ in a neighborhood of $(\sqrt{\varepsilon},y_0)$ where $y_0=\Pi_{Cd}^{-1}(\Pi_{L}^{-1}(\Phi_{3z}(x_0),{\bm \eta}),{\bm \eta})$. The image through the Poincaré map of $y_1$, that is $\Pi(y_1,{\bm \eta})$, will be exponentially close to the second component of $ \mathbf{q}_1^R,$ that is, $$\label{derpda1} \Pi(y_1,\bm{\eta})=(m+\lambda_R^s)(\sqrt{\varepsilon}-a)+\chi(y_1,\bm{\eta}),$$ with function $\chi(y_1,\bm{\eta})$ and its derivatives $O(\exp(-c/\varepsilon))$ small, where $c$ is a positive constant depending on $y_1.$ Hence, $$\label{derpda2} \frac{\partial \Pi}{\partial a}(y_1,\bm{\eta})=-(m+\lambda_R^s)+\frac{\partial \chi}{\partial a}(y_1,\bm{\eta})\simeq -m+\varepsilon, $$ and $\frac{\partial \Pi}{\partial a}(y_1,\bm{\eta})\neq 0$ as $m=\pm \sqrt{\varepsilon}\neq 0.$ Proof of Theorem \[th:supercritico\_LP\] ---------------------------------------- In Figure \[dib:FunsSyh\](a) we draw the graph of the function $R_{3z}(h,k,\sqrt{\varepsilon};m)$ as a function of $h$ by fixing parameter $k=2.5$ and parameter $\varepsilon\in\{0.05,0.01,0.005\}$. As it has been obtained in the proof of Proposition \[th:nhcond3z\], $R_{3z}(h,k,\sqrt{\varepsilon};m)$ tends to $ -e^{\frac {2\pi}{\sqrt{3}}}$ as $h$ decrease to $h_s$, and it tends to $\infty$ as $h$ tends to $h_M$. In panel (a) we also represent the zero, $h^\in (h_s,h_M]$, at which the function change sign. As it can be observed this zero tends to zero as $\varepsilon$ tends to zero. From Theorem \[th:nhcond3z\](c), in Figure \[dib:FunsSyh\](b) we draw function $h^(k,\sqrt{\varepsilon})$ for different values of $k<1$. As it can be observed, for a fixed $k<1$, as $\varepsilon$ tends to zero, so tends the zero $h^(k,\sqrt{\varepsilon})$, that is the height of the saddle-node tends to zero assuming $\varepsilon$ does. We conclude that fixed $k$ the singular limit of every saddle-node is the equilibrium point at the fold. Nevertheless, for any $h_0\in (h_s,h_M] \cup (h_r,h_u)$, we prove that there exists a suitable election of the parameter $k$ such that $$\lim_{\varepsilon\searrow 0} h^(k(\varepsilon),\varepsilon;m)=h_0,$$ which means that we can chose parameter $k$ in such a way that the singular limit of the saddle-note to be a singular cycle of prefixed height, $h_0$. Next, we address the supercritical case $m=-\sqrt{\varepsilon}$ and $h_0\in(h_s,h_M]$. The remainder cases follow by similar arguments. In order to obtain the parameter value $k$ allowing the existence of a saddle-node canard cycle with height $h_0$, we use implicit equation $h^(k,\varepsilon;m)=h_0$, where the function $h^(k,\varepsilon;m)$ is given in Proposition \[th:nhcond3z\](a-2-3). Since the partial derivative $$\frac{\partial h^}{\partial k}(k,\varepsilon;m)=\frac{2 \sqrt{\varepsilon} k^{\frac{k^2}{k^2-1}+1} e^{\frac{\pi \left(k^2-2 \varepsilon\right)}{\sqrt{3} \left(k^2-1\right)}} \left(2 \sqrt{3} \pi (2 \varepsilon-1)+3 k^2-6 \ln(k)-3\right)}{3 \left(1+e^{\frac{\pi }{\sqrt{3}}}\right) \left(k^2-1\right)^2},$$ is different from zero when $k$ is greater but close to $1$ and $\varepsilon>0$, the Implicit Function Theorem can be applied, and there exists $k(h,\varepsilon;m)$ such that the differential system - with parameters $k=k(h,\varepsilon;m)$ and $a=\hat{a}(k(h,\varepsilon;m),\varepsilon,x_0;m)$ exhibits the saddle-node canard $\Gamma_{x_{0}}$, with $x_0=\Phi_{3z}^{-1}(h)$, whose existence has been stated in Theorem \[th:supercritico\]. Conclusions and perspectives {#sec_conclusions} ============================ In this work we have analyzed the existence of saddle-node bifurcation of canard cycles in PWL systems. We have revised the results in the smooth framework obtained in [@KS01], but in the PWL context. As we have already commented, there, the authors considered two different scenarios, depending whether the Hopf bifurcation where the family of cycles is born is supercritical or subcritical. Here, we point out the similarities and differences that we have found in this study in both cases: - In [@KS01], canard cycles develop along a branch born at a Hopf bifurcation, at $a=a_H$, and the canard explosion takes place at a value which is at a distance of $O(\varepsilon)$ from the $a_H.$ In the PWL context, we have checked that the canard explosion takes place at a value which is at a distance of $O(\sqrt{\varepsilon})$ from the $a_H.$ - Supercritical case, $m=-\sqrt{\varepsilon}$: System - is able to reproduce the dynamics in the smooth case with $k \leq 1$, that is, the existence of a family of stable canard cycles. By letting $k$ to increase, we find new scenarios that have not been reported in the smooth framework. In particular, when $k>1,$ we find situations where two saddle-node bifurcations of canard cycles take place, one of headless canards and another one of canards with head, see Figure \[dib:hk34\], panel (a). In this case, three canard limit cycles can coexist, see Figure \[fig3canards\]. - Subcritical case, $m=\sqrt{\varepsilon}$: In this case, system - is able to reproduce the dynamics in the smooth case, with the advantage that in the PWL case we can easier control the different behaviors that appear. In particular, we have proved the existence of saddle-node bifurcation of headless canards for $k<1,$ and of canards with head for $k>1,$ see Figure \[dib:hk34\], panel (b). Moreover, in Theorem \[th:supercritico\_LP\] we have stated that, both in subcritical and supercritical cases, for every height between the smallest canard cycle and the relaxation oscillation cycle there exist parameters $k$ and $\varepsilon$ such that a saddle-node canard limit cycle with this height exists. ![Curves of saddle-node canard cycles of system - with $\varepsilon=1e-5$. Pointed/blue curves correspond with three zonal saddle-node canard cycles (headless canards). Solid/red curves correspond with four zonal saddle-node canard cycles (canards with head). The diagonal is the value of the height $h_M$ as a function of $k$. The part of the curves over the diagonal corresponds to zeros of the function $R_{3z}$ and $R_{4z}$ which are not saddle-node canards. Panel (a) represents the two saddle-node canard cycles appearing in the supercritical case $m=-\sqrt{\varepsilon}$ for $k>1 $. Panel (b) represents the saddle-node canard cycles appearing in the subcritical case $m=\sqrt{\varepsilon}$, in this case for each value of $k$ only one saddle-node limit cycle can appears.[]{data-label="dib:hk34"}](h34z-n.eps "fig:")![Curves of saddle-node canard cycles of system - with $\varepsilon=1e-5$. Pointed/blue curves correspond with three zonal saddle-node canard cycles (headless canards). Solid/red curves correspond with four zonal saddle-node canard cycles (canards with head). The diagonal is the value of the height $h_M$ as a function of $k$. The part of the curves over the diagonal corresponds to zeros of the function $R_{3z}$ and $R_{4z}$ which are not saddle-node canards. Panel (a) represents the two saddle-node canard cycles appearing in the supercritical case $m=-\sqrt{\varepsilon}$ for $k>1 $. Panel (b) represents the saddle-node canard cycles appearing in the subcritical case $m=\sqrt{\varepsilon}$, in this case for each value of $k$ only one saddle-node limit cycle can appears.[]{data-label="dib:hk34"}](h34z-p.eps "fig:") (0,0) (-365,125)[(a)]{} (-362,100)[$h^$]{} (-283,5)[$1$]{} (-220,5)[$k$]{} (-170,125)[(b)]{} (-167,100)[$h^$]{} (-87,5)[$1$]{} (-25,5)[$k$]{} (-325,620)[$h_M(k)$]{} (-140,53)[$h_M(k)$]{} (-94,15)(0,5)[10]{}[(0,1)[2]{}]{} (-103,5)[$k_1$]{} (-78,15)(0,5)[14]{}[(0,1)[2]{}]{} (-73,5)[$k_2$]{} ![Simulation of three canard limit cycles close to a saddle-node configuration for the supercritical case $m=-\sqrt{\varepsilon}$. According with Figure \[dib:hk34\](a) two configuration are possible when $k>1$: a saddle-node canard cycle without head and saddle-node canard cycle with head. Panel (a) depicts a configuration close to the saddle-node canard cycle without head for $\varepsilon=0.1$, $k=2.5$ and $a=a_1=0.2305968812$. The three canard cycles have initial conditions $(0,0.595), (0,0.642), (0,2.12361)$. Panel (b) depicts a configuration close to the saddle-node canard cycle with head for $\varepsilon=0.1$, $k=2.5$ and $a=a_2=0.23059688315966$, the three canard cycles have initial conditions $(0,0.561), (0,2.10673), (0,2.12)$. Note that $\|a_1-a_2\|\approx 2e-9\approx e^{-\frac 2{\varepsilon}}$, then both saddle-node canard cycles take place exponentially close.[]{data-label="fig3canards"}](SN_Small.eps "fig:"){width="45.00000%"}![Simulation of three canard limit cycles close to a saddle-node configuration for the supercritical case $m=-\sqrt{\varepsilon}$. According with Figure \[dib:hk34\](a) two configuration are possible when $k>1$: a saddle-node canard cycle without head and saddle-node canard cycle with head. Panel (a) depicts a configuration close to the saddle-node canard cycle without head for $\varepsilon=0.1$, $k=2.5$ and $a=a_1=0.2305968812$. The three canard cycles have initial conditions $(0,0.595), (0,0.642), (0,2.12361)$. Panel (b) depicts a configuration close to the saddle-node canard cycle with head for $\varepsilon=0.1$, $k=2.5$ and $a=a_2=0.23059688315966$, the three canard cycles have initial conditions $(0,0.561), (0,2.10673), (0,2.12)$. Note that $\|a_1-a_2\|\approx 2e-9\approx e^{-\frac 2{\varepsilon}}$, then both saddle-node canard cycles take place exponentially close.[]{data-label="fig3canards"}](SN_Big.eps "fig:"){width="45.00000%"} Furthermore, note that, in Figure \[dib:hk34\] we have represented the height of the saddle-node canard versus the parameter $k$ for a particular value of $\varepsilon$. The straight line corresponds with the height, $h_M$, of the canard orbit through the tangent point $\mathbf{p}_{LL}$, and coincides with the maximum height of a 3-zones saddle-node canard orbit. Hence, for $k_1<k<k_2$ the saddle-node canard orbits predicted by function $h^(k,\sqrt{\varepsilon};m)$ do not correspond neither with 3-zones nor 4-zones saddle-node canard orbits. It is clear that there must be a transition between both types of saddle-node canard orbits and, as we have mentioned along the manuscript, such a transition can not be obtained through the functions $R_{3z}$ and $R_{4z}$, see and respectively. In fact, similar situation happens in the smooth context, see [@DDR14], where slow divergence integral results to be not enough for the analysis of these transitory orbits. Although the results in this article do not apply to transitory canard cycles, from them, we can establish the following conjecture: For $\varepsilon=0$, consider the transitory singular canard $\Gamma$, see [@DDR14], formed by the critical manifold in $[-1,1]$ and the segment $\{(x,k):x\in[-1,1]\}$. There exist values of $\varepsilon>0$ and $k \approx 1$ such that corresponding system - exhibits one, two or three canard limit cycles in a neighborhood of $\Gamma$, even more, the system also exhibits zero, one or two saddle-node canard cycles in a neighborhood of $\Gamma$. Finally, in the subcritical case, the existence of saddle-node bifurcation of headless canards for $k<1,$ and of canards with head for $k>1,$ leads us to come up with: System - in the subcritical case ($m=\sqrt{\varepsilon}$) possesses a saddle-node bifurcation of transitory canard cycles for $k=1.$ The use of this simpler family of slow-fast systems to reproduce canard dynamics bring us some information which could be interesting when revisiting the smooth context. In particular, the conditions $k<1$ and $k>1$ organizing the dynamics in the main results, suggest the importance of the ratio between the slopes of the fast nullcline in order to exhibit or not saddle-node canard cycles with head. Assuming this idea, it seems that only saddle-node canard cycles with head can appear when the slope of the repelling branch of the critical manifold is greater than the slope of the attracting branches of the critical manifold. As this is not the case in the Van der Pol system, it can be expected that only headless saddle-node canard cycles are possible in the Van der Pol system. We finally point out that some quantitative information obtained in the manuscript could be relevant for applications. As example of such information we highlight the period of the canard cycles, see Lemma \[lem:times\], and the location of the saddle-node canards in terms of the parameter, see propositions \[prop:R3z\] and \[prop:R4z\]. The dependence between the height of a canard cycle and the bifurcation parameter $a$ at which it appears could be approximated from the estimation $\|\tilde{a}-\hat{a}\|$ appearing in Theorem \[th:exist\_Gamma\]. Acknowledgments =============== The three authors want to thank M. Desroches for introducing us into the problem and for his continuous support. They also thanks M. Krupa and S. Rodrigues for their fruitful conversations. First author is supported by Ministerio de Ciancia, Innovación y Universidades, through the project PGC2018-096265-B-I00. Second author is supported by Ministerio de Ciencia, Innovación y Universidades through the project RTI2018-093521-B-C31. Third author is supported by Ministerio de Economía y Competitividad through the project MTM2017-83568-P (AEI/ERDF, EU). About Poincaré half-maps and times of flight ============================================ In this section we summary the main technical results on piecewise linear dynamics that we use along the manuscript. In particular we provide approximations for the Poincaré half-map and for the time of flight of the solutions inside the regions of linearity. Consider the Poincaré half-maps $\Pi_R$, $\Pi_L$ and $\Pi_{LL}$ having the switching lines $\{x=\sqrt{\varepsilon}\},$ $\{x=-\sqrt{\varepsilon}\}$ and $\{x=-1\}$, respectively, as crossing section. By using the contact points $\mathbf{p}_R,\mathbf{p}_L$ and $\mathbf{p}_{LL}$ and the vector field on these points $\dot{\mathbf{p}}_R,\dot{\mathbf{p}}_L$ and $\dot{\mathbf{p}}_{LL}$, respectively, the Poincaré half-maps can be parametrized by $\Pi_R(\mathbf{p}_R-u\dot{\mathbf{p}}_R)= \mathbf{p}_R+v\dot{\mathbf{p}}_R$, $\Pi_L(\mathbf{p}_L-u\dot{\mathbf{p}}_L)= \mathbf{p}_L+v\dot{\mathbf{p}}_L$, and $\Pi_{LL}(\mathbf{p}_{LL}-u\dot{\mathbf{p}}_{LL})= \mathbf{p}_{LL}+v\dot{\mathbf{p}}_{LL}$, where $u$ and $v$ are positive values. Moreover, let ${\Pi}_{Ld}$ be the point transformation through the flow of the linear system $\dot{\mathbf{x}}=A_L\mathbf{x}+\mathbf{b}_L$, such that ${\Pi}_{Ld}(\mathbf{p}_{L}-u\dot{\mathbf{p}}_{L})= \mathbf{p}_{LL}-v\dot{\mathbf{p}}_{LL}$. \[lem:Pmaps\] Fix $\varepsilon_0>0$ small enough and $\nu \in (0,1)$, for $\varepsilon \in (0,\varepsilon_0)$ it follows that $$\begin{aligned} \Pi_L^{-1}\left(\begin{pmatrix}-\sqrt{\varepsilon} \\ h\end{pmatrix}\right)- \mathbf{q}_0^L &\approx \begin{pmatrix} 0 \\ h e^{-\frac {k h}{\varepsilon(\sqrt{\varepsilon}-a)}} \end{pmatrix}, \quad\text{for } h>(-m+\varepsilon^{-\nu}\lambda_L^s)(\sqrt{\varepsilon}+a),\\ \Pi_R\left(\begin{pmatrix}\sqrt{\varepsilon} \\ h\end{pmatrix}\right)-\mathbf{q}_1^R&\approx \begin{pmatrix} 0 \\ h e^{-\frac {h}{\varepsilon(\sqrt{\varepsilon}-a)}} \end{pmatrix},\quad\text{for } h>(m-\varepsilon^{-\nu}\lambda_R^s)(\sqrt{\varepsilon}-a),\\ \mathbf{q}_1^{LL}-\Pi_{LL}\left(\begin{pmatrix}-1 \\ h\end{pmatrix}\right) &\approx \begin{pmatrix} 0 \\ (k-h) e^{-\frac {k(k-h)}{\varepsilon(1+a)}} \end{pmatrix}, \quad\text{for } h< p_2+ \varepsilon^{-\nu} \lambda_{LL}^s(1+a), \\ \mathbf{q}_0^L-{\Pi}_{Ld}^{-1}\left(\begin{pmatrix}-1 \\ h\end{pmatrix}\right) &\approx \begin{pmatrix} 0 \\ (k-h) e^{-\frac{k^2}{\varepsilon} \ln\left(\frac {1+a}{a+\sqrt{\varepsilon}}\right) } \end{pmatrix}, \quad\text{for } h\in(p_2-\lambda_L^q(1+a),p_2),\\ $$ where $p_2=k(1-\sqrt{\varepsilon})-m(\sqrt{\varepsilon}+a)$ is the second coordinate of the point $\mathbf{p}_{LL}$. Next, we compute the expressions of the Poincaré half-maps $\Pi_R$ and ${\Pi}_{Ld}$, the remainder expressions in the lemma follows in a similar way. Following Chapter 3 in [@LT14], the coordinates $u$ and $v$ of the points $\mathbf{p}$ and $\Pi_R(\mathbf{p})$ in the Krylov base $\{\mathbf{p}_R,\dot{\mathbf{p}}_R\}$, that is $\mathbf{p}=\mathbf{p}_R-u \dot{\mathbf{p}}_R$ and $\Pi_{R}(\mathbf{p})=\mathbf{p}_R+v\dot{\mathbf{p}}_R$, are invariant through translations and linear transformation. Therefore, $u$ and $v$ satisfy that $v\in\left(0,-\frac {1}{\lambda_R^q}\right)$ and $$\left( \frac {1+v\lambda_R^s}{1-u\lambda_R^s} \right)^{\frac {\lambda_R^q}{\lambda_R^s}} = \frac {1+v\lambda_R^q}{1-u\lambda_R^q},$$ where $\lambda_R^s$ and $\lambda_R^q$ are the slow and the fast eigenvalues of the matrix $A_R$. For $\varepsilon$ small enough, $\lambda_R^s$ tends to zero and hence, taking the limit of the left side term in the previous identity as $\lambda_R^s$ tends to zero, we obtain that $v$ can be implicitly approximated by $$v=\frac 1{\lambda_R^q} \left(-1 + (1-u\lambda_R^q)e^{\lambda_R^q(u+v)}\right).$$ Since the $u$-coordinate of $\mathbf{p}=(\sqrt{\varepsilon},h)^T$ in the Krylov base is $$u=\frac {h-m(\sqrt{\varepsilon}-a)}{\varepsilon(\sqrt{\varepsilon}-a)}>0,$$ we conclude that $$\begin{aligned} \Pi_R(\mathbf{p}) &=\mathbf{p}_R+\frac{1}{\lambda_R^q}\left(-1+(1-u\lambda_R^q)e^{\lambda_R^q(u+v)}\right)\dot{\mathbf{p}}_R \\ &=\begin{pmatrix} \sqrt{\varepsilon} \\ \\ (m+\lambda_R^s)(\sqrt{\varepsilon}-a)-\left(\lambda_R^s(\sqrt{\varepsilon}-a) -u\varepsilon(\sqrt{\varepsilon-a})\right)e^{\lambda_R^q(u+v)} \end{pmatrix}\\ &=\begin{pmatrix} \sqrt{\varepsilon} \\ \\ (m+\lambda_R^s)(\sqrt{\varepsilon}-a)-\left((m+\lambda_R^s)(\sqrt{\varepsilon}-a) -h\right)e^{\lambda_R^q(v+\frac {h-m(\sqrt{\varepsilon}-a)}{\varepsilon(\sqrt{\varepsilon}-a)})} \end{pmatrix}\\ &= \mathbf{q}_1^R+\begin{pmatrix} 0 \\ \\ \left(h-(m+\lambda_R^s)(\sqrt{\varepsilon}-a) \right)e^{\lambda_R^q v+\frac {h-m(\sqrt{\varepsilon}-a)}{\lambda_R^s (\sqrt{\varepsilon}-a)}} \end{pmatrix}.\end{aligned}$$ Taking into account that $-1<\lambda_R^q v<0$, the exponent in the previous expression can be approximated by $\frac {h}{\lambda_R^s(\sqrt{\varepsilon}-a)}$ provided that $$\frac {h-m(\sqrt{\varepsilon}-a)} {\lambda_R^s(\sqrt{\varepsilon}-a)}<-\varepsilon^{-\nu},$$ for a $\nu$ value in $(0,1)$. From this we conclude that $h>(m-\varepsilon^{-\nu}\lambda_R^s)(\sqrt{\varepsilon}-a)$, what finish the proof of the the lemma. Consider now the transformation $\Pi_{Ld}$, from points of the form $\mathbf{p}_L-u\dot{\mathbf{p}}_L$ to points of the form $\mathbf{p}_{LL}-v\dot{\mathbf{p}}_{LL}$. Even when the relationship between the coordinates $u$ and $v$ of these points is not explicitly computed in [@LT14], we can use the same arguments than there to obtain that $v\in \left(0,\frac {1}{\lambda_L^s}\right)$ and $$\left( \frac {1-v\lambda_L^s}{1-u\lambda_L^s} \right)^{\frac {\lambda_L^q}{\lambda_L^s}}= r^{1-\frac {\lambda_L^q}{\lambda_L^s}} \left( \frac {1-v\lambda_L^q}{1-u\lambda_L^q} \right),$$ where $r=\frac{1+a}{\sqrt{\varepsilon}+a}$ satisfies that $\mathbf{p}_{LL}-\mathbf{e}_L=r(\mathbf{p}_{L}-\mathbf{e}_L)$. Taking the limit of the left side term in the previous identity as $\lambda_L^s$ tends to zero, we obtain a new implicit relation between the coordinates $u$ and $v$ given by $$u=\frac 1{\lambda_L^q} \left(1-r(1-v\lambda_L^q)e^{-\frac {\lambda_L^q}{\lambda_L^s}\ln(r)+\lambda_L^q (v-u)}\right).$$ Since the $v$ coordinate of a point $(-1,h)=\mathbf{p}_{LL}-v\dot{\mathbf{p}}_{LL}$ satisfies that $$v=\frac {k(1-\sqrt{\varepsilon})-m(\sqrt{\varepsilon}+a)-h}{\varepsilon(1+a)}\in \left(0,\frac 1{\lambda_L^s}\right),$$ it follows that $$k(1-\sqrt{\varepsilon})-m(\sqrt{\varepsilon}+a)-\lambda_L^q(1+a)<h<k(1-\sqrt{\varepsilon})-m(\sqrt{\varepsilon}+a),$$ and therefore, the preimage by ${\Pi}_{Ld}$ of the point $(-1,h)$ is $$\mathbf{p}_L-u\dot{\mathbf{p}}_L=\mathbf{q}_0^L + \begin{pmatrix} 0\\ \left( (1+a)\lambda_L^s+m(a+\sqrt{\varepsilon})-k(1-\sqrt{\varepsilon})+h \right) e^{-1-\frac{\lambda_L^q(1+a) \ln(r)-k(1-\sqrt{\varepsilon})+m(\sqrt{\varepsilon}+a)+h} {\lambda_L^s(1+a)}} \end{pmatrix}.$$ For $\varepsilon$ small enough previous expression can be rewriten as $$\mathbf{p}_L-u\dot{\mathbf{p}}_L=\mathbf{q}_0^L + \begin{pmatrix} 0\\ \left( (m+k)(a+\sqrt{\varepsilon})-\lambda_L^q(1+a)+h \right) e^{-\frac{k}{\varepsilon} \ln\left(\frac{1+a}{\sqrt{\varepsilon}+a}\right)} \end{pmatrix}.$$ The lemma for $\Pi_{Ld}$ follows by considering $\varepsilon$ small enough. \[rem:interv\] From Lemma \[lem:Pmaps\], the preimage by $\Pi_L$ of a point is exponentially close to $\mathbf{q}_0^L$ if there exists $\nu\in(0,1)$ such that, its second coordinate, $h$, is at a distance $d\geq\varepsilon^{-\nu}\lambda_L^s(\sqrt{\varepsilon}+a)$ from the point $\mathbf{p}_L$, in particular when $d> \lambda_L^s(\sqrt{\varepsilon}+a)$. Let $\tau_R(h)$ be the time of flight of the solution between the points $(\sqrt{\varepsilon},h)^T$ and $\Pi_R(\sqrt{\varepsilon},h)$, let $\tau_L(h)$ be the time of flight between $(-\sqrt{\varepsilon},h)^T$ and $\Pi_L(-\sqrt{\varepsilon},h)$, let $\tau_{LL}(h)$ be the time of flight between $(-1,h)^T$ and $\Pi_{LL}(-1,h)$, let ${\tau}_{Ld}(h)$ be the time of flight between $(-\sqrt{\varepsilon},h)^T$ and $\Pi_{Ld}(-\sqrt{\varepsilon},h)$ and let $\tau_{RR}$ be the time of flight between $\mathbf{q}_0^{RR}$ and $\mathbf{q}_1^R$, see Figure \[dib:geommetry\], that is, $\tau_{RR}=\tau_R(h_0)$ where $h_0$ is the second coordinate of $\mathbf{q}_1^{LL}$, \[lem:times\] For $\varepsilon>0$ and small enough it follows that $$\begin{aligned} \tau_R(h)&\approx-\frac 1{\lambda_R^s} \ln\left( 1 + \frac {(m+\lambda_R^s)(\sqrt{\varepsilon}-a)-h} {(\lambda_R^q-\lambda_R^s)(\sqrt{\varepsilon}-a)} \right),\\ \tau_L(h)&\approx\frac 1{\lambda_L^s} \ln\left( 1 + \frac {h+(m+\lambda_L^s)(\sqrt{\varepsilon}+a)} {(\lambda_L^q-\lambda_L^s)(\sqrt{\varepsilon}+a)} \right),\\ {\tau}_{Ld}(h)&\approx\frac 1{\lambda_L^s} \ln\left( 1 + \frac {h+m(\sqrt{\varepsilon}+a)+\lambda_L^s(2\sqrt{\varepsilon}+a-1)} {(\lambda_L^q-\lambda_L^s)(\sqrt{\varepsilon}+a)} \right),\\ \tau_{LL}(h)&\approx-\frac1{\lambda_{LL}^s} \ln\left( 1+ \frac{ h+m(\sqrt{\varepsilon}+a)+k(\sqrt{\varepsilon}-1)+\lambda_{LL}^s(1+a)} { (\lambda_{LL}^q-\lambda_{LL}^s)(1+a) } \right),\\ \tau_{RR}&\approx - \frac1{\lambda_R^s} \ln\left( 1+ \frac{ \lambda_R^s(\sqrt{\varepsilon}-a)+\lambda_{LL}^s(1+a)+k(\sqrt{\varepsilon}-1)+2m\sqrt{\varepsilon} } { (\lambda_R^q-\lambda_R^s)(\sqrt{\varepsilon}-a)} \right).\end{aligned}$$ Consider a point $\mathbf{p}=(\sqrt{\varepsilon},h)^T$ and its image by the Poincaré map $\Pi_R(\mathbf{p})$. From Lemma \[lem:Pmaps\], since $\varepsilon$ is small enough, we can substitute the point $\Pi_R(\mathbf{p})$ by the exponentially close point $\mathbf{q}_1^R$. In an equivalent way, we approximately compute the time $\tau_R(h)$ as the time of flight of the solution for travelling from $\mathbf{p}$ to $\mathbf{q}_1^R$. To do that we project the point $\mathbf{p}$ onto the point $\mathbf{p}_s$ contained in the slow manifold $\mu_R$, see , by following the fast eigenvector, $\mathbf{v}_R^q$. The point $\mathbf{p}_s$ is obtained by solving with respect the unknowns $r_s,r_q\in \mathbb{R}^+$ the linear system of equations $$\mathbf{p}_s=\mathbf{p}-r_q\mathbf{v}_R^q=\mathbf{e}_R-r_s \mathbf{v}_R^s.$$ We conclude that $$r_s=\frac{(\sqrt{\varepsilon}-a)(m-1-\lambda_R^s)-h}{\lambda_R^s(\lambda_R^s-\lambda_R^q)} =\frac{(\sqrt{\varepsilon}-a)(m+\lambda_R^q)-h}{\lambda_R^s(\lambda_R^s-\lambda_R^q)}.$$ Then, we compute $\tau_R(h)$ as the time of flight of the solution to travel from the projected point $\mathbf{p}_s$ to $\mathbf{q}_1^R$, that is $$e^{\lambda_R^s \tau_R(h)}=\frac {\\|\mathbf{q}_1^R-\mathbf{e}_R\\|}{\\|\mathbf{p}_s-\mathbf{e}_R\\|} =\frac{\frac {a-\sqrt{\varepsilon}}{\lambda_R^s} \\|\mathbf{v}_R^s\\|}{r_s \\|\mathbf{v}_R^s\\|} =\frac {(\sqrt{\varepsilon}-a)(\lambda_R^q-\lambda_R^s)}{(\sqrt{\varepsilon}-a)(m+\lambda_R^q)-h}.$$ The lemma follows by isolating $\tau_R(h)$. The remainder of the functions are computed by following similar arguments. In the computation of function $\tau_R(h)$ there is only one approximation, which consists in using the exponentialy close point $\mathbf{q}_1^R$, instead of the point $\Pi_R(\mathbf{p})$. For the purposes of this manuscript it is also important the knowledge of the map $\Phi(x_0)$ defined by the flow through the points on the fast nullcline $(x_0,f(x_0))$. Since the nullcline is piecewise, so is the map $\Phi$, which domain depends on $x_0<-1$ and $x_0>-1$. In particular, when $x_0\in (x_r,x_u)$ the map $\Phi$ is defined as $\Phi_{4z}(x_0)$ by the second coordinate of the point on the switching line $\{x=-1\}$ defined backward through the flow, and when $x_0\in [-1,x_s)$ the map $\Pi$ is defined as $\Phi_{3z}(x_0)$ by the second coordinate of the point on the switching line $\{x=-\sqrt{\varepsilon}\}$ defined forward through the flow. In the next result we relate the map $\Phi$ with the Poincaré maps analysed in Lemma \[lem:Pmaps\]. This relationship is established through the values $x_r$, $x_s$ and $x_u$ introduced in and , respectively. \[lem:dom\] For $\varepsilon>0$ and $\Phi(x_0)=h$ it follows that: - If $x_0\in(x_r,x_u)$ there exists $\nu\in (0,1)$ such that $h_r<h<h_u$, where $h_r=-(m+\lambda_L^s)(\sqrt{\varepsilon}+a)$ and $h_u=-m(\sqrt{\varepsilon}+a)+k(1-\sqrt{\varepsilon}). +\varepsilon^{-\nu}\lambda_{LL}^s(1+a),$ - If $x_0\in[-1,x_s)$ there exists $\nu\in (0,1)$ such that $h_s<h<h_M$, where $h_s=-(m-\varepsilon^{-\nu}\lambda_{L}^s)(\sqrt{\varepsilon}+a)$ and $h_M=-m(\sqrt{\varepsilon}+a)+k(1-\sqrt{\varepsilon}).$ We restrict ourselves to the proof of statement (b). The statement (a) of the lemma follows by using similar arguments. Let $r$ and $v$ be the coordinates of the points $(x_0,f(x_0))$ and $(-\sqrt{\varepsilon},\Phi(x_0))$, respectively, in the Krylov base $\{\mathbf{p},\dot{\mathbf{p}}\}$ with $\mathbf{p}=\mathbf{p}_L-\mathbf{e}_L$. Thus, there exists $\tau_f>0$ such that $e^{A_L \tau_f}r\mathbf{p}=\mathbf{p}+v\dot{\mathbf{p}}$,where $$r=\frac {a-x_0}{a+\sqrt{\varepsilon}},$$ is obtained from $(x_0,f(x_0))^T-\mathbf{e}_L=r(\mathbf{p}_L-\mathbf{e}_L)$. Following similar arguments than those in Chapter 3 in [@LT14], we obtain next relation between the coordinates $r$ and $v$, $$\label{aux:lem_dom} {r^{\lambda_L^s-\lambda_L^q}}(1+v\lambda_L^s)^{\lambda_L^q}= (1+v\lambda_L^q)^{\lambda_L^s},$$ In particular, when $x_0=x_s$, it follows that $r=1+\lambda_L^s$ and the corresponding coordinate $v$ satisfies $$\label{aux:lem3} e^{\lambda_L^s-\lambda_L^q}=e^{-v\lambda_L^q}(1+v\lambda_L^q),$$ where we have taken into account that $(1+v\lambda_L^s)^{\frac {\lambda_L^q}{\lambda_L^s}}$ and $(1+\lambda_L^s)^{\frac {\lambda_L^s-\lambda_L^q}{\lambda_L^s}}$ tend to $e^{v\lambda_L^q}$ and $e^{\lambda_L^s-\lambda_L^q}$, respectively, as $\lambda_L^s$ tends to zero. From expression and for $\varepsilon$ small enough it follows that $v>1$ and, hence, $\mathbf{p}+v\dot{\mathbf{p}}=(-\sqrt{\varepsilon},h)^T-\mathbf{e}_L$ with $h>-m(\sqrt{\varepsilon}+a)+\varepsilon(\sqrt{\varepsilon}+a)$. Therefore, for a fixed $\varepsilon>0$ there exists $\nu\in(0,1)$ such that $h>-m(\sqrt{\varepsilon}+a)+\varepsilon^{-\nu}\lambda_L^s(\sqrt{\varepsilon}+a)$, which proves the lower bound in the statement (b). For the upper bound, we note that when $x_0=-1$ then $r=\frac {a+1}{a+\sqrt{\varepsilon}}$, which tends to infinity as $\varepsilon^{-1/2}$ when $\varepsilon$ tends to zero. From the relation between the variables $v$ and $r$, see , we conclude that for $\varepsilon$ small enough, expression $\frac {1+v\lambda_L^s}{r}$ tends to 1 if $\varepsilon$ tends to 0. 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--- abstract: - 'Using a simple one-dimensional Frenkel-Kontorowa type model, we have demonstrated that finite commensurate chains may undergo the commensurate-incommensurate (C-IC) transition when the chain is contaminated by isolated impurities attached to the chain ends. Monte Carlo (MC) simulation has shown that the same phenomenon appears in two-dimensional systems with impurities located at the peripheries of finite commensurate clusters.' - '=3000Використовуючи просту одновимірну типову модель Френкеля-Конторової, ми показали, що скінченні співвимірні ланцюжки можуть зазнавати співвимірний-неспіввимірний (C-IC) перехід, якщо ланцюжок забруднений ізольованими домішками, дочепленими до кінців ланцюжка. За допомогою методу Монте Карло (MC) показано, що таке ж явище виникає в двовомірних системах з домішками, розміщеними на периферії скінчених співвимірних кластерів.' address: - \| Department for the Modelling of Physico-Chemical Processes, Faculty of Chemistry,\ Maria Curie-Sklodowska University, 20031 Lublin, Poland - 'Відділ моделювання фізико-хімічних процесів, Університет Марії Кюрі-Склодовської, Люблін, Польща' author: - 'A. Patrykiejew, S. Soko[ł]{}owski' - 'А. Патрикєєв, С. Соколовскі' date: 'Received February 24, 2014, in final form May 6, 2014' title: 'Формування домен-стінки в бінарних наноскопічних скінченних системах' --- Introduction ============ In modern nanotechnologies one often deals with very small systems of countable numbers of atoms or molecules. In such cases, the finite size and boundary effects are large and bound to significantly affect the properties of the system with respect to its bulk counterpart [@nano1]. Another important problem is the purity of small systems. While tiny amounts of impurities may be unimportant in macro-scale, the behavior of nanoscopic systems is considerably influenced even by a small number of impurity atoms [@imp1]. Among the systems in which the presence of impurities may be of importance are those exhibiting the C-IC transition. The C-IC transitions have been experimentally observed in a variety of systems including adsorbed films, [@cicex1; @cicex2; @cicex4; @cicex8] intercalated compounds [@cicex9; @cicex11] composite crystals [@cicex13] and magnetically ordered structures of rare-earth compounds [@cicex15]. Theoretical studies of C-IC transitions have focused on the domain wall description of incommensurate phases [@cic2; @cic4; @cic6; @cic7]. According to the domain wall formalism, the IC phase is a collection of C domains separated by domain walls. The density within the domain walls may be lower or higher than the density of commensurate domains. In the former case, the walls are light and superlight, while in the latter the walls are heavy and superheavy [@cic6]. The simplest theoretical approach which predicts the formation of domain walls is the one-dimensional Frenkel-Kontorova model [@FK1]. The original FK model assumes that an infinite chain of atoms interacting via harmonic potential at zero temperature is subjected to a periodic (sinusoidal) external field. Depending on the misfit between the equilibrium distance of the harmonic potential, the period and amplitude of the external field, the FK model is capable of describing the C-IC transition. The FK model has been extended to two-dimensional systems [@FK2D-1; @FK2D-2] to mixtures [@FK-mix1], systems with disorder [@FK-dis2] and has also been used to study finite chains [@FK-fin2; @FK-fin3; @FK-fin4]. The NPT Monte Carlo simulation has demonstrated [@phillips2005; @phillips2008] that finite one-dimensional chains, either uniform or subjected to periodic field, exhibit structures that cannot appear in infinite chains. In particular, it has been shown that the chain experiences very large density fluctuations. In the case of chains on a periodic substrate, a number of different structures (registered, free floating, domain-wall incommensurate and resulting from the chain fragmentation) have been found to appear during a single run. In one of our recent papers [@myArKr], we have shown that finite two-dimensional clusters of Kr adsorbed on graphite undergo the C-IC phase transition when contaminated by small amounts of Ar atoms. Computer simulation has demonstrated that the transition occurs already when the boundaries of a finite krypton island are covered with a single layer of argon atoms. In this paper we address the issue of the influence of impurities on the behavior of finite one- and two-dimensional (1D and 2D) systems. We are interested in the effects of impurities located at the peripheries of finite 1D chains and 2D clusters of atoms subjected to the periodic external field. The field is assumed to be strong enough to enforce the formation of commensurate structures in pure systems and we consider the possibility of the C-IC transition driven by the presence of impurities. The paper is organized as follows. In the next section we discuss the behavior of one-dimensional systems in the framework of a modified Frenkel-Kontorova model. Then, in the third section we consider two-dimensional finite systems studied by Monte Carlo simulation. In the final section we summarize our findings. One-dimensional Frenkel-Kontorova model ======================================= At first, we have considered a simple 1D finite chain of atoms at zero temperature and used the Frenkel-Kontorova (FK) model [@FK1]. The energy of a finite chain consisting of $N$ atoms subjected to a periodic external potential and containing impurities can be written as follows: $$E = \frac{1}{2}\left\{\sum_{i=1}^{N-1}K_{i}[x_{i+1}-x_{i}-b_{i}]^2 + \sum_{i=1}^Nv_{i}[1-\cos(2\pi x_i/a)]\right\}, \label{eq:Kon1}$$ where $K_{i}$ and $b_{i}$ are the elastic constant and the equilibrium distance for the pair $(i,i+1)$ and $v_i$ is the amplitude of the external field for the $i$-th particle. Having introduced the displacements $u_i = x_i/a-pi$ ($i=1,2,\ldots N$) with $p$ being a positive integer (in this work we set $p=2$), the energy given by eqn.(\[eq:Kon1\]) can be rewritten in units of $K_{0}a^2/2$ ($K_{0}$ being the elastic constant for a pure chain) as follows: $$E=\sum_{i=1}^{N-1}\hat{K}_{i}[u_{i+1}-u_{i}-m_{i}]^2 + \sum_{i=1}^N\hat{v}_{i}[1-\cos(2\pi u_i)]. \label{eq:Kon2}$$ In general, the elastic constants $\hat{K}_{i}$ (the misfits $m_{i}$) can assume one of three possible values $k_{0,0}\equiv 1$, $k_{0,\textrm{im}}$ or $k_{\textrm{im},\textrm{im}}$ ($m_{0,0}$, $m_{0,\textrm{im}}$ or $m_{\textrm{im},\textrm{im}}$) depending on the composition of the pair $(i,i+1)$, and the amplitude $\hat{v}_i = v_i/a^2K_{0}$ is equal either to $v_0$ or to $v_\textrm{im}$. In order to find the equilibrium configuration of the chain, the energy should attain its minimum value, specified by the condition stating that the forces $f_i=-\partial E/\partial u_i = 0$ for all $i$. We consider the systems in which a single impurity atom is located at one end of the chain (class I) and the systems with two impurity atoms located at both ends of the chain (class II) and put $k_{0,\textrm{im}}=k_\textrm{im}$ and $m_{0,\textrm{im}}=m_\textrm{im}$. The behavior of pure chains depends on $m_{0}$ and $v_0$. For the assumed value of $m_{0}=-0.1$, pure chains are commensurate when $v_0$ exceeds the critical value $v_{0,\textrm{c}}\approx 0.004950$. The calculations have been done for $N$ between 21 and 401 and $v_0=0.006$. The first series of calculations have been carried out for $k_\textrm{im}=1.0$, while $v_\textrm{im}$ and $m_\textrm{im}$ have been varied. Figure \[fig1\] shows the example of the results obtained for the chains with $N=41$, $m_\textrm{im}=-0.25$ and different values of $v_\textrm{im}$. The lower and upper panels of figure \[fig1\] show the results for the systems of class I and II, respectively. It is evident that the systems belonging to both classes exhibit a qualitatively similar behavior. Of course, the systems of the class I lack the symmetry of atomic displacements with respect to the central atom. For $v_\textrm{im}$ lower (higher) than $v_{\textrm{im},\textrm{l}}$ ($v_{\textrm{im},\textrm{u}}$), the chain remains commensurate, while for intermediate values of $v_\textrm{im}$, there appear incommensurate structures with domain walls. When $v_\textrm{im}$ is lower than $v_{\textrm{im},\textrm{l}}$, the energy cost to put the impurity out of registry position is low and the impurity can exhibit large displacements from the commensurate position, while the rest of the chain assumes a commensurate structure due to the domination of the surface over the elastic interaction. On the other hand, when $v_\textrm{im}>v_{\textrm{im},\textrm{u}}$, the impurity is strongly pinned by the surface potential and the chain retains the commensurate structure. Figure \[fig1\] shows that when $v_\textrm{im}>v_{\textrm{im},\textrm{u}}$, the displacements of impurities are considerably lower than when $v_\textrm{im}<v_{\textrm{im},\textrm{l}}$. ![(Color online) Atomic displacements vs. atomic positions for the systems of class I (part a) and II (part b) and different values of $v_\textrm{im}$ (given in the figure). The calculations have been done $N=41$, $m_\textrm{im}=-0.25$ and $k_\textrm{im}=1.0$. The insets show the changes of the average nearest-neighbor distance vs. $v_\textrm{im}$ The regions marked by $C$ and $ks$ ($k=1,2,3,4$) correspond to the commensurate structure to the incommensurate structures with $k$ domain walls, respectively.[]{data-label="fig1"}](fig1_){width="60.00000%"} For intermediate values of $v_\textrm{im}$, the gain in elastic energy due to transition into the incommensurate structure is larger than the loss of surface energy. The insets to figure \[fig1\] show the changes of the average nearest-neighbor distance with $v_\textrm{im}$. In both classes of systems, we have found a series of transitions characterized by a different number of domain walls. The calculations for chains with $N$ up to 401 have shown that in longer chains a larger number of structures appear, and it seems that in the limit of $N\rightarrow\infty$ the transitions form a harmless staircase [@cic2]. The values of $v_{\textrm{im},\textrm{l}}$ and $v_{\textrm{im},\textrm{u}}$ change with $m_\textrm{im}$ and there is a critical value of $m_{\textrm{im},\textrm{c}}$ for which the difference $\Delta v_\textrm{im} = v_{\textrm{im},\textrm{u}}-v_{\textrm{im},\textrm{l}}$ goes to zero (figure \[fig2\]). In the particular examples considered here ($N=41$ and $k_\textrm{im}=1.0$), $m_{\textrm{im},\textrm{c}}\approx -0.1856$ for the systems of class I and II. The inset to figure \[fig2\] shows that $\Delta v_\textrm{im}$ scales with $m_{\textrm{im},\textrm{c}} - m_\textrm{im}$ as follows: $$\Delta v_\textrm{im}\propto (m_{\textrm{im},\textrm{c}} - m_\textrm{im})^{1/2}, \qquad k=1,2. \label{eq:Kon10}$$ The same scaling appears for the transitions between different incommensurate structures, but the value of $m_{\textrm{im},\textrm{c}}$ for each transition is different (figure \[fig2\]). We have then investigated the effects due to changes in the magnitude of $k_\textrm{im}$. Figure \[fig3\] gives the example of results for the systems of class I. We see that the values of $v_\textrm{im}$ at the transitions $C-1s$, $1s-2s$ and $2s-3s$ are nearly independent of $k_\textrm{im}$, while those corresponding to transitions $3s-2s$, $2s-1s$ and $1s-C$ (leading to the recovery of commensurability), exhibit a logarithmic dependence on $k_\textrm{im}$. ![The phase diagrams showing the dependence of the difference between the upper and lower values of $v_\textrm{im}$ at the transition points ($\Delta v_\textrm{im}(\alpha,\beta)$) between different structures ($\alpha,\beta$) and the impurity misfit $m_\textrm{im}(\alpha,\beta)$. The calculations have been done for $N=41$ and $k_\textrm{im}=1.0$ and for the classes I and II. The inset shows the scaling plot for the C-IC transition.[]{data-label="fig2"}](fig2_){width="60.00000%"} ![(Color online) The phase diagrams showing the dependence of $v_\textrm{im}$ on $k_\textrm{im}$ at the transition points between different structures. The calculations have been done for $N=41$ and $m_\textrm{im}=-0.24$. The inset shows the atomic displacements versus actual atomic positions in the commensurate phase for $v_\textrm{im}=0.003$ and $v_\textrm{im}=0.017$ when $k_\textrm{im}=1.0$. Filled symbols mark the impurity atom.[]{data-label="fig3"}](fig3_){width="60.00000%"} The logarithmic dependence of $v_\textrm{im}$ on $k_\textrm{im}$ at transitions $3s-2s$, $2s-1s$ and $1s-C$ has the same origin as the C-IC transition in a pure FK model [@cic2]. On the other hand, the mechanism of transitions leading to commensurability when $v_\textrm{im}$ becomes very low is different. For sufficiently low $v_\textrm{im}$, the energy cost to put the impurity out of registry position and to restore the commensurate positions in the rest of the chain is low. Consequently, the impurity can exhibit a large displacement from the commensurate position, while the rest of the chain assumes a commensurate structure due to the domination of surface energy over the elastic energy. This is illustrated in the inset to figure \[fig3\], which shows atomic displacements for the systems with $k_\textrm{im}=1.0$ and $v_\textrm{im}= 0.003$ and $v_\textrm{im}=0.017$, while the rest of the parameters have been kept the same as in the main figure. Of course, there is an asymmetry of displacements in the chain but it is very small. In the commensurate phase at high values of $v_\textrm{im}=0.017$, the displacements at both ends are nearly the same. Another question is whether a single impurity can drive the incommensurate system into commensurability? The answer is no. On the other hand, two impurities located at both ends of the chains do lead to the recovery of commensurability when $v_\textrm{im}$ is sufficiently high. This is just the same as with the rope pinned either to one or two walls. In the first case, the rope hangs down freely. In the latter, the rope pinned to the opposite walls can be expanded to some extent. In the harmonic approximation, the chain always retains integrity, although when the interaction potential allows for dissociation, the chain may rupture [@markov; @phillips2005; @phillips2008] rather than restore the C structure. Two-dimensional finite clusters =============================== The phenomenon of reentrant commensurability is not restricted to the above discussed simple 1D model. It also appears in more realistic 2D systems in the ground state and at finite temperatures. To demonstrate this, we have performed MC simulation in the canonical ensemble for finite clusters decorated with impurities at the boundaries. The particles of $A$ and $B$ (impurity) have been placed in the substrate field $$v_i(x,y)=-V_{i}\{\cos({{\mbox{\bf q}}}_1{{\mbox{\bf r}}})+\cos({{\mbox{\bf q}}}_2{{\mbox{\bf r}}})+\cos[({{\mbox{\bf q}}}_1-{{\mbox{\bf q}}}_2){{\mbox{\bf r}}}]\} \label{eq:graphite}$$ corresponding to the graphite basal plane, where $V_i$ is the amplitude of the potential for the $i$th component and ${{\mbox{\bf q}}}_1$ and ${{\mbox{\bf q}}}_2$ are the reciprocal lattice vectors of the graphite basal plane. The initial configurations have been a hexagon of a perfectly ordered $(\sqrt{3}\times\!\!\sqrt{3})R30^\circ$ structure consisting of $N_Az$ atoms of $A$ and $N_B$ atoms of $B$ placed along the one edge (case I), the adjacent three edges (case II) or along all six edges (case III) of the cluster. The interaction between the particles has been represented by the LJ(12,6) potential with the fixed $\sigma_{AA}=1.46a$ ($a=2.46$ [Å]{} is the graphite lattice constant taken as the unit of length) and $\varepsilon_{AA}/k_\textrm{B}=170$ K (assumed to be a unit of energy) and $V^{\ast}_A=V_A/\varepsilon_{AA}= 0.12$. In order to reduce the tendency of $B$ atoms towards aggregation, we have put $\varepsilon_{BB}=0.5\varepsilon_{AA}$, while $\varepsilon_{AB}$ has been assumed to be equal to $\varepsilon_{AA}$. The parameters $\sigma_{BB}=\sigma_{AB}$ and $V_B$ have been varied. The simulations have been carried out for the systems with $N_A= 271$ and $N_B=11$ (case I), $N_B=33$ (case II) and $N_B=66$ (case III). Besides, we have also performed some runs for a larger system corresponding to case III, with $N_A=811$ and $N_B=114$ as well as with $N_A=1189$ and $N_B=138$. The simulation has been performed at reduced temperatures $T^{\ast}=k_\textrm{B}T/\varepsilon_{AA}$ between 0.005 and 0.3. The formation of domain walls in the system has been monitored using the order parameter [@myArKr; @HoulLan] $$\phi({{\mbox{\bf r}}})=\cos({{\mbox{\bf q}}}_1{{\mbox{\bf r}}})+\cos({{\mbox{\bf q}}}_2{{\mbox{\bf r}}})+\cos[({{\mbox{\bf q}}}_1-{{\mbox{\bf q}}}_2){{\mbox{\bf r}}}], \label{eq:local}$$ and assuming that the atom is commensurate (incommensurate) when $\phi>0$ ($\phi\leqslant 0$). We have calculated the average numbers of incommensurate atoms $A$ and $B$ neglecting the atoms located at the patch boundaries, i.e., those with less than 5 nearest neighbors. It has been found that for any value of $\sigma^{\ast}_{BB}=\sigma_{BB}/a$ between 1.34 and 1.37 and for sufficiently low and sufficiently high values of $V^{\ast}_B$, the systems remain commensurate, while for the intermediate values of $V^{\ast}_B$, there appear networks of heavy walls separating commensurate domains. In general, the results at low temperatures are qualitatively very similar to those obtained for 1D FK model. In particular, we have observed the formation of IC structures in the cases I, II and III, though the structure of the domain-wall networks is different in all cases. In general, the walls tend to assume the orientations perpendicular to the edges decorated with the impurity atoms. Therefore, in case III, we have found regular networks like that given in the leftmost part of figure \[fig4\]. On the other hand, the systems with only one or three adjacent edges decorated by impurities form irregular networks of domain walls (see the middle and rightmost panels to figure \[fig4\]). The region of $V^{\ast}_B$ over which the IC phase is stable at low temperatures gradually decreases when the size of impurity atoms increases, so that for sufficiently large impurity atoms, only the C phase occurs independently of the magnitude of $V^{\ast}_B$ (see figure \[fig5\]). The calculations for the larger system with $N_A = 811$ and $N_B=114$ (case III) have shown that maximum value of $\sigma^{\ast}_{AB}$ above which the C-IC transition does not appear slightly increases with the system size. Moreover, the simulations performed for larger systems have also demonstrated that at low temperatures the size of C domains does not change and only their number increases. This result is quite similar to that obtained for 1D FK model for the chains of different length. This is demonstrated by the snapshots given in the leftmost part of figure \[fig4\] and in figure \[fig6\]. ![(Color online) The examples of snapshots for the system with $\sigma^{\ast}_{BB} =1.36$ and $V^{\ast}_B =0.12$ at $T^{\ast}= 0.02$ corresponding to the three cases considered. Shaded circles represent $B$ atoms. Open (filled) circles correspond to the commensurate (incommensurate) $A$ atoms.[]{data-label="fig4"}](fig4_){width="70.00000%"} ![(Color online) The phase diagram showing the locations ($V^{\ast}_B$ versus $\sigma^{\ast}_{AB}$) of C-IC transition in the limit of $T^{\ast}\rightarrow 0$ in the systems with $N_A=271$ and different concentration of impurity atoms.[]{data-label="fig5"}](fig5_){width="48.00000%"} The impurity induced changes of the surface stress are responsible for the C-IC transition, which leads to the compression of the finite island. When the amplitude of the impurity surface potential, $V^{\ast}_B$, is very low, the presence of impurities at the cluster boundaries does not appreciably affect the strain of the atoms in the cluster. Therefore, the surface stress of the impurity decorated island does not match differently from that of the pure system. For sufficiently large values of $V^{\ast}_B$, the impurities are strongly pinned over the surface potential minima as well as do not enlarge the strain in the island. Therefore, for sufficiently low and high $V^{\ast}_B$, the system retains the C structure. For intermediate values of $V^{\ast}_B$, the competing atom-atom and atom-external field interactions lead to a sufficiently large increase of a surface stress in order to trigger the C-IC transition. One should note a similarity of the herein reported impurity driven C-IC transition to the behavior of metal nanowires [@nwir1] and to the formation of IC phases in the subjected to uniaxial compression self-assembled monolayer gold nanoparticles supported on a fluid [@gold]. ![(Color online) The example of snapshot for the system with $N_A=1189$ and $N_B=138$, $\sigma_{AB} =1.365$ and $V^{\ast}_B=0.08$ recorded at $T^{\ast}=0.01$ K. Shaded circles represent $B$ atoms. Open and filled circles correspond to the commensurate and incommensurate $A$ atoms.[]{data-label="fig6"}](fig6_){width="50.00000%"} The results presented here describe only the low temperature behavior of the systems. The simulation has demonstrated that upon an increase of temperature, the IC structure gradually changes and the C structure is restored at temperatures still below the melting point of the cluster. The effects of thermal excitations on the domain-wall networks will be discussed elsewhere. Summary ======= In this work we have studied the impurity driven commensurate-incommensurate transitions in one- and two-dimensional finite systems at zero temperature. In the case of one-dimensional finite chains, we have considered the situations in which the impurity is located at one and two ends of the chain. It has been shown that in both situations, the C-IC transition occurs when the amplitude of the external field experienced by the impurity atoms falls into the region between the lower and upper threshold values. These limiting values of $v_\textrm{im}$ depend upon the parameters characterizing the interaction between the atoms in the main chain, the amplitude of external field acting on the main chain atoms and the interaction between the main chain and the impurities. The number of solitons (domain walls) in the IC structure is different for the chains with one and two ends decorated with the impurity atoms and also depends upon the $v_\textrm{im}$. This behavior is a consequence of different symmetry properties of the system. In the case of two-dimensional finite clusters, a very similar C-IC transition has been found. In particular, we have observed that the transition occurs when only a part of the cluster boundary is covered with a single row of impurity atoms. Also, by analogy to the results of one-dimensional FK model, the transition occurs only between the lower and upper threshold values of $v_\textrm{im}$. It should be noted that the C-IC transition observed in two-dimensional systems occurs only when the parameters entering the potential describing the interactions between $AA$, $AB$, and $BB$ atoms are suitably chosen. First of all, the components $A$ and $B$ should not exhibit the tendency towards mixing. Otherwise, the impurity atoms $B$ are likely to penetrate the patch and different scenarios are possible. One, is the formation of a mixed commensurate phase. This has been found in the case of Ar-Kr finite patches at sufficiently high temperatures [@myArKr], when the $AB$ interaction potential parameters were obtained using the standard Lorentz-Berthelot mixing rules. The same mixture exhibits a different behavior at low temperatures, and does undergo the C-IC transition, although with the domain walls preferentially made of Ar atoms. Another requirement is that atoms $A$ should tend to order into the C phase, while atoms $B$ should order into the IC phase. This imposes certain restrictions on the choice of $\sigma_{AA}^{\ast}$, $\sigma_{BB}$ and the values of $V_A^{\ast}$ and $V_{B}^{\ast}$. In particular, $\sigma_{AA}^{\ast}$ and $V_A^{\ast}$ should assume the values ensuring that a pure $A$ patch is commensurate, but are likely to undergo the C-IC transition when the density exceeds the monolayer capacity or when subjected to an external force due to the presence of impurity atoms along the patch boundaries. Thus, $V_{A}^{\ast}$ cannot be too low or to high and the misfit between the sizes of surface lattice and adsorbate atoms is rather small. A good example of such a system is krypton adsorbed on graphite [@cicex4]. In fact, the parameters $\sigma_{AA}^{\ast}$ and $V_A^{\ast}$ used here are rather close to those describing the krypton adsorbed on the graphite basal plane [@myArKr]. On the other hand, the values of $\sigma_{BB}$ and $V_{B}^{\ast}$ should favor the formation of IC phase by pure component $B$. Thus, the misfit between the size of B atoms and the surface lattice should be sufficiently large (it can be positive as well as negative) and $V_{B}^{\ast}$ should be sufficiently small. However, there is still another requirement for the appearance of C-IC transition in finite patches. The $AB$ interaction should be sufficiently strong so that the layer of $B$ atoms along the patch boundary is stable. Otherwise, the atoms $B$ would prefer to form clusters rather than stay at the patch boundary. Besides, this condition is also necessary to exert a sufficiently large force upon the atoms $A$ inside the finite patch to trigger the C-IC transition. Usually, the C-IC transition occurs only when the film density exceeds the density of a fully filled commensurate phase. For the C-IC transition to occur in finite patches, the stability of the C phase should be low enough so that a rather small force exerted by a thin layer of impurity atoms along the patch boundary is able to drive the C-IC transition. From our earlier studies of adsorption of Ar-Kr, Ar-Xe and Kr-Xe mixtures [@myArKr; @myArXe; @myKrXe; @mymix] on graphite it follows that only in the case of Ar-Kr mixture the C-IC transition occurs at submonolayer densities in finite patches of adsorbed phase. One expects that at higher temperatures, thermal excitations may considerably influence the effects observed. This problem is currently under study and the results will be published elsewhere. Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported by ERA under the Grant PIRSES 268498. [99]{} Hwang I.-S., Fang C.-K., Chang S.-H., [Phys. Rev. B]{}, 2011, 83, 134119. . Jaubert M., Glachant A., Bienfait M., Boato G., [Phys. Rev. Lett.]{}, 1981, 46, 1679; . Krim J., Suzanne J., Shechter H., Wang R., Taub H., [Surface Sci.]{}, 1985, 162, 446; . Usachov D., Dobrotvorskii A.M., Varykhalov A., Rader O., [Phys. Rev. B]{}, 2008, 78, 085403;\ . Li L.J., Lu W.J., Zhu X.D., Ling L.S., Qu Z., Sun Y.P., [Europhys. Lett.]{}, 2012, 97, 67005;\ . Nuss J., Pfeiffer S., van Smallen S., Jansen M., [Acta Crystalogr. B]{}, 2010, 66, 27; . Vokhmyanin A.P., Lee S., Jang K.-H., Podlesnyak A.A., Keller L., Prokeš K., Sikolenko V.V., Park J.-G., Skryabin Yu.N., Pirogov A.N., [J. Magn. Magn. Mater.]{}, 2006, 300, e411; . Bak P., [Rep. Prog. Phys.]{}, 1982, 45, 587; . 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--- abstract: \| We consider the nonlinear Dirac equation, also known as the Soler model: $$i{\partial}\sb t\psi=-i\bm\alpha\cdot\bm\nabla\psi +m\beta\psi -(\psi\sp\ast\beta\psi)^k\beta\psi, \quad m>0, \quad \psi(x,t)\in{\mathbb{C}}^{N}, \quad x\in\R^n, \quad k\in{\mathbb{N}}.$$ We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $\omega\to m$, proving that if $k>2/n$, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with $\omega$ sufficiently close to $m$, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh–Schrödinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov–Kolokolov stability criterion. [Résumé.]{} Nous considérons l’équation de Dirac non linéaire, aussi connu comme le modèle de Soler. Nous étudions le spectre ponctuel des linéarisations aux ondes solitaires des petites amplitudes dans la limite $\omega\to m$, et montrons que si $k>2/n$, ensuite une valeur propre positive et une négative sont présents dans le spectre des linéarisations à ces ondes solitaires lorsque $\omega$ est suffisamment proche de $m$, ensuite ces ondes solitaires sont linéairement instable. L’approche est basée sur l’application de théorie de la perturbation de Rayleigh–Schrödinger à la limite non relativiste de l’équation. Les résultats sont en accord formel avec le critère de stabilité de Vakhitov–Kolokolov. author: - \| [Andrew Comech]{}\ [Texas A&M University, College Station, TX 77843, U.S.A.]{}\ [Institute for Information Transmission Problems, Moscow 101447, Russia]{}\ \ [Meijiao Guan, Stephen Gustafson]{}\ [University of British Columbia, Vancouver V6T 1Z2, Canada]{} bibliography: - 'all.bib' title: On linear instability of solitary waves for the nonlinear Dirac equation --- Introduction ============ A natural simplification of the Dirac–Maxwell system [@MR0190520] is the nonlinear Dirac equation, such as the massive Thirring model [@MR0091788] with vector-vector self-interaction and the Soler model [@PhysRevD.1.2766] with scalar-scalar self-interaction (known in dimension $n=1$ as the massive Gross–Neveu model [@PhysRevD.10.3235; @PhysRevD.12.3880]). These models with self-interaction of local type have been receiving a lot of attention in particle physics (see e.g. [@rebbi1984solitons]), as well as in the theory of Bose–Einstein condensates [@MR2542748; @PhysRevLett.104.073603]. There is an enormous body of research devoted to the nonlinear Dirac equation, which we can not cover comprehensively here. The existence of standing waves in the nonlinear Dirac equation was studied in [@PhysRevD.1.2766], [@MR847126], [@MR949625], and [@MR1344729]. The question of stability of solitary waves is of utmost importance: perturbations ensure that we only ever encounter stable configurations. Recent attempts at asymptotic stability of solitary waves in the nonlinear Dirac equation [@MR2259214; @MR2466169; @MR2985264; @MR2924465; @2012arXiv1203.2120C] rely on the fundamental question of spectral stability: > [Consider the Ansatz $\psi(x,t)=(\phi\sb\omega(x)+\rho(x,t))e^{-i\omega t}$, with $\phi\sb\omega(x)e^{-i\omega t}$ a solitary wave solution. Let ${\partial}\sb t\rho=A\sb\omega\rho$ be the linearized equation for $\rho$. Does $A\sb\omega$ have eigenvalues in the right half-plane? ]{} If $\sigma(A\sb\omega)\subset i\R$, we say that the solitary wave $\phi\sb\omega e^{-i\omega t}$ is spectrally stable. Otherwise, we say that the solitary wave is linearly unstable. Let us note that the “spectral stability” $\sigma(A\sb\omega)\subset i\R$ does not guarantee stability. One of the possibilities which may still lead to instability is the presence of eigenvalues of higher algebraic multiplicity. This occurs e.g. at $\lambda=0$ in the case $dQ(\omega)/d\omega=0$ (where $Q(\omega)$ is the charge of $\phi_\omega$– see below) at a particular value of $\omega$, leading to instability of the corresponding solitary wave; see [@MR1995870]. In [@MR2571965], to reflect this situation in the context of the nonlinear Schrödinger equation, linear instability is defined as either the presence of eigenvalues with positive real part or the presence of particular eigenvalues with higher algebraic multiplicity. There is a very clear picture of the spectral stability for nonlinear Schrödinger and Klein–Gordon equations [@VaKo; @MR723756; @MR783974; @MR804458] and general results for abstract Hamiltonian systems with $\mathbf{U}(1)$ symmetry [@MR901236], which stimulated many attempts at spectral stability in the nonlinear Dirac context. We mention the numerical simulations [@PhysRevLett.50.1230] and the analysis of the energy minimization under charge-preserving dilations and similar transformations [@MR592382; @PhysRevD.34.644; @MR848095; @PhysRevE.82.036604]. In spite of this, the question of spectral stability of solitary waves of nonlinear Dirac equation is still completely open. Numerical results [@MR2892774] show that in the 1D Soler model (cubic nonlinearity) all solitary waves are spectrally stable. We also mention the related numerical results in [@MR2217129; @chugunova-thesis]. According to [@VaKo], if $\phi\sb\omega(x)e^{-i\omega t}$ is a family of solitary wave solutions to the nonlinear Schrödinger equation $$i\dot u=-\frac{1}{2m}\Delta u-f({\vert u \vert}^2)u, \qquad u(x,t)\in{\mathbb{C}}, \quad x\in\R^n, \quad n\ge 1,$$ where $f$ is smooth and real-valued, and if $\phi\sb\omega$ have no nodes (such solitary waves are called ground states), then the linearization at the solitary wave corresponding to a particular value of $\omega$ has a positive eigenvalue if and only if at this value of $\omega$ one has $dQ(\omega)/d\omega>0$, where $Q(\omega)={\Vert \phi\sb\omega \Vert}\sb{L^2}^2$ is the charge (or mass) of the solitary wave. The opposite condition, $$\label{vk} \frac{d}{d\omega}Q(\omega)<0,$$ is called the Vakhitov–Kolokolov stability criterion; it ensures the absence of eigenvalues with positive real part. In the case of the nonlinear Dirac equation, the condition gives a less definite answer about the spectral stability. All we know is that at the value of $\omega$ where ${\partial}\sb\omega Q(\omega)$ vanishes, with $Q(\omega)$ being the charge of the solitary wave $\phi\sb\omega e^{-i\omega t}$, two eigenvalues of the linearized equation collide at $\lambda=0$, but we do not know where these eigenvalues are located when ${\partial}\sb\omega Q(\omega)\ne 0$ (see e.g. [@dirac-vk]). Yet, it is natural to expect that the condition remains meaningful in the nonrelativistic limit, as it was suggested in [@PhysRevE.82.036604]. While it is a common practice to obtain solitary wave solutions for relativistic equations as bifurcations from the solitary waves to the equation corresponding to the nonrelativistic limit (see e.g. the review [@MR2434346]), we show that the spectrum of the linearization at a solitary wave could also be learned from the nonrelativistic limit. More precisely, we develop the idea that the family of real eigenvalues of the linearization at a solitary wave of the nonlinear Dirac equation bifurcating from $\lambda=0$ is a deformed family of eigenvalues of the linearization of the corresponding nonlinear Schrödinger equation. As a result, we prove that if the Vakhitov–Kolokolov stability criterion [@VaKo] guarantees linear instability for the NLS, then the same conclusion also holds for solitary waves with $\omega\lesssim m$ in the nonlinear Dirac equation. Let us mention that our results only apply to the solitary wave solutions which we obtain from the solitary waves of the nonlinear Schrödinger equation in the nonrelativistic limit of the nonlinear Dirac. The model and the main results are described in Section \[sect-results\]. The necessary constructions in the context of the nonlinear Schrödinger equation are presented in Section \[sect-nls\]. The existence and asymptotics of solitary waves of the nonlinear Dirac equation is covered in Section \[sect-linearization\]. The main result (Theorem \[main-theorem-dirac-vbk\]) follows from Lemma \[lemma-xx-ff-3\] (existence of solitary wave solutions and their asymptotics) and Proposition \[prop-k3\] (presence of a positive eigenvalue in the spectrum of the linearized operator), which we prove using the Rayleigh–Schrödinger perturbation theory. Main result {#sect-results} =========== We consider the nonlinear Dirac equation $$\label{nld-nd} i{\partial}\sb t\psi=-i\bm\alpha\cdot\bm\nabla\psi +m\beta\psi -f(\psi\sp\ast\beta\psi)\beta\psi, \qquad \psi(x,t)\in\mathbb{C}^N, \qquad x\in\R^n,$$ with $m>0$ and $\psi\sp\ast$ being the Hermitian conjugate of $\psi$. We assume that the nonlinearity $f(s)$ is smooth and real-valued, and that $$\label{f-zero} f(0)=0.$$ Above, $ \bm\alpha\cdot\bm\nabla =\sum\limits\sb{j=1}\sp{n} \alpha\sb j\frac{{\partial}}{{\partial}x\sb j}, $ and the Hermitian matrices $\alpha\sb j$ and $\beta$ are chosen so that $$(-i\bm\alpha\cdot\bm\nabla+\beta m)^2=(-\Delta +m^2)I\sb{N},$$ where $I\sb{N}$ is the $N\times N$ unit matrix. That is, $\alpha_j$ and $\beta$ are to satisfy $$\label{eq:def-Dirac-matrices} \alpha_j \alpha_k + \alpha_k \alpha_j = 2 \delta_{jk} I\sb{N}, \qquad \beta^2=I\sb{N}; \qquad \alpha_j \beta+\beta\alpha_j=0.$$ The generalized massive Gross–Neveu model (the scalar-scalar case with $k>0$ in the terminology of [@PhysRevE.82.036604]) corresponds to the nonlinearity $f(s)={\vert s \vert}^k$. According to the Dirac–Pauli theorem (cf. [@1928RSPSA.117..610D; @vanderwaerden-1932; @MR1508031] and [@thaller Lemma 2.25]), the particular choice of the matrices $\alpha_j$ and $\beta$ does not matter: \[lemma-pauli\] Let $n\in{\mathbb{N}}$. For any sets of Dirac matrices $\alpha_j$ $\beta$ and $\tilde\alpha_j$, $\tilde\beta$ of the same dimension $N$, with $1\le j\le n$, there is a unitary matrix $S$ such that $$\label{pauli-odd} \tilde\alpha_j=S^{-1} \alpha_j S, \quad 1\le j\le n, \qquad \tilde\beta=S^{-1}\beta S$$ if $n$ is odd, and such that $$\label{pauli-even} \tilde\alpha_j=\sigma S^{-1}\tilde\alpha_j S, \quad 1\le j\le n, \qquad \tilde\beta=\sigma S^{-1}\beta S, \qquad \sigma=\pm 1,$$ if $n$ is even. For more details, see [@dirac-spectrum]. Lemma \[lemma-pauli\] allows one, by a simple change of variable, to transform the nonlinear Dirac equation, changing the set of Dirac matrices. Thus, when studying the spectral stability, we can choose the Dirac matrices at our convenience. We use the standard Pauli matrices, $$\label{def-pauli-matrices} \sigma\sb 1=\left(\begin{matrix}0&\,\,1\\1&\,\,0\end{matrix}\right), \qquad \sigma\sb 2=\left(\begin{matrix}0&-i\\i&0\end{matrix}\right), \qquad \sigma\sb 3 =\left(\begin{matrix}1&0\\0&-1\end{matrix}\right),$$ to make the following choice: $$\label{dm-1} n=1: \quad N=2, \quad \alpha = -\sigma_2;$$ $$\label{dm-2} n=2: \quad N=2, \quad \alpha\sb j = \sigma_j, \quad 1\le j\le 2;$$ $$\label{dm-3} n=3: \quad N=4, \quad \alpha\sb j =\left(\begin{matrix}0&\sigma\sb j\\ \sigma\sb j&0\end{matrix}\right), \quad 1\le j\le 3;$$ and in all these cases we choose $$\label{dm-beta} \beta =\left(\begin{matrix}I\sb{N/2}&0\\0&-I\sb{N/2}\end{matrix}\right).$$ If $n$ is even, Lemma \[lemma-pauli\] may only allow us to transform the equation with a particular set of the Dirac matrices to the set of the Dirac matrices as in , , , and , but with the opposite signs (this corresponds to $\sigma=-1$ in ). In this case, the signs of $\alpha_j$ are flipped by taking the spatial reflections, while $\beta$ being opposite to corresponds to considering the nonrelativistic limit $\omega\to -m$, with appropriate changes to Theorem \[main-theorem-dirac-vbk\]. For a large class of nonlinearities $f(s)$, there are solitary wave solutions of the form $$\label{sw} \psi(x,t)=\phi\sb\omega(x)e^{-i\omega t}, \qquad \phi\sb\omega \in H^1(\R,{\mathbb{C}}^{N}), \qquad {\vert \omega \vert}<m.$$ In dimension $n=1$, one can take $$\label{sol-forms-1} \phi\sb\omega(x) =\begin{bmatrix}v(x,\omega)\\u(x,\omega)\end{bmatrix},$$ with $v(x,\omega)$ positive and even and $u(x,\omega)$ real-valued and odd; under these conditions, the solitary wave $\phi_\omega(x)$ is unique (see Section \[sect-solitary-waves-1d\] for details). For $n = 2$ and $n=3$, respectively, $$\label{sol-forms-23} \phi\sb\omega(x) =\begin{bmatrix}v(r,\omega)\\ i e^{i\phi} u(r,\omega)\end{bmatrix}, \qquad \phi\sb\omega(x) = \begin{bmatrix} v(r,\omega) \begin{pmatrix} 1\\0 \end{pmatrix} \\ i u(r,\omega) \begin{pmatrix} \cos\theta \\ e^{i\phi}\sin\theta \end{pmatrix} \end{bmatrix},$$ where $v(r,\omega)$ and $u(r, \omega)$ are real-valued, radially-symmetric functions; $(r,\phi)$ are standard polar coordinates in $\R^2$, and $(r,\theta,\phi)$ are standard spherical coordinates in $\R^3$. The existence of solitary waves of this form is proved, for example, in [@MR847126]. The particular solutions we consider here, however, are those constructed for $\omega$ close to $m$, as outlined in Section \[sect-sw-hd\], from the nonrelativistic limit $\omega \to m$. Due to the $\mathbf{U}(1)$-invariance, for solutions to the value of the charge functional $$Q(\psi)=\int\sb{\R^n}{\vert \psi(x,t) \vert}^2\,dx$$ is formally conserved. For brevity, we also denote by $Q(\omega)$ the charge of the solitary wave $\phi\sb\omega(x)e^{-i\omega t}$: $$\label{def-q-omega} Q(\omega)=\int\sb{\R^n}{\vert \phi\sb\omega(x) \vert}^2\,dx.$$ We are interested in the spectrum of linearization of the nonlinear Dirac equation at a solitary wave solution . \[main-theorem-dirac-vbk\] Let $n\le 3$. Assume that $f(s)=s^k$, where $k\in{\mathbb{N}}$ satisfies $k > 2/n$ (and $k < 2$ for $n=3$). Then there is $\omega\sb 1<m$ such that the solitary wave solutions $\phi\sb\omega e^{-i\omega t}$ to described above are linearly unstable for $\omega\in(\omega\sb 1,m)$. More precisely, let $A\sb\omega$ be the linearization of the nonlinear Dirac equation at a solitary wave $\phi\sb\omega(x)e^{-i\omega t}$. Then for $\omega\in(\omega_1,m)$ there are eigenvalues $$\pm\lambda\sb\omega\in\sigma\sb{p}(A\sb\omega), \qquad \lambda\sb\omega>0, \qquad \lambda\sb\omega=O(m-\omega).$$ See Figure \[fig-spectrum\]. \[remark-nonlinear\] The existence of an eigenvalue with positive real part in the linearization at a particular solitary wave generally implies the dynamic, or nonlinear, instability of this wave. We expect that this could be proved following the argument of [@MR2916078] given in the context of the nonlinear Schrödinger equation. (0,180)(0,-90)=cmr10 at 10pt (6,58)[$i(m+{\omega})$]{} (5,19)[$i(m-{\omega})$]{} ( 23,-11)[$\lambda\sb\omega$]{} (-36,-11)[$-\lambda\sb\omega$]{} (-20,0) ( 20,0) ( 0,0) (-60, 0)[(1,0)[120]{}]{} ( 0,-80)[(0,1)[160]{}]{} ( 1,-80)[(0,1)[60]{}]{} (-1,-80)[(0,1)[20]{}]{} ( 1,80)[(0,-1)[20]{}]{} (-1,80)[(0,-1)[60]{}]{} Theorem \[main-theorem-dirac-vbk\] extends easily to nonlinearities $f\in C\sp 2(\R)$ of the form $$f(s)=a s^k+O(s^{k+1}), \qquad a>0.$$ \[remark-enough\] In Theorem \[main-theorem-dirac-vbk\], the value $\omega\sb 1<m$ could be taken to be the smallest point such that there is a $C^1$ family of solitary waves $\omega\mapsto \phi\sb\omega$ for $\omega\in(\omega\sb 1,m)$ and moreover ${\partial}\sb\omega Q(\omega)$ does not vanish on $(\omega\sb 1,m)$. Indeed, by [@dirac-vk], the positive and negative eigenvalues remain trapped on the real axis, not being able to collide at $\lambda=0$ for $\omega\in (\omega\sb 1,m)$ as long as ${\partial}\sb\omega Q(\omega)$ does not vanish on this interval. These eigenvalues cannot leave into the complex plane, either, since they are simple, while the spectrum of the operator is symmetric with respect to the real and imaginary axes. If the family of solitary waves $\omega\mapsto\phi\sb\omega$ is defined for $\omega\in(\omega\sb 0,m)$ with $\omega\sb 0<\omega\sb 1$ and ${\partial}\sb\omega Q(\omega)$ vanishes at $\omega\sb 1$, we do not know what happens for $\omega\lesssim\omega\sb 1$. If ${\partial}\sb\omega Q(\omega)$ changes the sign at $\omega\sb 1$, then, generically, either the pair of real eigenvalues, having collided at $\lambda=0$ when $\omega=\omega\sb 1$, turn into a pair of purely imaginary eigenvalues (linear instability disappears), or instead two purely imaginary eigenvalues, having met at $\lambda=0$, turn into the second pair of real eigenvalues (linear instability persists). If ${\partial}\sb\omega Q(\omega)$ vanishes at $\omega\sb 1$ but does not change the sign, then generically the eigenvalues touch and separate again, remaining on $\R\sb{+}$ and $\R\sb{-}$. More details are in [@dirac-vk]. Theorem \[main-theorem-dirac-vbk\] is in formal agreement with the Vakhitov–Kolokolov stability criterion [@VaKo], since for $\omega\lesssim m$ one has $Q'(\omega)>0$ for $k>\frac{2}{n}$. Let us mention that the sign of the stability criterion, $Q'(\omega)<0$, differs from [@VaKo] because of their writing the solitary waves in the form $\varphi(x) e^{+i\omega t}$. It has been shown that in the 1D case with $k=1$, the small amplitude solitary waves are spectrally stable [@dirac-spectrum]. We expect that in the 1D case with $k=2$ (“quintic nonlinearity”) the small solitary wave solutions of the nonlinear Dirac equation in 1D are spectrally stable. For the corresponding nonlinear Schrödinger equation (quintic nonlinearity in 1D), the charge is constant, thus the zero eigenvalue of a linearized operator is always of higher algebraic multiplicity. For the Dirac equation, this degeneracy is “resolved”: using the expression for the charge $Q(\omega)$ from [@PhysRevE.82.036604 Section 2A], one can see that the charge is now a decaying function for $\omega\lesssim m$ (with nonzero limit as $\omega\to m$), suggesting that there are two purely imaginary eigenvalues $\pm\lambda\sb\omega$ in the spectrum of $A\sb\omega$, with $\lambda\sb\omega=o(m-\omega)$, but no eigenvalues with nonzero real part. \[remark-3d\] Let us notice that in the 3D case for the cubic nonlinearity $f(s)=s$ (this is the original Soler model from [@PhysRevD.1.2766]), based on the numerical evidence from [@PhysRevD.1.2766; @PhysRevLett.50.1230], the charge $Q(\omega)$ has a local minimum at $\omega\sb 1\approx 0.936 m$, suggesting that the solitary waves with $\omega\sb 1<\omega<1$ are linearly unstable, but then at $\omega=\omega\sb 1$ the real eigenvalues collide at $\lambda=0$, and there are no nonzero real eigenvalues in the spectrum for $\omega\lesssim \omega\sb 1$. Incidentally, this agrees with the “dilation-stability” results of [@MR848095] (one studies whether the energy is minimized or not under the charge-preserving dilation transformations). We can not rule out the possibility that the eigenvalues with nonzero real part could bifurcate directly from the imaginary axis into the complex plane. Such a mechanism is absent for the nonlinear Schrödinger equation linearized at a ground state, for which the point eigenvalues always remain on the real or imaginary axes. At present, though, we do not have examples of such bifurcations in the context of nonlinear Dirac equation. \[remark-bl\] For $n=3$, we only consider the case $k=1$, and we do not consider dimensions $n>3$. This is because of the fact that the equation $-\Delta u+u={\vert u \vert}^{2k}u$ in $\R^n$ has nontrivial solutions in $H^1(\R^n)$ if and only if $0<k<2/(n-2)$, as follows from the virial identities; see [@MR0192184] and [@MR695535 Example 1]. This is why our method does not allow us to construct solitary wave solutions to the nonlinear Dirac equation in $\R^n$, $n\ge 3$, with $k\ge 2/(n-2)$. \[remark-noninteger\] We consider here only integer powers $k$ (and only dimensions $n=1$, $2$, and $3$), being physically the most important cases. Mathematically, this merely avoids some minor technical complications associated with a non-smooth nonlinearity, and the instability argument can be extended to handle the corresponding equation with $f(s) = \|s\|^k$, in any dimension $n\ge 1$, under the condition $k > 2/n$. (The restriction $k<2/(n-2)$ is needed so that there are nontrivial solitary waves in NLS; see the previous remark.) Nonlinear Schrödinger and its solitary waves {#sect-nls} ============================================ We are going to use the fact that the nonrelativistic limit of the nonlinear Dirac equation yields the nonlinear Schrödinger equation, $$\label{nls} i{\partial}\sb t\psi=-\frac{1}{2m} \Delta \psi - \|\psi\|^{2k} \psi, \quad \psi(x,t)\in{\mathbb{C}}, \quad x\in\R^n, \quad k>0, \quad n\in{\mathbb{N}}.$$ Solitary waves -------------- The properties of solitary wave solutions $$\psi(x,t)=\phi\sb\omega(x)e^{-i\omega t}, \qquad \phi\sb\omega\in H^1(\R^n),$$ with the amplitude $\phi_\omega(x)$ satisfying the stationary equation $$\label{nls-soleq} -\frac{1}{2m} \Delta \phi(x) -\|\phi\|^{2k} \phi = \omega \phi, \qquad x\in\R^n,$$ are well-known [@MR0454365; @MR695535]. For any $k>0$ when $n\le 2$ and for $0<k<2/(n-2)$ when $n\ge 3$, for each $\omega \in (-\infty,0)$, there is a unique positive, radially symmetric solution $$\phi_{\omega}(x) = \phi_{\omega}(\|x\|) > 0,$$ which decays exponentially. This family of solitary waves, known as the [ground states]{}, is generated by rescaling a single amplitude function: $$\label{nls-scaling} \phi_\omega(x) = {\vert \omega \vert}^{\frac{1}{2k}} F (\sqrt{2m{\vert \omega \vert}} x), \qquad \omega<0,$$ where $F(x) = F(\|x\|) > 0$ solves $$\label{F-such-that} -\Delta F - F^{2k+1} = -F, \qquad x\in\R^n.$$ In one space dimension ($n=1$), for $k>0$, $F(x)$ is given by the explicit formula $$F(x) = \left( \frac{k+1}{\cosh^2{k x}} \right)^{\frac{1}{2k}}.$$ Linearization at a solitary wave -------------------------------- To derive the linearization of the nonlinear Schrödinger equation at a solitary wave $\psi(x,t)=\phi\sb\omega(x)e^{-i\omega t}$, we use the Ansatz $$\psi(x,t)=(\phi\sb\omega(x)+\rho(x,t))e^{-i\omega t}, \qquad \rho(x,t)\in{\mathbb{C}}, \qquad x\in\R^n,$$ and arrive at the linearized equation $$\label{nls-lin} {\partial}\sb t \bm{\uprho} = \eub{j}\eub{l}(\omega) \bm{\uprho}, \qquad \bm{\uprho}(x,t)=\begin{bmatrix}{\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho(x,t)\\{\mathop{\rm{I\hskip -1pt m}}\nolimits}\rho(x,t)\end{bmatrix},$$ where $$\label{def-jl} \eub{j}= \begin{bmatrix}0&1\\-1&0\end{bmatrix}, \qquad \eub{l}= \begin{bmatrix}\eur{l}\sb{-}&0\\0&\eur{l}\sb{+}\end{bmatrix},$$ with $\eur{l}\sb\pm$ self-adjoint Schrödinger operators $$\label{def-lpm} \eur{l}\sb{-}(\omega)=-\frac{1}{2m}\Delta -\|\phi\sb\omega\|^{2k}-\omega, \qquad \eur{l}\sb{+}(\omega)=\eur{l}\sb{-}(\omega) -2k\|\phi\sb\omega\|^{2k}.$$ Since the solitary wave amplitudes $\phi_\omega(x) = \phi_\omega(\|x\|)$ we take here are radially symmetric, we may consider the operators $$\label{def-lr} \eub{l}_{rad}, \,\; \eur{l}_{\pm,rad} \; := \; \eub{l}, \, \eur{l}_{\pm} \mbox{ restricted to radially symmetric functions}.$$ The linear stability theory of NLS ground states is well understood, and can be summarized, in terms of their charge , as follows: \[lemma-vk\] For the linearization at a ground state solitary wave $\phi\sb\omega(x)e^{-i\omega t}$, there are real nonzero eigenvalues $\pm\lambda\in\sigma\sb{d}(\eub{j}\eub{l})$, $\lambda>0$, if and only if $\frac{d}{d\omega}Q(\omega)>0$ at this value of $\omega$. If so, then $\pm\lambda\in\sigma\sb{d} (\eub{j}\eub{l}_{rad})$ are simple eigenvalues, and moreover $\ker \eur{l}_{+,rad} = \{ 0 \}$. Using , we compute: $$\label{qp} Q(\omega) =\int\sb{\R^n} \|\phi\sb\omega(x)\|^2\,dx ={\vert \omega \vert}^{\frac 1 k} \int\sb{\R^n} F^2(\sqrt{2m{\vert \omega \vert}} x) \,dx = C {\vert \omega \vert}^{\frac 1 k - \frac n 2}, \quad \omega<0,$$ where $C=\int\sb{\R^n} F^2(\sqrt{2m}y) \,dy>0$. We see from that for $\omega<0$ one has $Q'(\omega)<0$ for $k < 2/n$, $Q'(\omega)=0$ for $k = 2/n$, and $Q'(\omega)>0$ for $k > 2/n$. Thus: \[lemma-nls-k\] Let $n\in{\mathbb{N}}$. If $k > n/2$ (and $k < 2/(n-2)$ if $n \geq 3$), then $\sigma\sb{p}(\eub{j}\eub{l}_{rad})\ni\{\pm\lambda\}$, for some $\lambda>0$, and in particular the NLS ground states are linearly unstable. Nonlinear Dirac and its solitary waves {#sect-linearization} ====================================== Solitary waves are solutions to of the form $$\psi(x,t)=\phi\sb\omega(x)e^{-i\omega t}{{\rm ;}\ }\phi\sb\omega\in H^1(\R,{\mathbb{C}}^{N}),\ \omega\in\R$$ and as such the amplitude $\phi_\omega(x)$ must satisfy $$\label{nld-sol} \omega \phi_\omega =-i\bm\alpha\cdot\bm\nabla\psi +m\beta\psi -f(\psi\sp\ast\beta\psi)\beta\psi, \qquad x\in\R^n.$$ Solitary waves in one dimension {#sect-solitary-waves-1d} ------------------------------- We first give a simple demonstration of the existence and uniqueness of solitary waves in one dimension, following the article [@MR2892774], and allowing for more general nonlinearities $f(s)$. \[lemma-existence-nld-1d\] Let $f(0)=0$. Denote $g(s)=m-f(s)$, and let $G(s)$ be the antiderivative of $g(s)$ such that $G(0)=0$. Assume that there is $\omega\sb 0<m$ such that for given $\omega\in(\omega\sb 0,m)$ there exists $\varGamma\sb\omega>0$ such that $$\label{def-Xi} \omega\varGamma\sb\omega=G(\varGamma\sb\omega), \quad \omega\ne g(\varGamma\sb\omega), \quad\mbox{and}\quad \omega s<G(s)\quad{\rm for} \ s\in(0,\varGamma\sb\omega).$$ Then there is a solitary wave solution $\psi(x,t)=\phi\sb\omega(x)e^{-i\omega t}$ to , where $$\label{psi-v-u} \phi\sb\omega(x) =\begin{bmatrix}v(x,\omega)\\u(x,\omega)\end{bmatrix}, $$ with both $v$ and $u$ real-valued, belonging to $H^1(\R)$ as functions of $x$, $v$ being even and $u$ odd. More precisely, for $x\in\R$ and $\omega\in(\omega\sb 0,m)$, let us define $\mathscr{X}(x,\omega)$ and $\mathscr{Y}(x,\omega)$ by $$\label{def-xi-eta} \mathscr{X}=v^2-u^2,\qquad \mathscr{Y}=v u.$$ Then $\mathscr{X}(x,\omega)$ is the unique positive symmetric solution to $${\partial}\sb x^2\mathscr{X} =-{\partial}\sb\mathscr{X}(-2G(\mathscr{X})^2+2\omega^2\mathscr{X}^2), \qquad \lim\sb{x\to\pm\infty}\mathscr{X}(x,\omega)=0, $$ and $\mathscr{Y}(x,\omega)=-\frac{1}{4\omega}{\partial}\sb x\mathscr{X}(x,\omega)$. This solution satisfies $\mathscr{X}(0,\omega)=\varGamma\sb\omega$. Substituting $\phi\sb\omega(x)e^{-i\omega t}$, with $\phi\sb\omega$ from , into (\[nld-sol\]), we obtain: $$\label{omega-v-u} \left\{ \begin{array}{ll} \omega v={\partial}\sb x u+g({\vert v \vert}^2-{\vert u \vert}^2)v, \\ \omega u=-{\partial}\sb x v-g({\vert v \vert}^2-{\vert u \vert}^2)u. \end{array} \right.$$ Since we assume that both $v$ and $u$ are real-valued, we may rewrite (\[omega-v-u\]) as the following Hamiltonian system: $$\label{stat-eqn} \left\{ \begin{array}{ll} {\partial}\sb x u=\omega v-g(v^2-u^2)v={\partial}\sb{v} h(v,u), \\ -{\partial}\sb x v=\omega u+g(v^2-u^2)u={\partial}\sb{u} h(v,u), \end{array} \right.$$ where the Hamiltonian $h(v,u)$ is given by $$\label{def-h} h(v,u)=\frac{\omega}{2}(v^2+u^2)-\frac{1}{2}G(v^2-u^2).$$ The solitary wave with a particular $\omega\in(\omega\sb 0,m)$ corresponds to a trajectory of this Hamiltonian system such that $$\lim\sb{x\to\pm\infty}v(x,\omega) =\lim\sb{x\to\pm\infty}u(x,\omega)=0,$$ hence $\lim\sb{x\to\pm\infty}\mathscr{X}=0$. Since $G(s)$ satisfies $G(0)=0$, we conclude that $ h(v(x),u(x))\equiv 0, $ which leads to $$\label{phi-phi-g} \omega(v^2+u^2)=G(v^2-u^2).$$ We conclude from that solitary waves may only correspond to ${\vert \omega \vert}<m$, $\omega\ne 0$. The functions $\mathscr{X}(x,\omega)$ and $\mathscr{Y}(x,\omega)$ introduced in are to solve $$\label{xi-eta-system} \left\{ \begin{array}{ll} {\partial}\sb x\mathscr{X}=-4\omega\mathscr{Y}, \\ {\partial}\sb x\mathscr{Y}=-(v^2+u^2)g(\mathscr{X})+\omega\mathscr{X} =-\frac{1}{\omega}G(\mathscr{X})g(\mathscr{X})+\omega\mathscr{X}, \end{array} \right.$$ and to have the asymptotic behavior $\lim\sb{{\vert x \vert}\to\infty}\mathscr{X}(x)=0$, $\lim\sb{{\vert x \vert}\to\infty}\mathscr{Y}(x)=0$. In the second equation in (\[xi-eta-system\]), we used the relation (\[phi-phi-g\]). The system (\[xi-eta-system\]) can be written as the following equation on $\mathscr{X}$: $$\label{xi-p-p} {\partial}\sb x^2\mathscr{X} =-{\partial}\sb\mathscr{X}(-2 G(\mathscr{X})^2+2\omega^2\mathscr{X}^2) =4 \big(G(\mathscr{X})g(\mathscr{X})-\omega^2\mathscr{X}\big).$$ This equation describes a particle in the potential $-2 G(s)^2+2\omega^2 s^2$. The condition is needed so that $s=\varGamma\sb\omega$ is the turning point for the zero energy trajectory in this potential. The existence of a positive solution $\mathscr{X}(x,\omega)$ follows. This solution is unique up to a translation, and it will be made symmetric in $x$ by requiring $\mathscr{X}(0,\omega)=\varGamma\sb\omega$. Solitary waves in the nonrelativistic limit {#sect-sw-hd} ------------------------------------------- In dimensions $n=1$, $2$ and $3$, we consider solitary wave amplitudes $\phi_\omega(x)$ of the forms given in and . Substituting these into the nonlinear Dirac equation , a straightforward calculation results in the system $$\label{soleq-nd} \left\{ \begin{array}{l} \omega v = {\partial}_r u + \frac{n-1}{r} u + m v-f(v^2-u^2)v \\ \omega u = -{\partial}_r v - m u+f(v^2-u^2)u \end{array} \right.$$ for the pair of real-valued functions $v = v(r,\omega)$, $u = u(r,\omega)$. Notice that equation includes the $1$-dimensional case if we interpret $r=x$. Recalling that $f(s)=s^{k}$, we arrive at $$(\omega-m) v = {\partial}_r u + \frac{n-1}{r} u - f v, \qquad (\omega+m) u = - {\partial}_r v + f u,$$ with $$f:=(v^2-u^2)^{k}.$$ To consider the nonrelativistic limit, we set $$m^2 - \omega^2 = \epsilon^2, \quad 0 < \epsilon \ll m,$$ and rescale $v(r,\omega)$ and $u(r,\omega)$ as follows: $$v(r,\omega) =\epsilon^{\frac 1 k} V(\epsilon r,\epsilon), \qquad u(r,\omega) =\epsilon^{1+\frac 1 k} U(\epsilon r,\epsilon).$$ Then $V$, $U$ should satisfy $$\left\{ \begin{array}{l} (\omega-m)\epsilon^{\frac 1 k} V =\epsilon^{2+\frac 1 k} ({\partial}\sb RU + \frac{n-1}{R} U) - \epsilon^{1/k} f V \\ (\omega+m)\epsilon^{1+\frac 1 k} U =-\epsilon^{1+\frac 1 k} {\partial}\sb R V + \epsilon^{1 + \frac 1 k} f U \end{array} \right.,$$ where $R = \epsilon r$ denotes the “rescaled variable". Using $\omega = m - \frac{1}{2m} \epsilon^2 + O(\epsilon^4)$, and taking into account that $$f = (\epsilon^{\frac{2}{k}} V^2 - \epsilon^{2+\frac{2}{k}} U^2 )^{k} = \epsilon^2 V^{2k} + \epsilon^4 O(U^{2k} + V^{2k}),$$ we re-write the system as $$\label{zero-is-phi1} \left\{ \begin{array}{l} (-\frac{1}{2m} + O(\epsilon^2)) V = {\partial}\sb R U + \frac{n-1}{R} U - V^{2k+1} + \epsilon^2 O((U^{2k} + V^{2k})\|V\|) \\ (2m + O(\epsilon^2)) U = -{\partial}\sb R V + \epsilon^2 V^{2k} U + \epsilon^4 O((U^{2k} + V^{2k})\|U\|) \end{array} \right..$$ The rescaled system has an obvious limit as $\epsilon\to 0$. Formally (for now) setting $$\hat V(r)=\lim\sb{\epsilon\to 0}V(r,\epsilon), \qquad \hat U(r)=\lim\sb{\epsilon\to 0}U(r,\epsilon),$$ we arrive at $$\label{def-hat-phi} -\frac{1}{2m} \hat V = {\partial}\sb R \hat U + \frac{n-1}{R} \hat U - \hat V^{2k+1}, \qquad 2m \hat U = -{\partial}\sb R\hat V.$$ Substituting the second equation into the first one yields $$-\frac{1}{2m} ({\partial}\sb R^2+\frac{n-1}{R}{\partial}\sb R)\hat V - \hat V^{2k+1} = -\frac{1}{2m} \hat V, \qquad \hat{U} = - \frac{1}{2m} {\partial}\sb R \hat V.$$ This equation for $\hat V(r)$ is precisely the equation for NLS solitary wave amplitudes $\phi_\omega$ with $\omega = -\frac{1}{2m}$. Thus we let $\hat V(r)$ be the (unique) NLS ground state: $$\label{Vhatdef} \hat V(r) := (2m)^{-\frac{1}{2k}} F(r), \qquad \hat U(r) := -(2m)^{-\frac{1}{2k}-1} F'(r),$$ with $F(r)$ the unique positive spherically symmetric solution to . We can use this nonrelativistic limit to construct nonlinear Dirac solitary waves for $\epsilon^2 = m^2 - \omega^2 \ll m^2$: \[lemma-xx-ff-3\] There is $\omega\sb 0 < m$ such that for $\omega = \sqrt{m^2-\epsilon^2} \in (\omega\sb 0,m)$, there are solutions of of the form $$v(r,\omega) = \epsilon^{\frac 1 k} \left[ \hat V(\epsilon r) + \tilde V(\epsilon r) \right], \quad u(r,\omega)=\epsilon^{1+\frac 1 k} \left[ \hat U (\epsilon r) + \tilde U(\epsilon r) \right],$$ $$\\| \tilde V \\|_{H^2} + \\| \tilde U \\|_{H^2} = O(\epsilon^2).$$ In the one-dimensional case, since the solitary waves are unique (up to symmetries), it follows that these asymptotics describe [every]{} solitary wave for $\omega$ close to $m$, or equivalently every small amplitude solitary wave. The argument parallels that of [@2008arXiv0812.2273G], where the (more general) nonlinearity $f(s) = {\vert s \vert}^{\theta}$, $0 < \theta < 2$ is considered for $n=3$. Writing $$V(R,\epsilon) = \hat{V}(R) + \tilde{V}(R,\epsilon), \qquad U(R,\epsilon) = \hat{U}(R) + \tilde{U}(R,\epsilon),$$ and subtracting equations and , we arrive at $$\begin{split} &-\frac{1}{2m}\tilde V + O(\epsilon^2) V = ({\partial}_R + \frac{n-1}{R}) \tilde{U} - (2k+1) \hat V^{2k} \tilde{V} \\ &\qquad\qquad + O(\|\hat V\|^{2k+1} \tilde V^2 + \|\tilde V\|^{2k+1}) + \epsilon^2 O((U^{2k}+V^{2k})\|V\|), \end{split}$$ $$2m \tilde U + O(\epsilon^2) U = - {\partial}\sb R\tilde V + \epsilon^2 V^{2k} U + \epsilon^4 O((U^{2k}+V^{2k})\|U\|),$$ which, setting $$\Xi(R,\epsilon) := \left[ \begin{array}{c} \tilde V (R,\epsilon) \\ \tilde U (R,\epsilon) \end{array} \right],$$ we may re-write as $$\label{HH} \eub{H} \Xi = O_{H^1} (\epsilon^2) + O(\epsilon^2 \|\Xi\| + \|\Xi\|^2 + \|\Xi\|^{2k+1}),$$ where $$\label{def-h-large} \eub{H} := \left[ \begin{array}{cc} -\frac{1}{2m} + (2k+1) \hat V^{2k} &\ \ - ({\partial}_R + \frac{n-1}{R}) \\ {\partial}_R & 2m \end{array} \right].$$ Since $$\left[ \begin{array}{c} \xi \\ \eta \end{array} \right] \in \ker \eub{H} \;\; \iff \;\; \xi \in \ker\eur{l}_+, \;\; \eta = -\frac{1}{2m} {\partial}\sb R \xi,$$ and $\ker\eur{l}_{+,rad} = \{ 0 \}$ (Cf. definitions , ), we see that $\ker \eub{H} = \{ 0 \}$. It then follows from the fact that $\eur{l}_{+,rad}^{-1}$ is bounded from $L^2_r(\R^n,{\mathbb{C}})$ to $H^2(\R^n,{\mathbb{C}})$, that $\eub{H}^{-1}$ is bounded from $H^1_r(\R^n,{\mathbb{C}}^2)$ to $H^2(\R^n,{\mathbb{C}}^2)$ (here $L^2_r$, $H^1_r$ are the corresponding subspaces of spherically symmetric functions). Hence $$\label{Xi} \Xi = \eub{H}^{-1} \left\{ O_{H^1} (\epsilon^2) + O(\epsilon^2 \|\Xi\| + \|\Xi\|^2 + \|\Xi\|^{2k+1}) \right\} ,$$ and since $\\| \Xi \\|_{L^\infty} \leq C \\| \Xi \\|_{H^2}$ (recall that $n \leq 3$), we arrive easily at $$\begin{split} \\| R.H.S. \eqref{Xi} \\|_{H^2} &\leq C \\| O_{H^1} (\epsilon^2) + O(\epsilon^2 \|\Xi\| + \|\Xi\|^2 + \|\Xi\|^{2k+1} ) \\|_{H^1} \\ &\leq C \left\{ \epsilon^2 + \epsilon \\|\Xi\\|_{H^2} + \\| \Xi \\|_{H^2}^2 + \\| \Xi \\|_{H^2}^{2k+1} \right\}, \end{split}$$ and so we see that for small enough $\epsilon$, the map on the r.h.s. of maps the ball of radius $\epsilon$ in $H^2$ into itself. A similar estimate shows that this map is a contraction, and hence has a unique fixed point $\Xi$ in this ball. Finally, we see from that $\\| \Xi \\|_{H^2} = O(\epsilon^2)$. Linear instability of small amplitude solitary waves {#sect-instability} ==================================================== Our first observation here is that on spinor fields of the form $$\psi(x,t) =\begin{bmatrix} \Psi_1(x,t) \\ \Psi_2(x,t) \end{bmatrix}, \qquad \psi(x,t) =\begin{bmatrix} \Psi_1(r,t)\\ i e^{i\phi} \Psi_2(r,t)\end{bmatrix},$$ $$\psi(x,t) = \begin{bmatrix} \Psi_1(r,t) \begin{pmatrix} 1\\0 \end{pmatrix} \\ i \Psi_2(r,t) \begin{pmatrix} \cos\theta \\ e^{i\phi}\sin\theta \end{pmatrix} \end{bmatrix}$$ in dimensions $n=1$, $n=2$, and $n=3$ respectively, the nonlinear Dirac equation reduces to the system $$\label{reduced} \left\{ \begin{array}{l} i {\partial}_t \Psi_1 = ({\partial}_r + \frac{n-1}{r}) \Psi_2 + m \Psi_1-f(\|\Psi_1\|^2-\|\Psi_2\|^2) \Psi_1 \; , \\ i {\partial}_t \Psi_2 = -{\partial}_r \Psi_1 - m\Psi_2+f(\|\Psi_1\|^2-\|\Psi_2\|^2) \Psi_2 \end{array} \right.$$ (with the convention $r=x$ for $n=1$). The solitary waves considered in and lie in this class of fields, corresponding to $$\label{sol-forms-all} \Psi_1(r,t)=v(r,\omega)e^{-i\omega t}, \qquad \Psi_2(r,t)=u(r,\omega)e^{-i\omega t}.$$ To prove the instability of these solitary waves, it suffices to show that they are unstable as solutions of . Linearization at a solitary wave -------------------------------- To derive the linearization of system at a solitary wave we consider solutions in the form of the Ansatz $$\left[ \begin{array}{c} \Psi_1(r,t) \\ \Psi_2(r,t) \end{array} \right] = \left( \left[ \begin{array}{c} v(r,\omega) \\ u(r,\omega) \end{array} \right] + \left[ \begin{array}{c} \rho_1(r,t) \\ \rho_2(r,t) \end{array} \right] \right) e^{-i\omega t},$$ where $$\rho(r,t) := \left[ \begin{array}{c} \rho_1(r,t) \\ \rho_2(r,t) \end{array} \right] \in {\mathbb{C}}^2.$$ Inserting this Ansatz into system with $f(s)=s^k$ and recalling that $u(r,\omega)$ and $v(r,\omega)$ are real-valued, we find that the linearized system for $\rho$ is $$\begin{aligned} \label{nld-1d-lin} i {\partial}_t \rho = \begin{bmatrix} m- \omega - (v^2-u^2)^k & {\partial}_r + \frac{n-1}{r} \\ -{\partial}_r & -m-\omega+ (v^2-u^2)^k \end{bmatrix} \rho \nonumber \\ \qquad - 2k(v^2-u^2)^{k-1} \begin{bmatrix} v^2 &-u v \\ -u v &u^2 \end{bmatrix} {\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho.\end{aligned}$$ We note that the above equation is $\R$-linear but not ${\mathbb{C}}$-linear, due to the presence of the term with ${\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho=\begin{bmatrix}{\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho_1\\{\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho_2\end{bmatrix}$. We rewrite equation in terms of ${\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho\in\R^2$ and ${\mathop{\rm{I\hskip -1pt m}}\nolimits}\rho\in\R^2$: $$\label{nld-1d-lin-2} {\partial}\sb t \begin{bmatrix}{\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho\\{\mathop{\rm{I\hskip -1pt m}}\nolimits}\rho\end{bmatrix} = {\eub{J}}{\eub{L}}(\omega) \begin{bmatrix}{\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho\\{\mathop{\rm{I\hskip -1pt m}}\nolimits}\rho \end{bmatrix}$$ where ${\eub{J}}$ corresponds to $1/i$: $${\eub{J}}= \begin{bmatrix} 0&I\sb 2\\-I\sb 2&0 \end{bmatrix},$$ and the $4 \times 4$ matrix operator ${\eub{L}}(\omega)$ is defined by $${\eub{L}}(\omega) = \begin{bmatrix} {\eur{L}}_{+}(\omega) & 0 \\ 0 & {\eur{L}}_{-}(\omega) \end{bmatrix} ,$$ where, writing $$f := f(\phi_\omega^* \beta \phi_\omega) = (v^2(r,\omega) - u^2(r,\omega))^k, \quad f' := f'(\phi_\omega^* \beta \phi_\omega),$$ we have $$\label{defD-} {\eur{L}}_{-}(\omega) = \begin{bmatrix} m-\omega-f& {\partial}_r + \frac{n-1}{r} \\ -{\partial}_r & -m-\omega+f \end{bmatrix} $$ and $$\label{defD+} \begin{split} &{\eur{L}}_{+}(\omega) = {\eur{L}}_{-}(\omega) -2 f' \begin{bmatrix} v^2 & -u v \\ -u v & u^2 \end{bmatrix}. \end{split}$$ Let us remind the reader that $v = v(r,\omega)$ and $u = u(r,\omega)$ in - both depend on $\omega$. Then equation which describes the linearization of the reduced system at the solitary wave $\phi\sb\omega e^{-i\omega t}$, takes the form $${\partial}\sb t \begin{bmatrix}{\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho\\{\mathop{\rm{I\hskip -1pt m}}\nolimits}\rho\end{bmatrix} = {\eub{J}}{\eub{L}}(\omega) \begin{bmatrix}{\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho\\{\mathop{\rm{I\hskip -1pt m}}\nolimits}\rho\end{bmatrix} = \begin{bmatrix}0&{\eur{L}}\sb{-}(\omega)\\-{\eur{L}}\sb{+}(\omega)&0\end{bmatrix} \begin{bmatrix}{\mathop{\rm{R\hskip -1pt e}}\nolimits}\rho\\{\mathop{\rm{I\hskip -1pt m}}\nolimits}\rho\end{bmatrix}.$$ For the sake of completeness we record here the essential spectrum of the linearized operator: \[lemma-cont\] $ \sigma\sb{\!\rm ess}({\eub{J}}{\eub{L}}(\omega)) = i\R\backslash i({\vert \omega \vert}-m,m-{\vert \omega \vert}). $ The proof follows from noticing that, due to the exponential spatial decay of $v(r,\omega)$, $u(r,\omega)$ and the Weyl theorem on the essential spectrum [@MR0493421 Theorem XIII.14, Corollary 2], which leads to $$\sigma\sb{\!\rm ess}({\eub{J}}{\eub{L}}(\omega)) =\sigma\sb{\!\rm ess}({\eub{J}}({\eub{D}}\sb m-\omega)),$$ where $${\eub{D}}\sb m=\begin{bmatrix}D\sb m&0\\0&D\sb m\end{bmatrix}, \qquad D\sb m=i\sigma\sb 2{\partial}\sb r+\sigma\sb 3 m.$$ At the same time, since $D\sb m^2=-\Delta+m^2$, $\sigma\sb{\!\rm ess}(D\sb m)=\R\backslash(-m,m)$, while ${\eub{J}}$ commutes with ${\eub{D}}\sb m$ and $\sigma({\eub{J}})=\{\pm i\}$, one concludes that $$\sigma\sb{\!\rm ess}({\eub{J}}({\eub{D}}\sb m-\omega)) = \sigma\sb{\!\rm ess}(i(D\sb m-\omega)) \cup \sigma\sb{\!\rm ess}(-i(D\sb m-\omega)) =i\R\backslash i({\vert \omega \vert}-m,m-{\vert \omega \vert}).$$ Unstable eigenvalue of ${\eub{J}}{\eub{L}}$ for $\omega\lesssim m$ {#sect-5.2} ------------------------------------------------------------------ \[prop-k3\] Let $k\in{\mathbb{N}}$ satisfy $k>2/n$. There is $\omega\sb 1<m$ (which depends on $n$ and $k$) such that for $\omega\in(\omega\sb 1,m)$ there are two families of eigenvalues $$\pm\lambda\sb\omega\in\sigma\sb p({\eub{J}}{\eub{L}}(\omega)), \qquad \mbox{with} \quad \lambda\sb\omega>0, \quad\lambda\sb\omega=O(m-\omega).$$ The relation $ \begin{bmatrix}0&{\eur{L}}\sb{-}\\-{\eur{L}}\sb{+}&0\end{bmatrix} \begin{bmatrix}\varphi\\\vartheta\end{bmatrix} = \lambda \begin{bmatrix}\varphi\\\vartheta\end{bmatrix} $, with $\varphi,\,\vartheta\in{\mathbb{C}}^2$, can be written explicitly as follows: $$\begin{bmatrix} -\lambda & 0 & m-\omega-f & {\partial}_r + \frac{n-1}{r} \\ 0 & -\lambda & -{\partial}_r & f-m-\omega \\ \omega-m+f+2f'v^2 & -{\partial}_r-2f'v u & -\lambda & 0 \\ {\partial}_r-2f'v u & m+\omega-f +2 f'u^2 & 0 & -\lambda \end{bmatrix} \begin{bmatrix}\varphi\sb 1\\\varphi\sb 2\\\vartheta\sb 1\\\vartheta\sb 2\end{bmatrix} =0.$$ We divide the first and the third rows by $\epsilon^2= m^2 - \omega^2$, the second and the fourth rows by $\epsilon$, and substitute $R=\epsilon r$, $\varphi\sb 2=\epsilon\varPhi\sb 2$, $\vartheta\sb 2=\epsilon\varTheta\sb 2$, to get $$\label{lhpm4} \begin{bmatrix} -\frac{\lambda}{\epsilon^2} &0&\frac{m-\omega-f}{\epsilon^2} & {\partial}_R + \frac{n-1}{R} \\ 0 & -\lambda & -{\partial}_R &-m-\omega \\ \frac{\omega-m+f+2f'v^2}{\epsilon^2} &-({\partial}_R + \frac{n-1}{R}) -\frac{2f'v u}{\epsilon} &-\frac{\lambda}{\epsilon^2} & 0 \\ {\partial}_R-\frac{1}{\epsilon}2f'v u & m+\omega-f +2f'u^2 & 0 & -\lambda \end{bmatrix} \begin{bmatrix}\varphi\sb 1\\ \varPhi\sb 2\\\vartheta\sb 1\\ \varTheta\sb 2\end{bmatrix} =0.$$ Anticipating the $\epsilon \to 0$ limit, formally set $\Lambda=\lim\limits\sb{\epsilon\to 0}\frac{\lambda}{\epsilon^2}$, and introduce the matrices $$\label{def-akk} \eub{A}\sb\Lambda=\begin{bmatrix} -\Lambda &0& \frac{1}{2m}-\hat{V}^{2k}(R) & {\partial}_R + \frac{n-1}{R} \\ 0&0&-{\partial}_R&-2m \\ -\frac{1}{2m} +(2k+1)\hat{V}^{2k}(R) &-({\partial}_R + \frac{n-1}{R}) &-\Lambda &0 \\ {\partial}_R &2m &0&0 \end{bmatrix},$$ $$\label{def-kk} \eub{K}\sb 1=\mathop{\rm diag}[1,0,1,0], \qquad \eub{K}\sb 2=\mathop{\rm diag}[0,1,0,1],$$ where $\hat{V}(R)$, the NLS ground state, was introduced in . We write in the form $$\label{lhpm5} \eub{A}\sb\Lambda \eta = \Big(\frac{\lambda}{\epsilon^2}-\Lambda\Big)\eub{K}\sb 1\eta +\lambda \eub{K}\sb 2\eta +W\eta, \qquad \eta=\begin{bmatrix}\varphi\sb 1\\\varPhi\sb 2\\\vartheta\sb 1\\\varTheta\sb 2\end{bmatrix} \in{\mathbb{C}}^{4},$$ where $W(R,\epsilon)$ is a zero order differential operator with $L\sp\infty$ coefficients. \[lemma-bounds-w\] $ {\Vert W(\cdot,\epsilon) \Vert}\sb{L\sp\infty(\R^+,{\mathbb{C}}^{4}\to{\mathbb{C}}^{4})} \le O(\epsilon^{2}). $ By Lemma \[lemma-xx-ff-3\] (and the Sobolev inequality), one has $$v(r,\omega) =\epsilon^{\frac 1 k}V(\epsilon r) =\epsilon^{\frac 1 k} (\hat{V}(\epsilon r) + \tilde{V}(\epsilon r,\epsilon)),$$ $$u(r,\omega) = \epsilon^{1+\frac 1 k}U(\epsilon r) =\epsilon^{1+\frac 1 k} (\hat{U}(\epsilon r) + \tilde{U}(\epsilon r,\epsilon)),$$ $$\\| \tilde{V} \\|_{L^\infty} + \\| \tilde{U} \\|_{L^\infty} = O(\epsilon^2).$$ Then $$f(v^2-u^2) =\epsilon^{2}(V^2 - U^2)^k,$$ $$f'(v^2-u^2) = k \epsilon^{2 - \frac 2 k}(V^2 - \epsilon^2 U^2)^{k-1},$$ $$\begin{split} -\frac{1}{2m} +\frac{m - \omega-f - 2 f' v^2}{\epsilon ^2} & = O(\epsilon^2) - (V^2-\epsilon^2 U^2)^k - 2k V^2(V^2-\epsilon^2 U^2)^{k-1} \\ & = - (1+2k)\hat{V}^{2k} + O_{L^\infty}(\epsilon^2), \end{split}$$ $$\frac{f' v u}{\epsilon} = k \epsilon^2 UV(V^2-\epsilon U^2)^{k-1} = O_{L^\infty}(\epsilon^2),$$ $$m-\omega+f = O(\epsilon^2) + \epsilon^2(V^2-\epsilon^2 U^2)^k = O_{L^\infty}(\epsilon^2),$$ $$f' u^2 = k \epsilon^4 U^2 (V^2 - \epsilon U^2)^{k-1} = O_{L^\infty}(\epsilon^4),$$ and $$\frac{1}{2m} + \frac{\omega-m+f}{\epsilon^2} = + O(\epsilon^2) + (V^2 - \epsilon^2 U^2)^k = \hat{V}^{2k} + O_{L^\infty} (\epsilon^2),$$ and the Lemma follows directly from this list of estimates. \[lemma-ker-a\] $\dim\ker \eub{A}\sb\Lambda = \dim \ker (\eub{j} \eub{l}_{rad} - \Lambda)$, where $$\eub{j}\eub{l}_{rad}=\begin{bmatrix} 0 & \eur{l}_{+,rad} \\ -\eur{l}_{-,rad} & 0 \end{bmatrix},$$ and where, we recall, $$\eur{l}_{-,rad} = -\frac{1}{2m}({\partial}_R + \frac{n-1}{R}) {\partial}_R +\frac{1}{2m}-\hat{V}(R)^{2k},$$ $$\eur{l}_{+,rad} = -\frac{1}{2m} ({\partial}_R + \frac{n-1}{R}) {\partial}_R + \frac{1}{2m} - (2k+1)\hat{V}(R)^{2k}.$$ Moreover, if $k > n/2$ ($k=1$ if $n=3$), there is $\Lambda>0$ such that $\pm\Lambda\in\sigma\sb{d}(\eub{j}\eub{H})$ are simple eigenvalues. Here $\eub{H}$ is the operator defined in . An easy computation shows that $ \Phi = \left[ \begin{array}{c} \Phi_1 \\ \Phi_2 \\ \Phi_3 \\ \Phi_4 \end{array} \right] \in \ker \eub{A}_\Lambda $ if and only if $$ 2m\Phi_2 = -{\partial}\sb R\Phi_1, \quad 2m\Phi_4 = -{\partial}\sb R\Phi_3, \quad (\eub{j} \eub{l}_{rad} - \Lambda) \left[ \begin{array}{c} \Phi_3 \\ -\Phi_1 \end{array} \right] = 0,$$ and the first statement of the Lemma follows from this observation. The second statement then follows from Lemma \[lemma-nls-k\]. So we may assume that there is $\Lambda>0$ such that $\pm \Lambda \in \sigma_d (\eub{A}_\Lambda)$, with eigenfunctions $$\ker \eub{A}_{\pm \Lambda} \ni \Phi_{\pm \Lambda} = \left[ \begin{array}{c} \pm \Phi_{1} \\ \mp \frac{1}{2m} {\partial}\sb R\Phi_1 \\ \Phi_3 \\ -\frac{1}{2m}{\partial}\sb R\Phi_{3} \end{array} \right], \;\; \eur{l}_{+,rad} \Phi_1 = -\Lambda \Phi_3, \;\; \eur{l}_{-,rad} \Phi_3 = \Lambda \Phi_1.$$ We will use the Rayleigh–Schrödinger perturbation theory to show that there are eigenvalues $\pm\lambda\in\sigma\sb{d}({\eub{J}}{\eub{L}})$ with $\lambda=\epsilon^2\Lambda+o(\epsilon^2)$. Writing $$\eub{A}_\Lambda = {\eub{J}}{\eub{L}}_0 - \Lambda \eub{K}_1, \quad {\eub{L}}_0 = \begin{bmatrix} {\eur{L}}_{0,+} & 0 \\ 0 & {\eur{L}}_{0,-} \end{bmatrix},$$ with $${\eur{L}}_{0,+} = \begin{bmatrix} \frac{1}{2m} - (2k+1) \hat{V}^{2k} & {\partial}_R + \frac{n-1}{R} \\ -{\partial}_R & -2m \end{bmatrix}, \quad {\eur{L}}_{0,-} = \begin{bmatrix} \frac{1}{2m} - \hat{V}^{2k} & {\partial}_R + \frac{n-1}{R} \\ -{\partial}_R & -2m \end{bmatrix}$$ self-adjoint, we see that $$\eub{A}_\Lambda^* = - {\eub{L}}_0 {\eub{J}}- \Lambda \eub{K}_1 = \eub{F} \eub{A}_\Lambda \eub{F}, \quad \eub{F} := \begin{bmatrix} 0 & I_2 \\ I_2 & 0 \end{bmatrix}.$$ Hence $\ker \eub{A}^_\Lambda$ is spanned by $\Phi_\Lambda^ := \eub{F} \Phi_\Lambda$. Let $\eub{P}_\Lambda$ denote the orthogonal projection onto $\Phi_\Lambda^$. Seeking $\eta$ and $\lambda$ in the form $$\eta=\Phi\sb\Lambda+\zeta, \quad \zeta \perp \Phi_\Lambda, \qquad \lambda = \epsilon^2(\Lambda + \mu),$$ then becomes $$\label{eveq} \eub{A}_\Lambda \zeta = \mu \eub{K}_1 ( \Phi_\Lambda + \zeta ) + \epsilon^2 (\Lambda + \mu) \eub{K}_2 (\Phi_\Lambda + \zeta) + W (\Phi_\Lambda + \zeta).$$ Applying $\eub{P}_\Lambda$ and $1- \eub{P}_\Lambda$ to , one has: $$\label{eq1} 0= \mu \langle \Phi\sb\Lambda^,\eub{K}\sb 1(\Phi_\Lambda + \zeta)\rangle + \epsilon^2(\Lambda + \mu) \langle \Phi\sb\Lambda^,\eub{K}\sb 2(\Phi_\Lambda+\zeta)\rangle +\langle \Phi\sb\Lambda^,W(\Phi+\zeta)\eta\rangle,$$ $$\label{eq2} \eub{A}_\Lambda \zeta= (1-\eub{P}\sb\Lambda) \Big( \mu \eub{K}\sb 1 + \epsilon^2(\Lambda + \mu) \eub{K}\sb 2 +W \Big)(\Phi_\Lambda+\zeta).$$ \[lemma-nondegen\] $$\langle \Phi\sb\Lambda^, \eub{K}\sb 1\Phi\sb\Lambda \rangle \ne 0.$$ Note that $$\langle \Phi_\Lambda^, \eub{K}_1 \Phi_\Lambda \rangle= \langle \eub{F} \Phi_\Lambda, \eub{K}_1 \Phi_\Lambda \rangle = 2 {\mathop{\rm{R\hskip -1pt e}}\nolimits}\langle \Phi_3, \Phi_1 \rangle.$$ Now the fact that $\Phi_\Lambda \in \ker \eub{A}_\Lambda$ means in particular that $L_{-} \Phi_3 = \Lambda \Phi_1$. Hence, since $L_{-}$ is self-adjoint, $$\R \ni \langle \Phi_3, L_{-} \Phi_3 \rangle = \Lambda \langle \Phi_3, \Phi_1 \rangle,$$ and so ${\mathop{\rm{R\hskip -1pt e}}\nolimits}\langle \Phi_3, \Phi_1 \rangle= 0$ only if $\langle \Phi_3, L_{-} \Phi_3 \rangle =0$. As is well-known, since $L_{-} \hat{V} = 0$ and $\hat{V}(r) > 0$, we have $L_{-} \geq 0$. Thus ${\mathop{\rm{R\hskip -1pt e}}\nolimits}\langle \Phi_3, \Phi_1 \rangle= 0$ only if $L_{-} \Phi_3 = 0$. This, in turn, would imply that either $\Lambda = 0$ or $\Phi_\Lambda = 0$, both of which are false. This finishes the proof. Denote by $L^2_r(\R^n,{\mathbb{C}}^4)\subset L^2(\R^n,{\mathbb{C}}^4)$ the subspace of spherically symmetric functions. Now using Lemma \[lemma-nondegen\] and the existence of the bounded inverse $\eub{A}_\Lambda^{-1} : {\mathop{\rm Range\,}}(1 - \eub{P}_\Lambda) \to \Phi_\Lambda^{\perp}$, equations , can be written as $$\mu=M(\mu,\zeta), \qquad \zeta=Z(\mu,\zeta),$$ with functions $ M:\R\times L^2_r \to\R, \;\; $ $ Z:\R\times L^2_r \to L^2_r $ given by $$\label{def-map-m} M(\mu,\zeta)= -\frac{1} {\langle \Phi\sb\Lambda^,\eub{K}\sb 1\Phi\sb\Lambda\rangle} \Big[ \mu \langle \Phi\sb\Lambda^,\eub{K}\sb 1\zeta\rangle + \langle \Phi_\Lambda^*, \epsilon^2 (\Lambda+\mu) \eub{K}\sb 2(\Phi\sb\Lambda+\zeta) + W \rangle \Big],$$ $$\label{def-map-z} Z(\mu,\zeta)= \eub{A}\sb\Lambda^{-1}(1-\eub{P}\sb\Lambda) \Big( \mu \eub{K}\sb 1 +\epsilon^2(\Lambda+\mu)\eub{K}\sb 2 +W \Big)(\Phi\sb\Lambda+\zeta).$$ Pick $\Gamma\ge 1$ such that $$\label{def-gamma} \Gamma \ge 2{\Vert \eub{A}\sb\Lambda^{-1}(1-\eub{P}\sb\Lambda)\eub{K}\sb 1\Phi\sb\Lambda \Vert}\sb{L\sp 2}.$$ \[lemma-mz-contraction\] Consider $\R\times L^2_r$ endowed with the metric $${\Vert (\mu,\zeta) \Vert}\sb{\Gamma} =\Gamma{\vert \mu \vert}+{\Vert \zeta \Vert}\sb{L^2}.$$ There is $\omega\sb 1\in(\omega\sb 0,m)$ such that for $\omega\in(\omega\sb 1,m)$ the map $$\label{def-m-z} M\times Z:\;\;\R\times L^2_r \to \R\times L^2_r, \qquad (\mu,\zeta)\mapsto\big(M(\mu,\zeta),Z(\mu,\zeta)\big),$$ restricted onto the set $$\mathcal{B}\sb{\epsilon} =\{(\mu,\zeta)\in\R\times L^2(\R,{\mathbb{C}}^{4}) {{\rm ;}\ }{\Vert (\mu,\zeta) \Vert}\sb{\Gamma}\le\epsilon \} \subset\R\times L^2_r$$ is an endomorphism and a contraction with respect to ${\Vert \cdot \Vert}\sb{\Gamma}$. Assuming $\Gamma \|\mu\| + \\| \zeta \\|_{L^2} \leq \epsilon < 1$, and using Lemma \[lemma-bounds-w\], we have the estimates $$\|M(\mu,\zeta)\| \leq C \left( \|\mu\| \\|\zeta\\|_{L^2} +\epsilon^2(1+\|\mu\|)( 1 + \\| \zeta \\|_{L^2}) + \\| W \\|_{L^\infty} \right) \leq C \epsilon^2,$$ $$\begin{split} &\\| Z(\mu,\zeta) \\|_{L^2} \\ & \leq \frac{1}{2} \Gamma \|\mu\| + C \left( \|\mu\| \\|\zeta\\|_{L^2} + \epsilon^2(1 + \|\mu\|)(1 + \\|\zeta\\|_{L^2}) + \\| W \\|_{L^\infty} (1 + \\|\zeta\\|_{L^2}) \right) \\ & \qquad \leq \frac{1}{2} \epsilon + C \epsilon^2, \end{split}$$ which show that for all sufficiently small $\epsilon$, $M \times Z$ maps $\mathcal{B}_\epsilon$ into itself. Similar estimates show that $(M\times Z){\vert\sb{\sb{\mathcal{B}\sb{\epsilon}}}}$ is a contraction in the metric ${\Vert \cdot \Vert}\sb{\Gamma}$. According to Lemma \[lemma-mz-contraction\], by the contraction mapping theorem, the map has a unique fixed point $ (\mu\sb 0(\omega),\zeta\sb 0(\omega)) \in \mathcal{B}\sb{\epsilon} \subset\R\times L^2_r $ (as long as $\omega\in(\omega\sb 1,m)$). Thus, we have $ \pm\epsilon^2(\Lambda+\mu\sb 0(\omega)) \in\sigma\sb p({\eub{J}}{\eub{L}}(\omega)), $ $ \omega\in(\omega\sb 1,m), $ with $\Gamma{\vert \mu\sb 0(\omega) \vert}\le\epsilon$, finishing the proof of the proposition. By Remark \[remark-enough\], Proposition \[prop-k3\] finishes the proof of Theorem \[main-theorem-dirac-vbk\].
--- abstract: 'We present total-intensity and linear-polarization observations made with the Very Large Array at $\lambda$20 and 6cm of a representative sample of 42 radio galaxies and quasars selected from the Molonglo Complete Sample. The sources have been chosen to be of large size to probe the depolarizing medium on these scales using our present data and later with observations at lower frequencies with the Giant Metrewave Radio Telescope. The $\lambda$20 and 6cm data are of similar resolutions and show that depolarization between these two wavelengths is seen largely in only those lobes which are within about 300 kpc of the parent galaxy. Examination of the depolarization of the lobes with arm-length asymmetry shows that depolarization is observed predominantly for the lobe which is closer to the nucleus. There is also a trend for the lobe closer to the nucleus to be brighter, consistent with the scenario that the nearer lobe is interacting with a denser environment which is responsible for the higher depolarization and greater dissipation of energy. We have also examined the depolarization asymmetry of the lobes on opposite sides of the nucleus for galaxies and quasars. This shows that the depolarization asymmetry for quasars is marginally higher than that for galaxies. The depolarization properties of our sample are possibly determined by an asymmetric environment as well as the effects of orientation.' author: - \| C.H. Ishwara-Chandra$^{1,2}$, D.J. Saikia$^1$, V.K. Kapahi$^1$ and P. J. McCarthy$^3$\ $^1$National Centre for Radio Astrophysics, TIFR, Post Bag 3, Ganeshkhind, Pune 411 007, India\ $^2$ Joint Astronomy Program, Department of Physics, Indian Institute of Science, Bangalore, 560 012, India\ $^3$The Observatories of the Carnegie Institution of Washington, 813 Santa Barbara st., Pasadena, CA 91101, USA\ title: A Polarization Study of Radio Galaxies and Quasars selected from the Molonglo Complete Sample --- -40pt galaxies: active - galaxies: nuclei - galaxies: jets - quasars: general - radio continuum: galaxies - polarization Introduction ============ The discovery by Laing (1988) and Garrington et al. (1988) that double radio sources depolarize less rapidly on the side with the radio jet than on the opposite side provides the strongest evidence that the observed jet is on the approaching side, and its apparent asymmetry is due to relativistic beaming. The approaching side is seen through less of the depolarizing medium, and the Laing-Garrington effect can be understood as an orientation effect (cf. Garrington, Conway & Leahy 1991; hereinafter referred to as GCL91; Garrington & Conway 1991). In addition, Liu & Pooley (1991) found the more depolarized side of the source to have a steeper radio spectrum. A detailed study of a sample of quasars by Dennett-Thorpe et al. (1997) shows that the spectrum of the high-brightness features is indeed flatter on the jet side, while the low-brightness features have a flatter spectrum on the side with the longer lobe. They suggest that this is due to relativistic bulk motion in the high brightness features and differential synchrotron ageing in the extended emission. The correlation of depolarization asymmetry with jet sidedness is also relevant for testing the unified scheme in which quasars and BL Lac objects are believed to be inclined at small angles to the line-of-sight while radio galaxies lie close to the plane of the sky (Barthel 1989; Antonucci 1993; Urry & Padovani 1995). In this scheme, the radio galaxies should exhibit a similar correlation of depolarization asymmetry with jet sidedness but of a smaller magnitude since the galaxies are inclined at large angles to the line of sight and the differential path length between the two lobes is smaller (cf. Holmes 1991; Saikia, Garrington & Holmes 1997). In addition to the effects of orientation, the depolarization of the lobes will also be affected by any asymmetry in the distribution of gas in the vicinity of the radio source. The possibility of an intrinsic asymmetry in the distribution of gas was suspected from the fact that the lobe on the jet side, which is approaching us, is often closer to the nucleus (Saikia 1981). An intrinsic asymmetry was demonstrated clearly by McCarthy, van Breugel & Kapahi (1991) who showed that there was invariably more emission-line gas on the side of the source which is closer to the nucleus. For a sample of 12 radio galaxies observed with the Very Large Array (VLA), Pedelty et al. (1989) found the arm-length ratios to be correlated with the amount of depolarization and emission line gas. For the radio galaxies which are at large angles to the line-of-sight, it appears that the correlation of depolarization with arm-length asymmetry is stronger than with jet asymmetry, while the reverse is true for quasars (cf. Laing 1993). In this paper we have studied the depolarization properties of a matched sample of radio galaxies and quasars selected from the Molonglo Reference Catalogue, to examine the effects of their environment as well as orientation on structural and depolarization asymmetries of these objects. Our objects were chosen to be of large angular and linear size to probe the depolarizing medium on these scales using high-frequency observations with the VLA and lower-frequency observations with the Giant Metrewave Radio Telescope (GMRT) in the future. We have also started a programme to make optical narrow-band images to study the emission-line gas as well as broad-band continuum images to study the environments of these objects. In this paper we present our total-intensity and linear polarization observations with the VLA BnA and CnB arrays at the L- and C-bands respectively. In Section 2 we describe the sample of sources, while the observations and observational results are presented in Sections 3 and 4 respectively. We then present our estimates of the depolarization of the lobes and hotspots, discuss their relationship with linear size and arm-length asymmetry, and any difference in depolarization asymmetry between radio galaxies and quasars. A study of the RM distributions in these sources and any relationship with line-emitting gas will be presented in a later paper. ---------- ---- ------- ------------ ------------ ------ ------ Source Id z S$_{1365}$ P$_{1365}$ LAS LLS Name mJy W/Hz/sr $''$ kpc 0017-207 Q 0.545 467 25.76 98 722 0058-229 Q 0.706 396 25.95 63 505 0133-266 Q 1.53 348 26.61 53 455 0137-263 G 1.1 509 26.51 79 679 0148-297 G 0.41 2778 26.28 148 961 0325-260 G 0.638 286 25.70 58 447 0346-297 G 0.413 620 25.65 142 929 0428-281 G 0.65 956 26.25 79 617 0437-244 Q 0.84 459 26.19 128 1059 0454-220 Q 0.533 1993 26.35 94 683 0551-226 G 0.8 330 26.00 53 438 0937-250 G 0.9 445 26.28 72 607 0938-205 G 0.371 444 25.39 72 445 0947-249 G 0.854 1487 26.75 71 595 0955-283 G 0.8 470 26.14 96 791 1022-250 G 0.34 338 25.20 61 363 1023-226 G 0.586 298 25.64 66 495 1025-229 Q 0.309 489 25.28 198 1105 1026-202 G 0.566 664 25.95 61 456 1029-233 G 0.611 403 25.80 80 609 1052-272 Q 1.103 553 26.56 89 760 1107-218 G 1.5 302 26.53 62 531 1107-227 G 2.0 781 27.40 72 590 1126-290 G 0.41 1086 25.89 111 725 1224-208 G 1.5 268 26.57 61 519 1226-297 Q 0.749 435 26.00 74 597 1232-249 Q 0.352 1988 25.97 110 660 1247-290 Q 0.77 696 26.27 60 489 1257-230 Q 1.109 788 26.65 53 458 1358-214 G 0.5 317 25.34 96 681 2035-203 Q 0.516 752 25.87 71 514 2040-236 Q 0.704 428 25.89 64 510 2042-293 G 1.9 414 26.99 79 655 2045-245 G 0.73 639 26.21 76 611 2118-266 G 0.343 403 25.24 89 526 2132-235 G 0.81 333 26.02 58 482 2137-279 G 0.64 455 25.93 57 442 2213-283 Q 0.946 820 26.53 75 633 2311-222 G 0.434 965 25.85 92 616 2325-213 G 0.58 986 26.17 84 632 2338-290 Q 0.446 427 25.51 79 532 2348-235 G 0.952 503 26.34 67 570 ---------- ---- ------- ------------ ------------ ------ ------ : The sample of sources The sample of sources ===================== Our sample has been selected from a complete sample of sources, the MRC/1Jy sample (McCarthy et al. 1996; Kapahi et al. 1998a, 1998b) which is a subset of the Molonglo Reference Catalogue (Large et al. 1981). The MRC/1 Jy sample consists of 558 sources which satsify the following criteria: $S_{408} \geq 0.95 $ Jy; lie in the declination range $-30^{\circ} < \delta < -20^{\circ}$ and the right asension ranges 20$^h$ 20$^m$ to 06$^h$ 15$^m$ and 09$^h$ 25$^m$ to 14$^h$ 03$^m$ in B1950 co-ordinates; and lie outside the Galactic plane ($\| b \| > 20^{\circ}$). We have chosen FR II (Fanaroff & Riley 1974) sources from this sample with an angular size larger than about an arcminute to probe the depolarizing medium on these scales using the VLA and GMRT. We have chosen all quasars with a measured redshift greater than about 0.3, and a matched sample of galaxies of similar luminosity and redshift. The final sample of sources consists of 27 radio galaxies and 15 quasars, which are listed in Table 1. The redshifts for 0137$-$263, 0551$-$226, 0937$-$250, 0955$-$283, 1107$-$218, 1107$-$227, 1224$-$208 and 2042$-$293 have been estimated from their K magnitudes and are listed to only the first decimal place. The flux density at 1365MHz and the spectral index between 1365 and 4935MHz used for calculating the luminosity have been estimated from our observations. The luminosity, P, is in units of WHz$^{-1}{\rm sr}^{-1}$ and log P$_{1365}$ is listed in the Table. The largest angular size (LAS) is expressed in arcsec and the corresponding largest linear size (LLS) is in kpc. The LAS has been estimated from the 1365MHz images and represents the separation between the high brightness peaks in the outermost regions of emission on opposite sides of the nucleus. The radio galaxies and quasars in our sample have similar distributions of redshift, luminosity and linear size. For the galaxies, the median values of redshift and luminosity are 0.64 and 26.02 WHz$^{-1}{\rm sr}^{-1}$ , compared to 0.71 and 26.00 WHz$^{-1}{\rm sr}^{-1}$ for the quasars. The corresponding values of angular and linear sizes are 72$''$ and 595 kpc for the galaxies and 75 $''$ and 597 kpc for the quasars. We have assumed an Einstein de-Sitter Universe with H$_{\circ}$ = 50 km s$^{-1}$ Mpc$^{-1}$. Observations and analyses ========================= The observations were made with scaled arrays of the Very Large Array of National radio Astronomy Observatory (Thompson 1980) in order to get similar resolutions at the L- and C-bands. A summary of the observations is presented in Table 2. Short scans of unresolved sources selected from the VLA Calibrator Manual (Perley 1996) were interspersed for phase calibraton. At both the bands two widely separated IFs with bandwidths of 50MHz were used except for the one centered at 1665MHz where the bandwidth was only 25MHz. The IFs were separated by about 300MHz so that the rotation measure (RM) could be determined reliably. One source, 1358$-$214, was not observed at the C band due to technical problems. Most sources were observed in two scans at different hour angles, each lasting about 10 min. ------- ---------- ------- ----------------- ------- Array Freq. of Band- Date Total Conf. obs. width time MHz MHz hr BnA 1365 50 1995 Sept. 20 16.0 1665 25 CnB 4635 50 1996 Jan. 20,31 10.5 4935 50 ------- ---------- ------- ----------------- ------- : Observing schedule ------------ ----- ----- ------- ------------- ------------- ------------- ------------- Source Name maj min PA $\sigma _I$ $\sigma _p$ $\sigma _I$ $\sigma _p$ 0017$-$207 4.5 3.5 $+$70 180 60 55 55 0058$-$229 4.2 3.5 $+$80 107 65 44 50 0133$-$266 5.0 3.0 $+$60 86 60 43 65 0137$-$263 5.0 3.0 $+$60 149 60 55 55 0148$-$297 4.5 3.5 $+$60 246 70 211 50 0325$-$260 4.2 4.2 00 119 65 40 50 0346$-$297 4.2 4.2 00 141 60 55 50 0428$-$281 4.5 3.5 $-$70 146 65 50 65 0437$-$244 4.5 3.2 $-$70 96 65 50 55 0454$-$220 4.5 3.0 $-$70 194 60 75 60 0551$-$226 6.0 3.5 $-$50 111 60 50 50 0937$-$250 8.0 4.5 $-$50 142 61 55 55 0938$-$205 7.5 5.0 $-$50 122 58 54 60 0947$-$249 8.0 4.0 $-$50 178 65 109 50 0955$-$283 6.0 6.0 00 180 66 65 55 1022$-$250 6.0 6.0 00 238 60 67 65 1023$-$226 8.0 4.5 $-$50 148 58 47 50 1025$-$229 8.0 4.5 $-$50 167 62 70 50 1026$-$202 8.0 4.2 $-$50 161 60 56 65 1029$-$233 6.0 4.5 $-$50 204 58 72 55 1052$-$272 6.0 5.0 $-$50 150 60 50 60 1107$-$218 6.5 4.5 $-$50 125 60 50 55 1107$-$227 6.5 4.5 $-$50 234 65 60 60 1126$-$290 6.0 5.0 $-$50 201 60 81 55 1224$-$208 7.0 4.2 $-$50 192 60 75 60 1226$-$297 7.0 4.5 $-$50 361 61 82 55 1232$-$249 7.0 4.5 $-$50 1352 85 87 55 1247$-$290 7.0 5.5 $-$50 250 60 75 60 1257$-$230 7.0 4.5 $-$50 545 70 53 50 1358$-$214 6.5 3.0 $-$50 178 60 $-$ $-$ 2035$-$203 7.0 3.0 $+$50 238 60 79 65 2040$-$236 6.4 3.0 $+$50 194 65 59 50 2042$-$293 8.4 3.0 $+$40 192 65 65 60 2045$-$245 7.5 3.0 $+$50 155 60 53 65 2118$-$266 7.5 3.0 $+$50 244 60 72 55 2132$-$236 7.5 3.0 $+$50 100 70 51 60 2137$-$279 7.5 3.0 $+$50 197 60 51 60 2213$-$283 6.5 3.0 $+$50 166 65 55 55 2311$-$222 5.0 3.0 $+$60 201 60 62 65 2325$-$213 5.0 3.0 $+$60 159 148 55 50 2338$-$290 5.0 3.5 $+$50 261 60 76 55 2348$-$235 5.0 3.0 $+$60 113 136 76 60 ------------ ----- ----- ------- ------------- ------------- ------------- ------------- : Image parameters The data for each IF at both bands were edited and calibrated separately using the NRAO [AIPS]{} package. All flux densities are on the Baars et al. (1977) scale, with 3C48 and 3C286 being the primary flux density calibrators. To determine the absolute position angle both 3C138 and 3C286 were observed while the instrumental polarization was determined by observing the unresolved calibtrators, over a range of parallactic angles. We have corrected for ionospheric Faraday rotation, which could be significant at the L-band, using the routines in the [AIPS]{} package (cf. Chiu 1975). Comparing the polarization position angles of 3C286 and 3C138 before and after applying the ionospheric Faraday correction based on this model, we find the difference in PA to be within about 5$^\circ$. After completing the continuum and polarization calibration, images of the sources were made with uniform weighting to all baselines using the [AIPS]{} task [MX]{} and [IMAGR]{}. Several iterations of self calibrations were also done to correct for residual phase errors. The final data set was used to make images in the Stokes parameters I, Q and U. A given source has been restored with a beam of the same size at both the L and C bands. We have also made circular polarization or Stokes V images of all sources to estimate the rms noise in our maps. The ratio of the peak intensity to the rms noise in the total-intensity images is typically more than about 1000:1 at the L band. The maps of polarized intensity, $p = (Q^2 + U^2)^{1/2}$, and position angle, $\chi = 0.5 {\rm tan}^{-1}(U/Q)$ were made by combining the Q and U maps. The resulting polarized intensity image has a positive bias which has been corrected using the [AIPS]{} task [POLCO]{}, and also all pixels of amplitude $\leq$ 2$\sigma$ were blanked, where $\sigma$ is the noise in the Q or U maps. The polarized flux density was estimated from this image, while the rms noise on the polarized intensity map has been estimated before applying the bias correction. The images of the sources at one IF in each of the two bands are presented in Figure 1, with the fractional polarization vectors superimposed on the total-intensity contours. The image of the source, 1358$-$214, for which we have data at only the L-band is shown at the end of Figure 1. The size of the restoring beam and the rms noise in the total-intensity and polarization images for these frequencies are listed in Table 3. The corresponding values for the other frequency in each band are similar. Observational results --------------------- We have estimated the total intensity and polarized flux density of each lobe of a source by specifying a rectangular box such that all the regions of the lobe with total flux density more than about 5 times the rms noise per beam in the total-intensity image are included in the box. Following Garrington et al. (1991), we have set the box based on the 20cm image and a box of the same size has been used for the $\lambda$6cm image. These values were used to estimate the degree of polarization and depolarization between these two wavelengths for the lobes. In addition, we have also estimated the degree of polarization and depolarization for the hotspots by specifying a small box of 5$\times$5 arcsec$^{2}$ centred on the pixel of maximum brightness in each lobe as seen on the $\lambda$20cm image. In our images the pixel size is one arcsec at both bands. The integrated and peak values are listed in Table 4 which is arranged as follows. Column 1: source name; column 2: component designation where N, S, W and E denote north, south, west and east respectively; columns 3 and 4: the total intensity, $S_I$, and the degree of scalar polarization, $m_l$=$(\Sigma p/\Sigma I)\times 100$ %, for the entire component at $\lambda$22cm; columns 5 and 6: the peak brightness of the lobe, $S_p$, and the degree of polarization of the hotspot, $m_{hs}$, at $\lambda$22cm; columns 7 to 10: same as columns 3 to 6, but at $\lambda$18cm; columns 11 to 14: same as columns 3 to 6, but at $\lambda$6.47cm; columns 15 to 18: same as columns 3 to 6, but at $\lambda$6.08cm. Error estimates ---------------- For our interpretation we need to estimate the error in the degree of polarization and the depolarization ratio between the L and C bands. The measurement errors in the total- and polarized- flux density are estimated from the off-source rms fluctuations. In the region over which we measure the total flux density, this error is $N_I^{1/2} \delta _I $, and the error in polarization maps is $N_p^{1/2} \delta _p $, where $N_I$ and $N_p$ are the number of non-blank beam areas and $\delta _I $ and $\delta _p $ are the off-source rms fluctuations per beam in the total- and polarized-intensity maps. The fractional polarization $m$ and the fractional error in $m$ are $m = {\Sigma p \over \Sigma I} \times 100\% ~$ and $~ \delta m = ({\sigma_p \over \Sigma p} + {\sigma_I \over \Sigma I}) m $ respectively. From the internal consistency of the data, we have estimated the uncertainity in polarization calibration to be better than about 0.3%. Combining this with the fractional error in $m$, the resultant uncertainity in $m$ is $\sigma_m = [(\delta m)^2 + 0.3^2]^{1/2} \%.$ The depolarization DP and the error in DP are as follows: DP$ = m_{20} / m_6 \ ; \ \sigma_{DP} = {\rm DP} \times [({\sigma_{m_{20}} \over m_{20}})^2 + ({\sigma_{m_{6}} \over m_6})^2 ]^{1/2}$, since the errors on $m_{20}$ and $m_6 $ are uncorrelated. The errors in our estimates of fractional polarization are in the range 0.3% - 0.6%, while the estimated errors in DP are less than about 0.08 in most cases. Results and discussion ====================== In our sample of 41 sources with information at both bands, there are a total of 82 lobes. However, 6 of these lobes, namely 0346$-$297N, 0938$-$205N, 1025$-$229S, 1226$-$297N, 2040$-$236W and 2118$-$266W, have been excluded from the analysis becasue they are very weak and diffuse and their polarization information could not be determined reliably. The derived parametes for the remaining 76 lobes are presented in Table 5 which is arranged as follows. Column 1: source name and letter designation identifying the component; column 2: the linear separation in kpc of the peak of the lobe, including the hotspot at the angular resolution of the image, from the radio core or the optical position, if a core has not been detected; column 3: the ratio of separation of the component from the core/optical position to that of the component on the opposite side; columns 4 and 5: the depolarization DP=m$_{20}$/m$_6$ and the error in DP for the entire lobe including the hotspot; columns 6 and 7: the depolarization and the error in DP for the hotspots. The distributions of the degree of polarization of the lobes, m$_l$, at $\lambda$6cm range from about 2.4 to 18.2%, with a median value of about 10%. This is similar to the sample of sources observed by GCL91, where the median value on the jet as well as counter-jet side is 10.7%. We have also examined the distributions for the sample of 40 strong lobes defined to be those with S$_{peak}\geq$25 mJy at $\lambda$6cm, and find their polarization to be similar to those of the weaker sources. The degree of polarization for the hotspots in the samples of strong and weak lobes are also similar. At $\lambda$20cm, the distribution of the degree of polarization, m$_{20}$, ranges from 2.8 to 18.4%, with a median value of again about 10%. The median values for the strong and weak source samples are about 9.3 and 11.0respectively. The weak sources have higher errors and the difference is margnial. There is a similar difference for the hotspot values. The degree of polarization in our lobes is only marginally higher than the jet sides at $\lambda$20cm of the GCL91 sample where the median value is 8%, although the counter-jet sides in GCL91 exhibit significant depolarization. The values of the depolarization parameter, DP = m$_{20}$/m$_6$, for the lobes range from about 0.54 to 2.95, with median values of about 1 for the strong lobes, and about 1.1 for the weaker ones. The values of DP for the hotspots are similar. ------------ ---- ------- ------- ------- --------- ------- ------- ------- --------- ------- ------- ------- --------- ------- ------- ------- --------- Name Cp $S_I$ m$_l$ S$_p$ m$_{h}$ $S_I$ m$_l$ S$_p$ m$_{h}$ $S_I$ m$_l$ S$_p$ m$_{h}$ $S_I$ m$_l$ S$_p$ m$_{h}$ 0017$-$207 N 260 13.8 170 15.6 191 14.3 140 16.1 75 13.4 51 16.0 70 14.1 48 17.2 S 202 9.4 99 8.6 173 8.9 85 8.3 62 9.7 33 6.6 58 7.6 32 6.7 0058$-$229 N 225 5.8 42 10.3 173 7.6 34 13.1 64 8.9 17 12.5 57 8.5 15 12.2 S 158 11.0 30 14.3 125 9.3 25 12.6 49 9.5 13 11.1 44 8.5 12 10.4 0133$-$266 N 196 4.2 133 4.2 155 4.2 110 4.2 58 3.0 44 3.2 53 4.5 41 3.7 S 134 8.3 67 6.4 102 7.4 54 6.8 30 6.9 18 6.3 26 7.7 17 7.5 0137$-$263 N 384 12.5 122 8.3 307 12.0 105 8.3 102 12.6 45 11.0 93 13.5 43 11.2 S 125 16.6 37 17.6 99 16.4 30 18.3 31 16.9 12 14.8 28 15.3 11 18.9 0148$-$297 N 1150 17.3 197 24.2 862 18.7 168 26.0 334 15.0 74 24.0 311 16.6 65 27.0 S 1628 18.3 251 21.3 1289 19.3 217 22.7 460 16.3 92 19.8 436 17.6 78 23.2 0325$-$260 N 168 6.3 84 3.7 140 5.6 70 3.2 53 4.0 29 2.1 48 3.9 27 2.0 S 117 9.9 24 9.8 93 9.6 19 10.8 35 10.0 8.1 7.8 30 8.2 7.1 8.2 0346$-$297 N 71 21.0 3.9 33.0 45 22.9 2.8 37.7 20 $-$ 1.2 $-$ 14 $-$ 0.97 $-$ S 549 18.4 55 17.1 424 17.7 43 16.6 142 15.3 18 13.5 128 16.1 16 14.5 0428$-$281 E 424 9.4 229 4.9 326 9.0 189 5.0 114 6.9 70 4.4 106 7.5 66 4.6 W 532 10.0 318 7.3 426 9.8 271 7.4 154 8.6 105 7.2 142 10.4 98 7.9 0437$-$244 N 280 8.0 66 12.9 197 7.8 53 12.0 73 8.2 22 13.0 66 7.8 20 13.0 S 160 12.2 44 6.8 108 9.2 36 7.0 41 10.3 15 5.7 37 6.7 14 4.2 0454$-$220 N 716 6.9 199 7.6 553 8.6 163 9.0 226 10.5 65 9.0 212 10.6 60 9.6 S 1006 10.5 380 10.9 792 10.5 312 11.4 283 10.9 107 11.7 264 10.7 99 11.8 0551$-$226 N 151 9.6 25 14.1 87 12.7 17 15.9 46 9.2 8.8 14.8 40 11.8 8.0 14.5 S 179 8.7 29 14.6 104 10.7 19 18.0 45 9.6 9.1 17.0 43 9.9 8.3 18.1 0937$-$250 N 175 8.8 56 9.1 129 11.0 46 9.3 31 7.4 17 4.1 30 10.7 17 5.4 S 270 5.8 100 5.8 215 5.0 81 4.7 70 5.2 29 4.3 64 5.4 27 5.2 0938$-$205 N 224 2.3 23 7.2 170 2.9 19 7.6 68 $-$ 8.7 $-$ 60 $-$ 8 $-$ S 220 11.9 40 9.8 167 12.0 32 9.1 66 7.0 16 5.7 59 11.2 16 8.1 0947$-$249 N 1087 10.4 602 13.1 845 12.3 487 13.4 243 15.1 159 13.0 227 16.2 149 14.0 S 336 6.6 223 6.1 267 6.4 180 5.7 89 6.4 64 5.7 83 6.5 60 5.8 0955$-$283 E 171 13.9 53 14.7 122 13.5 43 16.6 44 15.2 17 18.0 40 14.0 17 16.6 W 299 8.7 192 9.5 236 8.7 158 9.7 92 8.3 60 9.2 86 8.6 56 9.4 1022$-$250 E 127 5.6 37 8.7 87 5.1 28 8.2 33 4.5 12 8.6 29 6.4 11 7.2 W 212 12.6 52 17.2 160 12.7 41 17.1 62 12.4 19 13.8 57 12.2 18 13.1 1023$-$226 N 197 5.2 83 6.9 156 4.7 68 6.2 64 3.7 30 4.4 60 4.1 29 5.7 S 75 2.8 46 2.9 60 2.8 37 3.6 25 2.4 16 2.5 22 2.5 15 3.1 1025$-$229 N 250 9.0 78 5.8 181 8.8 67 5.9 59 7.1 27 5.8 52 8.5 24 5.9 S 176 10.7 20 9.4 122 10.5 17 12.4 38 $-$ 6.6 $-$ 33 $-$ 6.2 $-$ 1026$-$202 N 262 11.4 69 9.8 215 11.1 57 9.5 81 9.3 24 10.8 75 9.6 22 11.1 S 401 6.8 134 8.3 321 7.8 110 9.8 117 8.0 45 7.9 111 7.7 43 7.9 1029$-$233 E 290 14.2 116 11.4 220 15.1 94 12.7 91 14.4 40 13.9 87 14.3 38 14.3 W 86 11.0 38 9.4 66 10.3 30 9.9 28 9.9 13 9.2 25 10.7 12 10.8 1052$-$272 N 345 6.2 192 5.7 263 7.1 153 6.4 74 8.7 49 8.1 70 8.4 46 8.0 S 205 7.9 69 10.7 164 7.6 56 9.6 54 7.8 20 8.6 51 6.7 20 8.1 1107$-$218 E 105 5.1 56 4.1 83 6.9 48 5.0 33 5.3 21 4.2 30 5.8 20 4.2 W 197 7.2 124 10.5 153 7.8 103 11.3 60 8.0 43 10.7 55 10.0 40 12.0 1107$-$227 N 520 8.0 346 5.1 398 7.5 271 5.4 104 7.4 81 8.4 96 7.7 75 9.1 S 261 12.6 153 15.7 195 11.6 114 15.1 47 9.2 30 8.7 41 9.6 24 8.1 1126$-$290 N 352 11.6 73 6.5 275 11.3 60 6.8 71 14.4 24 5.8 65 13.7 23 5.8 S 734 12.1 169 13.1 565 11.9 146 13.6 193 12.4 64 13.4 176 12.2 62 13.2 1224$-$208 N 188 15.4 71 19.5 141 18.1 54 20.6 42 19.5 20 16.9 38 18.2 18 18.1 S 80 10.0 34 10.6 56 11.4 25 10.9 20 5.6 9.6 9.2 18 8.1 9.2 9.3 1226$-$297 N 17 $-$ 12 $-$ 12 $-$ 10 $-$ 6.8 $-$ 5.3 $-$ 6.8 $-$ 5.3 $-$ S 416 12.8 366 14.4 345 13.2 308 14.8 147 13.7 132 15.6 138 14.2 124 16.1 ------------ ---- ------- ------- ------- --------- ------- ------- ------- --------- ------- ------- ------- --------- ------- ------- ------- --------- ------------ ---- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- Name Cp $S_I$ m$_l$ S$_p$ m$_h$ $S_I$ m$_l$ S$_p$ m$_h$ $S_I$ m$_l$ S$_p$ m$_h$ $S_I$ m$_l$ S$_p$ m$_h$ 1232$-$249 N 1084 9.6 457 13.0 901 10.1 397 12.8 365 9.4 163 11.7 343 8.9 154 11.6 S 878 8.3 405 5.4 722 8.0 357 5.2 297 6.6 169 4.5 280 7.2 158 5.4 1247$-$290 N 416 11.9 290 12.3 335 13.1 234 13.4 121 12.7 88 13.2 115 11.9 80 12.9 S 253 11.9 133 8.6 205 11.5 110 8.7 75 5.9 46 4.7 69 8.3 42 6.8 1257$-$230 N 259 5.2 213 4.1 175 5.8 151 4.4 63 3.8 49 3.7 57 4.3 44 3.9 S 514 5.0 456 5.1 398 5.3 373 5.2 167 3.6 147 4.1 158 3.8 139 4.5 1358$-$214 N 211 15.9 58 5.1 163 16.4 45 6.3 $-$ $-$ $-$ $-$ $-$ $-$ $-$ $-$ S 172 13.7 39 11.3 155 11.6 33.6 12.2 $-$ $-$ $-$ $-$ $-$ $-$ $-$ $-$ 2035$-$203 E 563 13.8 417 13.1 487 13.5 363 13.8 225 13.6 175 14.7 213 14.8 167 15.5 W 154 17.4 21 18.1 126 15.5 17 15.9 43 15.7 7.2 16.6 39 10.8 6.7 14.4 2040$-$246 E 247 9.1 105 14.2 203 9.1 86 15.2 82 8.8 37 14.4 76 9.4 35 15.5 W 50 6.5 15 6.2 38 4.3 12 5.6 13 $-$ 4.9 $-$ 12 $-$ 4.2 $-$ 2042$-$293 N 286 9.6 48 9.1 209 10.4 39 8.7 71 4.8 16 6.6 65 6.3 15 6.6 S 128 12.8 23 13.7 88 14.0 17 14.5 32 10.7 7.4 15.4 26 12.2 6.8 12.5 2045$-$245 N 383 7.4 180 7.8 303 8.5 146 9.3 96 9.6 52 11.9 88 10.3 48 12.9 S 256 15.1 70 17.7 199 14.2 56 18.0 70 11.79 20 14.5 62 13.0 18 15.9 2118$-$266 E 277 7.8 78 4.1 212 7.2 65 4.5 84 5.0 28 4.4 77 4.9 26 3.5 W 81 8.5 5 12.0 48 9.7 3.8 13.7 16 $-$ 1.2 $-$ 18 $-$ 1.5 $-$ 2132$-$236 N 157 10.8 72 9.1 121 11.2 60 9.5 44 11.2 24 9.9 39 9.7 23 10.0 S 177 7.6 122 7.6 136 8.6 100 8.1 48 8.1 36 8.4 43 7.3 33 7.8 2137$-$279 N 170 8.1 46 10.5 132 8.1 37 9.9 46 6.4 14 9.5 42 6.1 13 8.5 S 285 11.4 168 12.6 215 13.5 135 13.8 71 13.1 45 13.8 67 14.3 42 14.4 2213$-$283 E 432 14.0 126 21.9 347 13.3 103 21.7 115 12.2 35 20.1 107 11.7 32 19.4 W 328 5.2 166 5.5 253 5.6 131 5.5 73 9.6 41 9.0 67 9.7 37 8.8 2311$-$222 E 816 11.8 577 9.6 666 12.0 505 9.9 278 11.4 205 10.7 262 11.2 194 10.8 W 144 7.0 64 4.0 113 5.8 53 3.9 46 3.8 23 3.1 44 2.4 22 2.9 2325$-$213 N 378 13.7 113 9.9 291 13.5 91 10.7 107 11.3 34 9.5 99 11.3 32 9.5 S 608 9.1 136 14.2 461 9.7 109 14.7 159 9.6 63 14.5 146 9.0 40 14.6 2338$-$290 N 166 11.5 58 15.4 128 11.0 47 15.5 56 8.9 21 13.9 49 10.2 21 14.4 S 216 15.0 17 16.3 146 15.6 14 15.2 59 13.8 5.8 15.0 51 12.7 5.2 17.0 2348$-$290 N 199 13.5 55 13.6 157 13.2 45 12.9 57 10.0 19 8.8 51 9.7 18 8.6 S 299 13.9 69 14.0 235 16.3 57 14.7 83 16.9 22 12.5 77 16.4 21 12.7 ------------ ---- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- GCL91 suggested that X-ray emitting gas associated with poor clusters of galaxies could produce the observed depolarization in their sample. Assuming similar parameters as those used by GCL91 for the depolarizing gas in an unresolved foreground Faraday screen (Burn 1966), and using the median values of size and redshift for our sources, we expect the depolarization between $\lambda$20 and $\lambda$6cm in the observed frame to be close to about 1 for galaxies and about 0.8 for quasars. However, there are significant uncertainties in the assumed parameters of GCL91, and the Faraday screens may also have partially resolved structures. We hope to place better constraints on these values after making observations over a longer wavelength range. Significantly stronger depolarization will be seen at longer wavelengths, say between $\lambda$20cm and either $\lambda$49cm or $\lambda$90cm. Since long-wavelength measurements are still being planned using the GMRT, we discuss in this paper any possible dependence of depolarization on linear size and lobe separation ratio using the present data. We also discuss the depolarization asymmetry of the oppositely directed lobes for both radio galaxies and quasars and comment on whether this is consistent with the unified scheme. In the following sections, we first present the data for the entire sample and then focus on the strong-lobe sample where the errors are smaller and the effect of poor signal to noise ratio will be minimum. ------------- ----- -------------- -------- -------------------- -------- -------------------- Source L $r_{\theta}$ DP$_l$ $\sigma_{_{DP_l}}$ DP$_h$ $\sigma_{_{DP_h}}$ Name kpc 0017$-$207N 348 0.93 0.98 0.04 0.91 0.03 0017$-$207S 374 1.08 1.24 0.08 1.84 0.19 0058$-$229N 201 0.66 0.68 0.05 0.80 0.04 0058$-$229S 305 1.52 1.29 0.09 1.27 0.07 0133$-$266N 189 0.71 0.94 0.10 1.11 0.13 0133$-$266S 266 1.41 1.08 0.09 0.98 0.07 0137$-$263N 323 0.89 0.92 0.03 0.59 0.04 0137$-$263S 363 1.13 1.09 0.06 0.99 0.05 0148$-$297N 509 1.11 1.04 0.03 0.93 0.02 0148$-$297S 458 0.90 1.04 0.03 1.02 0.02 0325$-$260N 241 1.17 1.63 0.19 1.93 0.39 0325$-$260S 206 0.85 1.21 0.10 1.27 0.13 0346$-$297S 383 0.70 1.14 0.04 1.22 0.07 0428$-$281E 352 1.32 1.24 0.07 1.02 0.08 0428$-$281W 266 0.76 0.96 0.04 0.87 0.06 0437$-$244N 415 0.64 1.02 0.07 1.13 0.06 0437$-$244S 645 1.55 1.83 0.20 1.82 0.28 0454$-$220N 239 0.54 0.65 0.03 0.60 0.04 0454$-$220S 445 1.86 0.98 0.04 0.95 0.06 0551$-$226N 258 1.43 0.82 0.05 0.67 0.05 0551$-$226S 180 0.70 0.87 0.05 0.96 0.07 0937$-$250N 270 0.80 0.82 0.06 1.74 0.32 0937$-$250S 339 1.25 1.07 0.09 0.98 0.13 0938$-$205S 328 2.79 1.06 0.05 1.07 0.17 0947$-$249N 242 0.69 0.64 0.02 0.92 0.03 0947$-$249S 353 1.46 1.01 0.07 0.86 0.08 0955$-$283E 375 0.90 1.02 0.05 1.08 0.06 0955$-$283W 417 1.11 1.00 0.05 0.99 0.05 1022$-$250E 251 2.15 0.88 0.11 1.32 0.07 1022$-$250W 116 0.46 1.03 0.04 1.28 0.19 1023$-$226N 214 0.75 1.24 0.13 1.33 0.17 1023$-$226S 285 1.34 1.10 0.24 1.03 0.33 1025$-$229N 568 1.05 1.05 0.08 1.03 0.12 1026$-$202N 241 1.11 1.19 0.06 1.03 0.07 1026$-$202S 217 0.90 0.88 0.06 0.86 0.07 1029$-$233E 299 0.97 0.99 0.03 0.82 0.03 1029$-$233W 310 1.03 1.03 0.07 1.01 0.12 1052$-$272N 197 0.35 0.74 0.05 0.66 0.05 1052$-$272S 564 2.86 1.18 0.08 0.94 0.10 1107$-$218E 269 1.02 0.87 0.09 1.10 0.16 1107$-$218W 264 0.98 0.72 0.04 0.86 0.04 1107$-$227N 274 0.87 1.03 0.06 0.45 0.05 1107$-$227S 316 1.15 1.31 0.06 1.78 0.14 1126$-$290N 401 1.24 0.85 0.04 0.59 0.12 1126$-$290S 325 0.81 0.99 0.04 1.04 0.05 1224$-$208N 156 0.43 0.85 0.03 0.92 0.04 1224$-$208S 363 2.32 1.25 0.11 1.20 0.14 1226$-$297S 309 1.07 0.90 0.03 0.90 0.03 ------------- ----- -------------- -------- -------------------- -------- -------------------- : Derived parameters of the lobes and hotspots ------------- ----- -------------- -------- -------------------- -------- -------------------- Source L $r_{\theta}$ DP$_l$ $\sigma_{_{DP_l}}$ DP$_h$ $\sigma_{_{DP_h}}$ Name kpc 1232$-$249N 356 1.17 1.08 0.05 0.98 0.08 1232$-$249S 304 0.85 1.16 0.07 1.05 0.08 1247$-$290N 195 0.66 1.00 0.04 0.97 0.04 1247$-$290S 297 1.52 1.43 0.08 1.22 0.09 1257$-$230N 217 0.88 1.22 0.12 1.21 0.15 1257$-$230S 245 1.13 1.32 0.13 1.42 0.16 2035$-$203E 279 1.13 0.93 0.03 0.85 0.03 2035$-$203W 246 0.88 1.60 0.09 1.26 0.10 2040$-$236E 153 0.43 0.97 0.05 0.97 0.05 2042$-$293N 383 1.40 1.52 0.11 2.70 0.64 2042$-$293S 274 0.71 1.05 0.07 1.37 0.14 2045$-$245N 269 0.78 0.73 0.04 0.70 0.04 2045$-$245S 343 1.27 1.16 0.05 1.14 0.04 2118$-$266E 247 0.88 1.61 0.15 1.20 0.19 2132$-$236N 275 1.33 1.11 0.07 0.92 0.05 2132$-$236S 207 0.75 1.05 0.07 0.98 0.07 2137$-$279N 250 1.30 1.33 0.11 1.16 0.15 2137$-$279S 193 0.77 0.80 0.03 0.91 0.03 2213$-$283E 469 2.86 1.20 0.04 1.15 0.03 2213$-$283W 164 0.35 0.54 0.04 0.62 0.04 2311$-$222E 281 0.84 1.05 0.04 0.94 0.04 2311$-$222W 335 1.19 2.95 0.47 1.33 0.21 2325$-$213N 355 1.27 1.21 0.05 1.11 0.06 2325$-$213S 278 0.78 1.01 0.05 0.97 0.04 2338$-$290N 275 0.98 1.13 0.07 1.07 0.04 2338$-$290S 282 1.03 1.18 0.07 1.02 0.12 2348$-$235N 285 1.00 1.40 0.07 1.32 0.11 2348$-$235S 286 1.00 0.84 0.03 1.09 0.05 ------------- ----- -------------- -------- -------------------- -------- -------------------- Linear separation of the lobe and depolarization ------------------------------------------------ We present the plot of the linear separation of each lobe from the core or optical position against the depolarization parameter, DP=m$_{20}$/m$_6$, of the entire lobe in Figure 2, and against the corresponding DP values of the hotspots in Figure 3. The upper panel in each figure shows the plot for the entire data while the bottom one shows only for the sample of strong lobes which have been defined to have S$_{peak}$(6cm) $> 25$ mJy. These figures show that evidence of depolarization is seen more frequently in the sources of smaller linear size. Considering the entire sample, 20 of the 43 lobes with a linear separation $L <$ 300 kpc have DP $<$ 1, while for those with $L >$ 300 kpc, only 7 of 33 have DP $<$ 1. A Kolmogorov-Smirnov test shows that the DP distributions for lobes with L $<$ 300 and $>$ 300 kpc are different at a significance level of about 99 per cent. A similar but weaker trend is seen for the hotspots. Concentrating only on those which show significant depolarization, say DP$<$ 0.9, there are 17 lobes, 15 of which have L$<$ 300 kpc, and 15 hotspots, 11 of which have L$<$ 300 kpc. These trends for the sample of strong lobes are seen clearly in the Figures. Above a linear separation, [L]{}, of 300 kpc, there is little evidence of depolarization, while lobes and hotspots below this value often have DP significantly less than 1. Here 13 of the 24 lobes with [L]{}$<$ 300 kpc have DP $<$ 1.0, while this is true for only 5 of 16 with [L]{}$>$300 kpc. All the 8 lobes with DP $<$ 0.9 have [L]{}$<$ 300 kpc. A similar trend is also seen while considering the values for the hotspots where 10 of 24 with [L]{}$<$300 kpc have DP$<$ 0.9 while this is true for only 2 of the 16 lobes with [L]{}$>$300 kpc. GCL91 reported that the depolarization parameter on the counter-jet side, DP$_{cj}$, decreases with increasing distance from the core, while for the jet side, DP$_{j}$, no such trend is seen. For a sample of 12 galaxies, observed by Pedelty et al. (1989) with the VLA at $\lambda$20 and 6cm, DP is about 0.3 for the smaller sources with linear separations $L$ within about 100 kpc, and increases to about 1 for a separation of about 300 kpc which is amongst their largest objects. Our source sizes are about a factor of two larger than those of Pedelty et al. The smallest separation in our sample of strong lobes is about 150 kpc and has a DP of about 0.54 $\pm$ 0.04. For $L >$ 300 kpc, almost all sources show no evidence of strong depolarization. However, a number of sources appear to have DP $>$ 1. We have examined the data for both strong and weak sources, and also for sources observed on different days and find no reasonable cause for any systematic error. A similar trend for dependence of DP on $L$ was also reported for relatively nearby radio galaxies by Jägers (1986) and Strom & Jägers (1988) from observations with the WSRT at $\lambda$20 and 49cm. They find DP$^{49}_{21}$ to be about 0.6 at $L$ of about 250 kpc, which is consistent with our DP$^{20}_{6}$ values at this separation. Considering the galaxies and quasars separately, there is no striking difference in their depolarization values. This is partly because we have selected large sources. The observed depolarization is likely to be affected by intrinsic asymmetries in addition to any path-length differences between quasars and radio galaxies. We examine the depolarization asymmetry between the oppositely directed lobes for radio galaxies and quasars in Section 4.4. Lobe separation ratio and depolarization ---------------------------------------- For a sample of 3CR radio galaxies, McCarthy, van Breugel & Kapahi (1991) showed that there is more emission-line gas on the side of the lobe which is closer to the nucleus, suggesting an environmental origin for the observed arm-length asymmetries rather than being due to light travel time across the source axes. For the sample of galaxies observed by Pedelty et al. (1989), the depolarization is almost always stronger for the lobe closer to the nucleus. In 5 of their 8 sources that have extended emission-line gas, there is a spatial correlation between depolarization and emission-line gas, while for the remaining three, the gas is more centrally located and the sources are not strongly depolarized. Our sample of sources has not been chosen on the basis of their emission-line properties which are being presently investigated. In Figure 4 (upper panel) we plot for each lobe the ratio, r$_\theta$ (arm-length ratio), of its separation from the nucleus to that of the opposite lobe, against the depolarization parameter of the lobe. As we are plotting the DP for each lobe or hotspot with significant polarized flux density at $\lambda$20 and 6 cm, r$_\theta$ is $<$ 1 for the nearer lobe or hotspot and $>$ 1 for the farther one. There is a trend for the lobes with DP$<$1 to have r$_\theta$ $<$1, i.e. they are closer to the nucleus compared to the lobes on the opposite side. For the entire sample of 76 lobes, 20 of 39 with r$_\theta$ $<$1 have DP $<$ 1, while of those with r$_\theta$ $>$1 this is true for only 7 of the 37 lobes. A Kolmogorov-Smirnov test shows that the DP distributions for sources with r$_\theta$ $<$ 1 is different from those with r$_\theta$ $>$ 1 at a significance level of $>$ 99 per cent. Concentrating on the sample of strong lobes, DP $<$ 1 for 15 out of 25 lobes with r$_\theta$ $<$1, but for only 3 out of 15 with r$_\theta$ $>$1. Among the eight lobes with significant depolarization, DP $<$ 0.90, all are on the nearer side with r$_\theta$ $<$1. A similar trend is also seen for the hotspots. There are 12 hotspots with DP $<$ 0.9, 10 of which have r$_\theta$ $<$1. The tendency for the shorter arm to be more depolarized can also be seen clearly if the arm-length ratio, defined to be the ratio of the length of the short arm to the length of the longer arm, is plotted against the depolarization of the short arm to the depolarization of the longer arm (Figure 5). It is seen that in 28 of the 35 sources, where the depolarization of the both the lobes have been determined, the nearer lobe is more depolarized than the farther one. There is a weak trend for the most asymmetric sources to have a lower depolarization ratio, similar to what was observed by Pedelty et al. (1989). This is seen more clearly in Figure 5 when one considers the galaxies and quasars with strong lobes. The dependence of depolarization ratio on arm-length ratio was studied by GCL91, but they did not find any such trend. Their sample consists largely of quasars with known radio jets and are of smaller angular size compared to our sources. In a sample of quasars selected to have known radio jets, the objects are likely to be close to the line of sight and effects of orientation should be dominant. The counter-jet side which exhibits significantly stronger depolarization is often farther from the nucleus. This is possibly due to intrinsic non-collinearities which may also appear amplified if the sources are at small angles to the line of sight. In our sample the sources are of large angular and linear size, reasonably collinear and the observed trend is possibly due to an asymmetric environment as well as effects of orientation. Comparison of the depolarization information with narrow-band images, and detection of radio jets in these objects should help clarify the situation. Flux density and brightness ratio of the lobes ---------------------------------------------- However, an examination of the brightness asymmetry of the lobes and its relationship to the arm-length asymmetry might also provide us with valuable insights. For example, if the source and the environment are intrinsically symmetric, the approaching component would be farther from the nucleus and brighter due to relativistic enhancement of the hot-spot flux density, provided the effects of evolution of individual components with age over the length scales of our sources are not signficant (Ryle & Longair 1967; Swarup & Banhatti 1981). On the other hand, if the source is evolving in an asymmetric environment, there will be a greater dissipation of energy on the side with the higher density which will also be closer to the nucleus (Eilek & Shore 1989; Gopal-krishna & Wiita 1991). Figure 6 shows the arm-length ratio, defined to be $\leq$1, plotted against the peak brightness ratio, which is the ratio of peak brightness of the nearer lobe to the farther one. There is a clear trend for the nearer component to be brighter, so that the ratio is $>$1 in 29 of the 41 objects in the sample. This is particularly true for the more asymmetric sources, say r$_\theta <$0.8, where 18 of the 22 sources have the brighter component closer to the nucleus. A similar trend is also seen while considering the total flux density ratio of the lobes. The sample of 3CR sources studied by McCarthy et al. (1991) shows a similar but weaker trend. This trend in our sample suggests that an asymmetric environment on opposite sides of the nucleus affects the flux density and arm-length ratios of the lobes, in addition to the observed depolarization asymmetry of the oppositely-directed lobes. Depolarization Asymmetry ------------------------ In the unified scheme for radio galaxies and quasars, the quasars are inclined at smaller angles to the line of sight. If the depolarization is caused by a magnetioionic halo associated with the parent optical object or a cluster of galaxies associated with it, the pathlength difference for the radiation from the two oppositely-directed lobes will be larger for quasars compared to the galaxies. Hence the quasars should exhibit a higher degree of depolarization asymmetry of the lobes or hotspots compared to the galaxies. We can attempt to infer an average statistical angle of inclination to the line of sight for our sample of galaxies and quasars using the fraction of emission from the core as a statistical indicator of orientation to the line of sight. All the quasars in our sample have detected radio cores, with the median fraction of emission from the core at an emitted frequency of 8GHz being about 12%. Most of the galaxies do not have detected cores, and the upper limit is generally less than about 0.1%. Assuming that the intrinsic fraction of emission from the nucleus is similar to that of 3CR sources (cf. Saikia & Kulkarni 1994), the core strength of galaxies and quasars are consistent with angles of inclination of about 65$^\circ$ and 30$^\circ$ respectively. The expected depolarization asymmetry between $\lambda$20 and 6cm for our sources using the typical parameters listed by GCL91 are close to about 1 and 0.8 for the galaxies and quasars respectively. However, our sources are much larger than those in GCL91, and we need to determine more reliably the parameters for the environment on these scales using X-ray and optical observations along with long-wavelength radio polarization measurements. Since we do not detect jets in almost all but one of our sources we examine the DP asymmetry for both radio galaxies and quasars, bearing in mind that the trend could be diluted by intrinsic asymmetries in the environment. In Figure 7, we present the depolarization asymmetry for both radio galaxies and quasars using the depolarization values for both the lobes and hotpots. We define the depolarization asymmetry as the ratio of the depolarization of one lobe to the other such that the ratio is always less than 1. There is a marginal trend for the quasars to be more asymmetric, consistent with the sense expected in the unified scheme. The median values for the entire sample of galaxies and quasars are about 0.78 and 0.67 respectively for the lobes as well as the hotspots. Considering the lobes and hotspots from the sample of strong sources, the median values of DP asymmetry of the lobes are 0.80 and 0.75 for the galaxies and quasars respectively, while the corresponding values for the hotspots are 0.89 and 0.72 respectively. Since the environments are likely to be asymmetric, as discussed earlier, and the sources are large, the effects of orientation are seen as a marginal trend. Laing-Garrington effect ----------------------- In the Laing-Garrington effect the counter-jet side depolarises more rapidly due to the extra pathlength through the magnetoionic medium. In our sample we detect a radio jet in only one object, namely the quasar 0454$-$220. The depolarization value for the lobe and the hotspot on the counter-jet side is 0.65$\pm$ 0.03 and 0.60$\pm$ 0.04 and for the jet side these values are 0.98$\pm$ 0.04 and 0.95$\pm$ 0.04 respectively. This is clearly consistent with the Laing-Garrington effect. The arm-length ratio, r$_\theta$ defined to be $<$1, is 0.53 with the counter-jet lobe being the nearer one, while the DP ratio of this lobe to the farther one is 0.67 $\pm$ 0.04 and 0.63 $\pm$ 0.06 for the hotspot. The component on the jet side is brighter, possibly due to mild relativistic beaming of the hotspot. In this case, since we can unambiguously identify the approaching component assuming the jet asymmetry to be due to relativistic beaming, the tendency for r$_\theta$ and DP to be $<$1, is largely due to the extra pathlength of the receding lobe. It is important to try and detect radio jets in a larger number of these objects to assess the relative importance of orientation and environmental effects. Concluding remarks ================== We have investigated the effects of environment and orientation on the observed depolarization properties of a sample of high-luminosity radio galaxies and quasars between $\lambda$ 20 and 6cm. We find that significant depolarization is usually seen in the lobes which are within about 300 kpc of the parent galaxy. Among the 17 lobes in the entire sample which show significant depolarization with DP$<$0.9, 15 are within 300 kpc from the parent galaxy. Comparing the depolarization on opposite sides of the source, we find that the side which is closer to the parent galaxy shows significant depolarization. Of these 17 lobes where significant depolarization is seen, 13 of them are closer to the parent galaxy compared to the lobe on the opposite side. The ratio of the depolarization parameter of the nearer lobe to the farther one is $<$1 for 28 of the 35 sources in the entire sample with reliable polarization information for both lobes. The nearer component is also brighter in 26 of these 35 objects, suggesting that the nearer component is advancing outwards through a denser environment which is responsible for stronger depolarization and greater dissipation of energy. The depolarization asymmetry of the lobes on opposite sides for galaxies and quasars shows that the latter are marginally more asymmetric, consistent with the trend expected in the unified scheme. In our sample of sources the polarization properties of the lobes appear to be due to an asymmetric environment on opposite sides of the parent optical object, as well as possibly due to different orientations of the sources to the line of sight. Detailed information on the distribution of the depolarizing medium from optical and X-ray observations, as well as detection of radio jets in these objects to identify the approaching and receding components, should enable us the clarify better the relative importance of the effects of an asymmetric environment and orientation. 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--- abstract: 'The accuracy of modern machine learning algorithms deteriorates severely on adversarially manipulated test data. Optimal adversarial risk quantifies the best error rate of any classifier in the presence of adversaries, and optimal adversarial classifiers are sought that minimize adversarial risk. In this paper, we investigate the optimal adversarial risk and optimal adversarial classifiers from an optimal transport perspective. We present a new and simple approach to show that the optimal adversarial risk for binary classification with $0-1$ loss function is completely characterized by an optimal transport cost between the probability distributions of the two classes, for a suitably defined cost function. We propose a novel coupling strategy that achieves the optimal transport cost for several univariate distributions like Gaussian, uniform and triangular. Using the optimal couplings, we obtain the optimal adversarial classifiers in these settings and show how they differ from optimal classifiers in the absence of adversaries. Based on our analysis, we evaluate algorithm-independent fundamental limits on adversarial risk for CIFAR-10, MNIST, Fashion-MNIST and SVHN datasets, and Gaussian mixtures based on them. In addition to the $0-1$ loss, we also derive bounds on the deviation of optimal risk and optimal classifier in the presence of adversaries for continuous loss functions, that are based on the convexity and smoothness of the loss functions.' bibliography: - 'ref.bib' --- [***]{} -- -- -- -- -- -- Department of Electrical & Computer Engineering\ University of Wisconsin-Madison December 2019 Introduction ============ Modern machine learning algorithms based on deep learning have had tremendous success in recent times, producing state-of-the-art results in many domains such as image classification, game playing, speech and natural language processing. Along with the success, it was also discovered that these algorithms exhibit surprising vulnerability to adversarial perturbations that are imperceptible to humans. Since the discovery by Szegedy et al. [@SzeEtal13], there has been a slew of [adversarial attacks]{} on neural network based classifiers [@AthCar18; @CarWag17; @GooEtal14] and defense methods against such attacks [@MadEtal18; @PapEtal16; @CisEtal17]. Often, the defense methods either fall short of new attacks or are computationally intractable for large neural networks. Recent work has focused on certifiable defenses that are provably robust against a pre-specified class of adversaries [@CohEtal19; @SinEtal17; @RagEtal18]. Theoretical investigation into the cause of adversarial examples has led to the hypothesis that adversarial examples may be inevitable in certain high-dimensional settings [@ShaEtal19; @MahEtal19]. Specifically, it has been shown that any classifier that works on data sampled from concentrated metric probability spaces is susceptible to a high adversarial risk. For instance, when the input distribution is uniform over a high dimensional sphere [@GilEtal18] or Boolean hypercube [@DioEtal18], or when the latent space of the data is a high dimensional Gaussian [@FawEtal18], the adversarial risk can be significantly higher than standard risk, even for small $\epsilon$. Moreover, it was recently proposed that adversarial risk may be fundamentally at odds with standard risk—a claim that finds support both in theory [@TsiEtal18] and in practice [@SuEtal18]. Our primary interest is in analyzing limits on the optimal adversarial risk without any dependence on the algorithm. The first question we investigate is the following: \[q\_1\] How much can the optimal adversarial risk differ from optimal standard risk? It is easy to see that the optimal adversarial risk is at least as large as optimal standard risk (see Section \[sec\_triv\_bound\]). Is it possible to derive a tighter lower bound for the optimal adversarial risk? Recent works in this direction have addressed this question by deriving upper and lower bounds on the optimal adversarial risk with respect to a fixed set of classifiers, by extending the PAC learning theory to encompass adversaries. For instance, Khim and Loh [@KhiLoh18] and Yin et al. [@YinEtal18] develop risk bounds based on a notion of adversarial Rademacher complexity, that is a function of both the data generating distribution and class of classifiers under consideration. In a similar vein, several works [@AttEtal18; @CulEtal18] derive sample-complexity bounds for robust learning. More recent works [@DioEtal19; @GouEtal19] specifically focus on lower bounds for sample-complexity, in order to characterize the hardness of robust learning. However, deriving lower bounds on the optimal adversarial risk that are classifier agnostic has not received much attention. Another related question is the following: [How much adversarial perturbation is sufficient to make the optimal adversarial risk significantly greater than the optimal standard risk?]{} Relevant works in this direction have again focused on developing robustness metrics that are specific to the classifier [@WenEtal18; @ZhaEtal18; @HeiEtal17]. In addition to Question \[q\_1\], one might also consider the nature of the optimal classifier under the standard and adversarial settings. This motivates the following question: \[q\_2\] Does the optimal classifier in the adversarial setting differ from that in the standard setting? If so, by how much? In a recent work, Moosavi-Dezfooli et al. [@MooEtal18] empirically observed that adversarial training significantly reduces the curvature of the loss function with respect to the input. Another line of work attempts to construct a provably robust classifier from a baseline classifier using randomized smoothing [@CohEtal19]. These works provide a clue that the optimal classifier may in fact be different in the presence of an adversary. Even so, many other interesting questions remain. For instance, is the optimal classifier without an adversary approximately the same as the optimal classifier with a small adversary (i.e., small $\epsilon$)? If decision boundaries change, do they change smoothly with increasing strength of an adversary, or do they change drastically? The closest work to ours is Bhagoji et al. [@BhaEtal19], who develop the first classifier-agnostic lower bounds for learning in the presence of an adversary. Specifically, [@BhaEtal19] contains a similar result to our Theorem \[th\_01bound\] which gives the optimal adversarial risk for binary classification with $0-1$ loss in terms of an optimal transport cost between the probability distributions of the two classes. We provide a new, simpler proof of this characterization by applying the Kantorovich duality of optimal transport for $0-1$ cost functions. We shall discuss the results from Bhagoji et al. [@BhaEtal19] and compare and contrast these with our results at appropriate points in the paper. Our contributions {#our-contributions .unnumbered} ----------------- In this paper, we consider two types of adversaries: (i) data perturbing and (ii) distribution perturbing. We primarily focus on the binary classification setting under $0-1$ loss function. We answer Question \[q\_1\] by providing [universal]{} bounds for adversarial risk under the two notions of adversaries that are agnostic to the classifier. We answer Question \[q\_2\] by deriving the optimal adversarial classifier in some special settings, and by providing bounds on the deviation of the optimal adversarial classifier from the standard optimal classifier in more general settings. Our contributions are listed below. 1. We provide a new and simple proof for the characterization of optimal adversarial risk for $0-1$ loss functions in terms of an optimal transport cost between the two data generating distributions, where the transport cost is given by $c_\epsilon(x, x') = \1\{d(x,x')>2\epsilon\}$ and $\epsilon$ is the perturbation budget of the adversary. This is analogous to the optimal risk (i.e., Bayes risk) for binary classification in the standard setting, which is a function of the total variation distance between the two data generating distributions, which in turn is also an optimal transport cost with transport cost $c_0(x, x') = \1\{d(x,x')>0\}$. This completely answers Question \[q\_1\] for this setting. Our proof establishes connections between adversarial machine learning and well-known results in the theory of optimal transport. 2. We propose a novel coupling strategy that achieves the proposed optimal transport cost between the two class-conditional densities for several univariate distributions like Gaussian, triangular and uniform. Using the analysis of optimal couplings, we obtain the optimal adversarial classifiers for these settings. This answers Question \[q\_2\] in these settings, and shows how the decision boundary of the optimal classifier changes in the presence of adversaries. In certain cases, we show that the decision boundary can change arbitrarily, even for small changes in the adversary budget $\epsilon$. 3. Using our analysis for $0-1$ loss, we obtain the exact optimal risk attainable for a range of adversarial budgets under $\ell_2$-norm and $\ell_\infty$-norm perturbation of data, for several real-world datasets, namely CIFAR10, MNIST, Fashion-MNIST, and SVHN. In addition, we analyze Gaussian mixture distributions based on these datasets and compute lower bounds on the optimal adversarial risk for them. These bounds indicate the optimal adversarial error achievable with data augmentation using Gaussian perturbations. 4. Addressing Questions \[q\_1\] and \[q\_2\] is more challenging for general continuous loss functions. We provide some preliminary results to address these that include upper and lower bounds on the optimal adversarial risk. These bounds depend on the convexity and smoothness of the loss function with respect to the data. For Question \[q\_2\], we prove a result that quantifies the maximum amount by which the optimal classifier can deviate in the presence of an adversary, in terms of the curvature of the loss function at the optimal classifier for standard risk. #### Structure: The rest of the paper is structured as follows: In Section \[sec: preliminaries\], we introduce the two types of adversaries: (1) data perturbing and (2) distribution perturbing, and show that the optimal risk in the data perturbing case is lower. We also present a simple example to show that the optimal risk and optimal classifier can deviate significantly in the presence of adversaries. In Section \[sec: opt risk 01 loss\], we discuss the optimal adversarial risk for binary classification with $0-1$ loss. We settle Question \[q\_1\] in this setting by introducing the $D_\epsilon$ optimal transport cost that completely characterizes the optimal risk. In Section \[sec: couplings\], we discuss the optimal adversarial classifier for binary classification with $0-1$ loss. We present a coupling strategy that achieves the optimal transport cost in special cases of interest. Using this coupling, we obtain the optimal adversarial classifier, thus settling Question \[q\_2\] for these special cases. In Section \[sec: opt risk gen loss\], we discuss the optimal risk for general loss functions and present our bounds on the optimal adversarial risk. In Section \[sec: opt classifier\], we discuss optimal classifiers for general loss functions and present our deviation bounds on the optimal adversarial classifier. Finally, in Section \[sec: expts\], we present adversarial risk lower bounds for real world datasets and evaluate our bounds for $0-1$ loss function. #### Notation: The complement of a set $A$ is denoted by $A^c$. Define $\1{\{C\}}$ to be the indicator function that maps all the inputs satisfying condition $C$ to $1$ and the rest to $0$. For a set $A$ in a Polish space $(\cX, d)$, the set $A^\epsilon$ denotes the $\epsilon$-expansion of $A$ in $\cX$. That is, $A^\epsilon = \{x\in\cX: d(x,x')\leq\epsilon \ \text{for some}\ x'\in A\}.$ For any two probability measures $\mu$ and $\nu$ defined over the Polish space $(\cX, d)$, we use $\Pi(\mu, \nu)$ to denote the set of all joint probability measures over $\cX\times\cX$ with marginals $\mu$ and $\nu$, respectively. We use $D_{TV}(\mu, \nu)$ and ${W_p}(\mu, \nu)$ to denote the total variation distance and $p$-Wasserstein distance between $\mu$ and $\nu$. We use $\lVert\cdot\rVert$ to denote a norm and $\lVert\cdot\rVert_$ to denote its dual norm. The cumulative distribution function (cdf) of the standard normal distribution is denoted by $\Phi$. Denote $Q(x) := 1- \Phi(x)$ to be the tail distribution function of a standard normal. Preliminaries {#sec: preliminaries} ============= In the framework of statistical learning, let $z=(x, y)$ denote labeled data points drawn i.i.d. from an unknown distribution $\rho$ over $\cZ=\cX\times\cY$, where $\cX$ is the input space and $\cY$ is the label space. Consider a hypothesis class parametrized by $w \in \cW$, where $\cW$ is a compact subset of $\mathbb{R}^p$. Let $\ell: \cZ\times\cW\to \mathbb{R}^+$ denote the loss function. Let $d: \cX\times\cX\to\mathbb{R}^+$ be a distance metric on the space $\cX$. Types of adversaries -------------------- To quantify the impact of an adversary, notions of adversarial risk have been proposed in the literature. We discuss two such notions here: (i) adversary perturbs data points, and (ii) adversary perturbs data distributions. #### Data perturbing adversary: Let the data distribution be $\rho$. The adversarial loss (or robust loss) incurred by a hypothesis $w\in \cW$ in the presence of a data perturbing adversary with a budget $\epsilon$ is defined as follows: $$\begin{aligned} \label{eq: data_loss} R_\epsilon(\ell, w) = \E_{(x,y)\sim\rho}\left[\sup_{d(x,x')\leq\epsilon}\ell((x', y),w)\right].\end{aligned}$$ Define the optimal adversarial loss attainable over the parameters $w\in\mathbb{R}^p$ as the [optimal adversarial risk]{} or [optimal robust risk]{}, $$\begin{aligned} R^_\epsilon = \inf_{w\in\cW} R_\epsilon(\ell, w).\end{aligned}$$ The classifier attaining the optimal adversarial risk is called the [optimal adversarial classifier]{} or [optimal robust classifier]{}. The parameter vector of such a classifier is denoted by $w^_\epsilon$. Note that for $\epsilon=0$, the adversarial loss reduces to the standard loss, and the adversarial risk reduces to the standard risk (or Bayes risk), $R^_0 = \inf_w \E_{z\sim\rho}\left[ \ell(z,w) \right]$. #### Distribution perturbing adversary: The adversarial loss incurred by a hypothesis $w\in \cW$ in the presence of a distribution perturbing adversary with a budget $\epsilon$ is defined as follows: $$\begin{aligned} \label{eq: distr_loss1} \widehat R_\epsilon(\ell, w) = \sup_{\rho' \in B_\epsilon(\rho)} \E_{z \sim \rho'} \ell(z,w),\end{aligned}$$ where $B_\epsilon(\rho)$ may be thought of as a ball of radius $\epsilon$ around $\rho$, the true data generating distribution. The Wasserstein distance has been one of the more popular metrics used to define $B_\epsilon(\cdot)$ in the space of distributions [@Woz14; @BlaMur16; @GaoKle16; @GaoEtal17; @EsfKuh18; @ZhuEtal19]. In this paper, we shall assume the distance being used is the 1-Wasserstein distance, unless otherwise stated. The optimal adversarial risk is defined as $$\begin{aligned} \label{eq: distr_loss2} \widehat R^_\epsilon = \inf_{w \in \cW} \widehat R^\epsilon(\ell, w).\end{aligned}$$ The optimizing $w$ for the adversarial risk is denoted by $\widehat w^_\epsilon.$ Note that $\widehat w^_0 = w^_0.$ A distribution perturbing adversary is considerably more powerful than a data perturbing adversary. For any fixed $w \in \cW$, and for each $z = (x,y)$, we may define $z' = (x',y)$, where $x'$ achieves the supremum in equation . It is easily checked that the 1-Wasserstein distance between the distributions of $Z$ and $Z'$ is upper-bounded by $\epsilon$. Thus, we have $$\begin{aligned} R_\epsilon(\ell, w) \leq \widehat R_\epsilon(\ell, w),\end{aligned}$$ and taking the infimum with respect to $w \in \cW$ yields $$\begin{aligned} R^_\epsilon \leq \widehat R^_\epsilon.\end{aligned}$$ #### A remark on the risk bounds for adversaries: All risk bounds proved in this paper are valid for both adversaries. Since the distribution perturbing adversary is stronger than the data perturbing adversary, any lower bound that holds for the latter holds for the former. Analogously, any upper bound for the distribution perturbing adversary holds for the data perturbing adversary. A trivial lower bound {#sec_triv_bound} --------------------- We start by presenting a trivial lower bound on the optimal adversarial risk. To derive the bound, we use the fact that the adversarial loss $R_\epsilon(\ell, w)$ for any classifier (i.e., for any $w$) is lower-bounded by the standard loss $R_0(\ell, w)$ for that classifier. \[th\_trivial\] The optimal adversarial risk is at least as large as the optimal standard risk, that is, $R^_\epsilon \geq R^_0$. We have the sequence of inequalities: $$\begin{aligned} R^_\epsilon = R_\epsilon(\ell, w^_\epsilon) \geq R_0(\ell, w^_\epsilon) \geq R_0(\ell, w^_0) = R^_0.\end{aligned}$$ The first inequality holds because $R_\epsilon(\ell, w)$ is a non-decreasing function of $\epsilon$ for any fixed $\ell$ and $w$. The second inequality follows from the fact that the adversarially optimal classifier $w^_\epsilon$ is sub-optimal for minimizing the standard risk. Note that the bound in Theorem \[th\_trivial\] does not depend on the strength of the adversary $\epsilon$, and hence it may not be very tight for large $\epsilon$. In what follows, we show tighter lower bounds for $R^_\epsilon$ that depend on $\epsilon$. A motivating example -------------------- Here, we present a simple binary classification problem with Gaussian class conditional densities which shows that the optimal classifier indeed differs from the Bayes optimal classifier, in the presence of an adversary. For this example, we are able to explicitly compute the optimal adversarial risk and the optimal adversarial classifier as a function of the adversarial budget $\epsilon$. The detailed proof of this fact is found in Theorem \[thm: same mean\]. Let $x\|(y=i)\sim \cN(0, \sigma_i^2)$ for $i=0,1$ ($\sigma_1<\sigma_0$). Let $\Phi_i$ denote the cumulative distribution function for $\cN(0, \sigma_i)$. Let the class labels $0$ and $1$ be equally likely. Consider classifiers parametrized by $\cW = \{w\in\mathbb{R}:w>0\}$ as follows: $f(x) = \1\{x\in [-w, w]\}$. Let $k = \left( \frac{\sigma_0^2 + \sigma_1^2}{\sigma_0^2 - \sigma_1^2} \right)$. Then the optimal adversarial risk and classifier are given by $$\begin{aligned} w^_\epsilon &= w^_0\sqrt{1+\frac{\epsilon^2(k^2-1)}{(w^_0)^2}}+\epsilon k \label{eq_tep},\\ R^_\epsilon &= \frac{1}{2}\left[1 - 2\left( \Phi_1(w^_\epsilon - \epsilon) - \Phi_0(w^_\epsilon + \epsilon) \right) \right] \label{eq_r_ep}.\end{aligned}$$ Equation shows that the optimal adversarial risk increases with increasing power of the adversary (i.e., increasing $\epsilon$). Moreover, we see from the optimal adversarial classifier can differ significantly from the Bayes optimal classifier when $\sigma_0$ is close to $\sigma_1$ (i.e., when $k$ is large). Figure \[fig: example gaussian\] shows a specific instance of this example. Optimal adversarial risk via optimal transport {#sec: opt risk 01 loss} ============================================== In this section, we present our results on adversarial risk under $0-1$ loss in the binary classification setting. We first define the optimal transport cost $D_\epsilon(\mu, \nu)$ between two probability measures $\mu$ and $\nu$ over a metric space $(\cX, d)$, as follows. For $\epsilon \ge 0$, define the cost function $c_\epsilon: \cX \times \cX \to \real$ as $c_\epsilon(x, y) = \1\{d(x, y) > 2\epsilon\}$. The optimal transport cost $D_\epsilon$ is defined as $$\begin{aligned} \label{eq_epsilon_Wasserstein} D_\epsilon(\mu, \nu) = \inf_{\pi\in\Pi(\mu, \nu)}\E_{(x,x')\sim\pi} c_\epsilon(x, x').\end{aligned}$$ For $\epsilon = 0$, the optimal cost is equivalent to the total variation distance, i.e., $D_0(\mu, \nu) = D_{TV}(\mu, \nu)$. For $\epsilon > 0$, this cost does not define a metric over the space of distributions. This is because $D_\epsilon(\mu, \nu) = 0$ does not imply $\mu$ and $\nu$ are identical. Moreover, it also does not define a pseudometric since the triangle inequality is not satisfied. To see this, observe that if $\mu_1$, $\mu_2$, and $\mu_3$ are unit point masses at $0$, $2\epsilon$, and $4\epsilon$, then $D_\epsilon(\mu_1, \mu_3) = 1 > 0 = D_\epsilon(\mu_1, \mu_2) + D_\epsilon(\mu_2, \mu_3).$ Next, we present the main theorem of this section that gives the optimal risk under the binary classification setup for a data perturbing adversary. \[th\_01bound\] Consider the binary classification setup with $\cY=\{0,1\}$, where the input $x\in\cX$ is drawn with equal probability from two distributions $p_0$ (for label $0$) and $p_1$ (for label $0$). We consider a set of binary classifiers of the form $\1\{x\in A\}$, where $A\subseteq\cX$ is a topologically closed set. That is, the classifier corresponding to $A$ assigns the label $1$ for all $x\in A$ and the label $0$ for all $x\notin A$. Consider the $0-1$ loss function $\ell((x,y),A) = \1\{x\in A, y=0\} + \1\{x\notin A, y=1\}$. The adversarial risk with the data perturbing adversary is given by $$\begin{aligned} \label{eq: 01 opt risk} R^_\epsilon = \frac{1}{2}\left[1- D_\epsilon(p_0, p_1)\right].\end{aligned}$$ Instantiating Theorem \[th\_01bound\] for $\epsilon=0$, we get $R^_0 = \frac{1}{2}\left[1- D_{0}(p_0, p_1)\right] = \frac{1}{2}\left[1- D_{TV}(p_0, p_1)\right]$, which is the Bayes risk. It is also possible to derive weaker bounds in terms of the $p$-Wasserstein distance between the distributions of the two data classes, as shown in the following corollary: \[cor\_dwp\_bound\] Under the setup considered in Theorem \[th\_01bound\], we have the following bound for $p\geq 1$: $$\begin{aligned} \label{eq_dwp_bound} R^_\epsilon \geq \frac{1}{2}\left[1- \left(\frac{{W_p}(p_0, p_1)}{2\epsilon}\right)^p \right].\end{aligned}$$ Let $A \subseteq \cX$ be a closed set such that the classifier declares $1$ on $A$ and $0$ on $A^c$. The robust risk over the hypothesis class of closed sets is given by $$\begin{aligned} R_\epsilon^ &= \min_{A\subseteq\cX} \frac{1}{2} \left( p_0(A^\epsilon) + p_1\left((A^c)^\epsilon\right)\right)\\ &= \frac{1}{2} \left( 1 - \sup_{A~\text{closed}} \left\{ p_1\left(A^{-\epsilon}\right)- p_0(A^\epsilon) \right\} \right),\end{aligned}$$ where we define $A^{-\epsilon} := \left((A^c)^\epsilon\right)^c$. For any two distributions, Strassen’s theorem (stated in Appendix \[app: strassen\]) states that $$\begin{aligned} D_\epsilon(p_0, p_1) = \sup_{A~\text{closed}} \left\{ p_0(A) - p_1(A^{2\epsilon}) \right\}.\end{aligned}$$ To prove the equality $R^_\epsilon = \frac{1}{2}[1-D_\epsilon(p_0, p_1)]$, notice that it is enough to prove that for measures $\mu$ and $\nu$, $$\begin{aligned} \label{eq: pm_epsilon} \sup_{A ~\text{closed}} \mu(A^{-\epsilon}) - \nu(A^{\epsilon}) = \sup_{A ~\text{closed}} \mu(A) - \nu(A^{2\epsilon}).\end{aligned}$$ To do so, we first we show that $A^\epsilon$ and $A^{-\epsilon}$ are also closed sets if $A$ is a closed set: \[lemma: closed\] If $A$ is closed, then $A^\epsilon$ is also closed. Similarly, if $A$ is closed, then $A^{-\epsilon}$ is also closed. Next, we show the order in which a set is thickened by $\epsilon$ and thinned by $\epsilon$ affects the size of the original set in opposing ways: \[lemma: thick and thin\] Let $A$ be a closed set. Then $(A^{-\epsilon})^{\epsilon} \subseteq A$ and $A \subseteq (A^\epsilon)^{-\epsilon}$. ![Illustration of $A, A^\epsilon, A^{-\epsilon}, (A^\epsilon)^{-\epsilon}$, and $(A^{-\epsilon})^{\epsilon}$ for a square in $(\mathbb{R}^2, \\|\cdot\\|_2)$. Observe that $(A^{-\epsilon})^{\epsilon} \subseteq A$ and $A \subseteq (A^\epsilon)^{-\epsilon}$.[]{data-label="fig: square"}](A_epsilon.PNG) Figure \[fig: square\] illustrates the above lemma when $A$ is a square in $\real^2$ with the Euclidean distance metric. Using the above lemmas, we now establish the desired equality from : We have the sequence of inequalities $$\begin{aligned} \sup_{A ~\text{closed}} \mu(A) - \nu(A^{2\epsilon}) &\stackrel{(a)}\geq \sup_{A ~\text{closed}} \mu(A^{-\epsilon}) - \nu((A^{-\epsilon})^{2\epsilon})\\ &\stackrel{(b)}\geq \sup_{A ~\text{closed}} \mu(A^{-\epsilon}) - \nu(A^{\epsilon}).\end{aligned}$$ Here, $(a)$ follows because $A^{-\epsilon}$ is contained in the set of all closed sets by Lemma \[lemma: closed\]. Inequality $(b)$ follows using Lemma \[lemma: thick and thin\], since $$(A^{-\epsilon})^{2\epsilon} = [(A^{-\epsilon})^{\epsilon}]^\epsilon \subseteq A^\epsilon,$$ and so $\nu((A^{-\epsilon})^{2\epsilon}) \leq \nu(A^\epsilon)$. For the other direction, notice that $$\begin{aligned} \sup_{A ~\text{closed}} \mu(A^{-\epsilon}) - \nu(A^{\epsilon}) &\stackrel{(a)}\geq \sup_{A ~\text{closed}} \mu((A^\epsilon)^{-\epsilon}) - \nu((A^\epsilon)^{\epsilon})\\ &\stackrel{(b)}\geq \sup_{A ~\text{closed}} \mu(A) - \nu(A^{2\epsilon}).\end{aligned}$$ Here, $(a)$ follows because $A^\epsilon$ is a closed set according to Lemma \[lemma: closed\]. To see $(b)$, first note that $(A^\epsilon)^\epsilon = A^{2\epsilon}$, and so $\nu((A^\epsilon)^\epsilon) = \nu(A^{2\epsilon})$. Moreover, Lemma \[lemma: thick and thin\] states that $$A \subseteq (A^{\epsilon})^{-\epsilon},$$ and so $\mu(A) \leq \mu((A^{\epsilon})^{-\epsilon})$. This completes the proof. #### Comparison with Bhagoji et al. [@BhaEtal19]: We point out that a similar result was obtained recently in [@BhaEtal19]. A key difference is that the proof in [@BhaEtal19] was established for a larger hypothesis class of measurable sets $A$; i.e., the following equality was established: $$\begin{aligned} \sup_{A ~\text{measurable}} \mu(A^{-\epsilon}) - \nu(A^{\epsilon}) = \sup_{A ~\text{measurable}} \mu(A) - \nu(A^{2\epsilon}).\end{aligned}$$ As pointed out in [@BhaEtal19], we may restrict to the smaller hypothesis class of closed sets $A$ to obtain an inequality $$\begin{aligned} \sup_{A ~\text{closed}} \mu(A^{-\epsilon}) - \nu(A^{\epsilon}) \leq \sup_{A ~\text{measurable}} \mu(A) - \nu(A^{2\epsilon}).\end{aligned}$$ Our result shows that we can obtain an equality if the supremum over the right hand side is also taken over closed sets. Using closed sets and directly using Strassen’s theorem allows us to considerably simplify the technical details as compared to the proof from [@BhaEtal19]. From Theorem \[th\_01bound\], we have $$\begin{aligned} R^_\epsilon = \frac{1}{2}\left[1- \inf_{\pi\in\Pi(\mu, \nu)}\E_{(x,x')\sim\pi}[\1{\{d(x,x')>2\epsilon\}}] \right].\end{aligned}$$ For $p\geq 1$ and any $\pi\in\Pi(\mu, \nu)$, we have the following: $$\begin{aligned} \E_{(x,x')\sim\pi}[\1{\{d(x,x')>2\epsilon\}}] &= \E_{(x,x')\sim\pi}[\1{\{d(x,x')^p>(2\epsilon)^p\}}] \leq \E_{(x,x')\sim\pi}\left[\left( \frac{d(x,x')}{2\epsilon} \right)^p\right],\end{aligned}$$ where the last inequality follows from Markov’s inequality. Therefore, $$\begin{aligned} R^_\epsilon = \frac{1}{2}\left[1- \inf_{\pi\in\Pi(\mu, \nu)}\E_{(x,x')\sim\pi}\left[\left( \frac{d(x,x')}{2\epsilon} \right)^p\right]\right] = \frac{1}{2}\left[1- \left(\frac{{W_p}(p_0, p_1)}{2\epsilon}\right)^p \right].\end{aligned}$$ Optimal adversarial classifiers via optimal couplings {#sec: couplings} ===================================================== In this section, we explicitly compute the optimal risk and optimal classifier for a data perturbing adversary in some special cases. Instead of using $D_\epsilon$, we have shown in Corollary \[th\_01bound\] that the optimal adversarial risk can be lower-bounded using other well-understood metrics such as the $W_p$ distances. However, these bounds are often too loose to use in practice, and this motivates us to study the optimal cost $D_\epsilon$ directly. In this section, we show that in certain special cases, the optimal coupling corresponding to calculating $D_\epsilon$ may be explicitly evaluated. Furthermore, in these cases, we can exactly characterize the optimal classifier and the optimal risk in the presence of an adversary. Given measures $\mu$ and $\nu$ corresponding to the two (equally likely) data classes, the general strategy we employ consists of the following steps: - Propose a coupling $\pi$ between $\mu$ and $\nu$. - Using this coupling, obtain the upper bound $$\begin{aligned} D_\epsilon(\mu, \nu) \leq \E_{(x, x') \sim \pi} c_\epsilon(x, x').\end{aligned}$$ - Identify a closed set $A$ and compute a lower bound using $$\begin{aligned} D_\epsilon(\mu, \nu) \geq \mu(A^{-\epsilon}) - \nu(A^{\epsilon}).\end{aligned}$$ - Show that the lower and upper bounds match. This shows that the proposed coupling is optimal, and the sets $A$ and $A^c$ define the two regions of the optimal robust classifier. In the examples we consider, guessing the set $A$ corresponding to the optimal robust classifier is easy. The challenging part is proposing a coupling and establishing its optimality. Although we shall focus on real-valued random variables, some of our results will also naturally extend to higher dimensional distributions. Gaussian distributions with identical variances {#section: same variance} ----------------------------------------------- \[thm: same variance\] Let $p_0 = \cN(\mu_0, \sigma^2)$ and $p_1 = \cN(\mu_1, \sigma^2)$ in the metric space $(\mathbb{R}, \|\cdot\|)$. Assume $\mu_0 < \mu_1$ without loss of generality. Then the following hold: 1. If $\epsilon \geq \frac{\abs{\mu_0 - \mu_1}}{2}$, the optimal robust risk is $1/2$. A constant classifier achieves this risk. 2. If $\epsilon < \frac{\abs{\mu_0 - \mu_1}}{2}$, the optimal classifier satisfies $A = \left[\frac{\mu_1+\mu_0}{2}, +\infty\right]$, where $A$ is the region where the classifier decides $1$. The optimal risk in this case is $\int_{ \frac{\mu_1+\mu_0}{2}-\epsilon}^\infty p_0(x) dx = Q\left(\frac{\frac{\mu_1-\mu_0}{2} - \epsilon}{\sigma}\right)$. The lower bound of $1/2$ on the adversarial risk is trivially achieved by the constant classifier. Part (1) of the theorem states that for large enough $\epsilon$, this is the best one can do. For smaller values of $\epsilon$, the above theorem shows that the most robust classifier is the same as the MLE classifier. Surprisingly, for larger values of $\epsilon$ the MLE classifier has a risk larger than* $1/2$; i.e., it is worse than the constant classifier. We shall prove (1) first. Note that if $\epsilon \geq \frac{\mu_1 - \mu_0}{2}$, the transport map $T$ defined by $T(x) = x+(\mu_1 - \mu_0)$ transports $p_0$ to $p_1$. Moreover, this coupling satisfies $\abs{T(x) - x} = \mu_1- \mu_0 \leq 2\epsilon$. Thus, the optimal transport cost for this coupling is 0, and therefore so is $D_\epsilon(p_0, p_1)$. This gives the lower bound $$\begin{aligned} R_\epsilon^* \geq \frac{1}{2}.\end{aligned}$$ However, since the constant classifier achieves the lower bound, we conclude $R_\epsilon^* = 1/2$. For part (2), we consider the following strategy. As shown in Figure \[fig: coupling\_same\_var\], consider the distribution $\tilde p_1$ obtained by shifting $p_1$ to the left by $2\epsilon$. It is evident that the overlapping area between $\tilde p_1$ and $p_0$ maybe be translated by $2\epsilon$ so that it lies entirely with $p_1$. This means that the overlapping area may be transported at 0 cost. It is easily verified that the overlapping area contains $2Q\left(\frac{\frac{\mu_1-\mu_0}{2} - \epsilon}{\sigma}\right)$ mass, and so the total cost of transportation is at most $1-2Q\left(\frac{\frac{\mu_1-\mu_0}{2} - \epsilon}{\sigma}\right)$. Plugging this into the lower bound, we see that $$R_\epsilon^* \geq Q\left(\frac{\frac{\mu_1-\mu_0}{2} - \epsilon}{\sigma}\right).$$ Since this risk is achieved by the MLE classifier, we conclude that this is the optimal robust risk and the MLE classifier is the optimal robust classifier. Theorem \[thm: same variance\] can be easily extended to $d$-dimensional Gaussians with the same identity covariances. Our results may be summarized in the following theorem: \[thm: same variance d-dim\] Let $p_0 = \cN(\mu_0, \sigma^2 I_d)$ and $p_1 = \cN(\mu_1, \sigma^2 I_d)$ in the metric space $(\mathbb{R}, \|\|\cdot\|\|_2)$. Then the following hold: 1. If $\epsilon \geq \frac{\|\|\mu_0 - \mu_1\|\|_2}{2}$, the optimal robust risk is $1/2$. A constant classifier achieves this risk. 2. If $\epsilon < \frac{\|\|\mu_0 - \mu_1\|\|_2}{2}$, the optimal classifier is given by the following halfspace: $$\begin{aligned} \label{eq: A d-dim} A = \left\{x ~:~ (\mu_1 - \mu_0) \left(x - \frac{\mu_0+\mu_1}{2} \right) \geq 0 \right\}.\end{aligned}$$ #### Comparison to Bhagoji et al. [@BhaEtal19]: Bhagoji et al. also explore optimal classifiers for multivariate normal distributions. In fact, they show a more general version of our Theorems \[thm: same variance\] and \[thm: same variance d-dim\] by considering data distributions $\cN(\mu_0, \Sigma)$ and $\cN(\mu_2, \Sigma)$, and an adversary that perturbs within $L_p$ balls. In the following subsections, we shall generalize Theorem \[thm: same variance\] in a different way by considering various interesting examples of univariate distributions and identifying optimal couplings for these. Gaussians with arbitrary means and variances {#section: same mean} -------------------------------------------- We shall introduce a general coupling strategy and apply it to the special case of Gaussian random variables. Given two probability measures $\mu$ and $\nu$ on $\real$, our strategy consists of the following steps: - Partition the real line into $K \ge 1$ intervals $S_i, 1 \leq i \leq K$, and let the restriction of $\mu$ to $S_i$ be $\mu_i$. - Partition the real line into $K \ge 1$ intervals $T_i, 1 \leq i \leq K$, and let the restriction of $\nu$ to $T_i$ be $\nu_i$. - Transport mass from $\mu_i$ to $\nu_i$ such that $D_\epsilon(\mu_i, \nu_i) = 0$. (We shall define an appropriate notion of mass transport between measures with unequal masses shortly.) The transport maps used in these $K$ problems may be arbitrary; however, we shall often use versions of the monotone optimal transport map [@Vil03]. Our first result identifies a necessary and sufficient condition for $D_\epsilon(\mu, \nu) = 0$ for arbitrary measures on $\real^d$. \[thm: zerozero\] Let $\mu$ and $\nu$ be finite positive measures on $\real^d$ that are absolutely continuous with respect to the Lebesgue measure and have bounded supports. Then $D_\epsilon(\mu, \nu) = 0$ if and only if $W_\infty(\mu, \nu) \leq 2\epsilon$. Here, $W_\infty(\mu, \nu) = \lim_{p \to \infty} W_p(\mu, \nu)$. An alternate description of $W_\infty(\mu, \nu)$ as per Givens and Shortt [@GivSho84] is $$\begin{aligned} W_\infty(\mu, \nu) = \inf\{\delta > 0~:~\mu(A) \leq \nu(A^\delta)\text{ for all measurable~} A\}.\end{aligned}$$ Naturally, if $W_\infty(\mu, \nu) \leq 2\epsilon$, then $\mu(A) \leq \nu(A^{2\epsilon})$ for all closed sets $A$. Hence, $$\begin{aligned} D_\epsilon(\mu, \nu) = \sup_{A~\text{closed}} \mu(A) - \nu(A^{2\epsilon}) \leq 0.\end{aligned}$$ Since $D_\epsilon(\mu, \nu) \geq 0$, we conclude that $D_\epsilon(\mu, \nu) = 0$. For the reverse direction, suppose that $D_\epsilon(\mu, \nu) = 0$. This means there exists a sequence of couplings $\{\pi\}_{i \geq 1}$ such that $\E_{\pi_i} c_\epsilon(x,x') \to 0$ where $\pi_i \in \Pi(\mu, \nu)$. Equivalently, the probability $\alpha_i := \prob_{\pi_i}(\\|x-x'\\| > 2\epsilon) \to 0$ as $i \to \infty$. For any fixed $p \geq 1$, we have the inequality $$\begin{aligned} W_p(\mu, \nu) &\leq \left(\E_{\pi_i} \\|x-x'\\|^p\right)^{1/p}\\ &\leq \left((2\epsilon)^p (1-\alpha_i) + C^p \alpha_i \right)^{1/p},\end{aligned}$$ where $C = \sup \\|x - x'\\|$ which is a constant since $\mu$ and $\nu$ are assumed to have bounded supports. Pick $i_0$ large enough so that $(2\epsilon)^p(1-\alpha_{i_0}) > C^p \alpha_{i_0}$. Now letting $p \to \infty$ and calculating the limits, we conclude that $W_\infty(\mu, \nu) \leq 2\epsilon.$ Calculating $W_\infty$ is non-trivial in general. However, in the univariate case the optimal coupling for all $W_p$’s is known to be the degenerate coupling given by the monotone transport map from $\mu$ to $\nu$. This allows us to present a more concrete condition for checking when $D_\epsilon(\mu, \nu) = 0$ for measures over $\real$. Our result is contained in the following theorem: \[thm: zero\] Let $\mu$ and $\nu$ be finite positive measures on $\real$ that are absolutely continuous with respect to the Lebesgue measure with Radon-Nikodyn derivatives $f(\cdot)$ and $g(\cdot)$, respectively. The cumulative distribution function (cdf) of $\mu$ is defined as $F(x) = \mu((-\infty, x])$, and for $t \in [0,1]$, the inverse cdf (or quantile function) is defined as $F^{-1}(t) = \inf\{x \in \real ~:~ F(x) \geq t\}$. The cdf $G(\cdot)$ and inverse cdf $G^{-1}(\cdot)$ are defined analogously. Suppose that $\mu(\real) = \nu(\real) = U$. Then $D_\epsilon(\mu, \nu) = 0$ if and only if $\\|F^{-1} - G^{-1}\\|_\infty \leq 2\epsilon$. Consider the monotone transport map from $\mu$ to $\nu$ given by $T(x) = G^{-1}(F(x))$ for $x \in \real$ [@Vil03]. We shall show that this map satisfies $\|T(x) - x\| \leq 2\epsilon$ for all $x \in \real$, and so the optimal transport cost $D_\epsilon$ must be 0. To see this, note that $$\begin{aligned} T(x) - x &= G^{-1}(F(x)) - x\\ &\leq F^{-1}(F(x)) + 2\epsilon - x\\ &= 2\epsilon,\end{aligned}$$ where the last equality is in the $\mu$-almost sure sense. A similar argument shows $x-T(x) \leq 2\epsilon$, and thus $\|T(x) - x\| \leq 2\epsilon$. For the converse, suppose that there exists a $t_0 \in (0,1)$ such that $G^{-1}(t_0) - F^{-1}(t_0) > 2\epsilon$. Equivalently, $ G^{-1}(t_0) > F^{-1}(t_0) + 2\epsilon.$ Applying the $G$ function on both sides, $$\begin{aligned} t_0 > G(F^{-1}(t_0) + 2\epsilon).\end{aligned}$$ Consider the set $\tilde A = (-\infty, F^{-1}(t_0)]$. For this set, notice that $$\begin{aligned} \nu(\tilde A^{2\epsilon}) &= \nu((-\infty, F^{-1}(t_0)+2\epsilon])= G(F^{-1}(t_0) + 2\epsilon).\end{aligned}$$ Thus, we have $$\begin{aligned} D_\epsilon(\mu, \nu) &= \sup_{A} \mu(A) - \nu(A^{2\epsilon})\\ &\geq \mu(\tilde A) - \nu(\tilde A^{2\epsilon})\\ &= t_0 - G(F^{-1}(t_0) + 2\epsilon)\\ &> 0.\end{aligned}$$ A similar argument may also be made for the case when $F^{-1}(t_0) - G^{-1}(t_0) > 2\epsilon$. The above argument shows that monotone transport maps are optimal when $D_\epsilon = 0$. But monotone maps are not always optimal for the cost function $c_\epsilon(\cdot, \cdot)$. Consider for example the two measures $\cN(0,1)$ and $\cN(1,1)$, and $\epsilon = 0.1$. The monotone map in this case is $T(x) = x+1$, which gives unit cost of transportation. However, Theorem \[thm: same variance\] shows that the optimal transport cost in this example is strictly smaller than 1. Checking the condition $\\|F^{-1} - G^{-1}\\| \leq 2\epsilon$ is not always easy. We identify a simple but useful characterization in the following corollary: \[cor: dominate\] Let $\mu$ and $\nu$ be as in Theorem \[thm: zero\]. Suppose that for every $x \in \real$, we have $F(x) \geq G(x)$ and $F(x) \leq G(x+2\epsilon)$. Then $D_\epsilon(\mu, \nu) = 0.$ Applying the $G^{-1}$ function to both sides of both inequalities, we arrive at $$\begin{aligned} T(x) \leq x, \quad \text{ and } \quad T(x) \geq x+2\epsilon.\end{aligned}$$ This gives $\abs{T(x) - x} \leq 2\epsilon$ for all $x$, which concludes the proof. Before we examine the case of Gaussian distributions with identical means but different variances, we define notions of transport for measures with unequal masses. \[def: unequal\]\[Optimal transport cost for general measures\] Let $\mu$ and $\nu$ be as in Theorem \[thm: zero\]. Suppose that $\mu(\real) = U$ and $\nu(\real) = V$ and $U \leq V$. Let $\nu'$ be a measure on $\real$ with Radon-Nikodyn derivative $g'$ such that $\nu'(\real) = U$. We say $\nu' \subseteq \nu$, or $\nu'$ is contained in $\nu$, if $g(x) \geq g'(x)$ $\nu$-almost surely. Then the optimal transport cost $D_\epsilon(\mu, \nu)$ is defined as $$\begin{aligned} D_\epsilon(\mu, \nu) = \inf_{\nu' \subseteq \nu} D_\epsilon(\mu, \nu').\end{aligned}$$ Note that the amount of mass being moved is $\min(U,V) = U$. \[lemma: shift\] Let $\mu$ and $\nu$ be as in Theorem \[thm: zero\]. Assume that $\mu(\real) = U$ and $\nu(\real) = V$. Suppose the following conditions hold: 1. The support of $g$ is $[a, +\infty)$ and the support of $f$ is $[a + 2\epsilon, +\infty) =: [a', +\infty)$. 2. For all $x \in \real$, we have $g(x) \leq f(x+2\epsilon)$. Then $D_\epsilon(\mu, \nu) = 0$. A similar result holds if the supports of $g$ and $f$ are $(-\infty, -a]$ and $(-\infty, -a-2\epsilon]$, and $f(-x-2\epsilon) \geq g(-x)$. ![Figure illustrating the conditions in Lemma \[lemma: shift\]. []{data-label="fig:lemma shift"}](lemma_11.PNG) Consider the transport map $T(x) = x+2\epsilon$ applied to $\nu$. This map has the effect of “translating” the measure $\nu$ by $2\epsilon$ to the right. Call this translated measure $\eta$. Since $f(x) \geq g(x-2\epsilon)$, it is immediate that $\eta \subseteq \mu$. Moreover, the transport cost is $D_\epsilon(\nu, \eta) = 0$. This shows that $D_\epsilon(\mu, \nu) = 0$. \[lemma: scrunch\] Let $\mu$ and $\nu$ be as in Theorem \[thm: zero\]. Assume that $\mu(\real) = \nu(\real) = U$. Suppose the following conditions hold (see Figure \[fig:lemma scrunch\] for an illustration): 1. Let $a, b \in \real$ be such that $b-a > 2\epsilon$. The support of $f$ is $[a, b]$ and the support of $g$ is $[a', b] := [a+2\epsilon, b]$. 2. There exists $t \in [a, b]$ such that $f(x) \geq g(x)$ for $x \in [a, t)$, and $f(x) \leq g(x)$ for $x \in (t, b]$. 3. Let $\tilde g(x) = g(x+2\epsilon)$. Note that $\tilde g$ is supported on $[a, b-2\epsilon]$. There exists $\tilde t \in [a, b-2\epsilon]$ such that $f(x) \leq \tilde g(x)$ for $x \in [a, \tilde t)$, and $f(x) \geq \tilde g(x)$ for $x \in (\tilde t, b-2\epsilon]$. Then $D_\epsilon(\mu, \nu) = 0$. A mirror image of this result also holds: $D_\epsilon(\mu, \nu) = 0$ when the support of $f$ is $[b, c+2\epsilon]$, that of $g$ is $[b, c]$, and $f(x) \leq g(x)$ for $x \in [b, t)$ and $f(x) \geq g(x)$ for $x \in [t, c+2\epsilon]$; and for $\tilde g(x) = g(x+2\epsilon)$ we have $f(x) \geq \tilde g(x)$ for $x \in [b+2\epsilon, \tilde t)$ and $f(x) \leq g(x)$ for $x \in [\tilde t, c+2\epsilon]$ . ![Figure illustrating the conditions in Lemma \[lemma: scrunch\].[]{data-label="fig:lemma scrunch"}](lemma_12.PNG) We first prove $F(x) \geq G(x)$. To see this, consider $H(x) = F(x) - G(x)$. Since the derivative of $H$ is $f-g$, it must be that $H$ is increasing from $[a, t)$ and decreasing from $[t, b]$. Also, we have $H(a) = H(b) = 0$, and so the function $H$ must be non-negative in $[a,b]$. Equivalently, we must have $F(x) \geq G(x)$ for $x \in \real$. We now prove $F(x) \leq G(x+2\epsilon)$. Consider $\tilde H(x) = F(x) - \tilde G(x)$. By condition $(3)$, the derivative of this function is negative from $[a, \tilde t]$ and positive from $[\tilde t, b]$. Thus, the function $\tilde H$ decreases on the interval $[a, \tilde t)$ and increases on the interval $[\tilde t, b]$. Note that since $\tilde H(a) = \tilde H(b) = 0$, the function $\tilde H$ must be non-positive in the interval $[a, b]$. Thus, we have $F(x) \leq G(x+2\epsilon)$. Applying Corollary \[cor: dominate\] concludes the proof. Our next lemma is specific to Gaussian pdfs: \[lemma: normal intersection\] Let $f$ and $g$ be Gaussian pdfs corresponding to $\cN(\mu_1, \sigma_1^2)$ and $\cN(\mu_2, \sigma_2^2)$, respectively. Assume $\sigma_1^2 > \sigma_2^2$. Then the equation $f(x)-g(x) = 0$ has exactly two solutions $s_1 < \mu_2 < s_2$. By scaling and translating, we may set $\mu_2 = 0$ and $\sigma_2^2 = 1$. Solving $f(x) - g(x) = 0$ is equivalent to solving the quadratic equation $$\begin{aligned} \frac{x^2}{2} - \frac{(x-\mu_1)^2}{2\sigma_1^2} = \log \sigma_1.\end{aligned}$$ Simplifying, we wish to solve $$\begin{aligned} x^2(\sigma_1^2 - 1) + 2\mu_1 x - (\mu_1^2 + 2\sigma_1^2 \log \sigma_1) = 0.\end{aligned}$$ Since $\sigma_1 > 1$, the above quadratic has two distinct roots: one negative and one positive. This proves the claim. We shall call the two points where $f$ and $g$ intersect as the left and right intersection points. \[thm: same mean\] Let $\mu$ and $\nu$ be the Gaussian measures $\cN(0,\sigma_1^2)$ and $\cN(0,\sigma_2^2)$, respectively. Assume $\sigma_1^2 > \sigma_2^2$ without loss of generality. Let $m >0$ be such that $f(m+\epsilon) = g(m-\epsilon)$. Let $A = (-\infty, -m] \cup [m, +\infty)$. Then the optimal transport cost between $\mu$ and $\nu$ is given by $$\begin{aligned} D_\epsilon(\mu, \nu) &= \mu(A^{-\epsilon}) - \nu(A^{\epsilon}) \\ &= 2Q\left(\frac{m+\epsilon}{\sigma_1}\right) - 2Q\left(\frac{m-\epsilon}{\sigma_2}\right).\end{aligned}$$ The corresponding robust risk is $$\begin{aligned} R_\epsilon^* = \frac{1 - \mu(A^{-\epsilon}) + \nu(A^{\epsilon})}{2}.\end{aligned}$$ Moreover, if $\mu$ corresponds to hypothesis 1, the optimal robust classifier decides 1 on the set $A$. We shall propose a map that transports $\mu$ to $\nu$. (See Figure \[fig: gaussians\_centered\] for an illustration.) Consider $r \in (0, m-\epsilon)$ whose value will be provided later. First, we partition $\real$ into the five regions for $\mu$ and $\nu$, as shown in Table \[table: partition\]. For $\mu$, these partitions are $(-\infty, -m-\epsilon]$, $(-m-\epsilon, -r]$, $(-r, +r)$, $[r, m+\epsilon)$, and $[m+\epsilon, \infty)$. Let $\mu$ restricted to these intervals be $\mu_{--}$, $\mu_{-}$, $\mu_0$, $\mu_{+},$ and $\mu_{++}$, respectively. The measure $\nu$ is also partitioned five ways, but the intervals used in this case are slightly modified to be $(-\infty, -m+\epsilon]$, $(-m+\epsilon, -r]$, $(-r, r)$, $[r, m-\epsilon)$, and $[m-\epsilon, +\infty)$. Call $\nu$ restricted to these intervals $\nu_{--}$, $\nu_{-}$, $\nu_0$, $\nu_+$, and $\nu_{++}$, respectively. $\mu_{--}$ $(-\infty, -m-\epsilon]$ ------------ -------------------------- $\mu_{-}$ $(-m-\epsilon, -r]$ $\mu_{0}$ $(-r, +r)$ $\mu_{+}$ $[r, m+\epsilon)$ $\mu_{++}$ $[m+\epsilon, \infty)$ : The real line is partitioned into five regions for $\mu$ and $\nu$, as shown in the table.[]{data-label="table: partition"} $\nu_{--}$ $(-\infty, -m+\epsilon]$ ------------ -------------------------- $\nu_{-}$ $(-m+\epsilon, -r]$ $\nu_{0}$ $(-r, +r)$ $\nu_{+}$ $[r, m-\epsilon)$ $\nu_{++}$ $[m-\epsilon, \infty)$ : The real line is partitioned into five regions for $\mu$ and $\nu$, as shown in the table.[]{data-label="table: partition"} The transport plan from $\mu$ to $\nu$ will consist of five maps transporting $\mu_{--} \to \nu_{--}$, $\mu_- \to \nu_-$, $\mu_0 \to \nu_0$, $\mu_+ \to \nu_+$, and $\mu_{++} \to \nu_{++}$. In each case, we plan to show that $D_\epsilon(\mu_{}, \nu_{}) = 0$, where $$ ranges over all possible subscripts. Note that these measures do not necessarily have identical masses, and thus by Definition \[def: unequal\], we are transporting a quantity of mass equal to the minimum mass among the two measures. For this reason, even though the transport cost is $D_\epsilon(\mu_, \nu_) = 0$, it does not mean $D_\epsilon(\mu, \nu) = 0$. Consider $\mu_{++}$ and $\nu_{++}$. We have $f(m+\epsilon) = g(m-\epsilon)$ by the choice of $m$. We argue that for any $t \geq 0$, we must have $f(m+\epsilon+t) \geq g(m-\epsilon+t)$. This is because any two Gaussian pdfs can intersect in at most two points. By Lemma \[lemma: normal intersection\], the $\epsilon$-shifted Gaussian pdfs $f(x+\epsilon)$ and $g(x-\epsilon)$ have $m$ as their right intersection point, and there are no additional points of intersection to the right of $m$. Since the tail of $f$ is heavier, it means that $f(m+\epsilon+t) \geq g(m-\epsilon + t)$ for all $t \geq 0$. By Lemma \[lemma: shift\], we can now conclude $D_\epsilon(\mu_{++}, \nu_{++}) = 0$. A similar argument also shows $D_\epsilon(\mu_{--}, \nu_{--}) = 0$. Before we consider $\mu_{-}$ and $\nu_{-}$, we first define $r$ as follows: Pick $r>0$ such that $\mu([-m-\epsilon, -r)) = \nu([-m+\epsilon, -r))$. To see that such an $r$ must exist, consider the functions $a(t) := \mu([-m-\epsilon, t))$ and $b(t):=\nu([-m+\epsilon, t))$ as $t$ ranges over $(-m+\epsilon, 0)$. When $t = -m+\epsilon$, we have $a(t) > b(t) = 0$. When $t=0$, we have $a(t) = 1/2 - \mu_{--}(\real) < b(t) = 1/2 - \nu_{--}(\real)$. Thus, there must exist a $t_0 \in (-m+\epsilon, 0)$ such that $a(t_0) = b(t_0)$. Pick the smallest (i.e., the leftmost) such $t_0$, and set $-r = t_0$. Call $f(\cdot)$ restricted to $[-m-\epsilon, -r)$ and $g(\cdot)$ restricted to $[-m+\epsilon, -r)$ as $f_-$ and $g_-$, respectively, and their corresponding cdfs $F_-$ and $G_-$, respectively. We claim that $\mu_-$ and $\nu_-$ satisfy all three conditions from Lemma \[lemma: scrunch\]. Since the supports of $f_-$ and $g_-$ are $[-m-\epsilon, -r)$ and $[-m+\epsilon, -r)$, condition $(1)$ is immediately verified. To check condition $(2)$, we break up the interval $[-m-\epsilon, -r)$ into two parts: $[-m-\epsilon, -s)$ and $[-s, -r)$, where $s$ is such that $f(-s) = g(-s)$. Observe that $f_- \geq g_-$ on $[-m-\epsilon, -s)$, whereas $f_- \leq g_-$ on $[-s, -r)$. This shows that condition $(2)$ is satisfied. We have $g_-(-m+\epsilon) = f_-(-m-\epsilon)$. Again, using Lemma \[lemma: normal intersection\] the $2\epsilon$-shifted Gaussian pdf $f(x-2\epsilon)$ and $g(x)$ have $-m+\epsilon$ as their left intersection point, and the right intersection point is to the right of 0. Thus, we have $f(x-2\epsilon) \leq g(x)$ for all $x \in [-m+\epsilon, 0] \supseteq [-m+\epsilon, r).$ Using this domination, we conclude that $f_- \leq \tilde g_-$ in the interval $[-m-\epsilon, -r-2\epsilon)$ and $f_- \geq g_- = 0$ in the interval $(-r-2\epsilon, -r]$, and so condition $(3)$ is satisfied. Applying Lemma \[lemma: scrunch\], we conclude $D_\epsilon(\mu_-, \nu_-) = 0$. An essentially identical argument may be used to show $D_\epsilon(\mu_+, \nu_+) = 0$. The minor difference being that $r$ is chosen to satisfy $\mu([r, m+\epsilon)) = \nu([r, m-\epsilon))$, and the mirror image of Lemma \[lemma: scrunch\] is applied. Finally, consider the interval $(-r, +r)$. In this interval, $f(x) \leq g(x)$ for every point. Hence, a transport map from $\mu_0$ to $\nu_0$ is obtained by simply considering the identity function. Any remaining mass in $\mu$ is moved to $\nu$ arbitrarily, incurring a cost of at most 1 per unit mass. The total cost of transport is then upper-bounded by $$\begin{aligned} D_\epsilon(\mu, \nu) &\leq 1 - \left[ \min(\mu_{--}, \nu_{--}) + \min(\mu_{-}, \nu_{-}) + \min(\mu_{0}, \nu_{0}) + \min(\mu_{+}, \nu_{+}) + \min(\mu_{++}, \nu_{++})\right]\\ &= 1 - \left[ \nu_{--} + \mu_{-} + \mu_{0} + \mu_{+} + \nu_{++}\right]\\ &= 1 - \mu([-m-\epsilon, m+\epsilon]) - 2\nu([m-\epsilon, \infty))\\ &= \mu(A^{-\epsilon}) - \nu(A^{\epsilon})\\ %&= 1 - \left(1 - 2Q\left(\frac{m+\epsilon}{\sigma_1}\right)\right) - 2Q\left(\frac{m-\epsilon}{\sigma_2}\right)\\ &= 2Q\left(\frac{m+\epsilon}{\sigma_1}\right) - 2Q\left(\frac{m-\epsilon}{\sigma_2}\right).\end{aligned}$$ where for brevity we have denoted $\mu_{}(\real)$ as $\mu_$. However, we also have $$\begin{aligned} D_\epsilon(\mu, \nu) &\geq \mu(A^{-\epsilon}) - \nu(A^{\epsilon}).\end{aligned}$$ The lower and upper bounds match and this concludes the proof. The robust risk $R_\epsilon^$ is given by Theorem \[th\_01bound\]. The robust risk of the classifier that decides $1$ on the set $A$ is easily seen to be $R_\epsilon^$. ![Optimal transport coupling for centered Gaussian distributions $\mu$ and $\nu$. As in the proof of Theorem \[thm: same mean\], we divide the real line into five regions. The transport plan from $\mu$ to $\nu$ consists of five maps transporting $\mu_{--} \to \nu_{--}$ (blue regions to the left), $\mu_- \to \nu_-$ (orange regions to the left), $\mu_0 \to \nu_0$ (green regions in the middle), $\mu_+ \to \nu_+$ (orange regions to the right), and $\mu_{++} \to \nu_{++}$ (blue regions to the right).[]{data-label="fig: gaussians_centered"}](gaussians_centered_transport_plan.PNG) We now extend the above proof strategy to demonstrate the optimal coupling for Gaussians with arbitrary means and arbitrary variances. Our main result is the following: \[thm: diff mean\] Let $\mu$ and $\nu$ be Gaussian measures $\cN(\mu_1, \sigma_1^2)$ and $\cN(\mu_2, \sigma_2^2)$ respectively. Assume $\sigma_1^2 > \sigma_2^2$ without loss of generality. Let $m_1, m_2 >0$ be such that $f(-m_1-\epsilon) = g(-m_1+\epsilon)$ and $f(m_2+\epsilon) = g(m_2-\epsilon)$. Let $A = (-\infty, -m_1] \cup [m_2, \infty)$. Then the optimal transport cost between $\mu$ and $\nu$ is given by $$\begin{aligned} D_\epsilon(\mu, \nu) = \mu(A^{-\epsilon}) - \nu(A^{\epsilon}).\end{aligned}$$ Consequently, the robust risk is given by $$\begin{aligned} R_\epsilon^ = \frac{1}{2} (1 - \mu(A^{-\epsilon}) + \nu(A^{\epsilon})).\end{aligned}$$ If $\mu$ corresponds to hypothesis 1, the optimal robust classifier decides 1 on the set $A$. As in the proof of Theorem \[thm: same mean\], we shall divide the real line into five regions as shown in Table \[table: diff\_meanvar\] where we define $r_1$ and $r_2$ shortly. Using an identical strategy as in Theorem \[thm: same mean\], we conclude $D_\epsilon(\mu_{--}, \nu_{--}) = D_\epsilon(mu_{++}, \nu_{++}) = 0$. Define $r_1$ as the leftmost point where $\mu([-m_1-\epsilon, r_1)) = \nu([-m_1+\epsilon, r_1))$. Similarly, define $r_2$ to be the rightmost point such that $\mu([r_2, m_2+\epsilon)) = \nu([r_2, m_2-\epsilon))$. We shall now prove $D_\epsilon(\mu_-, \nu_-) = 0$ by using Lemma \[lemma: scrunch\]. Verifying conditions $(1)$ and $(2)$ is exactly as in that of Theorem \[thm: same mean\]. The novel component of this proof is verifying condition $(3)$, since the domination used in the proof of Theorem \[thm: same mean\] does not work in this case due to the asymmetry. Consider the pdfs $f_-(x)$ and $g_-(x+2\epsilon)$. These two pdfs, being restrictions of Gaussian pdfs to suitable intervals, may only intersect in at most two points. One of these points of intersection is $-m_1-\epsilon$ by the choice of $m_1$, so there can be at most one other point of intersection in the interval $[-m_1-\epsilon, -r_1-2\epsilon]$. Note that there may be no point of intersection in this interval. However, the key observation is that in both cases, condition $(3)$ continues to be satisfied. To see this, suppose that there is a point of interaction $\tilde t$. In this case, $f_- \leq \tilde g_-$ in $[-m_1-\epsilon, \tilde t)$, and $f_- \geq g_-$ in $(\tilde t, -r_1]$. If there is no point of intersection, then $f_- \leq \tilde g_-$ in $[-m_1-\epsilon, -r_1-2\epsilon)$, and $f_- \geq g_- = 0$ in $(-r_1-2\epsilon, -r_1]$. This verifies condition $(3)$. Using Lemma \[lemma: scrunch\], we conclude $D_\epsilon(\mu_-, \nu_-) = 0$. An identical approach gives $D_\epsilon(\mu_+, \nu_+) = 0$. Since $f(x) \leq g(x)$ for all points in the interval $(-r_1, r_2)$, the identity map may be used to conclude $D_\epsilon(\mu_0, \nu_0) = 0$. Any remaining mass in $\mu$ is moved to $\nu$ arbitrarily, incurring a cost of at most 1 per unit mass. The total cost of transport is then upper-bounded by $$\begin{aligned} D_\epsilon(\mu, \nu) &\leq 1 - \left[ \min(\mu_{--}, \nu_{--}) + \min(\mu_{-}, \nu_{-}) + \min(\mu_{0}, \nu_{0}) + \min(\mu_{+}, \nu_{+}) + \min(\mu_{++}, \nu_{++})\right]\\ &= 1 - \left[ \nu_{--} + \mu_{-} + \mu_{0} + \mu_{+} + \nu_{++}\right]\\ &= 1 - \mu([-m_1-\epsilon, m_2+\epsilon]) - \nu((-\infty, -m_1+\epsilon)) - \nu([m_2-\epsilon, \infty))\\ &= \mu(A^{-\epsilon}) - \nu(A^{\epsilon}), %&= 1 - \left(1 - 2Q\left(\frac{m+\epsilon}{\sigma_1}\right)\right) - 2Q\left(\frac{m-\epsilon}{\sigma_2}\right)\\ %&= 2Q\left(\frac{m+\epsilon}{\sigma_1}\right) - 2Q\left(\frac{m-\epsilon}{\sigma_2}\right).\end{aligned}$$ where for brevity we have denoted $\mu_{}(\real)$ as $\mu_$. The rest of the proof is identical to that of Theorem \[thm: same mean\]. $\mu_{--}$ $(-\infty, -m_1-\epsilon]$ ------------ ---------------------------- $\mu_{-}$ $(-m_1-\epsilon, -r_1]$ $\mu_{0}$ $(-r_1, +r_2)$ $\mu_{+}$ $[r_2, m_2+\epsilon)$ $\mu_{++}$ $[m_2+\epsilon, \infty)$ : The real line is partitioned into five regions for $\mu$ and $\nu$ as shown in the table.[]{data-label="table: diff_meanvar"} $\nu_{--}$ $(-\infty, -m_1+\epsilon]$ ------------ ---------------------------- $\nu_{-}$ $(-m_1+\epsilon, -r_1]$ $\nu_{0}$ $(-r_1, +r_2)$ $\nu_{+}$ $[r_2, m_2-\epsilon)$ $\nu_{++}$ $[m_2-\epsilon, \infty)$ : The real line is partitioned into five regions for $\mu$ and $\nu$ as shown in the table.[]{data-label="table: diff_meanvar"} Beyond Gaussian examples ------------------------ The coupling strategy for Gaussian random variables can also be applied to other univariate examples that share some similarities with the Gaussian case. To illustrate, we describe the optimal classifier and optimal coupling for uniform distributions and triangular distributions. The precise proof details in these cases may be reconstructed from the proofs of Theorems \[thm: same mean\] and \[thm: diff mean\]. \[thm: uniform coupling\] Let $\mu$ and $\nu$ be uniform measures on closed intervals $I$ and $J$ respectively. Without loss of generality, we assume $\|I\| \leq \|J\|$. Then the optimal robust risk is $\nu(I^{2\epsilon})$ and the optimal classifier is given by $A = I^\epsilon$. Like in the proof for Theorem \[thm: same mean\], we prove Theorem \[thm: uniform coupling\] by partitioning the real line into several regions for $\mu$ and $\nu$, and transporting mass between these regions. Figure \[fig: coupling\_uniform\] shows the optimal coupling for the case when $I^{2\epsilon}\subseteq J$. We first prove a lower bound. Choose the set $A = I$, we have that $$\begin{aligned} D_\epsilon(\mu, \nu) \geq \mu(A) - \nu(A^{2\epsilon}) = 1 - \nu(I^{2\epsilon}).\end{aligned}$$ To establish the upper bound, we need to find a coupling that transports $\mu$ to $\nu$ such that the cost of transportation is bounded above by $1-\nu(I^{2\epsilon})$. Let $I = [i_1, i_2]$ and $J = [j_1, j_2]$. As shown in Figure \[fig: coupling\_uniform\], we pick $t_1 \in [i_1, i_2]$ such that $\nu([i_1 - 2\epsilon, t_1]) = \mu([i_1, t_1])$. Similarly, pick $t_2 \in [t_1, i_2]$ such that $\mu([t_2, i_2]) = \nu([t_2, i_2+2\epsilon])$. We now present a plan to transport $\nu$ to $\mu$. This plan consists of four mini-plans: 1. First, transport the mass $\nu([i_1-2\epsilon, t_1])$ to $\mu([i_1, t_1])$ using a monotone transport map. 2. Then transport the mass from $\nu([t_2, i_2+2\epsilon])$ to $\mu([t_2, i_2])$ using a monotone transport map. 3. Keep any mass in $[t_1, t_2]$ in its place. 4. Move any remaining mass in $\nu$ arbitrarily to the necessary places in $\mu$. The key point to note is that in maps (1) and (2), the total distance moved by every unit of mass is at most $2\epsilon$. The proof of this part is along similar lines to that of Theorem \[thm: diff mean\]. Thus, the transport cost in steps (1) and (2) is 0. Naturally, the transport cost in (3) is 0. This means that all the mass in the interval $[i_1-2\epsilon, i_2+2\epsilon]$ can be transported into $\mu$ for zero cost. The total cost of transportation is therefore at most $1-\nu([i_1-2\epsilon, i_2+2\epsilon])$, which matches our lower bound. It is easily checked that the error attained by the proposed classifier also matches the bound, which completes the proof. \[thm: triangle coupling\] Denote a triangular distribution with support $[m-\delta, \mu+\delta]$ as $\Delta(m, \delta)$. Let $\mu$ and $\nu$ correspond to the triangular distributions $\Delta(m_1, \delta_1)$ and $\Delta(\mu_2, \delta_2)$ with pdfs $f$ and $g$ respectively. Without loss of generality, assume $\delta_1 < \delta_2$. Let $l < m_1 < r$ be such that $f(l+\epsilon) = g(l-\epsilon)$ and $f(r-\epsilon) = g(r+\epsilon)$. (In case of multiple such points, pick $l$ to be the largest among all such points, and $r$ to smallest.) Let $A$ be the closed set $[l,r]$. Then $D_\epsilon(\mu, \nu) = \mu(A^{-\epsilon}) -\nu(A^\epsilon)$, the robust risk is $$\begin{aligned} R_\epsilon^* = \frac{1-\mu(A^{-\epsilon}) + \nu(A^\epsilon)}{2},\end{aligned}$$ and if $\mu$ corresponds to hypothesis 1, then the optimal robust classifier decides 1 on $A$. We omit all proof details and point to Figure \[fig: triangular\] which shows the coupling in a special case. ![Optimal transport coupling for triangular distributions $\mu$ and $\nu$. As in the proof of Theorem \[thm: diff mean\], we divide the real line into five regions. The transport plan from $\mu$ to $\nu$ consists of five maps transporting $\mu_{--} \to \nu_{--}$ (blue regions to the left), $\mu_- \to \nu_-$ (orange regions to the left), $\mu_0 \to \nu_0$ (green regions in the middle), $\mu_+ \to \nu_+$ (orange regions to the right), and $\mu_{++} \to \nu_{++}$ (blue regions to the right).[]{data-label="fig: triangular"}](triangular.PNG) Some results for continuous loss functions {#gen loss} ========================================== It is natural to ask if the results for $0-1$ loss may be generalized for continuous losses. In this section, we present adversarial risk bounds in regression-like settings with continuous losses, and investigate Questions \[q\_1\] and \[q\_2\] in light of these bounds. Throughout this section, we shall assume that parameters and data are real-valued vectors; i.e., $\cW \subseteq \real^{\dim(\cW)}$ and $\cZ \subseteq \real^{\dim(\cZ)}$. Note that the adversary’s perturbation ball around $x$ is according to a metric $d(\cdot, \cdot)$, which need not be the same as the $L_2$ metric. Optimal adversarial risk {#sec: opt risk gen loss} ------------------------ In this section, we prove a lower bound on optimal adversarial risk for for the data perturbing adversary and an upper bound optimal adversarial risk for for the distribution perturbing adversary. As noted earlier, the lower bound is also valid for distribution perturbing adversary and the upper bound is also valid for data perturbing adversary. For the lower bound, we consider loss functions that are convex with respect to the input $x$, as defined below. We say that the loss function $\ell:\cZ\times\cW\to\mathbb{R}^+$ is convex with respect to the input if it satisfies the following condition. $$\begin{aligned} \label{eq_convex} \ell((x', y), w) - \ell((x, y), w) &\geq \langle\nabla_x \ell(z, w), x'-x\rangle.\end{aligned}$$ \[th\_bound\_generalloss\] The adversarial risk for a loss function satisfying is bounded as follows. $$\begin{aligned} R^_\epsilon \geq R^_0 + \inf_{w \in \cW} \E_z\left[\sup_{d(x,x')\leq\epsilon} \langle \nabla_x\ell((x,y),w), x'-x \rangle \right].\end{aligned}$$ In the special case when the distance function $d$ corresponds to the norm $\lVert \cdot\rVert_\text{adv}$ i.e. $d(x,x')=\lVert x-x'\rVert_\text{adv}$, we can further tighten the result of Theorem \[th\_bound\_generalloss\] as follows. \[cor\_dualnorm\] In the setting of Theorem \[th\_bound\_generalloss\], if $d(x,x')=\lVert x-x'\rVert_\text{adv}$ for $x,x'\in\cX$, then the following bound holds: $$\begin{aligned} R^_\epsilon \geq R^_0 + \epsilon\inf_{w\in\mathcal{W}}\E_{z}[\lVert \nabla_x\ell((x,y),w) \rVert_\text{adv}],\end{aligned}$$ where $\lVert \cdot\rVert_\text{adv}$ is the dual norm of $\lVert \cdot\rVert_\text{adv}$. Since $w^_\epsilon$ is sub-optimal for minimizing standard risk, we have $$\begin{aligned} \E_z[\ell((x, y),w^_\epsilon)]\geq \E_z[\ell((x, y),w^_0)].\end{aligned}$$ Hence, $$\begin{aligned} R^_\epsilon - R^_0 &= \E_z\left[\sup_{d(x,x')\leq\epsilon}\ell((x', y),w^_\epsilon)\right] - \E_z[\ell((x, y),w^_0)]\\ &\geq \E_z\left[\sup_{d(x,x')\leq\epsilon}\ell((x', y),w^_\epsilon)\right] - \E_z[\ell((x, y),w^_\epsilon)]\\ &= \E_z\left[\sup_{d(x,x')\leq\epsilon}\ell((x', y),w^_\epsilon) - \ell((x, y),w^_\epsilon)\right]\\ &\geq \E_z\left[\sup_{d(x,x')\leq\epsilon} \langle \nabla_x\ell((x,y),w^_\epsilon), x'-x \rangle \right],\\ &\geq \inf_{w \in \cW} \E_z\left[\sup_{d(x,x')\leq\epsilon} \langle \nabla_x\ell((x,y),w^_\epsilon), x'-x \rangle \right].\end{aligned}$$ From the proof of Theorem \[th\_bound\_generalloss\], we have $$\begin{aligned} R^_\epsilon - R^_0 &\geq \E_z\left[\sup_{d(x,x')\leq\epsilon}\langle\nabla_x\ell((x', y),w^_\epsilon), x'-x \rangle \right]. \end{aligned}$$ Under the condition that $d(x,x') = \\|x-x'\\|_\text{adv}$, $$\begin{aligned} &\sup_{d(x,x')\leq\epsilon}\langle\nabla_x\ell((x', y),w^_\epsilon), x'-x \rangle\\ &= \sup_{\\|\delta\\|_\text{adv}\leq\epsilon}\langle\nabla_x\ell((x', y),w^_\epsilon), \delta \rangle\\ &= \epsilon \\|\nabla_x\ell((x', y),w^_\epsilon)\\|_\text{adv}.\end{aligned}$$ Now, we prove an upper bound for the adversarial risk for a distribution perturbing adversary. As noted earlier, this upper bound also holds for the data perturbing adversary. We make the following assumption on the loss function: We say that the loss function $\ell:\cZ\times\cW\to\mathbb{R}^+$ is $L_x(w)$-Lipschitz with respect to the input if it satisfies the following condition. $$\begin{aligned} \label{eq_lipschitz} \lvert\ell((x',y),w) - \ell((x,y), w)\rvert &\leq L_x(w)\lVert x'-x \rVert.\end{aligned}$$ \[th\_lowerbound\_risk\] The adversarial risk for a distribution perturbing adversary satisfies $\widehat R^_\epsilon \leq R^_0 + \epsilon L_x(w_0^).$ Naturally, we also have $R^_\epsilon \leq R^_0 + \epsilon L_x(w_0^).$ The proof of this result uses an optimal transport idea from [@TovJog18]. Suppose that the infimum in in equation is attained at $\widehat w^_\epsilon$, and the corresponding maximizer in equation is $\rho^_\epsilon$. Then $$\begin{aligned} \widehat R^_\epsilon - R^_0 &= \E_{Z \sim \rho^_\epsilon} \ell(Z, \widehat w^_\epsilon) - \E_{Z' \sim \rho_0} \ell(Z',w^_0)\\ &\stackrel{(a)}\leq \E_{Z \sim \rho^_\epsilon} \ell(Z, \widehat w^_0) - \E_{Z' \sim \rho_0} \ell(Z',w^_0)\\ &\stackrel{(b)}= \E_{(Z,Z') \sim \Pi(\rho^_\epsilon, \rho_0)} \ell(Z', w_0^) - \ell(Z, w_0^)\\ &\stackrel{(c)}\leq [\E_{(Z,Z') \sim \Pi(\rho^_\epsilon, \rho_0)} d(Z,Z')] \cdot L_{w_0^}.\end{aligned}$$ Here, (a) follows from the definition of $\widehat w^_\epsilon$, (b) follows from linearity of expectation since $\Pi(\rho_\epsilon^, \rho_0)$ is a coupling of $(X, Z')$ that preserves the marginals, and (c) follows from the Lipschitz assumption. Picking the optimal possible coupling corresponding to the Wasserstein distance between $\rho_0$ and $\rho^_\epsilon$, we conclude that $$\begin{aligned} \widehat R^_\epsilon - R^_0 \leq \epsilon L_x(w_0^).\end{aligned}$$ Optimal adversarial classifier {#sec: opt classifier} ------------------------------ In Sections \[sec: opt risk 01 loss\] and \[sec: opt risk gen loss\], we looked at Question \[q\_1\] and showed that the adversarial risk can be strictly lower-bounded as a function of adversarial budget $\epsilon$. In this section, we tackle Question \[q\_2\] and analyze how $w^_\epsilon$ or $\widehat w^_\epsilon$ may deviate from $w^_0$ with $\epsilon$. For the case of $0-1$ loss, the optimal classifier can change drastically even with small change in the adversarial budget $\epsilon$. For instance, consider the setting of Theorem \[thm: same variance\]. When $\epsilon$ changes from being less than $\frac{\abs{\mu_0 - \mu_1}}{2}$ to greater than $\frac{\abs{\mu_0 - \mu_1}}{2}$, the optimal classifier changes from a halfspace to a constant classifier. Studying the 0-1 loss is hard because closed sets are not parametrized easily. Hence we focus on the case of convex loss functions to derive bounds in this section. Since our proof technique uses the upper and lower bounds for adversarial losses obtained in Section \[sec: opt risk gen loss\], the bounds for deviation of $w^_\epsilon$ and $\widehat w^_\epsilon$ are identical. Now, we prove a theorem on how much the optimal classifier can change in the presence of an adversary. \[th\_opt\_clsfr\_deviation\] For a loss function $\ell$ that satisfies , the following result holds for sufficiently small $\epsilon$: $$\begin{aligned} \label{eq_opt_clsfr_deviation} \lVert w^_\epsilon - w^_0 \rVert \leq \sqrt{ \frac{\epsilon L_x(w^_0)} {\lambda_{min} \left(\nabla^2_w R_0(\ell, w^_0)\right)} }.\end{aligned}$$ With regards to Theorem \[th\_opt\_clsfr\_deviation\], we note that if $\ell$ is strongly convex with respect to $w$, then holds for any $\epsilon$. For instance, if $\ell_2$ regularization is used, then the overall loss function will naturally be strongly convex with respect to $w$. We take the Taylor expansion of $R_0(\ell, w)$ at $w^_0$. $$\begin{aligned} R_0(\ell, w^_\epsilon) &= R_0(\ell, w^_0) + \nabla_w R_0(\ell, w^_0)^T(w^_\epsilon - w^_0) \\ &+ (w^_\epsilon - w^_0) ^T \nabla^2_w R_0(\ell, w^_0) (w^_\epsilon - w^_0) \\ &+ \mathcal{O}\left( \lVert w^_\epsilon - w^_0 \rVert^3 \right).\end{aligned}$$ Now, $\nabla_w R_0(\ell, w^_0) = 0$ since $w^_0$ is the minimizer of $R_0(\ell, w^_0)$. Therefore for small enough $\epsilon$, we have the following. $$\begin{aligned} \epsilon L_x(w^_0) &\stackrel{(a)}\geq R^_\epsilon - R^_0\\ &\stackrel{(b)}\geq R_0(\ell, w^_\epsilon) - R_0(\ell, w^_0)\\ &\stackrel{(c)}\geq (w^_\epsilon - w^_0) ^T \nabla^2_w R_0(\ell, w^_0) (w^_\epsilon - w^_0)\\ &\geq \lambda_{min}(\nabla^2_w R_0(\ell, w^_0)) \lVert w^_\epsilon - w^_0 \rVert^2.\end{aligned}$$ Here, (a) follows from Theorem \[th\_lowerbound\_risk\], (b) follows from the fact that $w^_\epsilon$ is sub-optimal for minimizing $R_0(\ell, w)$, and (c) follows from the Taylor expansion presented above (for small enough $\epsilon$). Experiments {#sec: expts} =========== In this section, we present lower bounds on the optimal adversarial risk for empirical distributions derived from several real world datasets. [0.35]{} ![Lower bounds on adversarial risk computed using Theorem \[th\_01bound\]. The curves with $\sigma=0$ gives the exact optimal risk for empirical distributions, while the other cuvers give lower bounds on the optimal risk for Gaussian mixtures based on the empirical distributions using the coupling in Theorem \[thm: same variance d-dim\].[]{data-label="fig: datasets"}](noise_plots/cifar_l2.png "fig:"){width="\textwidth"} [0.35]{} ![Lower bounds on adversarial risk computed using Theorem \[th\_01bound\]. The curves with $\sigma=0$ gives the exact optimal risk for empirical distributions, while the other cuvers give lower bounds on the optimal risk for Gaussian mixtures based on the empirical distributions using the coupling in Theorem \[thm: same variance d-dim\].[]{data-label="fig: datasets"}](noise_plots/cifar_linf.png "fig:"){width="\textwidth"} [0.35]{} ![Lower bounds on adversarial risk computed using Theorem \[th\_01bound\]. The curves with $\sigma=0$ gives the exact optimal risk for empirical distributions, while the other cuvers give lower bounds on the optimal risk for Gaussian mixtures based on the empirical distributions using the coupling in Theorem \[thm: same variance d-dim\].[]{data-label="fig: datasets"}](noise_plots/mnist_l2.png "fig:"){width="\textwidth"} [0.35]{} ![Lower bounds on adversarial risk computed using Theorem \[th\_01bound\]. The curves with $\sigma=0$ gives the exact optimal risk for empirical distributions, while the other cuvers give lower bounds on the optimal risk for Gaussian mixtures based on the empirical distributions using the coupling in Theorem \[thm: same variance d-dim\].[]{data-label="fig: datasets"}](noise_plots/mnist_linf.png "fig:"){width="\textwidth"} [0.35]{} ![Lower bounds on adversarial risk computed using Theorem \[th\_01bound\]. The curves with $\sigma=0$ gives the exact optimal risk for empirical distributions, while the other cuvers give lower bounds on the optimal risk for Gaussian mixtures based on the empirical distributions using the coupling in Theorem \[thm: same variance d-dim\].[]{data-label="fig: datasets"}](noise_plots/fmnist_l2.png "fig:"){width="\textwidth"} [0.35]{} ![Lower bounds on adversarial risk computed using Theorem \[th\_01bound\]. The curves with $\sigma=0$ gives the exact optimal risk for empirical distributions, while the other cuvers give lower bounds on the optimal risk for Gaussian mixtures based on the empirical distributions using the coupling in Theorem \[thm: same variance d-dim\].[]{data-label="fig: datasets"}](noise_plots/fmnist_linf.png "fig:"){width="\textwidth"} [0.35]{} ![Lower bounds on adversarial risk computed using Theorem \[th\_01bound\]. The curves with $\sigma=0$ gives the exact optimal risk for empirical distributions, while the other cuvers give lower bounds on the optimal risk for Gaussian mixtures based on the empirical distributions using the coupling in Theorem \[thm: same variance d-dim\].[]{data-label="fig: datasets"}](noise_plots/svhn_l2.png "fig:"){width="\textwidth"} [0.35]{} ![Lower bounds on adversarial risk computed using Theorem \[th\_01bound\]. The curves with $\sigma=0$ gives the exact optimal risk for empirical distributions, while the other cuvers give lower bounds on the optimal risk for Gaussian mixtures based on the empirical distributions using the coupling in Theorem \[thm: same variance d-dim\].[]{data-label="fig: datasets"}](noise_plots/svhn_linf.png "fig:"){width="\textwidth"} For the case of empirical distributions, the computation of the optimal transport cost in can be formulated as a linear program and solved efficiently. Moreover, when the number of data points in the two empirical distributions is the same, the problem of finding the optimal coupling between the two distributions is reduced to an assignment problem (see Proposition 2.11 in [@Pey19]), wherein the task is to optimally match each data point from the first distribution to a distinct data point from the second distribution. Using this methodology, we evaluate the optimal risk for $\ell_2$ and $\ell_\infty$ adversaries for classes $3$ and $5$ in CIFAR10, MNIST, Fashion-MNIST and SVHN datasets. The results for other pairs of classes are very similar, and are therefore omitted for brevity. For MNIST, Fashion-MNIST and SVHN datasets, we evaluate the optimal adversarial risk given in Theorem \[th\_01bound\] by randomly sampling $5000$ data points from each class. The results are showing in Figure \[fig: datasets\] with the legend $\sigma=0$. Since a major fraction of the data points in the empirical distributions are well-separated in $\ell_2$ and $\ell_\infty$ metrics, the optimal risk bound remains $0$ even for high $\epsilon$. For instance, for CIFAR10 dataset, the optimal risk remains $0$ for $\epsilon$ as high as $40/255$ for $\ell_\infty$. Similar results were also obtained in Bhagoji et al. [@BhaEtal19]. However, the optimal risk bounds for the true distributions may not be $0$ for high $\epsilon$, as it is unreasonable to expect a perfectly robust optimal classifier under very strong adversarial perturbations. In addition, a common technique while training for a classifier is to augment the dataset with Gaussian perturbed samples for robustness and generalization [@HolKoi92; @GooEtal16]. Motivated by this, we also compute optimal risk lower bounds on Gaussian mixture distribution with the data points as the centers with scaled identity covariances. $\sigma = 0$ corresponds to the empirical distribution of the data points from the two classes. As $\sigma$ increases, the overlap in the probability mass between the two classes increases. This allows for the cost of optimal coupling that achieves $D_\epsilon$ to decrease, thus leading to a higher, possibly non-trivial bound for $R^_\epsilon$. To compute the optimal risk lower bound for Gaussian mixture, we use a coupling between the mixture distributions in two steps. In the first step, we solve for the optimal coupling that gives the exact optimal risk for the empirical distributions. This gives a pairwise matching of data points between the two empirical distributions. In the second step, we use the optimal coupling for multidimensional Gaussians from Theorem \[thm: same variance d-dim\] to transport the mass in the Gaussians within each pair. Overall, this transport map gives an upper bound on the $D_\epsilon$ optimal transport cost between the two mixture distributions. Using this, we obtain the lower bounds on adversarial risk shown in Figure \[fig: datasets\]. Figure \[fig: datasets\] shows the lower bounds for various values of the variance $\sigma$ used for the Gaussian mixture, where $\sigma^$ is half of the mean distance between data points from the two distributions. As explained previously, we see in Figure \[fig: datasets\] that the lower bound curves for higher values of $\sigma$ are above those for lower values. For instance, the optimal risk for CIFAR10 dataset under $\ell_2$ perturbation with $\epsilon=3$ is $0.25$ for $\sigma = \sigma^$. That is, the adversarial error rate for CIFAR10 with $\epsilon = 3$ for any algorithm cannot be less than $0.25$ even when trained with Gaussian data augmentation (with $\sigma = \sigma^$). In comparison, the lower bound obtained in Bhagoji et al. [@BhaEtal19] (which is equivalent to the case of $\sigma=0$) is $0$ for $\epsilon = 3$. Computation of non-trivial lower bounds for higher values of $\epsilon$ on adversarial error rate as in Figure \[fig: datasets\] is made possible by our analysis on the optimal coupling to achieve $D_\epsilon$ between multivariate Gaussians in section \[section: same variance\]. Discussion ========== In this paper, we have analyzed two notions of [adversarial risk]{} - one resulting from a distribution perturbing adversary ($\widehat{R}^_\epsilon$) and the other from a data perturbing adversary ($R^_\epsilon$). We have introduced the $D_\epsilon$ optimal transport distance between probability distributions. Through an application of duality in the optimal transport cost formulation (via Strassen’s theorem), we have shown that $D_\epsilon$ completely characterizes the optimal adversarial risk $R^_\epsilon$ for the case of binary classification under $0-1$ loss function. For general loss functions, we give lower bounds on $R^_\epsilon$ and upper bounds on $\widehat{R}^_\epsilon$ in terms of the smoothness properties of the loss function. Our analysis raises several interesting questions: How big is the gap between $\widehat{R}^_\epsilon$ and $R^_\epsilon$ for different kinds of loss functions? Is it possible to directly lower bound $\widehat{R}^_\epsilon$ without appealing to its dependence on $R^_\epsilon$? Does there exist an optimal transport distance akin to $D_\epsilon$ that characterizes $\widehat{R}^_\epsilon$? As evidenced by experiments, our bounds for general loss functions are not particularly tight. Furthermore, we need fairly strong assumptions such as convexity and Lipschitz property for the loss function to state these bounds. It would be interesting to study if these conditions may be relaxed and if tighter bounds could be obtained. In analysing the adversarial risk for $0-1$ loss functions, we give a novel coupling strategy based on monotone mappings that solves the $D_\epsilon$ optimal transport problem for symmetric unimodal distributions like Gaussian, triangular, and uniform distributions. Employing the duality in the optimal transport, we also obtain the adversarially optimal classifier under these settings. Our coupling analysis calls for an interesting open question: Is there a general coupling strategy, akin to the maximal coupling strategy to achieve the total variation transport cost, that works for a broader class of distributions? If yes, this gives us a handle on analyzing the nature of optimal decision boundaries in the adversarial setting. Optimal transport between measures with unequal mass has received attention in recent work [@ChiEtal18]. We plan to investigate if the version of transport from Definition \[def: unequal\] is useful in other contexts, and whether computational methods as in [@Pey19] may be used to compute it in practice. Our analysis for $0-1$ loss reveals how the optimal risk smoothly changes from Bayes risk as the data perturbing budget $\epsilon$ is increased. Somewhat more surprisingly, our analysis shows that in some cases, the optimal classifier can change abruptly in the presence of an adversary even for small changes in $\epsilon$. It remains to be seen if these observations on optimal risk and optimal classifier also hold for the distribution perturbing adversary. Using our characterization of $R^_\epsilon$ in terms of $D_\epsilon$, we obtain the optimal risk attainable for classification of real-world datasets like CIFAR10, MNIST, Fashion-MNIST and SVHN. Moreover, levaraging our optimal coupling strategy for Gaussian distributions, we also obtain lower bounds on optimal risk for Gaussian mixtures based on these datasets. These lower bounds have implications for the limits of data augmentation strategies using Gaussian perturbations. We note that our bounds on adversarial risk are classifier agnostic, and only depend on the data disributions. In addition, our bounds are efficiently computable for empirical/mixture distributions via reformulation as a linear program. Finally, we remark that analzing the $D_\epsilon$ optimal transport cost may be interesting in itself. The optimal transport cost $c_\epsilon(x, x') = \1\{d(x, x')>2\epsilon\}$ is discontinuous and does not satisfy triangle inequality. This makes it hard to analyse $D_\epsilon$ using standard techniques in optimal transport literature. For instance, it would be interesting to see how fast $D_\epsilon$ between empirical distributions converges to $D_\epsilon$ between the true data-generating distributions. This may be used to obtain finite-sample lower bounds for adversarial error. Proofs for Section \[sec: opt risk 01 loss\] ============================================ Strassen’s theorem {#app: strassen} ------------------ Strassen’s theorem is a special case for the Kantorovich duality in the case of a $0-1$ loss. The statement provided below is as in Villani [@Vil03 Corollary 1.28]: \[lem\_villani\] Let the input $X$ be drawn from a Polish space $\mathcal{X}$. Let $\Pi(p_0, p_1)$ be the set of all probability measures on $\mathcal{X}\times \mathcal{X}$ with marginals $p_0$ and $p_1$. Then for $\epsilon\geq 0$ and $A\subseteq\cX$, $$\begin{aligned} \inf_{\pi\in\Pi(p_0, p_1)} \pi[d(x, x' )>\epsilon] = \sup_{A\ \ closed} \left\{ p_0(A) - p_1(A^\epsilon) \right\}.\end{aligned}$$ Proof of Lemma \[lemma: closed\] {#app: lemma: closed} -------------------------------- Let $A$ be a closed set and let $B$ be the closed ball of radius $\epsilon$. Let $\{z_i\}_{i \geq 1}$ be a sequence of points in $A^\epsilon$ converging to a limit $z$. We shall show that $z \in A^\epsilon$ as well. Note that every $z_i$ admits an expression $z_i = a_i + b_i$, where $a_i \in A$ and $b_i \in B$. Since $B$ is a compact set, there exists a subsequence among the $\{b_i\}$ sequence that converges to $b^ \in B$. Call the subsequence $\{\tilde b_i\}_{i \geq 1}$ such that $\tilde b_i \to b^$ and $\|\tilde b_i - b^\| < \delta$ for some small $\delta > 0$. Since $z = \tilde a_i + \tilde b_i = \tilde a_i + \tilde b_i - b^* + b^.$ Thus, we see that $\tilde a_i$ is also bounded within a ball of radius $\delta$ around $z-b^$, and so there exists a convergence subsequence within the $\{\tilde a_i\}$ sequence. Let that subsequence converge to $a^$. We must have $a^\in A$ and $b^* \in B$ since $A$ and $B$ are closed. This means $z = a^* + b^*$ must lies in $A^\epsilon$, which shows that $A^\epsilon$ is closed. Recall that $A^{-\epsilon} = ((A^c)^{\epsilon})^c$. Since $A^c$ is an open set, it is enough to show that $C^\epsilon$ is open if $C$ is open. Let $z \in C^\epsilon$. We know that $z = c + b$ for some $c \in C$ and $b \in B$. Consider a small open ball of radius $\delta$ around $c$, called $N_\delta(c)$ that lies entirely in $C$. This is possible since $C$ is assumed to be open. Now observe that $N_\delta(z) \subseteq C^\epsilon$, since $N_\delta(z) = N_\delta(c) + b$. This shows that every point $z \in C^\epsilon$ admits a small ball around it that is contained in $C^\epsilon$, or equivalently, $C^\epsilon$ is open. This completes the proof. Proof of Lemma \[lemma: thick and thin\] {#app: lemma: thick and thin} ---------------------------------------- Notice that a point $x \in A^{-\epsilon}$ if and only if $N_\epsilon(x)$ (which is the ball of radius $\epsilon$ centered at $x$) lies entirely in $A$. If this were not the case, then we could find a $y \in A^c$ such that $d(x,y) \leq \epsilon$, and so $x \in (A^c)^\epsilon$, which implies $x \not\in ((A^c)^\epsilon)^c = A^{-\epsilon}$. This observation implies that $(A^{-\epsilon})^{\epsilon} \subseteq A$. Similarly, a point $x \in (A^\epsilon)^{-\epsilon}$ if and only if $N_\epsilon(x) \in A^\epsilon$. By definition of $A^\epsilon$, every point $x \in A$ satisfies $N_\epsilon(x) \in A^\epsilon$. Thus, if $x \in A$ then $x \in (A^\epsilon)^{-\epsilon}$. Equivalently, $A \subseteq (A^\epsilon)^{-\epsilon}$.
--- author: - \| By\ Ch. Aminul Islam --- Synopsis {#synopsis .unnumbered} ========
--- abstract: 'Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and $\alpha$ small or large, uniformly for unbounded real and complex values of $x$. The new expansions extend the range of computability of $L_n^{(\alpha)}(x)$ compared to previous expansions, in particular with respect to higher terms and large values of $\alpha$. Numerical evidence of their accuracy for real and complex values of $x$ is provided.' author: - 'T. M. Dunster' - 'A. Gil' - 'J. Segura' bibliography: - 'biblio.bib' date: 'Received: date / Accepted: date' title: 'Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions [^1]' --- Introduction {#intro} ============ In this paper we shall obtain computable asymptotic expansions for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, for the case $n$ large and ${\alpha}$ small or large. These are defined by $$L_{n}^{(\alpha)}(x)=\sum_{k=0}^{n}\left( \begin{array} [c]{c}% n+\alpha\\ n-k \end{array} \right) \frac{(-x)^{k}}{k!}.\label{eq0}%$$ In terms of the confluent hypergeometric functions we have $$L_{n}^{(\alpha)}(x)=\frac{\Gamma(n+\alpha+1)}{n!}\mathbf{M}(-n,\alpha+1,x) =\frac{(-1)^{n}}{n!}U(-n,\alpha+1,x),\label{eqLMU}%$$ where $\mathbf{M}$ denotes Olver’s scaled confluent hypergeometric function [@Olver:1997:ASF Chap. 7, sec. 9]. We will use the techniques described in [@Dunster:2017:COA] for computing uniform asymptotic expansions of turning point problems. These involve Airy function expansions for solutions of differential equations having a simple turning point, but in a form that differs from the classical Airy function expansions of [@Olver:1997:ASF Chap. 11]. Specifically, the coefficients in these new expansions are significantly easier to compute than previously, and close to the turning point the method utilises Cauchy’s integral formula. We remark that the Cauchy integral method of [@Dunster:2017:COA] has potential applications to other forms of differential equations, and also the method was subsequently used in [@SAPM:SAPM12172] to compute coefficients appearing in certain asymptotic expansions of integrals. In this paper we shall also obtain Liouville-Green (L-G) expansions for Laguerre polynomials and confluent hypergeometric functions in domains that do not contain turning points, and these involve the exponential function. The form of these expansions differs from the standard ones (see [@Olver:1997:ASF Chap. 10]), insomuch the coefficients in the expansions appear in the argument of the exponential. The advantage of this form is that the coefficients are again generally easier to compute. Indeed, L-G expansions of this form were used in [@Dunster:2017:COA] to obtain the new form of the Airy expansions described above. A recent reference on the computation of Laguerre polynomials using asymptotics is [@Gil:2017:ECL], where three different asymptotic approximations are used: two expansions in terms of Bessel functions (from [@Frenzen:1988:UAE] and [@Temme:2015:AMF Sect. 10.3.4]) and the Airy expansion of Frenzen and Wong [@Frenzen:1988:UAE]. These expansions, and in particular the Airy expansion, give accurate approximations for large $n$ when $\alpha$ is not large. For large $\alpha$ some expansions are available, like for instance the expansions for Whittaker functions of [@Dunster:1989:UAE] and [@Olver:1980:WFW], or some of the expansions in [@Temme:1990:AEF], but the coefficients of these expansions are very hard to compute. In this paper, we provide expansions that are computable and which can also be used for large $\alpha$, and we provide numerical evidence of their accuracy for real and complex values of $x$. The Laguerre polynomials satisfy the following form of the confluent hypergeometric equation $$x\frac{d^{2} y}{dx^{2}}+(\alpha+1-x)\frac{dy}{dx}+ny=0.\label{eq1}%$$ By the transformation $\tilde{y}=x^{(\alpha+1)/2}\exp\left( -\frac{1}% {2}x\right) y$ we can remove the first derivative in the usual manner, and we get $$\frac{d^{2}\tilde{y}}{dx^{2}}=\frac{x^{2}-2(2n+\alpha+1)x+\alpha^{2}-1}% {4x^{2}}\tilde{y}.\label{eq2}%$$ On replacing $x$ by $uz$, where $u=n+\frac{1}{2}$ (a choice being explained after (\[eq13a\]) below), this can then be re-written in the form $$\frac{d^{2} w}{dz^{2}}=\left\{ u^{2} f(a,z)+g(z)\right\} w,\label{eq3}%$$ with a solution $w=z^{(\alpha+1)/2}\exp\left( -\frac{1}{2} uz\right) L_{n}^{(\alpha)}(uz)$. On defining $a$ by $$\alpha=u\left( a^{2}-1\right) \ (a\geq0),\label{eq4}%$$ we have in (\[eq3\]) $$f(z) =\frac{(z-z_{1})(z-z_{2})}{4z^{2}},\ g(z) =-\frac{1}{4z^{2}},\label{eq5}%$$ where $$z_{1}=(a-1)^{2},\ z_{2}=(a+1)^{2}.\label{eq6}%$$ The latter are the turning points for large $u$, which is assumed here. These turning points coalesce with each other when $a=0$ ($\alpha=-u)$, and $z_{1}$ coalesces with the pole at $z=0$ when $a=1$ ($\alpha=0$). We shall consider the following cases separately. Case 1a: We obtain expansions for $z$ lying in domains containing the turning point $z_{1}$, which includes the interval $0\leq z\leq z_{2} -\delta$ (here and throughout $\delta>0$). The turning point $z=z_{2}$ is excluded, and this is covered in Case 2 below. We assume $1+\delta\leq a^{2}\leq a_{1}<\infty$ for fixed $a_{1}\in(1,\infty)$. Thus $z_{1}$ cannot coalesce with the pole at $z=0$ nor with the other turning point $z=z_{2}$, and $\alpha$ is positive and large, satisfying $u\delta\leq\alpha\leq u(a_{1} -1)$. Case 1b: This is the same as Case 1a, except now we consider $\alpha$ negative. In particular, we assume $0<a_{0}\leq a^{2}\leq1-\delta$ (for fixed $a_{0}\in(0,1)$), and in this case $-u(1-a_{0}) \leq\alpha\leq-u\delta<0$. Again $z_{1}$ cannot coalesce with the pole at $z=0$ nor with the other turning point $z=z_{2}$. Case 2: In this case expansions are derived for $z$ lying in domains containing the turning point $z_{2}$, including the interval $z_{1}+\delta\leq z<\infty$, but not $z_{1}$ or $0$. Here $a$ lies in a larger interval than Cases 1a and 1b, namely $0<a_{0}\leq a^{2}\leq a_{1}<\infty$, and hence $-u(1-a_{0})\leq\alpha\leq u(a_{1}-1)$. Note this can include large negative values of $\alpha$ if $a_{0}<1$. The turning points again cannot coalesce ($a=0$) in this case, but $z_{1}$ can coalesce with $z=0$ (since both points are excluded). The more general case $0\leq a<\infty$ with $0\leq z<\infty$ will require an application of asymptotic expansions valid for a coalescing turning point and double pole, and also for two coalescing turning points. This will be studied in a subsequent paper. The plan of this paper is as follows. In section 2 we consider Case 1a and construct L-G expansions for the functions. In this section a detailed description of the Liouvile transformation is provided. In section 3 we obtain Airy function expansions for Case 1a. In section 4 we obtain L-G and Airy expansions for Case 1b, and in section 5 we likewise do this for Case 2. Finally, in section 6 we present numerical results for the expansions of Cases 1a and 2 for Laguerre polynomials. L-G expansions: Case 1a ======================= We make the Liouville transformation $$\xi=\int_{z}^{z_{1}}f^{1/2}(t) dt. \label{eq7}%$$ On explicit integration (and noting that $\sqrt{z_{1} z_{2}}=\left\vert a^{2}-1\right\vert $) we obtain $$% \begin{array} [c]{ll}% \xi= & \dfrac{1}{2}\left( a^{2}+1\right) \ln\left\{ a^{2}+1-z-S\left( z\right) \right\} -\dfrac{1}{2}S(z) -\max\left\{ a^{2},1\right\} \ln(2a)\\ & \\ & +\dfrac{1}{2}\left\vert a^{2}-1\right\vert \ln\left\{ \dfrac{\left( a^{2}-1\right) ^{2}+\left\vert a^{2}-1 \right\vert S(z) -\left( a^{2}+1\right) z}{z}\right\} . \end{array} \label{eq8}%$$ In this $S(z)$ is given by $$S(z) =\left\{ (z_{1}-z)(z_{2}-z)\right\} ^{1/2}= \left\{ z^{2}-2\left( a^{2}+1\right) z+\left( a^{2}-1\right) ^{2}\right\} ^{1/2}, \label{eq9}%$$ where the principal logarithms are taken, and the branches of the square roots are chosen so that $S(z)>0$ for $-\infty<z<z_{1}$, and is continuous elsewhere in the plane having a cut on the interval $[z_{1},z_{2}] $. This means $S(z) <0$ for $z_{2}<z<\infty$. Thus $\xi$ is real and positive when $z\in(0,z_{1})$, and varies continuously in the $z$ plane having cuts along $(-\infty,0]$ and $[z_{1},\infty) $. Note that $z=z_{1}$ is mapped to $\xi=0$, and $z=z_{2}\pm i0$ (respectively above and below the cut) is mapped to $\xi=\pm\min\left\{ a^{2},1\right\} \pi i$. ![$z$ plane.[]{data-label="fig1zxi"}](Fig1.eps){height="6cm" width="8cm"} ![$\xi$ plane.[]{data-label="fig2zxi"}](Fig2.eps){height="6cm" width="8cm"} We find that $\xi\rightarrow+\infty$ as $z\rightarrow0^{+}$, such that $$\xi=\tfrac{1}{2}\left\vert a^{2}-1\right\vert \left\{ 2\ln\left\vert {a^{2}-1}\right\vert -\ln\left( z\right) -1\right\} -a^{2}\left\vert \ln(a)\right\vert +\mathcal{O}(z). \label{eq10}%$$ Furthermore, we have that $$\xi=\tfrac{1}{2}z-\tfrac{1}{2}\left( a^{2}+1\right) \left\{ \ln (z)+1\right\} +a^{2}\ln(a) +\min\left\{ a^{2},1\right\} \pi i +\mathcal{O}\left( {z^{-1}}\right) , \label{eq51b}%$$ as $z\rightarrow\infty$ in upper half plane. Figs. \[fig1zxi\] and \[fig2zxi\] depict the $z$-$\xi$ map for $0\leq \arg(z) \leq\pi$, with corresponding points labeled with the same letters. For the lower half $z$ plane we can use from the Schwarz reflection principle that $\xi(z) =\overline{\xi\left( \bar{z}\right) }$. With (\[eq7\]) and the new dependent variable given by $$V=f^{1/4}(z) w, \label{eq11}%$$ the differential equation (\[eq3\]) is transformed to $$\Frac{d^2 V}{d\xi^2}=\left\{ u^{2}+\phi(\xi)\right\} V, \label{eq12}%$$ where $$% \begin{array} [c]{ll}% \phi\left( \xi\right) & = \dfrac{4f\left( z\right) {f}^{\prime\prime }\left( z\right) -5{f}^{\prime2}\left( z\right) }{16f^{3}\left( z\right) }+\dfrac{g\left( z\right) }{f\left( z\right) }\\ & =-\dfrac{z\left\{ {4z^{3}-4\left( {3a^{2}-1}\right) \left( {a^{2}% -3}\right) z+8\left( {a^{2}+1}\right) \left( {a^{2}-1}\right) ^{2}% }\right\} }{4\left( {z-z_{1}}\right) ^{3}\left( {z-z_{2}}\right) ^{3}}. \end{array} \label{eq13}%$$ The function $\phi(\xi)$ is analytic in an unbounded domain $\Delta$ (say) which excludes $\xi=0$ and $\xi=\pm\min\left\{ a^{2},1\right\} \pi i$ (the singularities corresponding to the turning points). Then the part of $\Delta$ corresponding to $0\leq\arg(z)\leq\pi$ is the entire region depicted in Fig. \[fig2zxi\], except with the points $\xi=0$ and $\xi=\min\left\{ a^{2},1\right\} \pi i$ (labeled E and F, respectively) excluded. An important property is that $$\phi(\xi) =\mathcal{O}\left( \xi^{-2}\right) , \label{eq13a}%$$ as $\xi\rightarrow\infty$ in $\Delta$. We remark that the choice of the large parameter in the form $u=n+\tfrac{1}{2}$, and the subsequent partitioning of (\[eq3\]), resulted in the desired behaviour (\[eq13a\]). Asymptotic solutions of (\[eq12\]), accompanied by explicit error bounds, are given in [@Dunster:1998:AOT] by $$V_{n,1}(u,\xi) =\exp\left\{ u\xi+\sum\limits_{s=1}^{n-1}\frac{E_{s}(\xi )}{u^{s}}\right\} +\varepsilon_{n,1}(u,\xi), \label{eq14}%$$ and $$V_{n,2}(u,\xi) =\exp\left\{ -u\xi+\sum\limits_{s=1}^{n-1}(-1)^{s}\frac {E_{s}(\xi)}{u^{s}}\right\} +\varepsilon_{n,2}(u,\xi), \label{eq14a}%$$ where the coefficients $E_{s}(\xi)$ will be defined below. In (\[eq14\]) $\varepsilon_{n,1}(u,\xi)=e^{u\xi} \mathcal{O}\left( u^{-n}\right) $ uniformly in a certain domain $\Xi_{1}$, with $e^{-u\xi }\varepsilon_{n,1}(u,\xi)\rightarrow0$ as $\mathrm{Re} \xi\rightarrow-\infty$ in the domain, and likewise in (\[eq14a\]) $\varepsilon_{n,2}(u,\xi) =e^{-u\xi}\mathcal{O}(u^{-n})$ uniformly in a domain $\Xi_{2}$, with $e^{u\xi }\varepsilon_{n,2}(u,\xi)\rightarrow0$ as $\mathrm{Re} \xi\rightarrow\infty$ in the domain. By virtue of (\[eq13a\]), both $\Xi_{1}$ and $\Xi_{2}$ are unbounded, and are defined as follows. In general the domain of validity $\Xi_{1}$ comprises the $\xi$ point subset of $\Delta$ for which there is a path $\mathcal{P}_{1}$ (say) linking $\xi$ with $\alpha_{1}=-\infty-\frac{1}{2}\left\vert a^{2}-1\right\vert \pi i$ (corresponding to $z=\infty e^{\pi i}$) and having the properties (i) $\mathcal{P}_{1}$ consists of a finite chain of $R_{2}$ arcs (as defined in [@Olver:1997:ASF Chap. 5, sec. 3.3]), and (ii) as $t$ passes along $\mathcal{P}_{1}$ from $\alpha_{1}$ to $\xi$, $\mathrm{Re} (ut) $ is nondecreasing. The domain $\Xi_{2}$ comprises the $\xi$ point subset of $\Delta$ for which there is a path $\mathcal{P}_{2}$ (say) linking $\xi$ with $\alpha_{2} =+\infty$ (corresponding to $z=0$) and having the properties (i) $\mathcal{P}_{2}$ consists of a finite chain of $R_{2}$ arcs, and (ii) as $t$ passes along $\mathcal{P}_{2}$ from $\alpha_{2}$ to $\xi$, $\mathrm{Re}(ut)$ is nonincreasing. Let $\Xi_{j}^{+}$ ($j=1,2$) denote the subsets of $\Xi_{j}$ corresponding to $0\leq\arg(z) \leq\pi$. Then $\Xi_{1}^{+}$ coincides with the domain shown in Fig. \[fig2zxi\], but with the points $\xi=0$ and $\xi=\min\left\{ a^{2},1\right\} \pi i$ excluded. We denote by $D_{1}^{+}$ the $z$-domain corresponding to $\Xi_{1}^{+}$, and this is the entire upper half plane $\mathrm{Im}(z) \geq0$ excluding the turning points $z=z_{1},z_{2}$. On the other hand, due to the monotonicity requirement (ii), the domain $\Xi_{2}^{+}$ must also exclude the unbounded region FGH of Fig. \[fig2zxi\], as well as points on the segment EF. The corresponding $z$-domain $D_{2}^{+}$ (say) is the unshaded region depicted in Fig. \[fig3zxi\], where the interval $[z_{1},z_{2}]$ and the boundary curve FH must be excluded; the curve FH extends from $z=z_{2}$ to $z=\infty$ and is given parametrically by $$\int_{z_{2}}^{z}{f^{1/2}\left( t\right) dt=\tau i,\ 0\leq\tau<\infty.} \label{eq14b}%$$ We note that $z=+\infty$ is not contained in $D_{2}^{+}$ (but is contained in $D_{1}^{+}$). ![Domain $D_{2}^{+}$ in $z$ plane. The interval $\left[ z_{1},z_{2}\right] $ and boundary FH are excluded.[]{data-label="fig3zxi"}](Fig3.eps){height="6cm" width="8cm"} Returning to the asymptotic expansions (\[eq14\]) and (\[eq14a\]), the coefficients are given by $$E_{s}(\xi) =\int F_{s}(\xi) d\xi\ (s=1,2,3,\cdots), \label{eq16}%$$ where $$F_{1} (\xi) =\frac{1}{2}\phi(\xi) ,\ F_{2}(\xi) =-\frac{1}{4}\phi(\xi), \label{eq17}%$$ and $$F_{s+1}(\xi) =-\frac{1}{2}F_{s}^{\prime}(\xi) -\frac{1}{2}\sum\limits_{j=1}% ^{s-1} F_{j}(\xi) F_{s-j}(\xi)\,\, (s=2,3,\cdots). \label{eq18}%$$ Primes are derivatives with respect to $\xi$. The choice of integration constants in (\[eq16\]) must be the same for both of the L-G solutions (\[eq14\]) and (\[eq14a\]). As discussed in [@Dunster:2017:COA], the constants associated with the even terms ($s=2j$, $j=1,2,3\cdots$) are arbitrary, but those for the odd terms ($s=2j+1$, $j=0,1,2,\cdots$) must be precisely chosen, as described below. It is preferable to work in terms of $z$. Using $$\frac{d\xi}{dz}=f^{1/2}(z)=\frac{S(z)}{2z}, \label{eq19}%$$ and writing $\hat{E}_{s}(z) =E_{s}\left( \xi(z)\right) $ and $\hat{F}% _{s}(z)=F_{s}\left( \xi(z)\right) $, we have $$\hat{E}_{s} (z) =\int\frac{\hat{F}_{s} (z) S (z)}{2z}dz \ (s=1,2,3,\cdots), \label{eq20}%$$ where $$\hat{F}_{1} (z) =\frac{1}{2}\phi\left( \xi(z)\right) ,\ \hat{F}_{2} (z) =-\frac{z}{2 S(z)} \frac{d\phi\left( \xi(z)\right) }{dz}, \label{eq21}%$$ and $$\hat{F}_{s+1}(z) =-\frac{z}{S(z)}\hat{F}_{s}^{\prime}(z) -\frac{1}{2}% \sum\limits_{j=1}^{s-1} \hat{F}_{j} (z) \hat{F}_{s-j} (z)\ (s=2,3,\cdots). \label{eq22}%$$ We find by induction that $$\frac{\hat{F}_{s} (z) S(z)}{2z}= \frac{R_{2s+1}(z)}{S^{3s+2}(z)}, \label{eq23}%$$ where $R_{n}(z)$ is a polynomial of degree $n$. From this we can show that $$\hat{E}_{2j}(z) =\frac{zT_{4j-1}^{(e)}(z)}{S^{6j}(z)}, \label{eq24}%$$ and $$\hat{E}_{2j+1} (z) =\frac{T_{6j+3}^{(o)}(z)}{S^{6j+3}(z)}, \label{eq25}%$$ where $T_{n}^{(e)}(z)$ and $T_{n}^{(o)}(z)$ are also polynomials of degree $n$, provided the integration constants in (\[eq20\]) are chosen appropriately (which we assume). For example, for the coefficient $\hat{E}_{1}(z)$, we have $$% \begin{array} [c]{ll}% T_{3}^{(o)}(z) = & \dfrac{1}{48a^{2}}\left[ \left( a^{2}-1\right) ^{2}\left\{ \left( a^{2}-1\right) ^{2}-4a^{2}\right\} -3\left( a^{2}+1\right) ^{3} z \right. \\ & \\ & \left. +3\left( a^{4}+6a^{2}+1\right) z^{2} -\left( a^{2}+1\right) z^{3} \right] . \end{array} \label{eqT3}%$$ We note here that $\zeta^{1/2}\hat{E}_{2j+1}(z)$, regarded as a function of $\zeta=\left( \frac32\xi\right) ^{2/3}$, is meromorphic at $\zeta=0$. From [@Dunster:2017:COA] this is a requirement for the subsequent Airy function expansions to be valid at the turning point. We also note that $$\hat{E}_{s}(\infty)=\lambda_{s}, \label{eq26}%$$ (say), where $\lambda_{2j}=0$ ($j=1,2,3,\cdots$), whereas the odd terms are non-zero. The first three of the odd terms are found to be $$\lambda_{1} =-\frac{1+a^{2}}{48a^{2}},\lambda_{3} =\frac{7\left( 1+a^{6}\right) } {5760 a^{6}},\lambda_{5}=-\frac{31\left( 1+a^{10}\right) }{80640 a^{10}}. \label{eq27}%$$ We similarly find $$\hat{E}_{s}(0) =\mu_{s}, \label{eq28}%$$ (say), where $\mu_{2j}=0$ ($j=1,2,3,\cdots$), and again the odd terms are non-zero. The first two of these are found to be $$% \begin{array} [c]{l}% \mu_{1}=\Frac{a^{4}-6a^{2}+1}{48a^{2}\left\vert a^{2}-1\right\vert },\\ \\ \mu_{3}% =-\Frac{7a^{12}-21a^{10}+21a^{8}-30a^{6}+21a^{4}-21a^{2}+7} {5760a^{6}\left\vert a^{2}-1 \right\vert ^{3}}. \end{array} \label{eq29}%$$ Recall in this case we are assuming that $1+\delta\leq a^{2}\leq a_{1}<\infty $, for fixed $a_{1}\in( 1,\infty)$. Thus from (\[eq4\]) $\alpha$ is positive, and so using $$L_{n}^{(\alpha)}(x)=\frac{\Gamma(n+\alpha+1)}{n!\Gamma(\alpha+1)} \left\{ 1+\mathcal{O}(x)\right\} \ (x\rightarrow0) , \label{eq30}%$$ we therefore have the solution of (\[eq3\]) given by $$w_{0} (u,z) \equiv z^{(\alpha+1)/2} e^{-uz/2}L_{n}^{(\alpha)}(uz), \label{eq31}%$$ which has the unique behaviour as $z\rightarrow0$ $$w_{0} (u,z) \equiv\frac{\Gamma(n+\alpha+1)z^{(\alpha+1)/2}} {n!\Gamma (\alpha+1)} \left\{ 1+\mathcal{O}(z)\right\} . \label{eq32}%$$ We now match solutions that are recessive at $z=0$. From (\[eq11\]) we have by uniqueness (up to a multiplicative constant) of such solutions that$$w_{0} (u,z) \propto\dfrac{z^{1/2}V_{n,2}(u,\xi)} {\left\{ (z_{1}% -z)(z_{2}-z)\right\} ^{1/4}}. \label{eq32a}%$$ Then, using (\[eq10\]), (\[eq14a\]), (\[eq28\]), (\[eq31\]) and (\[eq32\]), we arrive at $$% \begin{array} [c]{l}% L_{n}^{(\alpha)}(uz) \sim\dfrac{\Gamma(n+\alpha+1)} {n!\Gamma(\alpha+1)} \left( \dfrac{\alpha}{u}\right) ^{1/2}\left( \dfrac{u}{u+\alpha}\right) ^{u/2} \left\{ \dfrac{\alpha^{2}}{(u+\alpha)uez}\right\} ^{\alpha/2}\\ \times\dfrac{1}{\left\{ (z_{1}-z)(z_{2}-z)\right\} ^{1/4}} \exp\left\{ \dfrac{1}{2}uz-u\xi+\displaystyle\sum_{s=1}^{\infty}(-1)^{s} \dfrac{\hat {E}_{s} (z) -\mu_{s}}{u^{s}}\right\} . \end{array} \label{eq33}%$$ This is uniformly valid for $z\in D_{2}^{+}\cup D_{2}^{-}$, where $D_{2}^{-}$ is the conjugate region of $D_{2}^{+}$. From [@Daalhuis:2010:CHF Eq. 13.2.24] a companion solution of (\[eq3\]) is given by $$w_{1}(u,z)\equiv(uz)^{(\alpha+1)/2}e^{uz/2}U(n+\alpha+1,\alpha+1,uze^{-\pi i}).\label{eq50}%$$ Now from [@Daalhuis:2010:CHF Eq. 13.7.3] $$U(n+\alpha+1,\alpha+1,uze^{-\pi i})=\left( uze^{-\pi i}\right) ^{-n-\alpha-1}\left\{ 1+\mathcal{O}\left( z^{-1}\right) \right\} \label{eq50b}%$$ as $z\rightarrow\infty$ for $\left\vert \arg\left( ze^{-\pi i}\right) \right\vert \leq\frac{3}{2}\pi-\delta$, and we see that $w_{1}(u,z)$ has the behaviour $$w_{1}(u,z)=(-1)^{n+1}e^{-\alpha\pi i}(uz)^{-u-\alpha/2}e^{uz/2}\left\{ 1+\mathcal{O}\left( z^{-1}\right) \right\} ,\label{eq51}%$$ and in particular this is the unique solution that is recessive in the half-plane ${\left\vert {\arg\left( {ze^{-\pi i}}\right) }\right\vert \leq{\frac{1}{2}}\pi}$. Thus we similarly deduce by matching recessive solutions that $$w_{1}(u,z)\propto\dfrac{z^{1/2}V_{n,1}(u,\xi)}{\left\{ (z_{1}-z)(z_{2}% -z)\right\} ^{1/4}},\label{eq51c}%$$ and hence using (\[eq51b\]) $$% \begin{array} [c]{l}% U(n+\alpha+1,\alpha+1,uze^{-\pi i}) \sim-\dfrac{i\exp\left\{ u+\tfrac{1}% {2}\alpha+\max(\alpha,0)\pi i\right\} } {u^{(u+1)/2}(u+\alpha)^{(u+\alpha )/2}(uz)^{\alpha/2}}\\ \\ \times\dfrac{1}{\left\{ (z_{1}-z)(z_{2}-z)\right\} ^{1/4}} \exp\left\{ -\dfrac{1}{2}uz+u\xi+\displaystyle\sum_{s=1}^{\infty} \dfrac{\hat{E}_{s}(z) -\lambda_{s}}{u^{s}}\right\} . \end{array} \label{eq51a}%$$ We remark that this expansion also holds for negative $\alpha$, and in particular for $0<a_{0}\leq a^{2}\leq1-\delta$ (for fixed $a_{0}\in(0,1)$). Likewise for its Airy expansion (\[eq43b\]) below. Airy expansions: Case 1a ======================== We now obtain asymptotic expansions which are valid at $z=z_{1}$. These involve Airy functions, and the standard Liouville transformation is given by $$\zeta=\left( \tfrac{3}{2}\xi\right) ^{2/3},\ W=\zeta^{-1/4}f^{1/4}(z) w, \label{eq34}%$$ where $\xi$ is again given by (\[eq8\]). Here $\zeta$ is defined to be analytic at $z=z_{1}$ (see [@Olver:1997:ASF Chap. 11]), and moreover $\zeta>0$ for $0<z<z_{1}$, and $\zeta<0$ for $z_{1}<z<z_{2}$. With this transformation the differential equation (\[eq3\]) takes the form $$\dfrac{d^{2}W}{d\zeta^{2}}=\left\{ u^{2}\zeta+\psi(\zeta)\right\} W, \label{eq35}%$$ where $\psi(\zeta)$ is analytic at $\zeta=0$ (i.e. $\xi=0$ and $z=z_{1}$), and is given explicitly by $$\psi(\zeta)=\tfrac{5}{16}\zeta^{-2}+\zeta\phi(\xi), \label{eq36}%$$ where $\phi(\xi) $ is given by (\[eq13\]). From [@Dunster:2017:COA] the following three exact solutions of (\[eq35\]) are given $$W_{j}(u,\zeta) =\mathrm{Ai}_{j}\left( u^{2/3}\zeta\right) A(u,z) +\mathrm{Ai}_{j}^{\prime}\left( u^{2/3}\zeta\right) B(u,z) \ (j=0,\pm1). \label{eq37}%$$ Here $\mathrm{Ai}_{j}(u^{2/3}\zeta)=\mathrm{Ai}(u^{2/3}\zeta e^{-2\pi ij/3})$, which are the Airy functions that are recessive in the sectors $\mathrm{\mathbf{S}}_{j}:=\left\{ \zeta:\|\arg(\zeta e^{-2\pi ij/3})\|\leq \pi/3\right\} $; see [@Olver:2010:AAR §9.2(iii)]. In our case, as functions of $z$, $W_{0}(u,\zeta) $ is recessive at $z=0$, $W_{1}(u,\zeta)$ is recessive at $z=\infty e^{\pi i}$, and $W_{-1}(u,\zeta) $ is recessive at $z=\infty e^{-\pi i}$. The coefficient functions $A(u,z)$ and $B(u,z)$ are analytic in a domain containing $z=z_{1}$, and in [@Dunster:2017:COA Theorem 2.1] it was shown that they possess the asymptotic expansions $$% \begin{array} [c]{l}% A(u,z)\sim\exp\left\{ \displaystyle\sum_{s=1}^{\infty} \dfrac{\hat{E}% _{2s}(z)+\tilde{a}_{2s}\xi^{-2s}/(2s)}{u^{2s}}\right\} \\ \times\cosh\left\{ \displaystyle\sum_{s=0}^{\infty} \dfrac{\hat{E}% _{2s+1}(z)-\tilde{a}_{2s+1}\xi^{-2s-1}/(2s+1)}{u^{2s+1}}\right\} , \end{array} \label{eq38}%$$ and $$% \begin{array} [c]{l}% B(u,z)\sim\dfrac{1}{u^{1/3}\zeta^{1/2}}\exp\left\{ \displaystyle\sum _{s=1}^{\infty} \dfrac{\hat{E}_{2s}(z)+a_{2s}\xi^{-2s}/(2s)}{u^{2s}}\right\} \\ \times\sinh\left\{ \displaystyle\sum_{s=0}^{\infty} \dfrac{\hat{E}% _{2s+1}(z)-a_{2s+1}\xi^{-2s-1}/(2s+1)}{u^{2s+1}}\right\} , \end{array} \label{eq39}%$$ where $a_{1}=a_{2}=\frac{5}{72}$, $\tilde{a}_{1}=\tilde{a}_{2}=-\frac{7}{72}$, and for $b=a$ and $b=\tilde{a}$ $$b_{s+1}=\frac{1}{2}(s+1)b_{s}+\frac{1}{2}\sum_{j=1}^{s-1}b_{j}b_{s-j} \quad(s=2,3,\cdots). \label{eq39a}%$$ ![Domain $\mathbf{D}$ in $z$ plane.[]{data-label="fig4zxi"}](Fig4.eps){height="6cm" width="8cm"} For our differential equation (\[eq35\]) these expansions are uniformly valid in an unbounded $z$ domain $\mathbf{D}$, which consists of all points which can be linked to each of the singularities $z=0,z=\infty e^{\pm\pi i}$ by a “progressive” path: that is, a finite chain of $R_{2}$ arcs that does not pass through $z_{2}$, and with the property that $\mathrm{Re}\xi(z)$ varies monotonically as the path is traversed from one end to the other. From Fig. \[fig2zxi\] it is straightforward to show that $\mathbf{D}$ consists of the intersection of $D_{1}^{+}$ and $D_{2}^{+}$ (i.e. $D_{2}^{+}$ itself), along with the conjugate of $D_{2}^{+}$, and in addition all points on the interval $[z_{1},z_{2})$. Thus $\mathbf{D}$ is the unshaded region shown in Fig. \[fig4zxi\]. In this $\left\vert \arg(z)\right\vert \leq\pi$, the boundary curve FH emanating from $z=z_{2}$ is defined by (\[eq14b\]), and FH’ is the conjugate curve. Points on both curves (including $z=z_{2}$) are excluded from $\mathbf{D}$, but the singularities $z=0$ and $z=\infty e^{\pm\pi i}$ lie in $\mathbf{D}$, as well as of course the turning point $z=z_{1}$. For each of the three solutions the respective domain of validity extends beyond $\mathbf{D}$. For example, for $W_{0}(u,\zeta)$ certain points accessible by crossing above or below the cut on the negative real axis can be included. For $W_{1}(u,\zeta)$ certain points crossing above this cut can be included, as well as points crossing the curve FH (but not FH’). For our purposes the common domain of validity $\mathbf{D}$ suffices, since an extension to a domain containing $z=z_{2}$ and $z=+\infty$ will be considered in the next section, and for the confluent hypergeometric function analytic continuation formulas can be used for values of $\arg(z)$ outside $[-\pi,\pi]$. We now match the Airy function expansions with the Laguerre polynomial and confluent hypergeometric function. Firstly we have, on matching the solutions recessive at $z=0$, namely $W_{0}(u,\zeta)$ and $\zeta^{-1/4}f^{1/4}(z) w_{0} (u,z)$ (see (\[eq31\]) and (\[eq37\])), that $$W_{0}(u,\zeta) =C_{0}(u) \left\{ \dfrac{(z_{1}-z)(z_{2}-z)}{\zeta}\right\} ^{1/4} z^{\alpha/2}e^{-uz/2}L_{n}^{(\alpha)}(uz), \label{eq40}%$$ for some constant $C_{0}(u)$. Now as $z\rightarrow0^{+}$ ($\zeta\rightarrow+\infty$) we find from (\[eq34\]), (\[eq38\]), (\[eq39\]), and the behaviour of the Airy function and its derivative for large argument (see [Olver:2010:AAR]{}), that the LHS behaves as $$W_{0}(u,\zeta)\sim\dfrac{1}{2\pi^{1/2}u^{1/6}\zeta^{1/4}}\exp\left\{ -u\xi-\sum\limits_{s=0}^{\infty}{\dfrac{\mu_{2s+1}}{u^{2s+1}}}\right\} . \label{eq41}%$$ On the other hand, from (\[eq30\]), the RHS of (\[eq40\]) has the behaviour $$% \begin{array} [c]{l}% C_{0}\left( u\right) \left\{ {\dfrac{\left( {z-z_{1}}\right) \left( {z-z_{2}}\right) }{\zeta}}\right\} ^{1/4}z^{\alpha/2}e^{-uz/2}L_{n}^{\left( \alpha\right) }\left( {uz}\right) \\ \sim\dfrac{C_{0}\left( u\right) \Gamma\left( {n+\alpha+1}\right) \left( {a^{2}-1}\right) ^{1/2}z^{\alpha/2}}{n!\Gamma\left( {\alpha+1}\right) \zeta^{1/4}}. \end{array} \label{eq42}%$$ Solving for $C_{0}\left( u\right) $ from (\[eq40\]) - (\[eq42\]), and on referring to (\[eq4\]) and (\[eq51b\]), we then arrive at $$% \begin{array} [c]{l}% L_{n}^{\left( \alpha\right) }\left( {uz}\right) \sim\dfrac{2\pi ^{1/2}u^{1/6}\Gamma\left( {n+\alpha+1}\right) }{n!\Gamma\left( {\alpha +1}\right) }\left( {\dfrac{\alpha}{u}}\right) ^{1/2}\left( {\dfrac {u}{u+\alpha}}\right) ^{u/2}\\ \\ \times\left( {\dfrac{\alpha^{2}}{\left( {u+\alpha}\right) uez}}\right) ^{\alpha/2}\left\{ {\dfrac{\zeta}{\left( {z_{1}-z}\right) \left( {z_{2}% -z}\right) }}\right\} ^{1/4}\exp\left\{ {\dfrac{1}{2}uz+\sum\limits_{s=0}% ^{\infty}{\dfrac{\mu_{2s+1}}{u^{2s+1}}}}\right\} \\ \\ \times\left\{ {\mathrm{Ai}\left( {u^{2/3}\zeta}\right) A\left( {u,z}\right) +}\mathrm{Ai}^{\prime}{\left( {u^{2/3}\zeta}\right) B\left( {u,z}\right) }\right\} . \end{array} \label{eq43}%$$ Similarly, on matching solutions recessive at $z=\infty e^{\pi i}$ we assert the existence of a constant $C_{1}\left( u\right) $ such that $$% \begin{array} [c]{l}% W_{1}\left( {u,\zeta}\right) =C_{1}\left( u\right) \left\{ {\dfrac {\left( {z_{1}-z}\right) \left( {z_{2}-z}\right) }{\zeta}}\right\} ^{1/4}\\ \times z^{\alpha/2}e^{uz/2}U\left( {n+\alpha+1,\alpha+1,uze^{-\pi i}}\right) . \end{array} \label{eq43aa}%$$ The constant is found by comparing both sides as $z\rightarrow\infty e^{\pi i}$, and as a result, on using $$% \begin{array} [c]{l}% \mathrm{Ai}_{1}\left( {u^{2/3}\zeta}\right) A\left( {u,z}\right) +\mathrm{Ai}_{1}^{\prime}\left( {u^{2/3}\zeta}\right) B\left( {u,z}\right) \\ \sim\dfrac{e^{\pi i/6}}{2\pi^{1/2}u^{1/6}\zeta^{1/4}}\exp\left\{ {u\xi +\sum\limits_{s=0}^{\infty}{\dfrac{\lambda_{2s+1}}{u^{2s+1}}}}\right\} , \end{array} \label{eq43a}%$$ we find that$$% \begin{array} [c]{l}% U\left( {n+\alpha+1,\alpha+1,uze^{-\pi i}}\right) \\ \\ \sim-\dfrac{2\pi^{1/2}e^{\pi i/3}}{u^{1/3}\left\{ {u\left( {u+\alpha }\right) }\right\} ^{\left( {u+\alpha}\right) /2}z^{\alpha/2}}\left\{ {\dfrac{\zeta}{\left( {z_{1}-z}\right) \left( {z_{2}-z}\right) }}\right\} ^{1/4}\\ \\ \times\exp\left\{ {-\dfrac{1}{2}uz+u+\dfrac{1}{2}\alpha+\max}\left( {\alpha,0}\right) {\pi i-\sum\limits_{s=0}^{\infty}{\dfrac{\lambda_{2s+1}% }{u^{2s+1}}}}\right\} \\ \\ \times\left\{ {\mathrm{Ai}_{1}\left( {u^{2/3}\zeta}\right) A\left( {u,z}\right) +}\mathrm{Ai}_{1}^{\prime}{\left( {u^{2/3}\zeta}\right) B\left( {u,z}\right) }\right\} . \end{array} \label{eq43b}%$$ By the matching of solutions that are recessive as $z\rightarrow\infty e^{-\pi i}$, we obtain corresponding L-G and Airy asymptotic expansions which are given by (\[eq51a\]) and (\[eq43b\]) respectively, with $i$ replaced by $-i$ in both, and ${\mathrm{Ai}_{1}}$ replaced by ${\mathrm{Ai}_{-1}}$ in the latter. Likewise for (\[eq61\]) and (\[eq63\]) below. L-G and Airy expansions: Case 1b ================================ Now let us consider the case $0<a_{0}\leq a^{2}\leq1-\delta$ ($-u\left( 1-{a_{0}}\right) \leq\alpha\leq-u\delta<0$). As mentioned earlier, in this case the expansions (\[eq51a\]) and (\[eq43b\]) remain valid for $U\left( {n+\alpha+1,\alpha+1,uze^{-\pi i}}\right) $. However, the same is not true for the corresponding expansions (\[eq33\]) and (\[eq43\]) of $L_{n}^{\left( \alpha\right) }\left( {uz}\right) $. This is because $w_{0}\left( {u,z}\right) $ (defined by (\[eq31\])) is in general not recessive at $z=0$ when $\alpha<0$ (except when $\alpha$ is an integer). Indeed, this time the appropriate solution recessive at the origin is given by$$w_{2}\left( {u,z}\right) \equiv z^{\left( {\alpha+1}\right) /2}% e^{-uz/2}\mathbf{N}\left( -n,\alpha+1,uz\right) , \label{eq431}%$$ where $\mathbf{N}\left( a,c,x\right) $ is Olver’s scaled confluent hypergeometric function defined in [@Olver:1997:ASF Chap. 7, sec. 9]. In particular we have for ${\alpha\neq1,2,3,\cdots}$ $$\mathbf{N}\left( -n,\alpha+1,x\right) =\frac{x^{-\alpha}}{\Gamma\left( {1-\alpha}\right) }\left\{ {1+\mathcal{O}\left( x\right) }\right\} \ \left( {x\rightarrow0}\right) , \label{eq432}%$$ and hence$$w_{2}\left( {u,z}\right) \equiv\frac{z^{\left( 1-{\alpha}\right) /2}% }{\Gamma\left( {1-\alpha}\right) u^{\alpha}}\left\{ {1+\mathcal{O}\left( z\right) }\right\} \ \left( {z\rightarrow0}\right) . \label{q433}%$$ Comparing this to (\[eq32\]) we see that $w_{2}\left( {u,z}\right) $ has the desired recessive behaviour at $z=0$ when $\alpha<0$. We remark that when $\alpha$ is a negative integer $w_{0}\left( {u,z}\right) $ and $w_{2}\left( {u,z}\right) $ are multiples of one another, since if $\alpha=-p$ for any integer $p\in\left[ 1,n\right) $ we have from (\[eq0\]) and (\[eq31\]) $$w_{0}\left( {u,z}\right) =\frac{\left( -u\right) ^{p}z^{\left( {p+1}\right) /2}}{p!}\left\{ {1+\mathcal{O}\left( z\right) }\right\} \ \left( {z\rightarrow0}\right) . \label{eq435}%$$ By identifying $w_{2}\left( {u,z}\right) $ and $f^{-1/4}\left( z\right) V_{n,2}\left( {u,\xi}\right) $ in a similar manner to (\[eq32a\]), we find for $-u\left( 1-{a_{0}}\right) \leq\alpha\leq-u\delta<0$ that $$% \begin{array} [c]{l}% \mathbf{N}\left( -n,\alpha+1,uz\right) \sim\dfrac{1}{\Gamma\left( 1-{\alpha}\right) }\left( {\dfrac{\left\vert \alpha\right\vert }{u}}\right) ^{1/2}\left( {\dfrac{u+\alpha}{u}}\right) ^{u/2}\left\{ {\dfrac{\left( {u+\alpha}\right) e}{\alpha^{2}uz}}\right\} ^{\alpha/2}\\ \\ \times\dfrac{1}{\left\{ {\left( {z_{1}-z}\right) \left( {z_{2}-z}\right) }\right\} ^{1/4}}\exp\left\{ {\dfrac{1}{2}uz-u\xi+\sum\limits_{s=1}^{\infty }{\left( {-1}\right) ^{s}\dfrac{\hat{{E}}_{s}\left( z\right) -\mu_{s}% }{u^{s}}}}\right\} , \end{array} \label{eqNLG}%$$ uniformly for $z\in D_{2}^{+}\cup D_{2}^{-}$. Likewise, analogously to (\[eq43\])$$% \begin{array} [c]{l}% \mathbf{N}\left( -n,\alpha+1,uz\right) \sim\dfrac{2\pi^{1/2}u^{1/6}}% {\Gamma\left( 1-{\alpha}\right) }\left( {\dfrac{\left\vert \alpha \right\vert }{u}}\right) ^{1/2}\left( {\dfrac{u+\alpha}{u}}\right) ^{u/2}\\ \\ \times\left\{ {\dfrac{\left( {u+\alpha}\right) e}{\alpha^{2}uz}}\right\} ^{\alpha/2}\left\{ {\dfrac{\zeta}{\left( {z_{1}-z}\right) \left( {z_{2}% -z}\right) }}\right\} ^{1/4}\exp\left\{ {\dfrac{1}{2}uz+\sum\limits_{s=0}% ^{\infty}{\dfrac{\mu_{2s+1}}{u^{2s+1}}}}\right\} \\ \\ \times\left\{ {\mathrm{Ai}\left( {u^{2/3}\zeta}\right) A\left( {u,z}\right) +}\mathrm{Ai}^{\prime}{\left( {u^{2/3}\zeta}\right) B\left( {u,z}\right) }\right\} , \end{array} \label{eqNAiry}%$$ uniformly for $z\in\mathbf{D}$. To obtain the desired expansions for the Laguerre polynomials we use (\[eqLMU\]) and the connection formula$$% \begin{array} [c]{l}% \mathbf{M}\left( -n,\alpha+1,z\right) =\dfrac{e^{\alpha\pi i}n!}% {\Gamma\left( {n+\alpha+1}\right) }\mathbf{N}\left( -n,\alpha+1,z\right) \\ -\dfrac{\sin\left( \pi\alpha\right) n!}{\pi}e^{z}U\left( {n+\alpha +1,\alpha+1,ze^{-\pi i}}\right) , \end{array} \label{eq436}%$$ to obtain$$% \begin{array} [c]{l}% L_{n}^{(\alpha)}\left( {uz}\right) =e^{\alpha\pi i}\mathbf{N}\left( -n,\alpha+1,z\right) \\ \\ -\pi^{-1}\sin\left( \pi\alpha\right) \Gamma\left( {n+\alpha+1}\right) e^{z}U\left( {n+\alpha+1,\alpha+1,ze^{-\pi i}}\right) . \end{array} \label{eq437}%$$ On inserting the expansions (\[eq51a\]) and (\[eqNLG\]) into (\[eq437\]) yields the desired expansion for $L_{n}^{\left( \alpha\right) }\left( {uz}\right) $ for $-u\left( 1-{a_{0}}\right) \leq\alpha \leq-u\delta<0$. Likewise, the corresponding Airy expansion comes from (\[eq43b\]), (\[eqNAiry\]), and (\[eq437\]). L-G and Airy expansions: Case 2 =============================== We note that ${S\left( z\right) <0}$ for $z\in\left( z_{2},\infty\right) $, where $S\left( z\right) $ is given by (\[eq9\]). Hence the same is true for ${f^{1/2}\left( z\right) =S\left( z\right) /}\left( 2z\right) ,$ and with this in mind we now define the L-G variable by $$\tilde{\xi}=-\int_{z_{2}}^{z}{f^{1/2}\left( t\right) dt,} \label{eq44}%$$ so that $\tilde{\xi}\geq0$ for $z\geq z_{2}$. Explicit integration yields $$% \begin{array} [c]{l}% \tilde{\xi}=-\dfrac{1}{2}\left( {a^{2}+1}\right) \ln\left\{ {z-S\left( z\right) -a^{2}-1}\right\} -\dfrac{1}{2}S\left( z\right) +\max\left\{ a^{2},1\right\} \ln\left( {2a}\right) \\ \\ -\dfrac{1}{2}\left\vert {a^{2}-1}\right\vert \ln\left\{ {\dfrac{\left( {a^{2}+1}\right) z-\left( {a^{2}-1}\right) ^{2}+\left\vert {a^{2}% -1}\right\vert S\left( z\right) }{z}}\right\} . \end{array} \label{eq45}%$$ We observe that $\tilde{\xi}{=}\xi-\min\left\{ a^{2},1\right\} \pi i$, where $\xi$ is the corresponding L-G variable from case 1. The L-G asymptotic solution that we employ this time is given by the expansion$$\tilde{V}_{n,2}\left( {u,\tilde{\xi}}\right) =\exp\left\{ -{u\tilde{\xi }+\sum\limits_{s=1}^{n-1}{\left( -{1}\right) ^{s}\frac{\hat{E}_{s}\left( z\right) }{u^{s}}}}\right\} +\tilde{\varepsilon}_{n,2}\left( {u,\tilde{\xi }}\right) , \label{eq52a}%$$ where the coefficients $\hat{E}_{s}\left( z\right) $ are the same as in Cases 1a and 1b. In (\[eq52a\]) $\tilde{\varepsilon}_{n,2}\left( {u,\tilde{\xi}}\right) =e^{-u{\tilde{\xi}}}\mathcal{O}\left( {u^{-n}}\right) $ uniformly in a certain unbounded domain $\tilde{\Xi}_{2}$, with $e^{u{\tilde{\xi}}}% \tilde{\varepsilon}_{n,2}\left( {u,\tilde{\xi}}\right) \rightarrow0$ as $\operatorname{Re}{\tilde{\xi}}\rightarrow\infty$ in the domain. Let $\tilde{\Delta}$ denote the domain in the $\tilde{\xi}$ plane corresponding to $\Delta$ in the $\xi$ plane: thus it is $\Delta$ shifted by the factor $-\min\left\{ a^{2},1\right\} \pi i$. Then the domain $\tilde{\Xi}_{2}$ comprises the $\tilde{\xi}$ point subset of $\tilde{\Delta}$ for which there is a path $\mathcal{\tilde{P}}_{2}$ (say) linking $\tilde{\xi}$ with $\tilde{\alpha}_{2}=+\infty$ (corresponding to $z=+\infty+i0$) and having the properties (i) $\mathcal{\tilde{P}}_{2}$ consists of a finite chain of $R_{2}$ arcs, and (ii) as $t$ passes along $\mathcal{\tilde{P}}_{2}$ from $\tilde{\alpha}_{2}$ to $\xi$, $\operatorname{Re}\left( ut\right) $ is nonincreasing. If $\tilde{\Xi}_{2}^{+}$ denotes the subset of $\tilde{\Xi}_{2}$ corresponding to $0\leq\arg\left( z\right) \leq\pi$, then due to condition (ii) above $\tilde{\Xi}_{2}^{+}$ must also exclude all points in the fourth quadrant of the $\tilde{\xi}$ plane. The corresponding $z$-domain $\tilde{D}_{2}^{+}$ (say) is thus unshaded region depicted in Fig. \[fig5zxi\], where the interval $\left[ z_{1},z_{2}\right] $ and the boundary curve EB must be excluded; the curve EB emanates from $z=z_{1}$ to a point on the upper part of the cut along $\left( -\infty,0\right) $, and is given parametrically by $$\int_{z_{1}}^{z}{f^{1/2}\left( t\right) dt=-\tau i,\ 0\leq\tau\leq\dfrac {1}{2}\left\vert {a^{2}-1}\right\vert .}%$$ We remark that $z=0$ is not contained in $\tilde{D}_{2}^{+}$. ![Domain $\tilde{D}_{2}^{+}$ in $z$ plane. The interval $\left[ z_{1},z_{2}\right] $ and boundary EB are excluded.[]{data-label="fig5zxi"}](Fig5.eps){height="6cm" width="8cm"} We now use the behaviour of the Laguerre polynomials at infinity to match it with the L-G expansion (\[eq52a\]). Specifically, we know that $$L_{n}^{\left( \alpha\right) }\left( {uz}\right) =\frac{\left( {-1}\right) ^{n}}{n!}U\left( {-n,\alpha+1,uz}\right) , \label{eq47}%$$ and hence the following solution of (\[eq3\]) $$\tilde{w}_{0}\left( {u,z}\right) \equiv\left( {uz}\right) ^{\left( {\alpha+1}\right) /2}e^{-uz/2}L_{n}^{\left( \alpha\right) }\left( {uz}\right) , \label{eq48}%$$ has, from [@Daalhuis:2010:CHF Eq. 13.7.3], the behaviour as $z\rightarrow\infty$ $$\tilde{w}_{0}\left( {u,z}\right) =\frac{\left( {-1}\right) ^{n}}% {n!}\left( {uz}\right) ^{\left( {2u+\alpha}\right) /2}e^{-uz/2}\left\{ {1+\mathcal{O}\left( {\frac{1}{z}}\right) }\right\} . \label{eq49}%$$ On comparing this to (\[eq51\]) we see that $\tilde{w}_{0}\left( {u,z}\right) $ is the unique solution that is recessive in right half plane. On matching, we therefore ascertain that there exits a constant $\tilde{c}% _{0}\left( u\right) $ such that $$\tilde{w}_{0}\left( {u,z}\right) =\tilde{c}_{0}\left( u\right) f^{-1/4}\left( z\right) \tilde{V}_{n,2}\left( {u,}\tilde{\xi}\right) , \label{eq52}%$$ where $\tilde{V}_{n,2}\left( {u,}\tilde{\xi}\right) $ is given by (\[eq52a\]). Using $$\tilde{\xi}={\tfrac{1}{2}}z-{\tfrac{1}{2}}\left( {a^{2}+1}\right) \left\{ {\ln\left( z\right) +1}\right\} +a^{2}\ln\left( a\right) +\mathcal{O}% \left( {z^{-1}}\right) \ \left( {z\rightarrow\infty}\right) , \label{eq53}%$$ we have from (\[eq52a\]), (\[eq26\]) and (\[eq49\]) $$% \begin{array} [c]{l}% \tilde{c}_{0}\left( u\right) =\lim\limits_{z\rightarrow\infty}\left\{ {\dfrac{f^{1/4}\left( z\right) \tilde{w}_{0}\left( {u,z}\right) }% {\tilde{V}_{n,2}\left( {u,}\tilde{\xi}\right) }}\right\} \\ \sim\dfrac{\left( {-1}\right) ^{n}u^{u/2}\left( {u+\alpha}\right) ^{\left( {u+\alpha}\right) /2}}{\sqrt{2}e^{u+\left( {\alpha/2}\right) }% n!}\exp\left\{ {-\sum\limits_{s=1}^{\infty}{\left( {-1}\right) ^{s}% \dfrac{\lambda_{s}}{u^{s}}}}\right\} . \end{array} \label{eq54}%$$ Hence from (\[eq48\]), (\[eq52\]) and (\[eq54\]) $$% \begin{array} [c]{l}% L_{n}^{\left( \alpha\right) }\left( {uz}\right) \sim\dfrac{\left( -1\right) ^{n}u^{\left( u-1\right) /2}\left( u+\alpha\right) ^{\left( u+\alpha\right) /2}}{e^{u+\left( \alpha/2\right) }n!\left( uz\right) ^{\alpha/2}\left\{ {\left( z-{z_{1}}\right) \left( z-{z_{2}}\right) }\right\} ^{1/4}}\\ \\ \times\exp\left\{ {\dfrac{1}{2}uz-u\tilde{\xi}+\sum\limits_{s=1}^{\infty }{\left( {-1}\right) ^{s}\dfrac{\hat{{E}}_{s}\left( z\right) -\lambda_{s}% }{u^{s}}}}\right\} , \end{array} \label{eq55}%$$ uniformly for $z\in\tilde{D}_{2}^{+}\cup\tilde{D}_{2}^{-}$, where $\tilde {D}_{2}^{-}$ is the conjugate region of $\tilde{D}_{2}^{+}$. We emphasise that this expansion is not valid on the interval $\left[ z_{1},z_{2}\right] $. We next construct an Airy function expansion, which is valid at $z=z_{2}$, similarly to (\[eq43\]). We have, again by matching solutions (\[eq37\]) with $j=0$ (all recessive at $z=+\infty$) with (\[eq48\]), that $$% \begin{array} [c]{l}% \tilde{C}_{0}\left( u\right) \left\{ {\dfrac{\left( {z-z_{1}}\right) \left( {z-z_{2}}\right) }{\tilde{\zeta}}}\right\} ^{1/4}z^{\alpha /2}e^{-uz/2}L_{n}^{\left( \alpha\right) }\left( {uz}\right) \\ \\ =\mathrm{Ai}\left( {u^{2/3}}\tilde{\zeta}\right) \tilde{A}\left( {u,z}\right) +\mathrm{Ai}^{\prime}\left( {u^{2/3}\tilde{\zeta}}\right) \tilde{B}\left( {u,z}\right) , \end{array} \label{eq56}%$$ for some constant $\tilde{C}_{0}\left( u\right) $. Here $\tilde{\zeta }=\left( {{\tfrac{3}{2}}}\tilde{\xi}\right) ^{2/3}$, where $\tilde{\xi}$ is given by (\[eq45\]), and the coefficient functions $\tilde{A}\left( {u,z}\right) \ $and $\tilde{B}\left( {u,z}\right) $ are analytic at $z=z_{2}$ ($\tilde{\zeta}=0$). These have the expansions (c.f. (\[eq38\]) and (\[eq39\])) $$% \begin{array} [c]{l}% \tilde{A}\left( {u,z}\right) \sim\exp\left\{ {\sum\limits_{s=1}^{\infty }{\dfrac{\hat{{E}}_{2s}\left( z\right) +\tilde{{a}}_{2s}\tilde{\xi}% ^{-2s}/\left( {2s}\right) }{u^{2s}}}}\right\} \\ \times\cosh\left\{ {\sum\limits_{s=0}^{\infty}{\dfrac{\hat{{E}}_{2s+1}\left( z\right) -\tilde{{a}}_{2s+1}\tilde{\xi}^{-2s-1}/\left( {2s+1}\right) }{u^{2s+1}}}}\right\} , \end{array} \label{eq56a}%$$ and $$% \begin{array} [c]{l}% \tilde{B}\left( {u,z}\right) \sim\dfrac{1}{u^{1/3}\tilde{\zeta}^{1/2}}% \exp\left\{ {\sum\limits_{s=1}^{\infty}{\dfrac{\hat{{E}}_{2s}\left( z\right) +a_{2s}\tilde{\xi}^{-2s}/\left( {2s}\right) }{u^{2s}}}}\right\} \\ \times\sinh\left\{ {\sum\limits_{s=0}^{\infty}{\dfrac{\hat{{E}}_{2s+1}\left( z\right) -a_{2s+1}\tilde{\xi}^{-2s-1}/\left( {2s+1}\right) }{u^{2s+1}}}% }\right\} . \end{array} \label{eq56b}%$$ uniformly for $z\in\mathbf{\tilde{D}}$; this domain is the unshaded region depicted in Fig. \[fig6zxi\], where EB’ is the conjugate curve of EB. All points on the boundary curve B’EB are excluded from $\mathbf{\tilde{D}}$. ![Domain $\mathbf{\tilde{D}}$ in $z$ plane.[]{data-label="fig6zxi"}](Fig6.eps){height="6cm" width="8cm"} Now as $\tilde{\zeta}\rightarrow\infty$ we find from (\[eq56a\]), (\[eq56b\]), and the behaviour of the Airy function and its derivative for large argument [@Olver:2010:AAR §9.7(ii)], that the RHS behaves as $$% \begin{array} [c]{l}% \mathrm{Ai}\left( {u^{2/3}\tilde{\zeta}}\right) \tilde{A}\left( {u,z}\right) +\mathrm{Ai}^{\prime}\left( {u^{2/3}\tilde{\zeta}}\right) \tilde{B}\left( {u,z}\right) \\ \\ \sim\dfrac{1}{2\pi^{1/2}u^{1/6}\tilde{\zeta}^{1/4}}\exp\left\{ {-u\tilde{\xi }-\sum\limits_{s=0}^{\infty}{\dfrac{\lambda_{2s+1}}{u^{2s+1}}}}\right\} . \end{array} \label{eq57}%$$ On the other hand, from (\[eq49\]), the LHS has the behaviour $$% \begin{array} [c]{l}% \tilde{C}_{0}\left( u\right) \left\{ {\dfrac{\left( {z-z_{1}}\right) \left( {z-z_{2}}\right) }{\tilde{\zeta}}}\right\} ^{1/4}z^{\alpha /2}e^{-uz/2}L_{n}^{\left( \alpha\right) }\left( {uz}\right) \\ \\ \sim\dfrac{\left( {-1}\right) ^{n}\tilde{C}_{0}\left( u\right) u^{n}z^{\left( {2n+\alpha+1}\right) /2}e^{-uz/2}}{n!\tilde{\zeta}^{1/4}}. \end{array} \label{eq58}%$$ Thus from (\[eq4\]), (\[eq53\]), (\[eq56\]) - (\[eq58\]) we can solve for $\tilde{C}_{0}\left( u\right) $, and from this deduce that $$% \begin{array} [c]{l}% L_{n}^{(\alpha)}\left( uz\right) \sim\dfrac{\left( {-1}\right) ^{n}% 2\pi^{1/2}u^{\left( {u}/2\right) -\left( {1/3}\right) }\left( {u+\alpha }\right) ^{\left( {u+\alpha}\right) /2}}{n!\left( uz\right) ^{\alpha/2}% }\\ \\ \times\left\{ {\dfrac{\tilde{\zeta}}{\left( {z-z_{1}}\right) \left( {z-z_{2}}\right) }}\right\} ^{1/4}\exp\left\{ {\dfrac{1}{2}uz-u-\dfrac {1}{2}\alpha+\sum\limits_{s=0}^{\infty}{\dfrac{\lambda_{2s+1}}{u^{2s+1}}}% }\right\} \\ \\ \times\left\{ {\mathrm{Ai}\left( {u^{2/3}\tilde{\zeta}}\right) \tilde {A}\left( {u,z}\right) +}\mathrm{Ai}^{\prime}{\left( {u^{2/3}\tilde{\zeta}% }\right) \tilde{B}\left( {u,z}\right) }\right\} . \end{array} \label{eq59}%$$ For the complementary solution we have, equivalent to (\[eq51a\]), the asymptotic expansion $$% \begin{array} [c]{l}% U\left( n+\alpha+1,\alpha+1,uze^{-\pi i}\right) \sim\dfrac{\left( {-1}\right) ^{n+1}e^{\alpha\pi i+u+\left( \alpha/2\right) }}{u^{\left( u+1\right) /2}\left( u+\alpha\right) ^{\left( u+\alpha\right) /2}}\\ \times\dfrac{1}{\left( uz\right) ^{\alpha/2}\left\{ \left( {z-z_{1}% }\right) \left( {z-z_{2}}\right) \right\} ^{1/4}}\\ \times\exp\left\{ {-\dfrac{1}{2}uz+u\tilde{\xi}+\sum\limits_{s=1}^{\infty }{\dfrac{\hat{{E}}_{s}\left( z\right) -\lambda_{s}}{u^{s}}}}\right\} . \end{array} \label{eq61}%$$ Similarly to the derivation of (\[eq59\]), using $$% \begin{array} [c]{l}% \mathrm{Ai}_{1}\left( {u^{2/3}\tilde{\zeta}}\right) \tilde{A}\left( {u,z}\right) +\mathrm{Ai}_{1}^{\prime}\left( {u^{2/3}\tilde{\zeta}}\right) \tilde{B}\left( {u,z}\right) \\ \sim\dfrac{e^{\pi i/6}}{2\pi^{1/2}u^{1/6}\tilde{\zeta}^{1/4}}\exp\left\{ {u\tilde{\xi}+\sum\limits_{s=0}^{\infty}{\dfrac{\lambda_{2s+1}}{u^{2s+1}}}% }\right\} , \end{array} \label{eq62}%$$ we arrive at$$% \begin{array} [c]{l}% U\left( {n+\alpha+1,\alpha+1,uze^{-\pi i}}\right) \sim\dfrac{\left( {-1}\right) ^{n+1}2\pi^{1/2}e^{-\pi i/6}e^{\alpha\pi i}}{u^{1/3}\left\{ {u\left( {u+\alpha}\right) }\right\} ^{\left( {u+\alpha}\right) /2}z^{\alpha/2}}\\ \\ \times\left\{ {\dfrac{\tilde{\zeta}}{\left( {z-z_{1}}\right) \left( {z-z_{2}}\right) }}\right\} ^{1/4}\exp\left\{ {-\dfrac{1}{2}uz+u+\dfrac {1}{2}\alpha-\sum\limits_{s=0}^{\infty}{\dfrac{\lambda_{2s+1}}{u^{2s+1}}}% }\right\} \\ \\ \times\left\{ {\mathrm{Ai}_{1}\left( {u^{2/3}\tilde{\zeta}}\right) \tilde{A}\left( {u,z}\right) +}\mathrm{Ai}_{1}^{\prime}{\left( {u^{2/3}\tilde{\zeta}}\right) \tilde{B}\left( {u,z}\right) }\right\} . \end{array} \label{eq63}%$$ Numerical results ================= Here we illustrate the accuracy of the new expansions for Cases 1a and 2. We concentrate on Laguerre polynomials with $n$ large and $\alpha$ non-negative, but analogous results are available for negative $\alpha$, as well as for the complementary confluent hypergeometric functions. Case 1a ------- In this case the expansions are valid in domains containing $z=0$ and $z=z_{1}$. The relevant L-G expansion is given by (\[eq33\]), and after truncating after $N\geq1$ terms we have $$% \begin{array} [c]{l}% L_{n}^{\left( \alpha\right) }\left( {uz}\right) =\dfrac{\Gamma\left( {n+\alpha+1}\right) }{n!\Gamma\left( {\alpha+1}\right) }\left( {\dfrac{\alpha}{u}}\right) ^{1/2}\left( {\dfrac{u}{u+\alpha}}\right) ^{u/2}\\ \times\left\{ {\dfrac{\alpha^{2}}{\left( {u+\alpha}\right) uez}}\right\} ^{\alpha/2}\dfrac{1}{\left\{ {\left( {z_{1}-z}\right) \left( {z_{2}% -z}\right) }\right\} ^{1/4}}\\ \times\exp\left\{ {\dfrac{1}{2}uz-u\xi+\sum\limits_{s=1}^{N}{\left( {-1}\right) ^{s}\dfrac{\hat{{E}}_{s}\left( z\right) -\mu_{s}}{u^{s}}}% }\right\} \left\{ 1+\mathcal{O}\left( \dfrac{1}{u^{N+1}}\right) \right\} . \end{array} \label{eq63aa}%$$ The order term is uniformly valid for $z\in D_{2}^{+}\cup D_{2}^{-}$, where $D_{2}^{+}$ is shown in Fig. \[fig3zxi\], and $D_{2}^{-}$ is the conjugate region of $D_{2}^{+}$. The Airy expansion is given by (\[eq43\]), and (\[eq38\]) and (\[eq39\]) are used to approximate the coefficient functions. Therefore uniformly for $z\in\mathbf{D}$ (the domain of validity depicted in Fig. \[fig4zxi\]) we have$$% \begin{array} [c]{l}% L_{n}^{\left( \alpha\right) }\left( {uz}\right) =\dfrac{2\pi^{1/2}% u^{1/6}\Gamma\left( {n+\alpha+1}\right) }{n!\Gamma\left( {\alpha+1}\right) }\left( {\dfrac{\alpha}{u}}\right) ^{1/2}\left( {\dfrac{u}{u+\alpha}% }\right) ^{u/2}\\ \times\left( {\dfrac{\alpha^{2}}{\left( {u+\alpha}\right) uez}}\right) ^{\alpha/2}\exp\left\{ {\dfrac{1}{2}uz+\sum\limits_{s=0}^{m-1}{\dfrac {\mu_{2s+1}}{u^{2s+1}}}}\right\} \\ \times\left[ {\mathrm{Ai}\left( {u^{2/3}\zeta}\right) }\left\{ \mathcal{A}_{m}{\left( {u,z}\right) }+\mathcal{O}\left( \dfrac{1}{u^{2m+1}% }\right) \right\} \right. \\ {+}\left. \mathrm{Ai}^{\prime}{\left( {u^{2/3}\zeta}\right) }\left\{ \mathcal{B}_{m}{\left( {u,z}\right) }+\mathcal{O}\left( \dfrac {1}{u^{2m+4/3}}\right) \right\} \right] . \end{array} \label{eq63a}%$$ In this, for any positive integer $m$, we have introduced the truncated expansions$$% \begin{array} [c]{l}% \mathcal{A}_{m}\left( {u,z}\right) =\chi\left( u,z\right) \exp\left\{ {\sum\limits_{s=1}^{m}{\dfrac{\hat{{E}}_{2s}\left( z\right) +\tilde{{a}% }_{2s}\xi^{-2s}/\left( {2s}\right) }{u^{2s}}}}\right\} \\ \times\cosh\left\{ {\sum\limits_{s=0}^{m-1}{\dfrac{\hat{{E}}_{2s+1}\left( z\right) -\tilde{{a}}_{2s+1}\xi^{-2s-1}/\left( {2s+1}\right) }{u^{2s+1}}}% }\right\} , \end{array} \label{eq64}%$$ and $$% \begin{array} [c]{l}% \mathcal{B}_{m}\left( {u,z}\right) =\dfrac{\chi\left( u,z\right) }% {u^{1/3}\zeta^{1/2}}\exp\left\{ {\sum\limits_{s=1}^{m}{\dfrac{\hat{{E}}% _{2s}\left( z\right) +a_{2s}\xi^{-2s}/\left( {2s}\right) }{u^{2s}}}% }\right\} \\ \times\sinh\left\{ {\sum\limits_{s=0}^{m-1}{\dfrac{\hat{{E}}_{2s+1}\left( z\right) -a_{2s+1}\xi^{-2s-1}/\left( {2s+1}\right) }{u^{2s+1}}}}\right\} , \end{array} \label{eq65}%$$ where$$\chi\left( u,z\right) =\left\{ {\dfrac{\zeta}{\left( {z_{1}-z}\right) \left( {z_{2}-z}\right) }}\right\} ^{1/4}. \label{eq65a}%$$ We remark that the term $\chi\left( u,z\right) $ has been absorbed into the scaled functions $\mathcal{A}_{m}\left( {u,z}\right) $ and $\mathcal{B}% _{m}\left( {u,z}\right) $ because it has a removable singularity at $z={z_{1}}$ ($\zeta=0$), and hence it is easier to compute via Cauchy’s integral formula when $z$ is close to ${z_{1}}$, as described next. Let $z\in\mathbf{D}$: if this point is not too close to the turning point $z_{1}$ we can use (\[eq64\]) and (\[eq65\]) directly for their numerically stable computation in (\[eq63a\]). On the other hand, if $z$ is close to $z_{1}$ these expansions are not stable, since each $\hat{{E}}% _{s}\left( z\right) $ is unbounded at this turning point. Instead we follow [@Dunster:2017:COA] to compute these functions via Cauchy’s integral formula. Now, neither $\mathcal{A}_{m}{\left( {u,z}\right) }$ nor $\mathcal{B}_{m}\left( {u,z}\right) $ are analytic at $z={z}_{1}$, but $\chi\left( u,z\right) A\left( u,z\right) $ and $\chi\left( u,z\right) B\left( u,z\right) $ are. Thus, we have$$% \begin{array}{ll} \chi\left( u,z\right) A\left( u,z\right) & = \dfrac{1}{2\pi i} \displaystyle\oint_{\mathcal{L}_{1}}\dfrac{\chi\left( u,t\right) A\left( u,t\right) }{t-z}dt\\ & \\ &=\dfrac{1}{2\pi i}\displaystyle\oint_{\mathcal{L}_{1}}\dfrac{\mathcal{A}% _{m}\left( u,t\right) +\mathcal{O}\left( u^{-2m-1}\right) }{t-z}dt\\ & \\ &=\dfrac{1}{2\pi i}\displaystyle\oint_{\mathcal{L}_{1}}\dfrac{\mathcal{A}_{m}\left( u,t\right) }{t-z}dt+\mathcal{O}\left( \dfrac{1}{u^{2m+1}}\right) , \end{array} \label{eq67}%$$ for some suitably-chosen bounded simple loop $\mathcal{L}_{1}$ in the $t$-plane that is positively orientated, lies in an equivalent domain to $\mathbf{D}$, and surrounds $t=z$ and $t=z_{1}$. Hence in (\[eq63a\]) it is legitimate (if necessary) to replace $\mathcal{A}_{m}{\left( {u,z}\right) }$ by $\left( 2\pi i\right) ^{-1}\oint\limits_{\mathcal{L}_{1}}\mathcal{A}% _{m}\left( u,t\right) \left( t-z\right) ^{-1}dt$, and similarly for $\mathcal{B}_{m}\left( {u,z}\right) $. We then compute these integrals using the trapezoidal rule, which has exponential convergence. In our computations we shall take $N$ even and $m=N/2$ so that the order terms in (\[eq63aa\]) and (\[eq63a\]) are of comparable magnitude (and likewise for Case 2 considered later). Note that for fixed ${\zeta}$$$\begin{array}{l} u^{1/6}{\mathrm{Ai}\left( {u^{2/3}\zeta}\right) =\exp}\left( -{u\xi }\right) \mathcal{O}\left( 1\right) ,\\ \\ \ u^{1/6}{\mathrm{Ai}}^{\prime }{\left( {u^{2/3}\zeta}\right) =\exp}\left( -{u\xi}\right) \mathcal{O}% \left( u^{1/3}\right) , \label{eq67a}% \end{array}$$ as $u\rightarrow\infty$ (see [@Olver:2010:AAR §9.7(ii)]). If we consider the natural choice of $\mathcal{L}_{1}$ being a circular path centred on $z_{1}$, it should have a radius smaller than $$r_{m}=\min\left\{ z_{1},z_{2}-z_{1}\right\} . \label{eq69}%$$ We have that $r_{m}=z_{1}$ when $a\leq3+2\sqrt{2}$ (that is, $\alpha/\left( n+1/2\right) \leq4\left( 4+3\sqrt{2}\right)$) and $r_{m}=z_{2}-z_{1}$ if $a\geq3+2\sqrt{2}$. When we have to use Cauchy’s integral formula (for $z$ close to $z_{1}$) it is clear that $\operatorname{Re}z>0$; however, we have shown that the validity of both the L-G approximation and the Airy expansion away from the turning point $z_{1}$ extends to $\operatorname{Re}z<0$. We first consider circular $\mathcal{L}_{1}$ as described above with a radius smaller that $r_{m}$, and later we give details for $\operatorname{Re}z<0$ where we can use (\[eq64\]) and (\[eq65\]) directly (without resorting to Cauchy integrals). Let the relative error of an approximation $f^{}$ of a function $f$ be defined in the usual manner by $$\epsilon_{rel}=\left\vert \frac{f-f^{}}{f}\right\vert . \label{eqRelErr}%$$ Then in Fig. \[fig5\] we show the relative errors of the L-G expansion (\[eq63aa\]) and the Airy expansion (\[eq63a\]) for the Laguerre polynomial. In both cases $\epsilon_{rel}$ is shown as a function of the angle when the upper half circle centered at $z_{1}$ is followed clockwise. As expected, we observe a step increase in the relative error for the L-G approximation as we cross a Stokes line; contrarily, the Airy expansion works well for all angles. In addition, we observe that the relative error for the Airy expansion is smaller than for the L-G expansion for all $\theta$. Fig. \[fig9\] also shows that this is true for real values $0<z<z_{1}$, while for negative $z$ both expansions give similar accuracies. In these and in most of the figures, we consider the case of Laguerre polynomials of degree $n=100$ and we use (\[eq63aa\]) with $N=16$ and (\[eq64\]) and (\[eq65\]) with $m=8$. If a higher degree $n$ or a larger number of coefficients is considered, the accuracy will improve (except, of course, if $N$ or $m$ is too large, because the series are asymptotic but not convergent). ![Relative errors $\epsilon_{rel}$ for the L-G expansion (\[eq63aa\]) and the Airy expansion (\[eq63a\]), with coefficients computed from (\[eq64\]) and (\[eq65\]), as a function of $\theta$, where $z=z_{1}-R e^{-i\theta}$, $\theta\in[0,\pi]$ and $R=0.5 r_{m}=0.5 z_{1}$.[]{data-label="fig5"}](f5.eps){height="5.5cm" width="7.5cm"} ![Relative errors $\epsilon_{rel}$ for the L-G expansion (\[eq63aa\]) and the Airy expansion (\[eq63a\]), with coefficients computed from (\[eq64\]) and (\[eq65\]), as a function of $z/z_{1}$ for real values of $z$, $z<z_{1}$.[]{data-label="fig9"}](f9.eps){height="5.5cm" width="7.5cm"} Fig. \[fig6\] shows the maximum relative error over circles centred at $z_{1}$ of different radii (but always contained in the strip $\{z:0<\operatorname{Re}z<z_{2}\}$). For the curves corresponding to $a<3+2\sqrt{2}$ ($\alpha=100,1000$) the circles do not get close to $z_{2}$, and this is why the relative error is monotonic as $r$ increases. In the other to cases, the circle touches $z_{2}$ when $r=1$ which explains the different pattern. We observe that the approximation works for $\alpha$ large comparable to $n$ but that when $\alpha$ is too large accuracy is lost. The circles considered in Fig. \[fig6\] are the only circles centered at $z_{1}$ that can be used for computing the coefficients of the Airy expansion by Cauchy integrals. Indeed, we can not consider $z>z_{2}$ because the expansions are meaningless there, and we can not consider $\operatorname{Re}% z<0$ because of the discontinuity branch at the negative real axis (which implies a discontinuity in the coefficients $A\left( u,z\right) $ and $B\left( u,z\right) $). ![The maximum relative error of the Airy expansion (\[eq63a\]), with coefficients computed from (\[eq64\]) and (\[eq65\]), over the circles given by $z=z_{2}+r r_{m} e^{i\theta}$, $\theta\in[0,2\pi)$, $r_{m}=\min\{z_{1}% ,z_{2}-z_{1}\}$, is plotted as a function of $r\in[0,1].$[]{data-label="fig6"}](f6.eps){height="5.5cm" width="7.5cm"} However, as commented earlier and shown in Fig. \[fig9\], the Airy expansion (with coefficients computed by L-G asymptotics) is valid for $\operatorname{Re}z<0$. In particular it is also valid for negative $z$, as Fig. \[fig7\] also shows. In this figure, we plot the relative error of the Airy expansion (with coefficients from asymptotics) for real $z$ as a function of $\tau=\left( z-z_{1}\right) /\left( z_{2}-z_{1}\right) $. This plot includes negative values of $z$ for the two smaller values of $\alpha$, for $\tau<\tau(0)\approx-0.0301$ when $\alpha=100$ and for $\tau<\tau (0)\approx-0.4028$ when $\alpha=1000$ (for the other two cases $\tau(0)<-1$ and therefore $z>0$ in the figure). We observe that the expansion is also accurate for negative $z$. As expected, the expansions fail close to the turning points ($\tau=0$, $\tau=1$). ![Relative error $\epsilon_{rel}$ of the Airy expansion (\[eq63a\]), with coefficients computed from (\[eq64\]) and (\[eq65\]), for real values of $z$ as a function of $\tau=(z-z_{1})/(z_{2}-z_{1}).$[]{data-label="fig7"}](f7.eps){height="5.5cm" width="7.5cm"} We notice that, as is well know, the Laguerre polynomials $L_{n}^{(\alpha )}(uz)$ have $n$ positive real zeros when $\alpha>-1$ and most of them in the interval $(z_{1},z_{2})$. Of course, the relative error (\[eqRelErr\]) at the zeros is meaningless, and loss of relative accuracy is unavoidable very close to these zeros. In the previous figures, this loss of accuracy is not clearly revealed because the function is sampled at values of $z$ which are not too close to the zeros (the sample values are not selected to avoid the zeros, simply happen to be not too close). Values of $z$ very close to the zeros are needed to observe a significant accuracy loss; however, the previous plots show some effect of the zeros because the relative error gives a relatively noisy plot for values of $z$ where zeros occur when compared to the cases without zeros. Compare, for example, the positive values of $\tau$ (for which there are zeros) with the negative values (no zeros) in Fig. \[fig7\]. When we discuss Case 2 in the next section, we will show a detailed example of computation close to the zeros. As commented before, even when the expansions are valid for $\operatorname{Re}% z<0$, these values can not be used for computing the coefficients by means of Cauchy integrals, due to the discontinuity at the branch cut. Fig. \[fig8\] shows this: we plot the imaginary part of the approximations $\mathcal{A}% _{m}{\left( {u,z}\right) }$ and $\mathcal{B}_{m}{\left( {u,z}\right) }$ as functions of the angle when the radius is such that the circle cuts the negative real axis ($r=1.1$), and when it does not ($r=0.9$). For the second case, both coefficients are real over the real line, but not in the first case, when they have a non-zero imaginary part when $z$ is real and negative, that is, when $\theta=0$ (we move along the circles clockwise). ![Imaginary parts of the coefficient function approximations $\mathcal{A}_{N/2}{\left( {u,z}\right) }$ and $\mathcal{B}_{N/2}{\left( {u,z}\right) }$ as a function of $\theta$, where $z=z_{1}-rz_{1}e^{-i\theta}$ for two values of $r$. []{data-label="fig8"}](f8a.eps){height="5.6cm" width="5.6cm"} ![Imaginary parts of the coefficient function approximations $\mathcal{A}_{N/2}{\left( {u,z}\right) }$ and $\mathcal{B}_{N/2}{\left( {u,z}\right) }$ as a function of $\theta$, where $z=z_{1}-rz_{1}e^{-i\theta}$ for two values of $r$. []{data-label="fig8"}](f8b.eps){height="5.6cm" width="5.6cm"} Finally, we notice that the relative error close to the turning point becomes large, as shown in Fig. \[fig7\]; this indicates that, as expected, the L-G expansions for the coefficients of the Airy expansion are not accurate close to the turning point. This loss of accuracy is repaired by computing these coefficients by Cauchy integrals over a contour encircling the turning point (but sufficiently away from it) and contained in the half-plane $\operatorname{Re}z>0$. Typically, the error that can be obtained around the turning point is close to the minimum error in Fig. \[fig6\] (choosing as Cauchy contour a circle with the radius corresponding to the minimum value of the error reached in that figure). We give more explicit examples on the application of Cauchy integrals for the Case 2 asymptotics we discuss next. Case 2 ------ Now we provide numerical evidence of the accuracy of the expansions for Laguerre polynomials in domains containing the turning point $z_{2}$. From (\[eq55\]) the L-G approximation with $N$ terms is now given by $$% \begin{array} [c]{l}% L_{n}^{\left( \alpha\right) }\left( {uz}\right) =\dfrac{\left( -1\right) ^{n}u^{\left( u-1\right) /2}\left( u+\alpha\right) ^{\left( u+\alpha\right) /2}}{e^{u+\left( \alpha/2\right) }n!\left( uz\right) ^{\alpha/2}\left\{ {\left( z-{z_{1}}\right) \left( z-{z_{2}}\right) }\right\} ^{1/4}}\\ \times\exp\left\{ {\dfrac{1}{2}uz-u\tilde{\xi}+\sum\limits_{s=1}^{N}{\left( {-1}\right) ^{s}\dfrac{\hat{{E}}_{s}\left( z\right) -\lambda_{s}}{u^{s}}}% }\right\} \left\{ 1+\mathcal{O}\left( \dfrac{1}{u^{N+1}}\right) \right\} . \end{array} \label{eq70}%$$ The order term is uniformly valid for $z\in\tilde{D}_{2}^{+}\cup\tilde{D}% _{2}^{-}$, where $\tilde{D}_{2}^{+}$ is shown in Fig. \[fig5zxi\], and $\tilde{D}_{2}^{-}$ is the conjugate region of $\tilde{D}_{2}^{+}$. The Airy expansion is given by (\[eq43\]), (\[eq56a\]) and (\[eq56b\]), uniformly for $z\in\mathbf{\tilde{D}}$ (the domain of validity depicted in Fig. \[fig6zxi\]). Hence truncating similarly to (\[eq63a\]) - (\[eq65\]) we have$$% \begin{array} [c]{l}% L_{n}^{\left( \alpha\right) }\left( {uz}\right) =\dfrac{\left( {-1}\right) ^{n}2\pi^{1/2}u^{\left( {u}/2\right) -\left( {1/3}\right) }\left( {u+\alpha}\right) ^{\left( {u+\alpha}\right) /2}}{n!\left( uz\right) ^{\alpha/2}}\\ \times\exp\left\{ {\dfrac{1}{2}uz-u-\dfrac{1}{2}\alpha+\sum\limits_{s=0}% ^{m-1}{\dfrac{\lambda_{2s+1}}{u^{2s+1}}}}\right\} \\ \times\left[ {\mathrm{Ai}\left( {u^{2/3}\tilde{\zeta}}\right) }\left\{ \widetilde{\mathcal{A}}_{m}{\left( {u,z}\right) }+\mathcal{O}\left( \dfrac{1}{u^{2m+1}}\right) \right\} \right. \\ {+}\left. \mathrm{Ai}^{\prime}{\left( {u^{2/3}\tilde{\zeta}}\right) }\left\{ \widetilde{\mathcal{B}}_{m}{\left( {u,z}\right) }+\mathcal{O}% \left( \dfrac{1}{u^{2m+4/3}}\right) \right\} \right] , \end{array} \label{eq71}%$$ where, for any positive integer $m$,$$% \begin{array} [c]{l}% \widetilde{\mathcal{A}}_{m}\left( {u,z}\right) =\tilde{\chi}\left( u,z\right) \exp\left\{ {\sum\limits_{s=1}^{m}{\dfrac{\hat{{E}}_{2s}\left( z\right) +\tilde{{a}}_{2s}\tilde{\xi}^{-2s}/\left( {2s}\right) }{u^{2s}}}% }\right\} \\ \times\cosh\left\{ {\sum\limits_{s=0}^{m-1}{\dfrac{\hat{{E}}_{2s+1}\left( z\right) -\tilde{{a}}_{2s+1}\tilde{\xi}^{-2s-1}/\left( {2s+1}\right) }{u^{2s+1}}}}\right\} , \end{array} \label{eq72}%$$ and $$% \begin{array} [c]{l}% \widetilde{\mathcal{B}}_{m}\left( {u,z}\right) =\dfrac{\tilde{\chi}\left( u,z\right) }{u^{1/3}\tilde{\zeta}^{1/2}}\exp\left\{ {\sum\limits_{s=1}% ^{m}{\dfrac{\hat{{E}}_{2s}\left( z\right) +a_{2s}\tilde{\xi}^{-2s}/\left( {2s}\right) }{u^{2s}}}}\right\} \\ \times\sinh\left\{ {\sum\limits_{s=0}^{m-1}{\dfrac{\hat{{E}}_{2s+1}\left( z\right) -a_{2s+1}\tilde{\xi}^{-2s-1}/\left( {2s+1}\right) }{u^{2s+1}}}% }\right\} , \end{array} \label{eq73}%$$ in which $\tilde{\chi}\left( u,z\right) =\left\{ {\dfrac{\tilde{\zeta}% }{\left( {z-z_{1}}\right) \left( {z-z_{2}}\right) }}\right\} ^{1/4}.$ As in Case 1a, if $z$ is close to the turning point (this time $z_{2}$) we can replace $\widetilde{\mathcal{A}}_{m}{\left( {u,z}\right) }$ by $\left( 2\pi i\right) ^{-1}\oint\limits_{\mathcal{L}_{2}}\widetilde{\mathcal{A}}% _{m}\left( u,t\right) \left( t-z\right) ^{-1}dt$ in (\[eq71\]), and similarly for $\widetilde{\mathcal{B}}_{m}\left( {u,z}\right) $. Here $\mathcal{L}_{2}$ is a bounded contour in the $t$-plane that is positively orientated, lies in an equivalent domain to $\mathbf{\tilde{D}}$, and surrounds $t=z$ and $t=z_{2}$. As in Case 1a we typically take this to be a circle. Figs. \[fig1\] and \[fig10\] compare the accuracy attainable for the L-G expansion (\[eq70\]) and the Airy expansion (\[eq71\]) - (\[eq73\]), both for real and complex variables. Figs. \[fig2\] and \[fig3\] illustrate the accuracy of this same Airy expansion but for different values of $\alpha$. And, finally, Fig. \[fig4\] shows the accuracy of the Airy expansion with the coefficients computed by Cauchy integrals. In all of these cases, the coefficients of the Airy expansion are computed using $N=16$ ($m=8$), and the degree of Laguerre polynomials is set to $n=100$. In Fig. \[fig1\], we compare the L-G expansion with the Airy expansion away from the turning point (with coefficient functions still computed with (\[eq72\]) and (\[eq73\])). The relative accuracy over a circle of radius $R=0.25\left( z_{2}-z_{1}\right) $ centered at $z_{2}$ is shown as a function of the angle $\theta\in\lbrack0,\pi]$; this half-circle is circulated counterclockwise. We observe that the L-G expansion (\[eq70\]) tends to fail for large $\theta$ in this interval, as the expansion loses meaning when we cross a Stokes line. Contrarily, the Airy type expansion (\[eq71\]) - (\[eq73\]) is valid in all the interval. In addition, we observe that the relative error for the Airy expansion is smaller than for the L-G expansion for all $\theta$. We further explore this fact for real variables $z>z_{2}$ in Fig. \[fig10\]. ![Relative errors $\epsilon_{rel}$ for the L-G expansion (\[eq70\]) and the Airy (\[eq71\]) expansion, with coefficients computed from (\[eq72\]) and (\[eq73\]), as a function of $\theta$, where $z=z_{2}+R e^{i\theta}$, $\theta\in[0,\pi]$ and $R=0.25 (z_{2}-z_{1})$. []{data-label="fig1"}](f1.eps){height="5.5cm" width="7.5cm"} ![Relative errors $\epsilon_{rel}$ for the L-G expansion (\[eq70\]) and the Airy expansion (\[eq71\]), with coefficients computed from (\[eq72\]) and (\[eq73\]), as a function of $z/z_{2}$ for real values of $z$, $z>z_{2}$. []{data-label="fig10"}](f10.eps){height="5.5cm" width="7.5cm"} Fig. \[fig2\] shows the maximum relative error of the Airy expansion (\[eq71\]) - (\[eq73\]), over circles centred at $z_{2}$. As expected, the relative error increases both when the radius is small (because we are too close to the turning point $z_{2}$) as well as when part of the circle becomes too close to $z_{1}$. ![The maximum relative error of the Airy expansion (\[eq71\]), with coefficients computed from (\[eq72\]) and (\[eq73\]), over the circles given by $z=z_{2}+Re^{i\theta}$, $\theta\in[0,2\pi)$ is plotted as a function of $r=R/(z_{2}-z_{1})$. The circles are sampled with $100$ points. []{data-label="fig2"}](f2.eps){height="5.5cm" width="7.5cm"} ![Relative error $\epsilon_{rel}$ of the Airy expansion (\[eq71\]), with coefficients computed from (\[eq72\]) and (\[eq73\]), for real values of $z$ as a function of $\rho=\left( z-z_{2}\right) /\left( z_{2}-z_{1}\right) $. []{data-label="fig3"}](f3.eps){height="5.5cm" width="7.5cm"} Fig. \[fig3\] shows again the relative error of the Airy expansion (\[eq71\]) - (\[eq73\]) with coefficients computed with asymptotic series, but for real variable. Again, the relative error increases close to the turning points. For $z>z_{2}$ the relative error decreases as $z$ increases, as can be expected. This figure shows that our alternative Cauchy integral method for the computation of the coefficients $\widetilde{\mathcal{A}}% _{m}{\left( {u,z}\right) }$ and $\widetilde{\mathcal{B}}_{m}\left( {u,z}\right) $ is needed around the turning point. ![Relative error $\epsilon_{rel}$ of the Airy expansion (\[eq71\]) for real values of $z$ as a function of $\rho=\left( z-z_{2}\right) /\left( z_{2}% -z_{1}\right) $, with the coefficient approximations $\widetilde{\mathcal{A}% }_{m}{\left( {u,z}\right) }$ and $\widetilde{\mathcal{B}}_{m}{\left( {u,z}\right) }$ computed by Cauchy integrals. The Cauchy contour is a circle of radius $R=0.7\left( z_{2}-z_{1}\right) $ centred at $z_{2}$.[]{data-label="fig4"}](f4.eps){height="5.5cm" width="7.5cm"} Fig. \[fig4\] provides this computation using the trapezoidal rule with a discretization of $100$ points in the upper half of the Cauchy contour (in the lower half we consider complex conjugation). Combining this computation with the results in Fig. \[fig3\] we observe that it is possible to compute accurately the Laguerre polynomials for $z>z_{1}$, but not too close to $z_{1}$. Observe that, according to Fig. \[fig2\] (see also Fig. \[fig3\]), over a circle of radius $0.7(z_{2}-z_{1})$ centered at $z_{2}$, the relative errors are of the order of $10^{-30}$, and we observe that the application of Cauchy integrals permits us to maintain this accuracy inside the circle (but not too close to the circle). As we see in Fig. \[fig4\], the relative error has little variation and it is of the order of $10^{-30}$ inside the circle $\left\vert z-z_{2}\right\vert <0.5\left( z_{2}-z_{1}\right) $ (the figure is only for real $z$ but the same is true for complex $z$). As we commented in the previous subsection, the relative accuracy unavoidably degrades very close to the zeros and this degration is, for example, responsible for the different appearence of the graphs for $\rho>0$ and $\rho<0$ in Figs. \[fig3\] and \[fig4\]. To illustrate the uniform accuracy of our approximations near the zeros we need to replace the denominator of (\[eqRelErr\]) with an “envelope” of the Laguerre polynomial, which mimics the amplitude of $L_{n}^{\left( \alpha\right) }\left( {x}\right) $ but does not vanish at its zeros. For polynomials having simple zeros, the envelope function $\mathrm{env}f(x) =\left\{ f^{2}(x)+f^{\prime2}(x)\right\} ^{1/2}$ serves this purpose. Then we define the modified relative error of an approximation $f^{}$ to a function $f$ as $$\hat{\epsilon}_{rel}=\Frac{\|f-f^\|}{\mathrm{env}f},$$ where in our case $f(x)=L_{n}^{(\alpha)}(x)$. In Fig. \[errze\] we then show similar results for the Airy expansion in a more restricted interval containing two zeros. In this both the relative error $\epsilon_{rel}$ (solid line) and the modified relative error $\hat{\epsilon }_{rel}$ (dashed line) are shown. In the figure we used many more sample points than in Fig. \[fig4\], so that the (unavoidable) degradation of the relative error $\epsilon_{rel}$ becomes more apparent, whereas in constrast the modified relative error $\hat{\epsilon}_{rel}$ remains bounded. As can be seen, the degradation of $\epsilon_{rel}$ only takes place very close to the zeros. Of course, this relative error degradation is unavoidable and common to any method of numerical computation. On the other hand, $\hat{\epsilon}_{rel}$ is small throughout the interval which illustrates the uniform absolute accuracy of our Airy expansion in the whole interval. ![Relative error $\epsilon_{rel}$ (solid line) and modified relative error $\hat {\epsilon}_{rel}$ (dashed line) of the Airy expansion (\[eq71\]) for real values of $z$ as a function of $\rho=\left( z-z_{2}\right) /\left( z_{2}-z_{1}\right) $, with the coefficient approximations $\widetilde {\mathcal{A}}_{m}{\left( {u,z}\right) }$ and $\widetilde{\mathcal{B}}% _{m}{\left( {u,z}\right) }$ computed by Cauchy integrals. The Cauchy contour is a circle of radius $R=0.7\left( z_{2}-z_{1}\right) $ centered at $z_{2}% $.[]{data-label="errze"}](ErrorZer.eps){height="5.5cm" width="7.5cm"} Finally, we give some additional results for other values of $n$, $\alpha$ and $N$ as a further illustration of the accuracy of the computation using Cauchy integrals. We pick a value of $z$ in the disc $\left\vert z-z_{2}\right\vert <0.5\left( z_{2}-z_{1}\right) $. In particular, we fix $z=z_{1}% +0.1(z_{2}-z_{1})$. We show the corresponding relative errors in Table \[table1\]. In the table, we take $300$ points in the upper half of the Cauchy contour because with the previous selection ($100$ points) the discretization error is not small enough in some cases; in particular for $n=1000$ and $N=16$ ($m=N/2=8$), when the relative error becomes of the order of $10^{-50}$. $$% \begin{array} [c]{\|c\|c\|c\|c\|c\|c\|}\hline & & & & & \\ N\Rightarrow & 2 & 4 & 8 & 12 & 16\\ \alpha,\, n & & & & & \\ \Downarrow & & & & & \\\hline & & & & & \\ 0,\,10 & 1.15\,10^{-6} & 2.88 \,10^{-9} & 2.20 \,10^{-13} & 1.28\,10^{-16} & 2.86\,10^{-19}\\\hline & & & & & \\ 0,\,100 & 1.46\,10^{-9} & 4.03 \,10^{-14} & 3.65 \,10^{-22} & 2.45\,10^{-29} & 6.24\,10^{-36}\\\hline & & & & & \\ 0,\,1000 & 1.50\,10^{-12} & 4.21 \,10^{-19} & 3.89 \,10^{-31} & 2.66\,10^{-42} & 6.85\,10^{-53}\\\hline\hline & & & & & \\ 100,\,10 & 4.90\,10^{-7} & 1.22 \,10^{-9} & 9.53 \,10^{-13} & 5.68\,10^{-17} & 1.30\,10^{-19}\\\hline & & & & & \\ 100,\,100 & 7.79\,10^{-10} & 1.94 \,10^{-14} & 1.69 \,10^{-22} & 1.13\,10^{-29} & 2.88\,10^{-36}\\\hline & & & & & \\ 100,\,1000 & 1.31\,10^{-12} & 3.41 \,10^{-19} & 2.76 \,10^{-31} & 1.70 \,10^{-42} & 4.06 \,10^{-53}\\\hline \end{array}$$ We observe in Table \[table1\] that the relative error shows little variation with the value of $\alpha$, as the previous figures also showed, and that the dependence on $n$ and $N$ is as expected for asymptotic approximations with an error of $\mathcal{O}\left( n^{-N-1}\right) $. This is shown more explicitly in Table \[table2\], where we give the values of the computational asymptotic error constants $n^{N+1}\epsilon_{rel}$, with $\epsilon_{rel}$ the relative errors in Table \[table1\]; we observe these experimental constants have a slow variation as a function of $n$ and $\alpha $. While we also expect these constants to be $\mathcal{O}\left( 1\right) $ they are in fact all quite small, which illustrates the uniform high accuracy of our Airy expansions. $$% \begin{array} [c]{\|c\|c\|c\|c\|c\|c\|}\hline & & & & & \\ N\Rightarrow & 2 & 4 & 8 & 12 & 16\\ \alpha,\, n & & & & & \\ \Downarrow & & & & & \\\hline & & & & & \\ 0,\,10 & 1.15\,10^{-3} & 2.88 \,10^{-4} & 2.20 \,10^{-4} & 1.28\,10^{-3} & 2.86\,10^{-2}\\\hline & & & & & \\ 0,\,100 & 1.46\,10^{-3} & 4.03 \,10^{-4} & 3.65 \,10^{-4} & 2.45\,10^{-3} & 6.24\,10^{-2}\\\hline & & & & & \\ 0,\,1000 & 1.50\,10^{-3} & 4.21 \,10^{-4} & 3.89 \,10^{-4} & 2.66\,10^{-3} & 6.85\,10^{-2}\\\hline\hline & & & & & \\ 100,\,10 & 4.90\,10^{-4} & 1.22 \,10^{-4} & 9.53 \,10^{-5} & 5.68\,10^{-4} & 1.30\,10^{-2}\\\hline & & & & & \\ 100,\,100 & 7.79\,10^{-4} & 1.94 \,10^{-4} & 1.69 \,10^{-4} & 1.13\,10^{-3} & 2.88\,10^{-2}\\\hline & & & & & \\ 100,\,1000 & 1.31\,10^{-3} & 3.41 \,10^{-4} & 2.76 \,10^{-4} & 1.70 \,10^{-3} & 4.06 \,10^{-2}\\\hline \end{array}$$ [^1]: The authors acknowledge support from Ministerio de Economía y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE).
--- abstract: 'We study the method for generating the initial data of black hole systems in Gauss-Bonnet (GB) gravity. The initial data are assumed to be momentarily static and conformally flat. Although the equation for the conformal factor is highly nonlinear, it is successfully solved by numerical relaxation for one-black-hole and two-black-hole systems. The common apparent horizon is studied in the two-black-hole initial data, and the result suggests that the Penrose inequalities are satisfied in this system. This is the first step for simulating black hole collisions in higher-curvature theories.' author: - Hirotaka Yoshino date: 'February 8, 2011' title: \| Black hole initial data in Gauss-Bonnet gravity:\ Momentarily static case --- Introduction ============ Higher-dimensional gravity has been attracting a lot of attentions motivated by the TeV-gravity scenarios [@ADD98-1; @ADD98-2; @RS99-1]. If the three-dimensional space is a $3$-brane in large or warped extra dimensions, the Planck energy could be of $O({\rm TeV})$, and the trans-Planckian collision will happen at accelerators such as the Large Hadron Collider (LHC) [@BF99; @DL01; @GT02]. If this is the case, the intermediate state of a collision with sufficiently high energy and small impact parameter is expected to be a mini black hole, and this motivated a lot of works (see [@Kanti:2009] for a review). At this time, no evidence for black hole signals has been found, and the restrictions for the Planck energy and the minimum black hole mass $M_{\rm BH}^{\rm (min)}$ (such that the produced object can be regarded as a black hole if $M>M_{\rm BH}^{\rm (min)}$) have been derived [@CMS:2010]. Another motivation for studying higher-dimensional gravity is the AdS/CFT correspondence, which conjectures the duality between gravity in anti-de Sitter (AdS) spacetime and the conformal field theory (CFT) on the boundary of that spacetime. If this conjecture is correct, the phenomena in CFT for which direct calculation is difficult due to strong coupling effect can be predicted by calculating the dual gravitational system. One of the important approaches for higher-dimensional gravity is to explore the nonlinear dynamics of higher-dimensional spacetimes by numerical relativity. Several formulations and codes of higher-dimensional numerical relativity have been developed so far [@YS09; @Zilhao:2010; @Sorkin:2009], and interesting simulations have been performed: the time evolution of Gregory-Laflamme instability [@Lehner:2010], slow-velocity collision of black holes [@Zilhao:2010; @Witek1:2010; @Witek2:2010], dynamics of complex scalar field minimally coupled to gravity in Kaluza-Klein spacetimes [@Sorkin:2009], and bar-mode instability of rapidly rotating Myers-Perry black holes [@SY09; @SY10]. In these works, time evolutions of spacetimes are simulated in the framework of general relativity (GR), which is the simplest theory of higher-dimensional gravity. One of the interesting extensions of higher-dimensional numerical relativity is to include the higher-curvature terms. In four dimensions, the Lagrangian density that leads to the second-order equation of the metric is just the Ricci scalar. However, in higher-dimensions, the Lagrangian density including higher-curvature terms also leads to the second-order equation if the combination of higher-curvature terms are chosen appropriately. These theories are Gauss-Bonnet (GB) gravity [@Lanczos] or, more generally, Lovelock gravity [@Lovelock]. Among these higher-curvature theories, we take attention to GB gravity in this paper. The GB term gives a quadratic correction with respect to the Riemann tensor to GR, and this GB correction arises in the low-energy limit of the heterotic string theory [@Zwiebach:1985; @Duff:1986; @Gross:1986]. Therefore, the GB term may have important effects on the mini black hole phenomena at accelerators. Also, the AdS/CFT correspondence in the context of GB gravity has been considered (e.g., [@Ogushi:2004; @Buchel:2009; @Hu:2010]). The black holes in GB gravity have interesting properties. Although some part of GB gravity resembles GR, the other part does not. For example, the solution of a static spherically symmetric black hole found in Ref. [@BD85] has two branches: the non-GR branch that is asymptotically AdS and the GR branch that is asymptotically flat (see [@TM05-1; @TM05-2] for a detailed study on causal structures of these solution). The GB version of Birkoff’s theorem [@Wiltshire:1988; @Charmousis:2002] (see also [@Zegers:2005] for Lovelock gravity) states that the spherically-symmetric vacuum spacetime (at least locally) corresponds to one of the two branches. The black hole of GR branch is shown to be unstable for $D=5$ against scalar mode and for $D=6$ against tensor mode if the coupling constant $\alpha_{\rm GB}$ of the higher-curvature term is sufficiently larger than appropriate power of $GM$, where $G$ is the gravitational constant and $M$ is the Arnowitt-Deser-Misner (ADM) mass of the black hole [@Dotti:2004; @Dotti:2005; @Gleiser:2005; @Beroiz:2007] (see also [@Takahashi:2010] for a study that proves the generic appearance of instability of a spherically-symmetric black hole in Lovelock gravity). The GB version of numerical relativity (say, numerical GB gravity), if it is developed, would play important roles to explore nonlinear phenomena in GB gravity. For example, the temporal evolution of the instability of a small spherically symmetric black hole could be followed by simulations to clarify the final end state. Also, by simulating the time evolution from initial data of a nonstationary rotating black hole, it would be possible to obtain an indication for the stationary rotating black holes whose analytic solutions have not been found to date (but see [@Brihaye:2010] for a numerical construction of the black hole with two equal rotational parameters). Of course, the simulation of high-velocity collision of black holes is an interesting issue to clarify the effect of higher-curvature terms on mini black hole phenomena at accelerators (see [@Rychkov:2004] for a partial result for a head-on collision of the Aichelburg-Sexl particles in GB gravity). There are few studies for numerical GB gravity. The $N+1$ formalism, which is the extension of the ADM formalism to GB gravity, has been done by Torii and Shinkai [@TS08]. Similarly to the ADM formalism, Einstein-Gauss-Bonnet equation is decomposed into the Hamiltonian and momentum constraints and the evolution equations. On the other hand, the numerically stable formalism of GB gravity which is analogous to, e.g., Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism [@Shibata:1995; @Baumgarte:1998] has not been developed yet, and also, no simulation of numerical GB gravity has been done. When one simulates a spacetime in GB gravity, similarly to the case of GR, the first step is to prepare an initial spacelike hypersurface (i.e., initial data) by solving the constraint equations. In this paper, we focus attention to the method for preparing initial data of black hole systems in GB gravity. As a first step, the initial data are assumed to be momentarily static (i.e., time symmetric). In GR, the Brill-Lindquist initial data [@BL63] are well known and widely used as the initial data of momentarily static multi black holes (see also [@YN04; @YSS05] for higher-dimensional studies). We discuss the extension of the Brill-Lindquist initial data to GB gravity, and successfully generate the initial data for one black hole and two equal-mass black holes. Using the two-black-hole initial data, we discuss whether the Penrose inequality [@Penrose] holds in this system. The Penrose inequality states that the area of an apparent horizon (AH) is not greater than that of the Schwarzschild-Tangherlini black hole with the same ADM mass. In the case of the momentarily static initial data, the Penrose inequality was proved for $4\le D\le 7$ using the conformal flow method [@Bray:2007]. It is of interest whether such a bound on the AH area holds or not in GB gravity. In addition to the original Penrose inequality, we also discuss whether the AH area is bounded from above by that of the spherically-symmetric black hole in GB gravity. The results suggest that both of the inequalities are held in this system. This paper is organized as follows. In the next section, we review GB gravity and $N+1$ formalism in this theory focusing attention to the part that is related to our study. In Sec. III, general framework for constructing the GB version of the Brill-Lindquist initial data is explained. The equation for the conformal factor $\Psi$ is a Laplace equation with formal source term which is highly nonlinear in $\Psi$, and we prove the regularity of the source term which is necessary for the existence of the solution. In Sec. IV, the one-black-hole initial data are constructed numerically, and we show the agreement of the data with the time-symmetric slice of a spherically symmetric black hole spacetime. In Sec. V, the two-black-hole initial data are constructed. Using those data, we analyze the condition for the AH formation, and the Penrose inequalities in this system are discussed. Sec. VI is devoted to summary and discussion. Throughout this paper, we use the unit $c=1$, while the gravitational constant $G$ is explicitly written. Constraint equations in Gauss-Bonnet gravity ============================================ In this section, we review GB gravity and the $N+1$ formalism of Ref. [@TS08] focusing attention to the part that is related to the setup of our study. Gauss-Bonnet action and equations --------------------------------- In this paper, $D$ denotes the dimensionality of the spacetime $\mathcal{M}$ with the metric $g_{\mu\nu}$. We also introduce $N = D-1$, which is the dimensionality of a spacelike hypersurface $\Sigma$ in the spacetime $\mathcal{M}$. Throughout this paper, we assume the spacetime to be vacuum. The Einstein-Gauss-Bonnet action [@Lanczos] in $D$-dimensional spacetime is $$S=\frac{1}{16\pi G}\int_{\mathcal{M}} \left(\mathcal{R}+\alpha_{\rm GB} \mathcal{L}_{\rm GB}\right)\sqrt{-g}d^Dx \label{action}$$ with $$\mathcal{L}_{\rm GB} = \mathcal{R}^2 -4\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu} +\mathcal{R}_{\mu\nu\rho\sigma}\mathcal{R}^{\mu\nu\rho\sigma},$$ where $\mathcal{R}$, $\mathcal{R}_{\mu\nu}$, and $\mathcal{R}_{\mu\nu\rho\sigma}$ are the Ricci scalar, the Ricci tensor, and the Riemann tensor, respectively, and $\alpha_{\rm GB}$ is the coupling constant with dimensionality of squared length. The action gives the gravitational equation as $$\mathcal{G}_{\mu\nu}+\alpha_{\rm GB}\mathcal{H}_{\mu\nu}=0,$$ where $$\mathcal{G}_{\mu\nu} = \mathcal{R}_{\mu\nu} -\frac{1}{2}\mathcal{R}g_{\mu\nu},$$ and $$\mathcal{H}_{\mu\nu}=2\left( \mathcal{R}\mathcal{R}_{\mu\nu} -2\mathcal{R}_{\mu\alpha}\mathcal{R}^{\alpha}_{~\nu} -2\mathcal{R}^{\alpha\beta}\mathcal{R}_{\mu\alpha\nu\beta} +\mathcal{R}_{\mu}^{~\alpha\beta\gamma}\mathcal{R}_{\nu\alpha\beta\gamma} \right) -\frac{1}{2}g_{\mu\nu}\mathcal{L}_{\rm GB}.$$ In the case $D=4$, the term $\mathcal{L}_{\rm GB}$ in the action gives a topological invariant as proved in the generalized Gauss-Bonnet theorem [@Chern], and hence $\mathcal{H}_{\mu\nu}=0$. The GB term becomes nontrivial only for higher dimensions, $D\ge 5$. The $N+1$ formalism ------------------- Here, we review the initial value equations in GB gravity based on the $N+1$ formalism developed in Ref. [@TS08]. We consider the spacelike hypersurface $\Sigma$ with the induced metric $\gamma_{\mu\nu}$ and the extrinsic curvature $K_{\mu\nu}$, and let $n^{\mu}$ be the future-directed timelike unit normal to the hypersurface. The projections of the gravitational equation $(\mathcal{G}_{\mu\nu}+\alpha_{\rm GB}\mathcal{H}_{\mu\nu})n^\mu n^\nu$, $(\mathcal{G}_{\mu\nu}+\alpha_{\rm GB}\mathcal{H}_{\mu\nu})n^\mu \gamma^\nu_{~\rho}$, $(\mathcal{G}_{\mu\nu}+\alpha_{\rm GB}\mathcal{H}_{\mu\nu}) \gamma^\mu_{~\rho}\gamma^\nu_{~\sigma}$ give the Hamiltonian constraint, the momentum constraint, and the evolution equations, respectively. The initial data should be prepared so that they satisfy the two constraints. The Hamiltonian constraint is written as $$M+\alpha_{\rm GB}(M^2-4M_{ab}M^{ab}+M_{abcd}M^{abcd}) = 0,$$ where $$M_{ijkl}=R_{ijkl}+(K_{ik}K_{jl}-K_{il}K_{jk}),$$ and $M_{ij}=\gamma^{ab}M_{iajb}$ and $M=\gamma^{ab}M_{ab}$. The explicit form of the other equations can be found in Ref. [@TS08]. Conformal approach ------------------ Hereafter, we focus our attention to the momentarily static case, or in other words, the time-symmetric case: $K_{ij}=0$. In this case, the momentum constraint is trivially satisfied, and $M_{ijkl}=R_{ijkl}$. We further assume the initial space $\Sigma$ to be conformally flat: $$\gamma_{ij} = \Psi^{4/(N-2)}\hat{\gamma}_{ij},$$ where $\hat{\gamma}_{ij}$ is the flat-space metric. Note that the authors of Ref. [@TS08] treated $\hat{\gamma}_{ij}$ as an arbitrary metric, and here we chose the very special case. Also, in Ref. [@TS08], the conformal factor was set to be $\Psi^{2m}$, where $m$ is an arbitrary number. Here $m=2/(N-2)$ is chosen, because in this case, we can treat the problem as a natural generalization of the GR studies. In this setup, the equation for the conformal factor is written as $$\hat{D}_a\hat{D}^a\Psi =\alpha_{\rm GB}\hat{S},$$ where $\hat{D}_a$ is the covariant derivative with respect to the flat-space metric. Here, $\hat{S}$ is written as $$\begin{gathered} \hat{S} = \frac{N-3}{(N-1)(N-2)}\Psi^{-\frac{N+2}{N-2}} \Bigg\{ 4(N-2)\left[(\hat{D}_a\hat{D}^a\Psi)^2 -(\hat{D}_a\hat{D}_b\Psi)(\hat{D}^a\hat{D}^b\Psi)\right] \\ -8\Psi^{-1}(\hat{D}\Psi)^2\hat{D}_a\hat{D}^a\Psi +8N\Psi^{-1}\hat{D}^a\Psi\hat{D}^b\Psi\hat{D}_a\hat{D}_b\Psi -\frac{4N(N-1)}{N-2}\Psi^{-2}(\hat{D}\Psi)^4 \Bigg\}.\end{gathered}$$ Method of initial data construction =================================== In this section, we give the formulation for generating the initial data with $N_{\rm BH}$ black holes in GB gravity. Decomposition of conformal factor --------------------------------- Since we would like to obtain the initial data of GB gravity as continuous extension of those of the GR case, we decompose $\Psi$ as $$\Psi = \Psi_0 + \alpha_{\rm GB} g. \label{decomposition}$$ Here, $\Psi_0$ is the solution in the GR case (i.e., $\hat{D}_a\hat{D}^a\Psi_0=0$), and the term $\alpha_{\rm GB}g$ represents the deviation from the GR case in the presence of $\alpha_{\rm GB}\neq 0$. The equation for $g$ becomes $$\hat{D}_a\hat{D}^a g = \hat{S} \label{Eq-g-original}$$ with $$\hat{S} = \frac{N-3}{(N-1)(N-2)}\Psi^{-\frac{N+2}{N-2}} \sum_{n=0}^4\alpha_{\rm GB}^n s^{(n)}, \label{source-decompose}$$ where $$\begin{gathered} s^{(0)} = -4(N-2)(\hat{D}_a\hat{D}_b\Psi_0)(\hat{D}^a\hat{D}^b\Psi_0) +8N\Psi^{-1}(\hat{D}_a\Psi_0)(\hat{D}_b\Psi_0)(\hat{D}^a\hat{D}^b\Psi_0) \\ -\frac{4N(N-1)}{N-2}\Psi^{-2}(\hat{D}\Psi_0)^4, \label{source0}\end{gathered}$$ $$\begin{gathered} s^{(1)} = -8(N-2)(\hat{D}_a\hat{D}_b\Psi_0)(\hat{D}^a\hat{D}^b g) -8\Psi^{-1}(\hat{D}\Psi_0)^2(\hat{D}_a\hat{D}^a g) \\ +8N\Psi^{-1}\left[ (\hat{D}_a\Psi_0)(\hat{D}_b\Psi_0)(\hat{D}^a\hat{D}^b g) +2(\hat{D}^a\Psi_0)(\hat{D}^bg)(\hat{D}_a\hat{D}_b\Psi_0) \right] \\ -\frac{16N(N-1)}{N-2}\Psi^{-2} (\hat{D}\Psi_0)^2(\hat{D}_a\Psi_0)(\hat{D}^ag), \label{source1}\end{gathered}$$ $$\begin{gathered} s^{(2)} = -4(N-2)\left[(\hat{D}_a\hat{D}_bg)(\hat{D}^a\hat{D}^bg) -(\hat{D}^a\hat{D}_ag)^2\right] -16\Psi^{-1}(\hat{D}_a\Psi_0)(\hat{D}^ag)(\hat{D}^b\hat{D}_bg) \\ +8N\Psi^{-1} \left[ (\hat{D}^ag)(\hat{D}^bg)(\hat{D}_a\hat{D}_b\Psi_0) +2(\hat{D}^a\Psi_0)(\hat{D}^bg)(\hat{D}_a\hat{D}_bg) \right] \\ -\frac{8N(N-1)}{N-2}\Psi^{-2} \left[ 2(\hat{D}_a\Psi_0\hat{D}^a g)^2 +(\hat{D}\Psi_0)^2(\hat{D}g)^2 \right], \label{source2}\end{gathered}$$ $$\begin{gathered} s^{(3)}= -8\Psi^{-1}(\hat{D}g)^2(\hat{D}^a\hat{D}_ag) +8N\Psi^{-1}(\hat{D}^ag)(\hat{D}^bg)(\hat{D}_a\hat{D}_bg) \\ -\frac{16N(N-1)}{N-2}\Psi^{-2}(\hat{D}_a\Psi_0)(\hat{D}^a g)(\hat{D}g)^2, \label{source3}\end{gathered}$$ and $$s^{(4)}=-\frac{4N(N-1)}{N-2}\Psi^{-2}(\hat{D}g)^4. \label{source4}$$ We call the right hand side $\hat{S}$ of Eq. the “source term” hereafter. Puncture initial data --------------------- Here, we specify the GR solution $\Psi_0$ of a system with $N_{\rm BH}$ black holes. For this purpose, we introduce the Cartesian coordinates $(x^a)$ to the flat space and specify $N_{\rm BH}$ points at which $\Psi_0$ diverges (i.e., punctures). The location of the $n$-th puncture is denoted by $x^a = \bar{x}_{(n)}^a$, and we define $x_{(n)}^a = x^a - \bar{x}_{(n)}^a$ and $R_{(n)} = \left\|x_{(n)}^a\right\|$. Then, the solution to the Laplace equation $\hat{D}_a\hat{D}^a\Psi_0=0$ is adopted as $$\Psi_0 = 1 + \sum_{n=1}^{N_{\rm BH}} \psi_{(n)}, \label{Psi0-Np-punctures}$$ with $$\psi_{(n)} = \frac{4\pi GM_0^{(n)}}{(N-1)\Omega_{N-1}R_{(n)}^{N-2}},$$ where $\Omega_{N-1}$ denotes the area of a $(N-1)$-dimensional unit sphere and $M_0^{(n)}$ is a mass parameter for $n$-th black hole. The Arnowitt-Deser-Misner (ADM) mass $M_0$ is given as $M_0=\sum_{n=1}^{N_{\rm BH}}M_0^{(n)}$. This solution was studied in Ref. [@BL63] in the $D=4$ case and is often called the Brill-Lindquist initial data (see [@YN04; @YSS05] for the studies in higher-dimensional cases). This space possesses the $N_{\rm BH}$ Einstein-Rosen bridges (say, throats) and each puncture corresponds to the asymptotically flat region beyond each throat. Finiteness of the source term {#Sec:sourceterm} ----------------------------- The conformal factor $\Psi_0$ of the Brill-Lindquist initial data in the GR case diverges at each puncture as $R_{(n)}^{-(N-2)}$. Then, a naive estimate gives a severely divergent behavior of the source term $\hat{S}\sim R_{(n)}^{-(N+4)}$ at each puncture. If this is the case, $g$ also has to diverge at each puncture, and then, the behavior of $\hat{S}$ would be further modified, implying further stronger divergence of $g$. This procedure would continue eternally, and hence the solution $g$ would not exist. However, this naive estimate is not correct, since the cancellation of divergent terms occurs. As a result, $\hat{S}$ becomes zero at the punctures, and hence the source term $\hat{S}$ is well behaved. Let us check this in the following. Assuming a regular behavior of $g$ at each puncture, the functions $s^{(2)}$, $s^{(3)}$, and $s^{(4)}$ obviously become zero at each puncture after multiplying the factor $\Psi^{-(N+2)/(N-2)}$ in Eq. . Therefore, we focus our attention only to $s^{(0)}$ and $s^{(1)}$. We substitute Eq. into Eqs. and , rewrite it with $R_{(n)}\psi_{(n)}^{\prime\prime} + (N-1)\psi_{(n)}^{\prime}=0$, and collect only terms for which divergence at the punctures is suspected. For example, $\Psi^{-2}(D\Psi_0)^4$ from the third term of Eq. is calculated as $$\begin{aligned} \Psi^{-2}(D\Psi_0)^4 &=& \Psi^{-2}\sum_{k,l,m,n} \psi_{(k)}^{\prime}\psi_{(l)}^{\prime}\psi_{(m)}^{\prime}\psi_{(n)}^{\prime} (\vec{n}_{(k)}\cdot\vec{n}_{(l)})(\vec{n}_{(m)}\cdot\vec{n}_{(n)}) \nonumber\\ &=& \Psi^{-2}\left[\sum_{k}\psi_{(k)}^{\prime 4} +4\sum_{k,l (k\neq l)} \psi_{(k)}^{\prime}\psi_{(l)}^{\prime 3} (\vec{n}_{(k)}\cdot\vec{n}_{(l)})+O(\psi^{\prime 2})\right],\end{aligned}$$ where $n_{(n)}^a = x_{(n)}^a/R_{(n)}$ is the unit vector and $(\vec{n}_{(k)}\cdot\vec{n}_{(l)})$ is the inner product. Here, $O(\psi^{\prime 2})$ are the terms that become zero at the punctures after multiplying the factor $\Psi^{-(N+2)/(N-2)}$ in Eq. , and thus, we do not consider these terms because we are interested in cancellation of divergent terms. In this manner, $s^{(0)}$ and $s^{(1)}$ are calculated as $$\begin{gathered} s^{(0)} = -\frac{4N(N-1)}{N-2}\sum_k \left(\frac{\psi_{(k)}^\prime}{R_{(k)}\Psi}\right)^2 \left[R_{(k)}\psi_{(k)}^{\prime} +(N-2)\Psi\right]^2 \\ +4N \sum_{k,l (k\neq l)} \frac{\psi_{(k)}^\prime\psi_{(l)}^\prime}{\Psi} \left\{ \left[R_{(k)}\psi_{(k)}^\prime+R_{(l)}\psi_{(l)}^\prime+(N-2)\Psi\right] \frac{1-N(\vec{n}_{(k)}\cdot\vec{n}_{(l)})^2}{R_{(k)}R_{(l)}} \right. \\ \left. -\frac{4(N-1)}{N-2} \left(\frac{\psi_{(l)}^\prime}{\Psi}\right) \left[R_{(l)}\psi_{(l)}^\prime+(N-2)\Psi\right] \frac{(\vec{n}_{(k)}\cdot\vec{n}_{(l)})}{R_{(l)}} \right\}+O(R_{(k)}^{-2}),\end{gathered}$$ $$\begin{gathered} s^{(1)} = -8\sum_{k} \left[R_{(k)}\psi_{(k)}^\prime + (N-2)\Psi\right] \left(\frac{\psi_{(k)}^\prime}{R_{(k)}\Psi}\right) \\ \times \left[\hat{D}^a\hat{D}_ag +Nn_{(k)}^an_{(k)}^b\hat{D}_b\hat{D}_ag +\frac{2N(N-1)}{N-2} \left(\frac{\psi_{(k)}^\prime}{\Psi}\right) n_{(k)}^a\hat{D}_ag \right] +O(R_{(k)}^{-2}).\end{gathered}$$ Now, we use the relation $$R_{(n)}\psi_{(n)}^{\prime} + (N-2)\psi_{(n)} = 0. \label{psi-relation2}$$ and find $s^{(0)}=O(R_{(n)}^{-4})$ and $s^{(1)} = O(R_{(n)}^{-3})$. Therefore, $\hat{S}$ behaves as $\sim R_{(n)}^{N-2}$ at each puncture, and no divergence of the source term occurs. For this reason, we can expect the existence of the solution of $g$ that is regular at each puncture. In Secs. IV and V, we explicitly construct the numerical solution of $g$ for one black hole and two black holes, respectively, assuming the regularity of $g$ at each puncture. There, it turns out that this numerical method works well. It is worth pointing out the relation between our method and the method for generating non-time-symmetric initial data of puncture-type boosted black holes in GR developed by Brandt and Brügmann [@BB97]. In their formalism, the space is assumed to be conformally flat, and the analytic solution of the extrinsic curvature found by Bowen and York [@BY80] is used. The conformal factor is decomposed as $\Psi=\Psi_0+\psi$, where $\Psi_0$ is the conformal factor for the Brill-Lindquist solution. The equation for $\psi$ becomes the Laplace equation with a formal source term depending on $\Psi$. Here, the source term is shown to become zero at each puncture, and hence, $\psi$ can be solved numerically assuming the regularity at each puncture. Therefore, our method is very analogous to the Brandt-Brügmann formalism. ADM mass {#Sec:ADM} -------- Here, we discuss how to calculate the total gravitational energy. The ADM mass $M$ is given by $$M = -\frac{(N-1)}{4\pi (N-2)G} \int_{\mathcal{S}} \hat{D}_a\Psi dS^a,$$ where $\mathcal{S}$ is the surface at infinity. Suppose $\Psi = \Psi_0$ is the conformal factor with the ADM mass $M_0$ in the GR case. Then, using the Gauss law and Eq. , we find that the ADM mass in the GB case is $$M = M_0 - \frac{(N-1)\alpha_{\rm GB}}{4\pi (N-2)G}\int_{\hat{\Sigma}}\hat{S}dx_1...dx_N, \label{ADM_mass}$$ where $\hat{\Sigma}$ is the whole flat space. It is important to point out that the ADM mass $M$ in the case $\alpha_{\rm GB}\neq 0$ is different from $M_0$, although $M_0$ is the ADM mass in the GR case $\alpha_{\rm GB}=0$. Therefore the parameter $M_0$ is (say) an artificial mass for $\alpha_{\rm GB}\neq 0$, and the true ADM mass $M$ is determined after $\Psi$ is solved. The similar phenomena can be found also in the Brandt-Brügmann formalism. One-black-hole initial data =========================== Let us begin our numerical analysis with the one-black-hole initial data. Here we assume the initial data to be spherically symmetric and introduce the radial coordinate $R$ in which the metric of the flat space becomes $$ds^2 = dR^2 + R^2d\Omega_{N-1}^2.$$ Here, $d\Omega_{N-1}^2$ is the line element of the $(N-1)$-dimensional unit sphere. We impose $g=g(R)$ and set the function $\Psi_0$ in Eq. to be $$\Psi_0 = 1 + \left(\frac{R_S(M_0)}{R}\right)^{N-2},$$ where we defined $R_S(M_0)$ as $$R_S(M_0) := \left[\frac{4\pi GM_0}{(N-1)\Omega_{N-1}}\right]^{1/(N-2)}. \label{def:R_S}$$ Note that $R_S(M_0)$ is different from the Schwarzschild radius $r_S(M_0)$, and they are related as $r_S(M_0) = 4^{1/(N-2)}R_S(M_0)$. In the case of $\alpha_{\rm GB} = 0$, the conformal factor $\Psi = \Psi_0$ gives the spacelike hypersurface which is known as the Einstein-Rosen bridge (i.e., the time-symmetric Cauchy slice in the Schwarzschild-Tangherlini spacetime). The coordinate $R$ is called the isotropic coordinate, and the minimal surface (or the apparent/event horizon) is located at $R=R_S(M_0)$. Because of the GB version of Birkoff’s theorem [@Wiltshire:1988; @Charmousis:2002], also in the case of $\alpha_{\rm GB} > 0$, the initial data is expected to give the time-symmetric Cauchy slice of the spherically-symmetric black hole spacetime [@BD85] whose metric is $$ds^2 = -f(r)dt^2 +\frac{dr^2}{f(r)}+r^2d\Omega_{N-1}^2, \label{spherically-symmetric1}$$ with $$f(r) = 1+\frac{r^2}{2\tilde{\alpha}_{\rm GB}} \left( 1-\sqrt{1+\frac{4\tilde{\alpha}_{\rm GB}\tilde{M}}{r^{N}}} \right), \label{spherically-symmetric2}$$ where $\tilde{\alpha}_{\rm GB} = (N-2)(N-3)\alpha_{\rm GB}$ and $\tilde{M} = \left[r_S(M)\right]^{N-2}$. If we can find the coordinate transformation from the Schwarzschild-like coordinate $r$ to the isotropic coordinate $R$, the conformal factor $\Psi(R)$ is obtained. However, since $f(r)$ has a complicated form in the cases $\alpha_{\rm GB}\neq 0$, it seems impossible to find the analytic formula for the coordinate transformation in contrast to the GR case. Therefore, this problem has to be solved numerically. Besides, we can obtain a lot of lessons for the numerical method from this problem as we will see later. Perturbative analysis --------------------- In the numerical calculation, $R_S(M_0)$ is adopted as the unit of the length, and also in the following, we use the unit $R_S(M_0)=1$ unless explicitly specified. Let us consider the case where the coupling constant $\alpha_{\rm GB}$ is small, and thus, the correction from the GR case can be treated as a perturbation. In this case, $\Psi$ in Eq. should be replaced by $\Psi_0$, and the equation is reduced to $$g_{,RR}+\frac{N-1}{R}g_{,R} =-4N(N-3)R^{-2}\Psi_{0,R}^2\Psi_0^{-3-\frac{4}{N-2}}.$$ Here, the cancellation of divergent terms has occurred in the right-hand side because of the relation $\Psi_{0,R}+(N-2)\Psi_0/R = (N-2)/R$, and as a result, the source term is $O(R^{N-2})$ in the neighborhood of the puncture. In the case of $D=5$ (i.e., $N=4$), this equation can be solved analytically: $$g=\frac{4\left(1+3R^2+R^4\right)}{3\left(1+R^2\right)^3}. \label{perturbative}$$ Thus we explicitly confirm the existence of the solution that is regular at the origin $R=0$. This result leads to the expectation that a regular solution of $g$ exists in more general cases including higher-order terms in $\alpha_{\rm GB}$. The function $g$ behaves as $\simeq (4/3)R^{-2}$ at the distant region, $R\gg 1$. This indicates that the ADM mass is shifted as $$M = M_0 + \frac{2\pi\alpha_{\rm GB}}{G}.$$ Therefore, we confirm the statement of Sec. \[Sec:ADM\] explicitly: If we impose the regularity of $g$ at the origin, the function $g$ also contributes to the mass, and hence, the ADM mass is determined after $g$ is solved. Numerical approach ------------------ Now, we study the numerical generation of $g(R)$ for the general cases of $\alpha_{\rm GB}\ge 0$. We write down the Laplace operator $\hat{D}_a\hat{D}^a$ and the source term $\hat{S}$ in terms of the radial coordinate $R$. Here, we adopt the formulas of $s^{(0)}$ and $s^{(1)}$ for which the divergent terms are canceled out by the relation $\Psi_{0,R}+(N-2)\Psi/R = (N-2)(1+\alpha_{\rm GB}g)/R$. Then, the right hand side becomes $O(R^{N-2})$ similarly to the perturbative case and the source term $\hat{S}$ is well behaved at the origin $R=0$. This confirms the general proof for the finiteness of $\hat{S}$ of Sec. \[Sec:sourceterm\]. The numerical calculation is done in a finite region, and hence, there is an outer boundary of the computation domain, $R=R_{\rm max}$. Here, we have to specify the outer boundary condition. As discussed in the previous subsection, the behavior of $g$ at $R\gg 1$ is expected to be $g\simeq C/R^{N-2}$. Since the value of $C$ is not known before solving $g$, we eliminate $C$ using the combination of $g_{,R}$ and $g$ as $$g_{,R}+(N-2)g/R=0, \label{BC:Robin}$$ and adopt this Robin boundary condition at $R=R_{\rm max}$. We used the finite differencing method with the fourth-order accuracy with respect to the grid size. Here, the method of nested hierarchical grids was adopted in the numerical calculation. Specifically, we located the outer boundary at $R_{\rm max}=1024$, and put 12 and 16 layers for $0\le\alpha_{\rm GB}< 10^2$ and $10^2\le \alpha_{\rm GB}$, respectively (remember that the unit of the length is $R_S(M_0)$). The boundary of the $n$-th layer is located at $R=1024/2^{n-1}$, and the grid size is $12.8/2^{n-1}$. Then, the solutions were obtained by using the successive-over-relaxation (SOR) method: For each calculation, a test surface is prepared initially and it is made slowly converge to the real solution until the difference from the finite difference equations normalized by the absolute value of $g$ becomes less than $10^{-12}$. Figure \[g-behave\] shows the behavior of $g$ as a function of $R$ for $\alpha_{\rm GB}=0$ and $10^k$ for $D=5$–$8$, where $k=1,...,4$ for $D=5$ and $6$ and $k=1,...,3$ for $D=7$ and $8$. The curve for $\alpha_{\rm GB}=0$ in the case $D=5$ agrees with Eq. . As $\alpha_{\rm GB}$ is increased, the curve becomes steeper around the origin. This is the reason why we increased the number of the layers for $100\le \alpha_{\rm GB}$: More grid number is required for larger $\alpha_{\rm GB}$. In order to check that the geometry of the generated initial data agrees with that of the time-symmetric Cauchy surface in the spherically-symmetric spacetime, Eqs. and , we calculate the minimal surface (or the AH) and compare the relations between the two nondimensional quantities, $\alpha_{\rm GB}/r_S^2(M)$ and $r_H/r_S(M)$, where $r_H$ is the horizon radius. The horizon of the analytic solution is given by $f(r_H)=0$, which leads to the relation $$\left(\frac{r_H}{r_S}\right)^{N-4} \left[\left(\frac{r_H}{r_S}\right)^{2} +\frac{\tilde{\alpha}_{\rm GB}}{r_S^2}\right]=1. \label{Eq:horizon_radius}$$ On the other hand, we find the minimal surface of the numerical data by searching the location $R=R_H$ at which $$\frac{2}{N-2}R\Psi_{,R}+\Psi=0$$ is satisfied, and calculate the horizon radius as $$r_H = R_H\Psi^{2/(N-2)}(R_H).$$ Then, the ADM mass $M$ is calculated by Eq. , and the value of $r_H/r_S(M)$ is evaluated. Figure \[alpha\_radius\] shows the relation between $\alpha_{\rm GB}/r_S^2(M)$ and $r_H/r_S(M)$ for $D=5$–$8$. The solid curve shows the analytic formula, and the circles are our numerical data points. They agree very well, and thus, we can confirm that the generated initial data is the time-symmetric slice in the spherically-symmetric spacetime. The cases for $\alpha_{\rm GB}/R_S^2(M_0)=0$, $n\times 10^{k-1}$ (where $n$ and $k$ are integers satisfying $1\le n\le 9$ and $0\le k\le k_{\rm max}$), and $10^{k_{\rm max}}$ are shown for all $D$. Here, $k_{\rm max}$ is chosen as $k_{\rm max}=4$ for $D=5$ and $6$, and $k_{\rm max}=3$ for $D=7$ and $8$. As seen from this figure, the value of $r_H/r_S(M)$ is decreased as $\alpha_{\rm GB}/R_S^2(M_0)$ is increased, but even at $\alpha_{\rm GB}/R_S^2(M_0)=10^4$, the value of $r_H/r_S(M)$ is $\simeq 0.36$ and $0.13$ for $D=5$ and $6$, respectively. Therefore, generating the initial data with a horizon radius $r_H$ that is much smaller than the Schwarzschild radius $r_S$ requires a very large number of $\alpha_{\rm GB}/R_S^2(M_0)$, and thus, it is a hard task at least in this approach. Here, we point out the importance of the location of the outer boundary $R_{\rm max}$. Figure \[boundaryerror\] shows the numerical results with the choices of $R_{\rm max}=16$ and $64$. The agreement with the analytic relation is fairly good for small $\alpha_{\rm GB}$, but the deviation becomes significant for $\alpha_{\rm GB}/R_S^2(M_0)\gtrsim 10$ and $100$ for $R_{\rm max}=16$ and $64$, respectively. This is because the integrand of Eq. cannot be ignored for $R>R_{\rm max}$ and therefore the ADM mass $M$ is evaluated to be a smaller value than the true value. We have to remember that the larger value of $R_{\rm max}$ is required for a larger number of $\alpha_{\rm GB}$ for accurate calculations. Now we summarize the lessons obtained in the analysis in this section: (i) The source term for the equation $g$ becomes zero at $R=0$ because of the cancellation of divergent terms, and thus, the regular solution can be constructed; (ii) The ADM mass $M$ has to be evaluated after the function $g$ is generated, because it contributes to the mass; (iii) The function $g$ becomes steeper around $R=0$ as $\alpha_{\rm GB}$ is increased, and therefore, the better resolution is required; and (iv) We have to take care of the location of the outer boundary $R_{\rm max}$ for large $\alpha_{\rm GB}$ values since the integrand of Eq. becomes large at the distant region. Two-black-hole initial data =========================== Now we turn our attention to the study on the two-black-hole initial data. After generating the conformal factor, we study the common AH and assess whether the Penrose-like inequalities are held in this system. Numerical calculation --------------------- We assume the two black holes to have the same mass and the initial space to be axisymmetric with $O(N-1)$ symmetry (i.e., there is the $z$-axis and the directions orthogonal to axis have the same structure). To be specific, we introduce the $(z, \rho)$ coordinates where the metric of the flat space is $$ds^2=dz^2 + d\rho^2+\rho^2 d\Omega_{N-2}^2,$$ and assume $\Psi=\Psi(z,\rho)$. The GR solution $\Psi_0$ is adopted as the Brill-Lindquist solution with two equal-mass black holes that is described as $$\Psi_0 = 1+\frac{1}{2}[R_S(M_0)]^{N-2} \left(\frac{1}{R_+^{N-2}} +\frac{1}{R_-^{N-2}}\right).$$ Here, the punctures are located at $z=\pm z_0$ on the $z$ axis and $$R_\pm := \sqrt{(z\mp z_0)^2+\rho^2},$$ and $R_S(M_0)$ is defined in Eq. . In this setup, the space possesses the two throats and three asymptotically flat regions (say, one upper sheet and two lower sheets). The input parameters in the numerical calculations are $z_0$ and $\alpha_{\rm GB}$ in the unit $R_S(M_0)=1$. In the numerical calculation of $g(z,\rho)$, we write down the Laplace operator $\hat{D}_a\hat{D}^a$ and the source term $\hat{S}$ in terms of $(z, \rho)$, where the divergent terms of Eqs. and (i.e., $s^{(0)}$ and $s^{(1)}$) are canceled out. Since the two black holes have the same mass, there is a mirror symmetry with respect to $z=0$. For this reason, we choose the computation domain as $0\le z\le z_0+\Delta z_{\rm max}$ and $0\le \rho\le \Delta\rho_{\rm max}$ where $\Delta z_{\rm max}$ and $\Delta\rho_{\rm max}$ are chosen as $\Delta z_{\rm max} = \Delta \rho_{\rm max} =1024$ in the unit $R_S(M_0)=1$. At the outer boundary, we impose the same boundary condition as the spherically symmetric case but rewritten in the $(z, \rho)$ coordinates. Similarly to the case of one-black-hole initial data, we used the fourth-order finite differencing and the method of nested hierarchical grids. We put 13 layers to the computational domain, where the $n$-th layer has the boundary at $z=z_0 \pm \Delta z_{\rm max}/2^{n-1}$ and $\rho = \Delta \rho_{\rm max}/2^{n-1}$. If the layer crosses $z=0$, the region $z<0$ is discarded. The grid number $N_{\rm grid}$ of $\rho$ coordinate of each layer is varied as $10$, $20$, and $40$ depending on the situation, and the grids of $z$ coordinate have the same size. Then, the solution is obtained by the SOR method. The relaxation is continued until the deviation from the finite difference equations normalized by the absolute value of $g$ becomes less than $10^{-12}$. The validity of the numerical computation is checked in three manners. First, it is checked that the numerical solution $g(z,\rho)$ in the case of $z_0=0$ agrees with that of one-black-hole initial data. Next, as $z_0$ is increased, the solution is expected to asymptote to that of one isolated black hole with half mass in the neighborhood of each puncture, and the numerical solution $g(z,\rho)$ is actually confirmed to show this behavior. Finally, we checked whether the numerical solution shows the appropriate convergence with respect to the grid size. Figure \[D5-error\] shows the typical numerical error in $g(z,\rho)$ for $\alpha_{\rm GB}=0$, $1$, $10$, and $100$ for the case $D=5$ and $z_0=1$ as a function of $\log_{10}(1/N_{\rm grid})$. Here, the error is evaluated for $N_{\rm grid}=10$, $20$, and $40$ by calculating the difference from the solution of the case $N_{\rm grid}=80$. The slope for the curves of $\alpha_{\rm GB}=0$ and $100$ is approximately four, reflecting the fact that the adopted method is the fourth-order accuracy scheme. On the other hand, the slope for the curves of $\alpha=1$ and $10$ is larger than four; the slope for $\alpha=1$ is approximately five. This is a somewhat strange result, because the five-order accuracy was obtained by using the scheme with the fourth-order accuracy. This is probably because the numerical error in $g$ changes the value of the right-hand side $\hat{S}$ of Eq. , and the change in $\hat{S}$ further modify the value of $g$. Because of this effect, the cancellation of the numerical error would have happened accidentally. Anyway, the numerical data shows at least the fourth-order convergence, and this supports the accuracy of our numerical calculation. Figure \[3D\_pic\] shows the 3D plot of the generated data $g(z,\rho)$ for $z_0=1$ and $\alpha_{\rm GB}=0$, $1$, and $10$ in the case $D=5$. Here, the data of the ninth layer (layer of $n=9$) are used to draw this figure. As the value of $\alpha_{\rm GB}$ is increased, the surface becomes steeper around the puncture. Common apparent horizon ----------------------- The AH is defined as the outermost marginally trapped surface, and it satisfies the equation of zero expansion, $\theta_+=\nabla_\mu k^\mu =0$, where $k^\mu$ is a tangent vector of the null geodesic congruence from the AH. In GR, the formation of an AH implies the existence of an event horizon (EH) outside of it assuming the cosmic censorship and the null energy condition for the energy-momentum tensor. On the other hand, in GB gravity, this statement does not hold because whether $-\mathcal{H}_{\mu\nu}$ obeys the null energy condition is quite uncertain. However, also in GB gravity, the formation of an AH at least implies the existence of a region where gravity is strong. Furthermore, many theorems, such as the area theorem and the second law for a future trapping horizon, has been shown to hold also in GB gravity for a spherically-symmetric system with matter of GR branch [@Nozawa:2007]. For this reason, the AH may imply the black hole formation also in GB in a certain condition. For this reason, it is interesting to study the formation of an AH also in GB gravity. Because the space is momentarily static in our setup, the equation of an apparent horizon is reduced to $D_is^i=0$, where $s^i$ is a unit normal to the horizon. Assuming the location of the AH to be $R=h(\theta)$, where $\theta=\arctan(\rho/z)$, the horizon equation becomes $$\begin{gathered} h_{,\theta\theta}- (D-2)(h^2+h_{,\theta}^2) \left(\frac{2}{D-3}\frac{\Psi_{,R}}{\Psi}+\frac{1}{h}\right) +\frac{h_{,\theta}^2}{h}\\ +h_{,\theta}\left(1+\frac{h_{,\theta}^2}{h^2}\right) \left[\frac{2(D-2)}{(D-3)}\frac{\Psi_{,\theta}}{\Psi} +(D-3)\cot\theta\right]=0.\end{gathered}$$ The common AH that encloses the two black holes is found using the so-called shooting method by solving this equation under the boundary condition $h_{,\theta}=0$ at $\theta=0$ and $\pi/2$. Figure \[AH-shape\] shows examples of coordinate shape of the common AH on the $(z,x^i)$ plane, where $x^i$ is one of the orthogonal directions to the $z$ axis. Here, the cases $D=5$ and $\alpha_{\rm GB}=0$, $1$, $10$, and $100$ are shown, and the values of $z_0$ are $0$, $0.2$,..., and $z_0^{\rm (crit)}$, where $z_0^{\rm (crit)}$ is the critical value for the AH formation. As the value of $z_0$ is increased, the AH becomes more distorted. There is the other solution to the AH equation, which corresponds to the inner boundary of the trapped region. At $z_0=z_0^{\rm (crit)}$, the two solutions degenerate and the solution vanishes for $z_0>z_0^{\rm (crit)}$. Figure \[Z0\_AGB\_UnitM0\] shows the region where the AH can be found on the $(z_0,\alpha_{\rm GB})$ plane for $D=5$–$8$. Here, the unit of the length is adopted as $R_S(M_0)$. In general dimensions, the AH is not formed for sufficiently large $z_0$. In the cases $D=5$ and $6$, the value of $z_0^{\rm (crit)}/R_S(M_0)$ becomes large as $\alpha_{\rm GB}$ is increased. At first glance, one may think that increasing the coupling constant $\alpha_{\rm GB}$ helps the AH formation. However, this interpretation is not correct, because in this figure, the artificial mass $M_0$ is used in the length unit $R_S(M_0)$. As we have seen in Sec. \[Sec:ADM\], the ADM mass $M$ is changed as $\alpha_{\rm GB}$ is increased following Eq. . Several contours of $M/M_0$ are shown in Fig. \[Z0\_AGB\_UnitM0\]. As the value of $\alpha_{\rm GB}$ is increased, the mass $M$ also becomes large. Figure \[Z0\_AGB\_UnitADM\] shows the region of AH formation on the $(z_0, \alpha_{\rm GB})$ plane, but now the Schwarzschild radius of the ADM mass, $r_S(M)$, is used as the unit of the length. In all dimensions $D=5$–$8$, the value $z_0^{\rm (crit)}$ decreases as the value of $\alpha_{\rm GB}$ is increased, indicating that the coupling constant $\alpha_{\rm GB}$ makes the AH formation difficult compared to the GR case. This is a natural result, because in the spherically symmetric case, the ratio of the horizon radius to the Schwarzschild radius, $r_H(M)/r_S(M)$, decreases as $\alpha_{\rm GB}/r_S^2(M)$ is increased. The border of the region of the AH formation is shown by a solid curve by interpolating the numerical data. The dashed line shows the expected border for large $\alpha_{\rm GB}$ for which numerical calculation has not been done in this paper. In the case $D=5$, it is naturally expected that the border line crosses $z_0$ axis at $\alpha_{\rm GB}/r_S^2(M)=0.5$, because for this value, the horizon radius of the spherically-symmetric black hole becomes zero. For the other dimensions, the border line would not cross $z_0$ axis, since $r_H(M)>0$ for arbitrary value of $\alpha_{\rm GB}$ (see Fig. \[alpha\_radius\]). Penrose inequalities -------------------- The Penrose inequality in GR, $$A_{\rm AH}\le \Omega_{D-2}\left[r_S(M)\right]^{D-2}, \label{Eq:penrose1}$$ conjectures that the AH area is bounded from above by the horizon area of a spherically-symmetric black hole (Schwarzschild-Tangherlini black hole) with the same mass. The reason for this conjecture as follows. If the cosmic censorship holds, an AH is formed in an EH, and since the EH is located outside of the AH, its area is expected to be larger than that of the AH. Because of the area theorem by Hawking, the area of the EH will increase and asymptote to that of a final stationary black hole. Since the horizon area of a rotating black hole is smaller than that of a Schwarzschild black hole with the same mass, and the final mass is smaller than the ADM mass because of the gravitational radiation, the inequality is expected to hold. On the other hand, if the system with an AH whose area is greater than the bound of , the validity of the cosmic censorship, one of the assumptions of the above discussion, may be suspected. The Penrose inequality has been attracting a lot of attentions, and it was proved for momentarily static initial data for $D=4$–$7$ [@Bray:2007]. On the other hand, a “counterexample” (but not in a strict sense) has been found in Ref. [@BenDov:2004] in a system consisting of combined portions of the Schwarzschild and Oppenheimer-Snyder spacetimes. This example does not contradict the cosmic censorship, but contradicts the assumption that the area of the EH is larger than that of the AH. But the Penrose inequality can be modified to match this counterexample as follows. The AH in a usual sense can be regarded as a “black hole AH” since it is formed in a black hole. On the other hand, we can consider a “white hole AH” formed in a white hole whose past-directed outgoing null geodesic congruence has zero expansion. In the above counterexample, the black hole AH is located inside of the white hole AH, and the area of the white hole AH satisfies the Penrose inequality. Therefore, if we adopt the outermost horizon, the Penrose inequality still holds. This Penrose inequality for the outermost horizon can be proved in spherically-symmetric case assuming weak energy condition [@Malec:1994]. In GB gravity, the relation between the Penrose inequality and the cosmic censorship becomes unclear because the above reasoning for the Penrose inequality may not hold since $-\mathcal{H}_{\mu\nu}$ may violate the energy condition. However, it is still of interest whether the universal relation like holds or not from a mathematical point of view. The proof for the GR case cannot be applied to the GB case at least straightforwardly, because it is unclear if $-\mathcal{H}_{\mu\nu}$ satisfies the energy condition. Therefore, it is interesting to test the inequality using the initial data constructed in this paper. Other than the original version of the Penrose inequality , we can consider another inequality that may be expected to hold in GB gravity.[^1] Namely, since the horizon radius $r_H(M)$ of a spherically-symmetric black hole is different from $r_S(M)$, the Penrose inequality may be modified as $$A_{\rm AH}\le \Omega_{D-2}\left[r_H(M)\right]^{D-2}, \label{Eq:penrose2}$$ where $r_H(M)$ is defined by Eq. . In the following, we test if these two inequalities and hold in our system. For this purpose, we define $P_1:=A_{\rm AH}/\Omega_{D-2}\left[r_S(M)\right]^{D-2}$ and $P_2:=A_{\rm AH}/\Omega_{D-2}\left[r_H(M)\right]^{D-2}$ and evaluate these values for selected values of $z_0$ and $\alpha_{\rm GB}$. Figure \[penrose1\] shows the behavior of $P_1$ as a function of $z_0$. Here the cases of $\alpha_{\rm GB}=0$, $1$, $10$, and $100$ are shown for $D=5$ and $6$, and the cases of $\alpha_{\rm GB}=0$, $0.2$, $2$, $20$ are shown for $D=7$ and $8$. In all cases, the values of $P_1$ are smaller than unity, suggesting that the inequality is kept in this system. We find that $P_1$ becomes smaller as $\alpha_{\rm GB}$ is increased, and therefore, the coupling constant tends to help the AH satisfy the inequality if $\alpha_{\rm GB}$ is positive. Figure \[penrose2\] shows the behavior of $P_1$ as a function of $z_0$. Here the cases of $\alpha_{\rm GB}=1$, $10$, and $100$ are shown for $D=5$ and $6$, and the cases of $\alpha_{\rm GB}=0.2$, $2$, $20$ are shown for $D=7$ and $8$. For all $D$, the value of $P_2$ is unity for $z_0=0$ because the space agrees with the time-symmetric slice of the spherically symmetric black hole spacetime. As $z_0$ is increased, the value of $P_2$ becomes smaller for all values of $\alpha$. Therefore, the distortion of the AH makes the AH area smaller, and the modified version of the Penrose inequality also holds in our system. To summarize, the black hole initial data in this paper satisfy both two Penrose inequalities and , and no counterexample has been detected. Therefore, the Penrose inequalities might hold also in GB gravity under appropriate conditions. Summary and discussion ====================== In this paper, we studied the method for generating the initial data for one-black-hole and two-black-hole systems in GB gravity. Assuming the initial space to be momentarily static and conformally flat, the highly nonlinear equation of the Hamiltonian constraint in the $N+1$ formalism was successfully solved numerically. Using the generated initial data, we studied the common AH that encloses the two black holes, and discussed the Penrose inequalities in GB gravity. The result suggests that both two inequalities and hold in this system. Here, let us discuss whether the proposed conjecture for the condition of the AH formation in GR can be generalized to GB gravity. In four-dimensional GR, the hoop conjecture [@Thorne:72] is well known as the condition of the horizon formation. The hoop conjecture states that a black hole with horizon forms when and only when the hoop length $C$ for a system satisfies $C\lesssim 2\pi r_S(M)$. This conjecture is loosely formulated in the sense that the definitions of the horizon (AH or EH), the hoop, and the mass are not explicitly specified. But this conjecture is known to give the approximate condition of the AH formation (under appropriate definitions of the mass and the hoop which may depend on researchers), although not explicitly proved. The point in this conjecture is that the typical one-dimensional length of the system is restricted from above if an AH is formed, implying that an arbitrarily long AH does not form. But this statement holds only for $D=4$, because the black hole can be arbitrarily long in higher dimensions as expected from the black string solution, and explicitly shown in [@IN02]. Instead, Ida and Nakao proposed the generalized hoop conjecture [@IN02] as $$C_{D-3}\lesssim \Omega_{D-3}[r_S(M)]^{D-3}, \label{hyperhoop}$$ where $C_{D-3}$ is the typical $(D-3)$-dimensional quantity (“hyperhoop”) in this system. This hyperhoop conjecture was discussed in several papers [@YN04; @IN02; @YN03; @Yoo:2005; @Yamada:2009], and the results support its effectiveness. Let us consider a two-black-hole system with total mass $M$, and suppose the two black holes to be momentarily at rest with a typical distance $L$. In four dimensions, the typical hoop length is estimated by $C\simeq 2\pi r_S(M/2)+2L$. In a similar manner, in $D$ dimensions, the typical hyperhoop quantity would be $C_{D-3}\simeq \Omega_{D-3}[r_S(M/2)]^{D-3} +\Omega_{D-4}[r_S(M/2)]^{D-4}L$. Substituting this formula into Eq. , one has $$L\lesssim \frac{\Omega_{D-3}}{2^{1/(D-3)}\Omega_{D-4}}r_S(M) \sim r_S(M).$$ This gives an approximate condition for the AH formation at least in a qualitative sense. Can the hyperhoop conjecture be further extended to GB gravity? One simple manner of generalization would be to change from $r_S$ to $r_H$ in Eq. as $$C_{D-3}\lesssim \Omega_{D-3}[r_H(M)]^{D-3}, \label{GB-hyperhoop}$$ where $r_H(M)$ is defined in Eq. . However, this simple generalization does not work for a two-black-hole system. Since the value of $C_{D-3}$ in this case is given by $C_{D-3}\simeq \Omega_{D-3}[r_H(M/2)]^{D-3} +\Omega_{D-4}[r_H(M/2)]^{D-4}L$, the condition gives $$L\lesssim \left(\frac{\Omega_{D-3}}{\Omega_{D-4}}\right) \frac{[r_H(M)]^{D-3}-[r_H(M/2)]^{D-3}}{[r_H(M/2)]^{D-4}}. \label{GB_condition_L}$$ Let us consider the case $D=5$ and suppose $\alpha_{\rm GB}$ and $M$ satisfy the relation $\alpha_{\rm GB}=(1/2)r_S^{2}(M/2)$. In this case, $r_H(M/2)=0$ and $r_H(M)=r_S(M)/\sqrt{2}$, and Eq. gives $L\lesssim \infty$. Therefore, the condition predicts that the common AH forms even for an arbitrarily long distance between the two black holes. This obviously contradicts our numerical result in Fig. \[Z0\_AGB\_UnitADM\]. Therefore, the condition for the AH formation cannot be obtained at least by a straightforward extension of the hyperhoop conjecture, and a further study is required. The numerical work in this paper is the first step toward simulations of black holes in GB gravity, and a lot of extensions can be considered. For example, it is necessary to extend our method of generating time-symmetric initial data to the method of generating time-asymmetric initial data like a boosted black hole. For this purpose, the extension of the Bowen-York method [@BY80] should be an interesting possibility. Also, the time evolution of black hole initial data could be done using the approximation analogous to close-limit or close-slow methods in GR [@PP94; @YSS05; @YSS06]. The final goal would be to develop numerically stable formulations of numerical GB gravity by extending $N+1$ formalism and simulate black hole systems fully numerically to clarify a lot of interesting phenomena such as time evolution of an unstable GB black hole, rotating systems, and high-velocity collision of black holes. 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--- abstract: 'Three models for nonlocal transport are here compared against Vlasov-Fokker-Planck (VFP) codes to assess their accuracy in situations relevant to both inertial fusion hohlraums and tokamak scrape-off layers. The models tested are (i) a moment-based approach using an eigenvector integral closure (EIC) originally developed by Ji, Held and Sovinec; (ii) the non-Fourier Landau-fluid (NFLF) model of Dimits, Joseph and Umansky; and (iii) Schurtz, Nicolaï and Busquet’s multigroup diffusion model (SNB). We find that while the EIC and NFLF models accurately predict the damping rate of a small-amplitude temperature perturbation (within $10\%$ at moderate collisionalities), they overestimate the peak heat flow by as much as 35% and do not predict preheat in case where there is a large temperature difference. The SNB model, however, agrees better with VFP results for the latter problem . Additionally, we present for the first time a comparison of the SNB model against a VFP code for a hohlraum-relevant problem with inhomogeneous ionisation and show that the model overestimates the heat flow in the helium gas-fill by a factor of $\mathord{\sim}2$ despite predicting the peak heat flux to within 16%.' author: - 'J. P. Brodrick' - 'R. J. Kingham' - 'M. M. Marinak' - 'M. V. Patel' - 'A. V. Chankin' - 'J. T. Omotani' - 'M. V. Umansky' - 'D. Del Sorbo' - 'B. Dudson' - 'J. T. Parker' - 'G. D. Kerbel' - 'M. Sherlock' - 'C. P. Ridgers' title: Testing for magnetic and inertial confinement fusion applications --- =1 Introduction ============ Performing full integrated simulations of large-scale fusion devices, such as the National Ignition Facility (NIF) or the ITER tokamak, is very challenging due to the wide range of scales over which the important physical processes occur. Codes, such as HYDRA[@HYDRA] and BOUT++[@BOUTframework], used to perform integrated simulations of inertial and magnetic confinement fusion (ICF/MCF) respectively, often include reduced models to capture complex aspects of the physics. The accuracy of the models used naturally affects the predictive capability of these codes. In this paper we compare three different models for kinetic (i.e. nonlocal) effects on thermal : : $$\label{braginskii_q} \bm{Q}^\mathrm{(B)} = -\kappa^\mathrm{(B)} n_\mathrm{e} v_\mathrm{T} \lambda_\mathrm{ei}^{\textrm{(B)}} \nabla k_\mathrm{B} T_\mathrm{e},$$ Here and for the entirety of this paper we assume there to be no drift velocity or current (hence the ion and electron rest frames are equivalent). Note that Epperlein and Haines[@EppHaines] have calculated $\kappa^\mathrm{(B)}$ to an increased accuracy and Epperlein and Short[@EppShort] later suggested that this can be approximated well by $\kappa^\mathrm{(B)} \approx \xi(Z) 128/3\uppi$, where $\xi(Z) = (Z+0.24)/(Z+4.2)$. Equation \[braginskii\_q\] breaks down if the collisionality of the electrons becomes low. This is due to the inadequacy of the assumption that the electron distribution function is close to Maxwellian; electrons with mfp’s larger than the temperature scalelength can in fact escape gradients before being scattered and depositing their energy into the plasma, excess heat to be deposited beyond the steepest part of the temperature profile (often referred to as ‘preheat’ in the literature [@EppShort]). A reduced population of suprathermals is left behind in the region of steep temperature gradient, the heat flux. These ‘nonlocal’ effects can become important even for temperature scalelengths as long as $\mathord{\sim}200$ thermal mfp’s.[@SNB] Such situations occur frequently in : In tokamaks, nonlocal thermal transport is thought to play a role in heat flow from the core plasma to the ‘divertor’[@Chodura], a region of the tokamak edge specifically designed to absorb and exhaust excess heat from the plasma. Thermal electrons entering the SOL at the separatrix have mfp’s ranging from 1% (C-Mod) to 20% (DIII-D/ (TdeV)) of the distance to the divertor target (connection length). In fact, the ratio of $\lambda_\mathrm{ei}^{\textrm{(B)}}$ to the local temperature scalelength $L_T = 1/\nabla_\parallel \log T_\mathrm{e}$ tends to vary along the SOL from approximately 1 (TdeV) or 0.1 (DIII-D) near the separatrix, to 0.01 near the colder divertor.[@Batishchev] rendering the heat transport inherently nonlocal. Furthermore, transient events—Edge Localised Modes (ELMs), disruptions and filaments—can create even higher temperatures and steeper gradients, with which the associated suprathermals would be almost collisionless.[@Omotani] Current state of the art codes, such as SOLPS[@SOLPS1; @SOLPS2] and UEDGE[@UEDGE], have been shown to significantly underestimate the outer divertor target electron temperature and overestimate its density compared to experiment in existing tokamaks, which in turn affects other plasma parameters. Chankin and Coster [@Chankin2D] have suggested that nonlocal effects in addition to treatment of neutrals (which has largely been ruled out by a sensitivity analysis) and inappropriate use of time-averaging could explain this discrepancy. The plausibility of this hypothesis is supported by recent gyrokinetic simulations performed by Churchill et al.[@Churchill]. Resolution of this discrepancy is critical as excessive heat loads could erode and severely limit the lifetimes of the divertor target plates.[@Turnyanskiy] A common approach to incorporate the flux reduction aspect of nonlocal transport is to restrict the local heat flow to some fraction $\mathfrak{f}_\mathrm{L}$ of the free-streaming limit $Q_\mathrm{fs} = n_\mathrm{e} k_\mathrm{B} T_\mathrm{e} v_\mathrm{T}$. However, at best the flux-limiter $\mathfrak{f}_\mathrm{L}$ must be tuned against previous experiments, limiting predictive capability—values ranging from 0.03 to for SOL modeling[@fundamenski]—and preheat effects cannot be predicted. A more complete way to take nonlocal effects into account, however, is with a fully kinetic approach. By solving the Vlasov-Fokker-Planck (VFP) equation for the electron distribution function directly we need not assume Maxwellian; nonlocal effects are calculated ab-initio. Such an approach typically assumes binary collisions and small-angle scattering limiting its applicability in regions where the plasma is strongly coupled ($\log \Lambda$ approaching unity) such as in ICF fuel capsules or the colder part of the partially ionised hohlraum wall. While it is possible for VFP codes to treat collisions between multiple ion species[@IFP0D2V] and even neutrals[@VFPNeutrals] (though the latter might require coupling to a neutral Monte Carlo code such as EIRENE[@KIPPDev; @KIPPBenchmarks; @B2Eirene] due to the importance of large-angle collisions) here we only consider collisions of electrons with a single stationary ion species. Quantum-mechanical effects such as Fermi degeneracy which could be of importance in solid density material are also typically negelcted[@VFPReview], nevertheless methods to incorporate these have been suggested[@FermiDirac] However, due to the extra dimensionality associated with solving in velocity-space, VFP codes are computationally intensive and difficult to incorporate into existing integrated modeling codes. Such demands of resolving the distribution function and collision time are especially restrictive in cold, dense regions such as deep in the hohlraum wall where a fluid treatment might even be sufficient. Therefore, alternative models that have more predictive capability than flux-limiters, and reduced computational requirements compared to a full kinetic simulation, are desirable. A dedicated experiment to measure nonlocal effects performed by Gregori et al.[@Gregori] has . A large number of reduced models for nonlocal electron thermal transport have been suggested for applications in inertial fusion and laser-plasmas [@EppShort; @LMV; @CMG; @SNB; @Batischev2002; @SNBmag; @Brantov2013; @Dario; @Dariomag; @Cao] and to the SOL [@Hammett; @JiHeld; @Omotani; @NFLFTheory; @NFLF2014; @Hermite; @NFLFImplementation; @Izacard; @Izacard2017]. This paper focuses on three of these models, here referred to as the SNB[@SNB; @SNBmag; @Dario; @Dariomag; @Cao], EIC[@JiHeld; @Omotani; @Hermite] and NFLF[@NFLFTheory; @NFLF2014; @NFLFImplementation] models , and compares them for accuracy against VFP simulations.While the SNB model has recently been compared to VFP results by Marocchino et al.[@Marocchino], this has shown that the two approaches agree well for $Z=1$ but begin to deviate from each other as the ionisation increases. This is surprising as the SNB model was originally derived in the high-$Z$ (Lorentz) limit and we observe here that such findings are sensitive to precise implementation details of the model. Additionally, while the EIC and NFLF models have been shown to predict similar heat-fluxes[@NFLFImplementation], they have not yet been validated against a full time-dependent VFP code. The first problem we investigate here is the damping of a small-amplitude sinusoidal temperature profile of various wavelengths in section \[sec:linear\]. This test will be used to justify a tuning parameter which can be applied to the SNB model to improve agreement with VFP simulations. We will additionally suggest a new analytic fit for the conductivity reduction and use this to obtain improved coefficients for the NFLF model. In section \[sec:homo\], we will then consider the case, more relevant , of a plasma with a large temperature variation. We will show that the SNB model agrees very well with VFP simulations using the same tuning factor as in the linearised problem described above and that the EIC and NFLF models overpredict the peak heat flux. While this suggests that the SNB model may also be useful in SOL simulations, we also consider potential improvements to the other models to improve their performance. Finally, we will show in section \[sec:varion\] that the SNB model can significantly overpredict the heat flow relative to VFP in the low-density helium gas-fill using a problem particularly relevant to the hohlraum wall. The importance of gradients in both average ionisation $Z$ and electron density $n_\mathrm{e}$ here could mean the findings could also be important for the detached divertor scenario. Vlasov-Fokker-Planck Modeling {#sec:vfp} ============================= The evolution of the distribution function $f_\mathrm{e}$, , can be by the Vlasov-Fokker-Planck equation[@IMPACT] $$\frac{\partial f_\mathrm{e}}{\partial t} + \bm{v}\cdot\nabla f_\mathrm{e}- \frac{e}{m_\mathrm{e}}\left(\bm{E}+\frac{\bm{v}\times\bm{B}}{c}\right)\cdot \frac{\partial f_\mathrm{e}}{\partial \bm{v}} = C(f_\mathrm{e}),$$ where $\bm{v}$ is the electron velocity, $\bm{E}$, $\bm{B}$ are the electric and magnetic fields respectively, $m_\mathrm{e}$ is the electron mass. In this work we consider only collisions of electrons with themselves and a single ion species using the standard Trubnikov-Rosenbluth[@Trubnikov; @Rosenbluth] form of the Fokker-Planck collision operator $C=C_\mathrm{ee}+C_\mathrm{ei}$, where $$\begin{split} \frac{C_{\mathrm{e}\beta}(f_\mathrm{e},f_\beta)}{\Gamma_{\mathrm{e}\beta}}=&-\frac{\partial}{\partial \bm{v}_i} \Bigg(\frac{m_\mathrm{e}}{m_\beta}f_\mathrm{e}\frac{\partial}{\partial \bm{v}_i} \int \frac{f_\beta}{\lvert \bm{v}-\bm{u}\rvert}\; \mathrm{d}^3 \bm{u}\\ &-\frac{1}{2}\frac{\partial f_\mathrm{e}}{\partial \bm{v}_j}\frac{\partial^2 }{\partial \bm{v}_i\partial\bm{v}_j}\int {f_\beta}\lvert \bm{v}-\bm{u}\rvert\; \mathrm{d}^3 \bm{u}\Bigg) \end{split}$$ (applying standard Einstein summation over repeated indices). Here we have defined $$\Gamma_{\mathrm{e}\beta} = \frac{4\uppi Z_\mathrm{e}^2Z_\beta^2e^4}{m_\mathrm{e}^2}\log \Lambda_{\mathrm{e}\beta},$$ where $Z_\mathrm{i}=Z$ is the average ionisation and $Z_\mathrm{e}=-1$, along with $m_\mathrm{i}$ the ion mass. The ion distribution function $f_\mathrm{i}$ is assumed by KIPP to be Maxwellian, here we enforce the ion temperature to be equal to the electron temperature but this is not necessary[@KIPPRelax], all other codes and models assume a cold ion population and neglect terms of order $m_\mathrm{e}/m_\mathrm{i}$, simplifying the electron-ion collision operator to: $$\frac{C_\mathrm{ei}(f,n_\mathrm{i}\delta(\bm{v}))}{n_\mathrm{i}\Gamma_{\mathrm{ei}}}=\frac{\partial}{\partial \bm{v}_i} \Bigg(\cancelto{0}{\frac{m_\mathrm{e}}{m_\mathrm{i}}\frac{\bm{v}_i}{v^3}f_\mathrm{e}} +\frac{v^2\delta_{ij}-\bm{v}_i\bm{v}_j}{2v^3}\frac{\partial f_\mathrm{e}}{\partial \bm{v}_j}\Bigg), $$ where $\delta(\bm{v})$ is the Dirac delta function and $\delta_{ij}$ is the Kronecker delta tensor. For the case where variations only occur along magnetic field lines, symmetry in the perpendicular direction allows for elimination of the magnetic field by ‘gyro-averaging’ ( this process is still valid even in the absence of magnetic fields); this yields the 1d2v equation $$\frac{\partial \langle f_\mathrm{e}\rangle}{\partial t} + v_\parallel\frac{ \partial\langle f_\mathrm{e} \rangle}{\partial s_\parallel} - \frac{e E_\parallel}{m_\mathrm{e}} \frac{ \partial \langle f_\mathrm{e}\rangle}{\partial v_\parallel} = \langle C(f_\mathrm{e})\rangle,$$ where $\langle\cdot\rangle$ represents a gyro-average (an explicit representation of $\langle C\rangle$ can be found in previous work by Xiong et al.[@Xiong] and Chankin et al.[@KIPPDev]), and $\parallel$ denotes components of vectors parallel to the magnetic field. The KIPP code[@KIPPDev] is designed to solve this equation using Cartesian spatial and velocity grids. The code uses an operator splitting method suggested by Shoucri and Gagne[@opSplit; @ShoucriBook] that treats the spatial derivative using a second-order explicit scheme followed by the electric field and collision operator terms using a first-order (in time, second-order in velocity) implicit scheme. The velocity grid covers the domain $ v_\parallel \in [-v_\mathrm{max}, v_\mathrm{max}], v_\perp \in [0, v_\mathrm{max}]$ where $v_\mathrm{max}$ is a user-defined parameter. The distribution function is simply taken to be zero at the exterior of the grid and reflected along the interior $v_\perp=0$ axis. A simplified approach is the diffusion approximation, which consists of expanding the distribution function in Cartesian tensors and truncating all but the first two terms ($f_\mathrm{e}=f_0(v)+\bm{v}\cdot\bm{f}_1(v)/v$). increasing efficiency and has been observed to correctly predict heat flows to within 5% for temperature scalelengths $L_T \lessapprox 10\lambda_\mathrm{ei}^\mathrm{(B)}$[@MatteVirmont]. The IMPACT code[@IMPACT] employs this approach and makes a further simplification of ignoring angular scattering due to electron-electron collisions, valid in the Lorentz limit. In order to recover the correct local thermal conductivity for lower-$Z$ plasmas the factor $\xi(Z)$ is used in the electron-ion collision frequency. Our comparisons between IMPACT and KIPP suggest that these approximations do not greatly affect the results for the problems studied in section \[sec:homo\]. $$\begin{aligned} \label{eq:f0_orig} \frac{\partial f_0}{\partial t}+\frac{v}{3}\nabla \cdot \bm{f}_1 - \frac{e\bm{E}}{3m_\mathrm{e}v^2}\cdot\frac{\partial v^2 \bm{f}_1}{\partial v} &= C_{\mathrm{ee0}}\left[f_0\right], \\ \label{eq:f1_orig} \frac{\partial \bm{f}_1}{\partial t}+v\nabla f_0 - \frac{e\bm{E}}{m_\mathrm{e}}\frac{\partial f_0}{\partial v}-\frac{e\left(\bm{B}\times\bm{f}_1\right) }{m_\mathrm{e} c}&= - \frac{\nu_\mathrm{ei}}{\xi}\bm{f}_1,\end{aligned}$$ where $$\begin{split} C_\mathrm{ee0}[f_0] = \frac{4\uppi \Gamma_\mathrm{ee}}{v^2}\frac{\partial}{\partial v}\bigg(\int_{\mathrlap{0}}^{\mathrlap{v}} f_0 u^2\; &\mathrm{d}u f_0\\ + \frac{1}{v}\int_{\mathrlap{0}}^{\mathrlap{v}} u^2 \int_{\mathrlap{u}}^{\mathrlap{\infty}} f_0 w \;&\mathrm{d} w \;\mathrm{d}u\frac{\partial f_0}{\partial v}\bigg) \end{split}$$ is the isotropic contribution of the electron-electron collision operator and $$\nu_\mathrm{ei} = \frac{n_\mathrm{i}\Gamma_\mathrm{ei}}{v^3}=\frac{4\uppi Zn_\mathrm{e} e^4 \log\Lambda_\mathrm{ei}}{m_\mathrm{e}^2v^3}$$ is the velocity-dependent electron-ion collision frequency. IMPACT is fully implicit and first order in time, and uses fixed-point/Picard iterations to handle nonlinear terms. Note that the implicit treatment of the electric field charge conservation at every iteration. The main reason for using IMPACT in section \[sec:varion\] is that KIPP has not yet been extended to spatially-varying ionisation along $s_\parallel$. Finally, we also include results previously obtained with the SPRING[@Epperlein94] VFP code which takes a Cartesian tensor expansion to arbitrary order and does not neglect/approximate electron-electron collisions. This code uses a linearised approach, i.e. the spatial gradient operator $\nabla$ is replaced by $\mathrm{i}k$, making it advantageous for analysing the featured in section \[sec:linear\], but not problems with large temperature perturbations. Nonlocal models {#sec:models} =============== Eigenvector Integral Closure ---------------------------- The first model investigated here was originally proposed by Ji, Held and Sovinec [@JiHeld] and is directly derived from simplifications of the VFP equation. Necessarily, the time-derivative term is to allow the heat flow to be calculated based on input density and temperature profiles only, and not the history of the distribution function; this assumption should be reasonable over ‘mean’ SOL profiles (i.e. averaged over time to eliminate fine-scalelength fluctuations), but could break down for transient events with faster timescales such as filaments and ELMs. The distribution function is expressed as $f_\mathrm{e} = f^{(0)} + \delta f$, where $\delta f$ is a perturbation to an initial, Maxwellian, guess $f^{(0)}$. Assuming the perturbation $\delta f$ is small, the nonlinear collision and electric field terms in the gyro-averaged VFP equation are linearised, which yields $$\label{eq:DKE} \frac{\partial \langle\delta f\rangle}{\partial s_\parallel} - \frac{\langle C_\mathrm{L}(\delta f)\rangle}{v_\parallel} = \frac{e E_\parallel}{m_\mathrm{e}v_\parallel}\frac{\langle\partial f^{(0)}\rangle}{\partial v_\parallel}-\frac{\partial \langle f^{(0)}\rangle}{\partial s_\parallel},$$ where $$C_\mathrm{L}(\delta f) = C_\mathrm{ee}(f^\mathrm{(0)},\delta f)+ C_\mathrm{ee}(\delta f,f^\mathrm{(0)})+C_\mathrm{ei}(\delta f,n_\mathrm{i}\delta(\bm{v}))$$ is the linearised collision operator. The next step is to attempt a separation of variables into $s_\parallel$ and $\bm{v}/v_\mathrm{T}$, where $\bm{v}$ is made up of parallel and perpendicular components $v_\parallel$ and $v_\perp$, by expressing $$\label{eq:efunc} \langle\delta f\rangle = \sum_n A_n(s_\parallel) \psi_n(\bm{v}/v_\mathrm{T}) \textrm{ such that } \frac{\langle C_\mathrm{L}(\psi_n)\rangle}{v_\parallel} = \frac{\psi_n}{\lambda_n},$$ where $\psi_n$ are eigenfunctions of the operator $\langle C_\mathrm{L}\rangle/v_\parallel$, which depends only on $\bm{v}/v_\mathrm{T}$, and $\lambda_n$ the inverse of their eigenvalue with dimensions of length. Substituting (\[eq:efunc\]) into (\[eq:DKE\]) and assuming that the dependence of $\psi_n$ on space through $v_\mathrm{T}$ is negligible (only valid when relative temperature perturbations are small globally) yields $$\label{JiHeldFinal} \sum_n \Big( \psi_n\frac{\partial A_n}{\partial s_\parallel } + \cancelto{0}{A_n \frac{\partial \psi_n}{\partial s_\parallel}} + \frac{A_n\psi_n}{\lambda_n} \Big) = \frac{e E_\parallel}{m_\mathrm{e}v_\parallel}\frac{\partial \langle f^{(0)}\rangle}{\partial v_\parallel}-\frac{\partial \langle f^{(0)}\rangle}{\partial s_\parallel}.$$ By similarly decomposing the right-hand side into (orthogonal) eigenfunction contributions, a set of independent first-order ODE’s for $A_n$ is formed that can be solved efficiently. Consequently, $\delta f$ can be reconstructed and the nonlocal correction to the heat flux computed through an integral (hence the nomenclature Eigenvector Integral Closure or EIC). The advantage of this approach is that it circumvents the need to solve in velocity-space at every timestep (as a VFP code must). The main challenge is identifying a discrete description of the eigenfunctions $\psi_n$ In practice, this is done by using an orthonormal polynomial moment-basis to express $\psi_n$ as a vector and the operator $C_\mathrm{L}/v_\parallel$ as a matrix. Ji et al. [@JiHeld] proposed a Legendre-Laguerre (LL) basis in pitch angle and total speed. This converges rapidly in the hydrodynamic limit but slowly in the collisionless limit. As an alternative, Omotani et al. [@Hermite] proposed a Hermite-Laguerre (HL) basis, decoupling parallel and perpendicular velocity components, which allows for easier implementation of sheath boundary conditions. Non-Fourier Landau-Fluid ------------------------ While there are a lot of computational benefits to the EIC model over a full VFP code, a large number of eigenfunctions (at least 120 according to Omotani et al.[@Hermite]) are needed for convergence. The NFLF model [@NFLFTheory; @NFLF2014; @NFLFImplementation] provides a cheaper approach, potentially solving as few as three second-order ODE’s, but without a direct link to the VFP equation. The popular Landau-fluid approach initially proposed by Hammett and Perkins [@Hammett; @H2; @H3] provides a closure for the nonlocal heat flux $\bm{\tilde{Q}}$ in Fourier space. This recovers the correct damping rate of a sinusoidal temperature perturbation in both the collisional and collisionless limits (where the wavelength is much longer/shorter than the thermal mfp). However, the Fourier transforms involved are inconvenient for complex SOL geometries and large temperature and density variations. The innovation by Dimits, Joseph and Umansky [@NFLFTheory; @NFLF2014] was to enable direct calculation of the nonlocal parallel heat flux in configuration space by approximating the closure as a sum of Lorentzians $$\label{eq:Lors} \bm{\tilde{Q}} \approx \frac{\bm{\tilde{Q}}^\mathrm{(B)}}{1+a\|k\|\lambda_\mathrm{ei}^{\textrm{(B)}}} \approx \sum_{j=1}^N \frac{\alpha_j\bm{\tilde{Q}}^\mathrm{(B)}}{\beta_j^2+\big(ak\lambda_\mathrm{ei}^{\textrm{(B)}}\big)^2} \equiv\sum_{j=1}^N \bm{\tilde{q}}_j,$$ where $\bm{\tilde{Q}}^\mathrm{(B)}$ is the (parallel) Braginskii heat flow in reciprocal space, $a$ parametrises the behaviour in the collisionless limit and is determined analytically, $k$ is the wavenumber , $N$ is the number of Lorentzians chosen by the user for the fit and $\alpha_j, \beta_j$ are fit parameters. Equating the contribution from each Lorentzian to a dummy contribution $\bm{q}_j$, rearranging and taking the inverse Fourier transform gives a set of $N$ second-order ODE’s for each spatial direction of interest that can be used to recover the nonlocal heat flow: $$\label{eq:NFLF} \Big(\beta_j^2+\big(ak\lambda_\mathrm{ei}^{\textrm{(B)}}\big)^2\Big)\bm{\tilde{q}}_j \rightarrow \left(\beta_j^2-a^2\lambda_\mathrm{ei}^{\textrm{(B)}2}\nabla^2\right)\bm{q}_j = \alpha_j\bm{Q}^\mathrm{(B)}.$$ Multigroup Diffusion (SNB) -------------------------- The final model being investigated is the multigroup diffusion or ‘SNB’ model named after the original authors Schurtz, Nicolaï and Busquet [@SNB]. It is widely used in inertial fusion codes such as Lawrence Livermore National Laboratory’s HYDRA [@HYDRA], CELIA laboratory’s CHIC [@CHIC] ; and it is applicable in multiple spatial dimensions. The SNB model is best explained starting from the diffusion approximation of the kinetic equations (see equations (\[eq:f0\_orig\]) and (\[eq:f1\_orig\]) above), . The isotropic part of the distribution function $f_0$ is then split into a Maxwell-Boltzmann distribution $f_0^{\mathrm{(mb)}}=n_\mathrm{e}\exp(-v^2/2v_\mathrm{T}^2)/(2\uppi v_\mathrm{T})^{3/2}$ and a deviation $\delta f_0=f_0-f_0^{\mathrm{(mb)}}$. The anisotropic part $\bm{f}_1$ is similarly split, but the ‘Maxwellian’ part $\bm{f}_1^{\mathrm{(mb)}}$, obtained from substituting $f_0^{\mathrm{(mb)}}$ into equation (\[eq:f1\_orig\]), is replaced by an alternative, $\bm{g}_1^{\mathrm{(mb)}}$: $$\begin{split} \bm{f}_1^{\mathrm{(mb)}} &= -\lambda_\mathrm{ei}^* \Big(\frac{m_\mathrm{e} v^2}{2 k_\mathrm{B} T_\mathrm{e}}-4\Big) f_0^{\mathrm{(mb)}} \frac{\nabla T_\mathrm{e}}{T_\mathrm{e}} , \\ \rightarrow \bm{g}_1^{\mathrm{(mb)}} &= -\lambda_\mathrm{ei}^* f_0^{\mathrm{(mb)}} \frac{\nabla T_\mathrm{e}}{T_\mathrm{e}}. \end{split}$$ This modification achieves positive-definiteness without affecting the integral used to calculate the heat flow, and is argued to be compensated by other approximations of the model [@SNB]. Here we have defined $\lambda_\mathrm{ei}^* = \xi\lambda_\mathrm{ei} = \xi v/\nu_\mathrm{ei}$. Note that the factor of 3 difference from the original paper in $\bm{f}_1^{\mathrm{(mb)}}$ is simply due to the use of spherical harmonics by Schurtz et al. while we use a Cartesian tensor expansion. where the local form for $\bm{E}=k_\mathrm{B}T_\mathrm{e}(\nabla \log n_\mathrm{e} + \gamma\nabla \log T_\mathrm{e})$ is used, with $\gamma=1+3(Z+0.477)/2(Z+2.15)$. Substituting $\bm{f}_1 = \bm{g}_1^{\mathrm{(mb)}} + \lambda_\mathrm{ei}^{(E)}\nabla\delta f_0$ into equation the stationary form of (\[eq:f0\_orig\]) obtains $$\label{eq:SNB} \frac{C_{\mathrm{ee0}}\left[\delta f_0\right]}{v} +\nabla\cdot \frac{\lambda_\mathrm{ei}^{(E)}}{3}\nabla \delta f_0 = \frac{\nabla\cdot \bm{g}_1^{\mathrm{(mb)}} }{3}.$$ This can be solved to compute the deviation from the local heat flow $$\delta \bm{Q} = \frac{2\uppi m_\mathrm{e}}{3}\int_{\mathrlap{0}}^{\mathrlap{\infty}}\delta \bm{f}_1 v^5\;\mathrm{d}v = -\frac{2\uppi m_\mathrm{e}}{3}\int_{\mathrlap{0}}^{\mathrlap{\infty}} v^5\lambda_\mathrm{ei}^{(E)} \nabla \delta f_0\; \mathrm{d}v.$$ The main computational advantage of this approach is through the use of efficient model collision operators that are local in velocity-space. This allows for a more effective discretisation into velocity/energy groups, and removes the need to store the entire distribution function in memory. The original authors suggested using a velocity-dependent Krook (BGK) operator due to its simplicity, but Del Sorbo et al. [@Dario] have also employed a more realistic operator suggested by Albritton, Williams, Bernstein and Swartz (AWBS) [@AWBS]: $$\begin{aligned} C_{\mathrm{ee0}}^\textrm{BGK}[\, \cdot \, ] &= -r\frac{\nu_\mathrm{ei}}{Z}\times \cdot \, , &C_{\mathrm{ee0}}^\textrm{AWBS}[\, \cdot \, ] &= \frac{\nu_\mathrm{ei}}{Z}v\frac{\partial}{\partial v}[\, \cdot\, ],\end{aligned}$$ where The original authors [@SNB] halved the geometrically averaged mfp$\lambda_\mathrm{e} = \sqrt{Z\lambda_\mathrm{ei}^\lambda_\mathrm{ei}}$ (see equation (23) of Schurtz et al.[@SNB] and also section III C of Del Sorbo et al.* [@Dario]), which is equivalent to setting $r=4$ . However, in a later section of the original paper[@SNB] (III F) as well as section II of a later publication [@SNBmag] this technicality is not restated when demonstrating a link to the kinetic equations, from which a value of $r=1$ could be interpreted. this value for $r$ in the SNB model along with neglecting corrections to angular scattering from electron-electron collisions (i.e. $\xi$ is set to one) happens to give good agreement with VFP codes for $Z=1$. However, this agreement is observed to get progressively worse as $Z$ increases. In this paper we show that using the BGK collision operator with a different value ($r=2$) and $\xi=(Z+0.24)/(Z+4.2)$ gives very good agreement of SNB with VFP across a wide range of problems (and values of $Z$). Note that, despite the differential form of the AWBS operator, its use does not actually require a significant increase in computational time unless an attempt to parallelise over energy groups is being made. This is because the velocity-space first-order differential equation is simply closed from above with the boundary condition $\delta f_0(v=\infty) = 0$. In a finite difference scheme this could simply be implemented by enforcing the highest energy group to be zero, and solving for the next highest group first. (Bear in mind that discretisation in velocity-space need not be uniform.) However, we identify other issues with the AWBS operator in section \[sec:hydro\] that limit its usefulness and the SNB model using this operator is therefore not explored further. Decay of a Small-Amplitude, Sinusoidal Temperature Perturbation {#sec:linear} =============================================================== First we consider the damping of a small-amplitude temperature $T_e = T_0 + \delta T\cos(kx)$ (often referred to as the Epperlein-Short [@EppShort] test) with a constant uniform background density and ionisation. Due to nonlocal effects as the wavelength is reduced, the dimensionless thermal conductivity $\kappa$ decreases from that predicted in the local limit, $\kappa^\mathrm{(B)}$. A with a relative amplitude of $\delta T/T_0 = 10^{-3}$ was initialised for the KIPP simulations. We used a uniform spatial grid of 127 cells over a half-wavelength with a non-uniform Cartesian velocity grid extending to $v_\mathrm{max} = 7v_\mathrm{T}$ (with parameters $mmax = 256, EPS = 1.01$ as defined by Chankin et al.[@KIPPDev]). The thermal conductivity obtained by both these methods fluctuated in time initially (due to initialising as a Maxwellian) and was tracked until both methods approached constant values. Due to incomplete convergence in timestep these values were slightly different and an average was then taken between the two final conductivity reductions. In order to improve accuracy without using unnecessary amounts of computational time due to a tiny timestep (KIPP is only first-order accurate in time), extrapolation to zero timestep was performed by fitting a straight line of conductivity reduction against timestep. Such complications were unnecessary when using the inherently stationary models: instead of evolving in time, it was possible to calculate heat flow (and thus conductivity reduction) from a single profile with a for each wavelength. table [data/linearised\_z1\_analytical]{}; table [data/linearised\_z1\_nflf3]{}; table [data/linearised\_z1\_kipp]{}; table [data/linearised\_z1\_legendre4040]{}; table [data/linearised\_z1\_snbr3]{}; table [data/linearised\_z1\_snbr2]{}; at (rel axis cs:.95,.95) \[anchor=north east\] [$Z=1$]{}; table [data/linearised\_z1\_analytical]{}; table [data/linearised\_z1\_hp]{}; table [data/linearised\_z1\_spring]{}; table [data/linearised\_z1\_legendre4040]{}; table [data/linearised\_z1\_hermite4040]{}; table [data/linearised\_z1\_kipp]{}; at (rel axis cs:.95,.95) \[anchor=north east\] [$Z=1$]{}; table [data/linearised\_z8\_analytical]{}; table [data/linearised\_z8\_impact]{}; table [data/linearised\_z8\_spring]{}; table [data/linearised\_z8\_legendre5050]{}; table [data/linearised\_z8\_snbr2]{}; at (rel axis cs:.95,.95) \[anchor=north east\] [$Z=8$]{}; Kinetic results from the linearised VFP code SPRING calculated by Epperlein[@Epperlein94] and provided numerically by Bychenkov et al.[@Bychenkov; @Bychenkov95] are also shown in Fig. \[linearhigh\] for comparison against the nonlocal models. Both the LL[@JiHeld] and the HL[@Hermite] bases for the EIC model were investigated using 40,40 moments to achieve convergence to within $1\%$ for $k\lambda_\mathrm{ei}^{\textrm{(B)}} < 1$. Figure \[linear\] shows that both bases agree well with KIPP (within 5 and 10% respectively) in this limit. For higher $k\lambda_\mathrm{ei}^\mathrm{(B)}$ , the SPRING VFP results begin to deviate from both the EIC and KIPP results for a number of reasons: Firstly, the onset of electron inertia effects at high $k\lambda_\mathrm{ei}^\textrm{(B)}$, ignored by the EIC model, introduces a phase shift between the heat flow and temperature perturbation making evaluation of the decay rate difficult with KIPP (the linearised formulation of the SPRING code makes this easier, and likely more accurate; note that it is the modulus of the complex thermal conductivity that has been provided in this case). Additionally, (see below and also Fig. 2 and 3 of Ji et al.[@JiHeld] whose results we have successfully replicated), this is about a factor of 1.8 less than the true collisionless heat flow predicted analytically and by the SPRING code (see section \[sec:cless\]). ----- ------ ------ ------ ----- ----- ----- ----- ----- ----- ----- ----- ----- ---------- $Z$ 1 2 3 4 6 8 10 12 14 20 30 500 $\infty$ $b$ 43.5 73.6 96.0 113 139 157 170 180 189 206 222 261 264 ----- ------ ------ ------ ----- ----- ----- ----- ----- ----- ----- ----- ----- ---------- Results for an ionisation of $Z=8$ are shown in Fig. \[linearz8\]. down around $k\lambda_\textrm{ei}^{\textrm{(B)}}\approx0.3$. Note that the thermal conductivity reduction the lower nonlocality parameter $k\lambda_\textrm{ei}^{\textrm{(B)}}$ at which this occurs is due to the reduced importance of electron-electron collisions at higher ionisations. Hydrodynamic Limit {#sec:hydro} ------------------- table [data/z\_xi\_EppShort]{}; table [data/z\_xi\_Sanmartin]{}; table [data/z\_xi\_Actual\_EppHaines]{}; table [data/z\_b\_newfit]{}; table [data/z\_b\_Actual\_EIC]{}; Bychenkov et al.* [@Bychenkov; @Bychenkov95] have shown that for long wavelength perturbations (i.e. in the hydrodynamic limit) $$\label{eq:bZ} \bm{\tilde{Q}} \sim \bm{\tilde{Q}}^\mathrm{(B)}\!\big(1-bZk^2\lambda_\mathrm{ei}^{\textrm{(B)}2}\big)$$ to second-order in $k\lambda_\mathrm{ei}^{\textrm{(B)}}$, where $b\approx264$ in the Lorentz limit ($Z=\infty$). As the assumptions of the EIC model are valid in this linear and collisional limit (except for neglection of electron inertia which would only introduce a component into the heat flow), and convergence of the LL basis is relatively rapid (only 2 Legendre modes are theoretically needed) we have used it to calculate the value of $b$ for various $Z$ . This was done by fitting a straight line on a graph of heat flow against $Zk^2\lambda_\mathrm{ei}^{\textrm{(B)}2}$ for $k\lambda_\mathrm{ei}^{\textrm{(B)}} <10^{-3}/\sqrt{Z}$ (the lower range of $k\lambda_\mathrm{ei}^{\textrm{(B)}}$ extended below $2\times 10^{-4}/\sqrt{Z}$ and there were typically at least six points on the graph). Results using the EIC model are summarised in Table \[tab:bZ\] and Fig. \[fig:bZ\], which also includes numerical results [@EppHaines] and rational approximations [@EppShort; @Sanmartin] for the low-Z conductivity correction . We find that the approximation $b(Z)/b(\infty)=Z/(Z+11/2)$ fits numerical results to within 7%, whereas simply using $\xi$ overestimates $b$ by as much as 43% at $Z=1$. However, the implications of this for the validity of using $\xi$ in IMPACT and the SNB model are not as serious as they seem because $b$ only quantifies the initial deviation from the local limit, and the total heat flux is not very sensitive to marginal errors in $b$ in the hydrodynamic limit. Table \[tab:SNB\] outlines the values of $b$ predicted by the SNB model depending on the model collision operator and choice of source term. This has been derived in the low-amplitude and continuum-velocity limit as detailed in Appendix A. Using the AWBS operator and the kinetic source term $\nabla \cdot \bm{f}_1^{\mathrm{(mb)}}$ on the right-hand side of equation (16) gives a priori* the closest value of $b=316.9\xi$ (top right) to within 20% of that predicted analytically in the Lorentz limit (Table 1) The ability of the AWBS collision operator to predict the deviation in the hydrodynamic limit fairly accurately might suggest that it provides an improvement to the original SNB model, however we find that coupling it with the original source term leads to negative values of the thermal conductivities at $k\lambda_\mathrm{ei}^{\textrm{(B)}}> 0.124/\sqrt{\xi Z}$ due to it not being positive-definite (see Appendix B). This should never occur in the linearised problem considered here (i.e. decay of a small-amplitude temperature perturbation) as it would result in instabilities at these wavelengths. However, this issue does not necessarily imply that the AWBS operator is an inappropriate choice for other nonlocal models. For example, the M1 model presented by Del Sorbo et al.[@Dario; @Dariomag] does not appear to exhibit this issue of positive-definitiveness; we leave a detailed analysis of this model for future work. Setting $r=2$ exactly in the original implementation of the SNB model (BGK collision operator with the source term $\nabla \cdot \bm{g}_1^{\mathrm{(mb)}}$) remarkably gives the same value of $b=316.9\xi$ as with the AWBS operator and the source term $\nabla \cdot \bm{f}_1^{\mathrm{(mb)}}$ (compare bottom left and top right entries of Table \[tab:SNB\]) and in fact the entire distribution function in this limit (see Appendix A). However, to match the kinetic results for $b$, a value of $r=2.4$ is required in the Lorentz limit and $r=3$ for $Z=1$. We suggest that matching coefficients to such accuracy is not necessary, and that using $r=2$ achieves much better agreement for problems involving large temperature variations (see below). Results using both $r=2$ and $r=3$ for $Z=1$ have been provided in figures to enable the reader to compare. RHS $C_{\mathrm{ee0}}^\textrm{BGK}$ $C_{\mathrm{ee0}}^\textrm{AWBS}$ ----------------------------------------- --------------------------------- ---------------------------------- $\nabla \cdot \bm{f}_1^{(\textrm{mb})}$ $3169\xi/r$ $316.9\xi$ $\nabla \cdot \bm{g}_1^{(\textrm{mb})}$ $633.8\xi/r$ $63.38\xi$ : \[tab:SNB\] Predictions for $b$ by the SNB model, depending on choice of collision operator (columns) and source term (rows) Faithfulness to kinetic results for $b$ can be guaranteed with the NFLF model by modifying the analytical Fourier closure and constraining the fit coefficients appropriately as described in the next section. Collisionless Limit {#sec:cless} -------------------- With decreasing wavelength, the heat flow is predicted to slowly approach a constant value. By fitting to the results of both the EIC and SPRING models. (we were unable to obtain meaningful KIPP results at low enough collisionalities due to issues mentioned above) we find that $$\label{eq:cless} \bm{\tilde{Q}} \sim \frac{3}{2}\sqrt{2} \chi_1 \bm{\tilde{Q}}_\mathrm{fs} \left(1-\frac{c_\infty}{ k^{\epsilon}}\right)\frac{\delta T}{T_0},$$ where $\bm{\tilde{Q}}_\mathrm{fs}$ is antiparallel to the wave-vector , is a reasonable description for the asymptotic behaviour in this limit for low-$Z$ plasmas (i.e. a graph of $\tilde{Q}$ against $\log(k)$ resembles a straight line). The extra factor $\sfrac{3}{2}$ compared to the formalism of Hammett and Perkins[@Hammett] (which inspired previous implementations of the NFLF model[@NFLFImplementation; @NFLFTheory; @NFLF2014]) was found to be necessary due to the isotropic definition of the electron temperature used here. [@Epperlein94; @Bychenkov; @Bychenkov95] Again, the LL basis was used, this time due to the nonconvergence of the HL basis explained above, however at least 40,40 moments were needed to achieve convergence above $k\lambda_\mathrm{ei}^\mathrm{(B)}\approx1$. As found by the original developers of the model[@JiHeld], the value of $\chi_1=1.2/\sqrt{\uppi}$ agreed with the value predicted by Chang and Callen[@ChangCallen]; this is exactly 40% less than the value predicted by Hammett and Perkins [@Hammett] ($\chi_1=2/\sqrt{\uppi}$) because of the quasi-stationary assumption. We have also calculated that an alternative value of $\chi_1=1.225$ can be obtained by numerically solving for zeroes of the response function. ------------ ------------- ------ ------ ------ ------------- $ Z$ 1 2 4 6 8 $\epsilon$ 0.32 (0.32) 0.28 0.23 0.22 0.20 (0.19) $c_\infty$ 0.6 (1.4) 0.7 0.7 0.75 0.75 (1.5) $c_1$ 1.9 (1.5) 2.2 2.7 3.1 3.4 (3.0) ------------ ------------- ------ ------ ------ ------------- : \[tab:cless\] Values for parameters appearing in equations (\[eq:cless\]) and (\[eq:c1\]) obtained with EIC model (using at least 40,40 LL moments) and available SPRING data from reference [Bychenkov,\Bychenkov95]{} (in parentheses), the latter is presumed to be more accurate. Calculated values of $\epsilon$ and $c_\infty$, as well as the $c_1$ referred to below, are summarised in Table \[tab:cless\] for $Z=1,2,4,6,8$. Simulations with EIC at higher $Z$ require a prohibitive number of moments for convergence at high $k\lambda_\mathrm{ei}^{\textrm{(B)}}$. Both the index $\epsilon$ and the coefficient $c_\infty$ vary weakly with $Z$ and have similar orders of magnitude to those predicted by Bychenkov et al[@Bychenkov; @Bychenkov95]. The values obtained here should be slightly closer to reality as Bychenkov et al.* assume complete stationarity (all time derivatives neglected) in their calculations, but there are large uncertainties in our numerical fit (approximately 10% for the EIC data). The limited numerical results available from the assumingly exact SPRING code[@Epperlein94; @Bychenkov; @Bychenkov95] infer a value for $\epsilon$ at $Z=1$ within $0.5\%$ of the EIC prediction, but the value for $c_\infty$ (=1.36) is larger by a factor of 2.2. Due to the combination of stationarity and diffusion approximations, the SNB model without the phenomenological mfp limitation to include electric fields predicts the collisionless heat flow to decrease as $\mathord{\sim}1/k$ to zero as the wavelength decreases[@SNB] . Incorporating the mfp limitation resolves the issue of numerical stability, but introduces an amplitude-dependence of $\chi_1$ that is not observed in VFP simulations. While the NFLF will also always predict a $\mathord{\sim}1/k$ decay of the heat flow for high enough $k\lambda_\mathrm{ei}^{\textrm{(B)}}$, increasing the number of Lorentzians used to improve the fit can progressively extend the validity into lower collisionality regimes. The fitting function we used interpolates behaviour in both the hydrodynamic and collisionless limits with a similar but slightly more robust method than used by Bychenkov et al.[@Bychenkov; @Bychenkov95]: $$\label{eq:c1} \frac{\kappa}{\kappa^\mathrm{(B)}} = \bigg(1+\bigg(\frac{1}{bZk^2\lambda_\mathrm{ei}^{\textrm{(B)}2}}+\frac{\sfrac{3}{2}\sqrt{2}\chi_1/\kappa^\mathrm{(B)}}{k\lambda_\mathrm{ei}^{\textrm{(B)}} (1+c_1/k^\epsilon)}\bigg)^{-1} \bigg)^{-1},$$ where $c_1$ differs from $c_\infty$ by optimising the fit for $k\lambda_\mathrm{ei}^{\textrm{(B)}} \leqslant 1$. Using the parameters as defined in Table \[tab:cless\] for $Z=1$, fits the KIPP and SPRING results to within $6$ and $10\%$ respectively for $k\lambda_\mathrm{ei}\leqslant 1$ and up to $26/20\%$ above this; altering the value of $c_1$ to 1.5 reduces the maximum discrepancy with SPRING results to 11%. ----------- -------------------------------- --------- --------- -------- -------- --------- $N$ 3 $ \alpha$ $2.176\mathord{\times}10^{-3}$ 0.06316 1.6823 $\beta$ 0.1020 0.3513 2.4554 $N$ 6 $ \alpha$ $1.575\mathord{\times}10^{-4}$ 0.01206 0.07960 0.5086 3.5041 49.3331 $\beta$ 0.06195 0.17684 0.5064 1.7432 7.0442 44.4953 ----------- -------------------------------- --------- --------- -------- -------- --------- : \[tab:3L\] NFLF fit parameters for $N=3,6$ ($Z=1$) This new fit is depicted in Fig. 1 with the simpler fit $1/(1+ak\lambda_\mathrm{ei}^{\textrm{(B)}})$ , which overestimates the thermal conductivity at moderate collisionalities around $k\lambda_\mathrm{ei}^{\textrm{(B)}} \approx 0.5$ by over $25\%$. Note that we have observed a recent closure in configuration space (thus convenient for convolution models) suggested by Ji and Held[@JiClosure] to closer fit the EIC results with one more fitting parameter (if the $\alpha$ used by Ji and Held is not considered a free parameter)—tuning of these parameters could probably also achieve an improved fit to the VFP results. We would also like to highlight a recent paper by Joseph and Dimits who have performed detailed analysis of closures for the response function that connects the collisionless and collisional regime[@Joseph]. Three Lorentzians (i.e. $N=3$ in equation \[eq:Lors\]) can approximate our new fit to within $2.5\%$ up to $k\lambda_\mathrm{ei}^{\textrm{(B)}}\approx1.6$; using six Lorentzians allows this to be extended up to $k\lambda_\mathrm{ei}^{\textrm{(B)}}\approx30$. The coefficients used are given in Table \[tab:3L\], and were obtained using the variable projection method [@varpro], constrained such that equation (\[eq:bZ\]) is obeyed to second-order in $k\lambda_\mathrm{ei}^{\textrm{(B)}}$. Comparison for large temperature variations =========================================== Homogeneous density and ionisation {#sec:homo} ---------------------------------- [ ]{} +\[thin,black,fill=themeGray!25!white,mark=none,on layer=pre main\] table [data/Fig5\_xmfp\_Teinit]{} ; table [data/Fig5\_xmfp\_qlocal]{}; table [data/Fig5\_xmfp\_qfl\_f0.05]{}; table [data/Fig5\_xmfp\_qfl\_f0.7]{}; fill between\[of=high and low\]; table [data/Fig5\_xmfp\_qEICHermite164]{}; table [data/Fig5\_xmfp\_qNFLF3]{}; table [data/Fig5\_xmfp\_qSNBr3]{}; table [data/Fig5\_xmfp\_qSNBr2]{}; table [data/Fig5\_xmfp\_qKIPP]{}; We now investigate the accuracy of the EIC, NFLF and SNB models in calculating the heat flow in the case where we have a large relative temperature variation. We consider the case of an initial temperature profile consisting of a ramp connecting two large hot and cold regions (‘baths’). This has the advantages of allowing simple reflective boundary conditions and not requiring any external heating/cooling mechanisms that would also need to be carefully calibrated between codes. Results and initial conditions are here presented in terms of reference quantities encouraging the translation of the problem to both ICF and MCF relevant situations. The hot and cold baths had temperatures of $T_0$ and $0.15 T_0$; these were connected by a cubic ramp given by $$T_\mathrm{e} /T_0 = \begin{dcases} 1 & n_\mathrm{c}' \leqslant -75\\ 0.575 - \frac{0.85}{300} n_\mathrm{c}'\bigg(3-\left(\frac{n_\mathrm{c}'}{75}\right)^2\bigg) & n_\mathrm{c}' \in [-75,75] \\ 0.15 & n_\mathrm{c}' \geqslant 75, \end{dcases}$$ where $n_\mathrm{c}' \in [-154,100]$ is the cell number counting from the centre of the temperature ramp. Cell size in mfp’s was varied between simulations to scan a range of collisionalities. The initial temperature profile is illustrated in Fig. \[heatandtemp\] . For these simulations the electron density, Coulomb logarithm and ionisation ($Z$ = 1) were all kept constant and uniform. table [data/Fig6\_xmfp\_qoverqlocal\_EICHermite206]{}; table [data/Fig6\_xmfp\_qoverqlocal\_NFLF6]{}; table [data/Fig6\_xmfp\_qoverqlocal\_KIPP]{}; table [data/Fig6\_xmfp\_qoverqlocal\_SNBr2]{}; KIPP simulations showed an evolution of the heat flow from the local (due to initialising as a Maxwellian) to the nonlocal, with a reduced peak, over an initial transient phase (over which the temperature ramp flattened somewhat). The transient phase was considered over when the ratio of the KIPP heat flow to the expected local heat flow stopped decreasing. After the transient phase this ratio begins to slowly increase as the thermal conduction flattens the temperature ramp and the ratio of the scalelength to mfp increases (i.e. the thermal transport slowly becomes more local). We then took the temperature profile from KIPP and used our implementation of the various nonlocal models to calculate the heat flow they would predict given this profile. Figure \[heatandtemp\] shows that the EIC and NFLF models agree well with each other (to within $10\%$ almost everywhere for the snapshot shown). However, agreement with KIPP is not nearly as good; the models overestimate the peak heat flux by 30–35% and do not predict the observed preheat into the cold region. The SNB model is shown to perform much better here, predicting the peak heat flux to within $6\%$ as well as the existence of preheat (although this is overestimated). Similar results are observed at all temperature ramp scalelengths investigated as illustrated in Fig. \[allscales\], which depicts the reduction in the peak heat flow compared to the local Braginskii prediction. Spatially-varying density and ionisation {#sec:varion} ---------------------------------------- While comparisons between the SNB model and VFP codes have previously been performed [@SNB; @Marocchino], none have included spatially-inhomogeneous ionisation. be tested in such an environment. Variations in ionisation may also be important in the ‘detached’ divertor scenario where a moderate-$Z$ gas is injected in front of the divertor to radiate excess heat; an investigation of this scenario is left as further work. For evaluating this, the IMPACT[@IMPACT] VFP code was used due to its ability to . table [data/gdhohlraum\_xmic\_init\_TekeV\_interp]{} \[yshift=8pt\] node\[pos=0.2,anchor=west\] [$T_{\textrm{e}}$(20 ns)]{}; table [data/gdhohlraum\_xmic\_5ps\_TekeV\_interp]{}\[yshift=-12pt\] node\[pos=0.2,anchor=west\] [$T_\mathrm{e}$(20.005 ns)]{};; table [data/gdhohlraum\_xmic\_init\_TekeV\_hydra]{}; table [data/gdhohlraum\_xmic\_Z\_interp]{}\[yshift=8pt\] node\[pos=0.85\] [$Z$]{}; table [data/gdhohlraum\_xmic\_Z\_hydra]{}; table [data/gdhohlraum\_xmic\_ne1e20cm3\_interp]{}\[yshift=8pt\] node\[pos=0.85\] [$n_{\textrm{e}}$]{}; table [data/gdhohlraum\_xmic\_ne1e20cm3\_hydra]{}; We performed a HYDRA simulation in 1D spherical geometry of a laser-heated gadolinium hohlraum containing a typical helium gas-fill. A leak source was implemented with an area equal to the laser entrance hole to reproduce the energy balance. Electron temperature ($T_\mathrm{e}$), density ($n_\mathrm{e}$) and ionisation ($Z$) profiles (shown in Fig. \[fig:initVar\]) at 20 nanoseconds were extracted and used as the initial conditions (along with the assumption that the electron distribution function is initially Maxwellian everywhere) for the IMPACT simulation (which was performed instead in planar geometry). At this point very steep gradients in all three variables were set up with a change from $T_\mathrm{e} = 2.5\ \mathrm{keV}, n_\mathrm{e} = 5\mathord{\times}10^{20}\ \mathrm{cm}^{-3}, Z=2$ to $T_\mathrm{e} = 0.3\ \mathrm{keV}, n_\mathrm{e} = 5\mathord{\times}10^{21}\ \mathrm{cm}^{-3}, Z=39$ across approximately 100 $\upmu$m at the helium-gadolinium interface. table [data/gdhohlraum\_xmic\_5ps\_LocalWcm2]{}; table [data/gdhohlraum\_xmic\_5ps\_separatedsnbWcm2]{} ; table [data/gdhohlraum\_xmic\_5ps\_averagedsnbWcm2]{}; table [data/gdhohlraum\_xmic\_5ps\_IMPACTWcm2]{}; Spline interpolation was used to increase the spatial resolution near the steep interface for the IMPACT simulations, helping both numerical stability and runtime due to needing a reduced number of nonlinear iterations. For simplicity, the Coulomb logarithm was treated as a constant $\log \Lambda_\mathrm{ei} = \log \Lambda_\mathrm{ee}= 2.1484$. Note that in reality the plasma is only strongly coupled in the colder region of the gadolinium bubble beyond $\mathord{\sim}$ and $\log\Lambda_\mathrm{ei}\approx8$ up to $\mathord{\sim}$ in the hotter corona. The $n_\mathrm{e}$ and $Z$ profiles did not evolve in the IMPACT simulation due to the treatment of the electric field discussed in section \[sec:vfp\] that ensures quasineutrality and the neglection of ion motion and ionisation models. As with the KIPP simulations in the previous section, there is an initial transient phase where the IMPACT heat flux gradually reduces from the Braginskii prediction as the distribution function rapidly moves away from Maxwellian. Once again this transient phase is considered to be over when the ratio of the peak heat flow to the Braginskii prediction stops reducing. This ratio is not observed to change by more than 5% after the first 5 ps of our 15.7 ps simulation. Therefore, we conclude that it safe to assume the transient phase is over after 5 ps, at which point the temperature front has advanced by approximately 8 $\mathrm{\upmu}$m leading to a as shown in Fig. \[fig:initVar\]. In comparing the IMPACT and SNB heat flow profiles we encountered an important subtlety concerning the implementation of the model. While more recent publications concerning the SNB model[@SNBmag; @Dario] use a formulation similar to that used here in section II C with separate electron-ion and electron-electron mfp’s or collision frequencies, the original paper[@SNB] used a geometrically averaged mfp $\lambda_\mathrm{e} = \sqrt{Z\lambda_\mathrm{ei}^\lambda_\mathrm{ei}}$. However, this averaging process is only valid for the case of homogeneous ionisation, and Fig. \[fig:Zvar\] shows the large effect this has on the heat flow when the ionisation varies. While using separated mfp’s provides a very good prediction of the preheat into the hohlraum, the peak heat flow to within 16% and an improved estimate of the thermal conduction in the gas-fill region, the latter is still too large by a factor of $\mathord{\sim}2$. This discrepancy could potentially lead to an overestimate of hohlraum temperatures and thus cause issues similar to those arising with using an overly restrictive flux limiter[@HYDRA]. Discussion and Further Work =========================== The capability of the NFLF to closely match the results of EIC for the case of homogeneous density and ionisation is fairly impressive, considering that only 6 Lorentzians were needed for convergence compared to EIC’s 64 moments (16,4 in the HL basis, chosen instead of LL as convergence is faster for this problem). This implies that the NFLF is about 5 times faster (assuming the NFLF’s second-order ODE’s take approximately twice the time to solve as EIC’s first-order). However, this result should not be too surprising as both models are based on some kind of linearisation procedure, causing them to fail in almost exactly the same way for a nonlinear problem. For example, the lack of preheat or spatial shift in peak location predicted by the models are both features observed in the linear problem studied in section \[sec:linear\]. The SNB model requires 25 groups for convergence resulting in an only slightly faster computation time the EIC model. Improving performance of the models for large temperature variations would require approaches that did not affect the desirable agreement in the linearised limit. For EIC, a simple method is nonlinear iteration; i.e., updating the right-hand side of equation (\[JiHeldFinal\]) by adding on nonlinear terms such as $\displaystyle \frac{eE_\parallel}{m_\mathrm{e}}\frac{\partial \delta f}{\partial v_\parallel}-\sum_n A_n \frac{\partial \psi_n}{\partial s_\parallel}$ from the initial calculation and repeating until convergence. However, the computational time to apply the differential operators and separate into eigenvector components would probably increase the computational time by an undesirably large factor on the order of the number of moments used. Conversely, a correct approach for improving the NFLF model is not immediately apparent, and probably requires deeper analysis of the link between the model and the VFP equation. However, it is conceivable that this could be done without additional computational expense; for example, replacing the $a^2 \lambda_\mathrm{ei}^2 \nabla^2$ term in equation (\[eq:NFLF\]) with $a^2 (\lambda_\mathrm{ei}\nabla)\cdot (\lambda_\mathrm{ei}\nabla)$ would affect only nonlinear behaviour. It is important to investigate sensitivity of divertor temperature to the errors in these models to confirm whether an accurate treatment of nonlocal transport can reconcile simulation and experiment. Furthermore, the discrepancies observed with the SNB model when ionisation gradients are steep could potentially have critical knock-on effects for integrated modeling; it needs to be determined whether further improvements to the SNB model are necessary to avoid this. One key neglection in this work is the effect of electron-electron collisions on the anisotropic part of the distribution function $\bm{f}_1$ for the case of spatially varying ionisation. It was shown in section \[sec:hydro\] that inclusion of this in KIPP and EIC predicts a noticeably different nonlocal deviation (consider, for example, the value $b$) than would be predicted by using the phenomenological collision fix $\xi$ (which incorrectly predicts $b(Z)/b(\infty) = \kappa^\mathrm{(B)}(Z)/\kappa^\mathrm{(B)}(\infty)=\xi$ as depicted in figure \[fig:bZ\]). But this did not seem to be the case for the more physically realistic large temperature variation studied in section \[sec:homo\], as using the value $r=2$ in the SNB model, derived in the linearised and Lorentz limits, seemed to be preferable to $r=3$. Nevertheless, the use of $\xi$ in IMPACT as an ad-hoc substitution for a more complete approach to anisotropic electron-electron collisions could still potentially lead to inaccuracies in the heat flow predictions depicted in section \[sec:varion\], and this should be further investigated. Less critical to our findings are the inaccuracies experienced by VFP codes in strongly coupled plasmas. While this could play a role in the cooler part of the hohlraum wall studied in section \[sec:varion\] where the Coulomb logarithm drops to $\mathord{\sim}2$ (theoretically rendering the effect of collisions in this region only accurate to $\mathord{\sim}50\%$) it does not affect the conclusion that the separated SNB model predicts the same heat flow into the wall as IMPACT while overpredicting that in the corona as both use the same treatment of $\log\Lambda$. We have simply shown quantitatively that reduced models can be an effective stepping stone between hydrodynamic and VFP approaches. However. this does act as a reminder that even a highly sophisticated VFP code could be faced with challenging inaccuracies in certain regions of the plasma (though it would surely still be an improvement to a purely hydrodynamic approach which would experience the same difficulties with strongly coupled plasmas); a potential method in overcoming this and incorporating large-angle collisions in a continuum code could be a Monte Carlo based approach[@Turrell]. Similar points can be made for other deficiencies, such as collisions with neutrals and Fermi degeneracy, although these are probably slightly easier to address and incorporate into models[@VFPNeutrals; @FermiDirac]. Following on from these basic test problems and sensitivity tests, there are still important questions on predictive modeling of fusion plasma heat flows that could be answered using VFP codes. Firstly, the distribution function predicted by the SNB model should be compared to that of a fully kinetic code to assess the former’s viability in predicting other transport coefficients or parameteric instabilities[@SherlockSNB]. Further modifications of the distribution function to a Dum-Langdon-Matte type super-Gaussian[@Dum1; @Dum2; @Langdon; @Matte] due to inverse bremsstrahlung by laser heating in inertial fusion could also significantly alter the transport processes[@InvBrem]. Furthermore, kinetic effects can still affect perpendicular transport (both heat flow and magnetic field advection rates) for moderate magnetisations[@reemerge; @Archis], this could be relevant to recent interest in magnetised hohlraums[@Montgomery; @Strozzi2015] or magnetic islands in tokamaks[@Islands]; and while a few reduced models have been suggested to capture some of these aspects[@Dariomag; @SNBmag] they still need to be properly validated with kinetic codes. Conclusions =========== In conclusion, we have optimal implementations, revealing potential subtleties in the description of the models. We have demonstrated that the SNB model—using the original BGK operator, but scaled according to an analysis of small-amplitude temperature sinusoids ($r=2$), along with the modified source term $\nabla \cdot \bm{g}_1^{\mathrm{(mb)}}$ appearing on the right-hand side of equation (\[eq:SNB\])—performs better than NFLF and EIC for the problems investigated with large temperature variations. Ensuring that the electron-electron and electron-ion collisionalities appear separately in this equation further improves agreement with VFP for a problem with spatially-varying ionisation. However, the problems studied with large temperature variation only reach a nonlocality parameter of $\mathord{\sim}15\%$, suggesting that SNB is most likely suitable for modeling hohlraum energetics problems (with the current exception of gas-fill heat flow, which is overestimated by a factor of $\mathord{\sim}2$) and mean SOL profiles but could break down at the even shorter scalelengths relevant to transient events. The NFLF and EIC models have been found to perform favourably against KIPP when predicting the rate of decay of a small-amplitude temperature pertubation over a wider range of collisionalities than the SNB. However, these models overestimate the peak heat flux by up to $35\%$ in the case of a large temperature variation as well as failing to predict preheat. Additionally, a new analytic fit to kinetic results for temperature sinusoids has been presented in equation (\[eq:c1\]) that could be useful in traditional Landau-fluid implementations. Acknowledgements ================ The authors would like to thank J Y Ji for sharing the numerical results of his closure[@JiClosure], and A V Brantov for assistance in understanding his work with Bychenkov et al.[@BrantovNoB; @Brantov2013] We would also like to express our gratitude towards A M Dimits, I Joseph, W Rozmus and L LoDestro for interesting discussions and pointing us towards a number of relevant references. We additionally appreciate valuable feedback on our manuscript from O Izacard and our Physics of Plasmas reviewer. Finally, thanks to M Zhao for sharing a script to analyse the KIPP results. All data used to produce the figures in this work, along with other selected supporting data, can be found at [dx.doi.org/10.15124/3de4e00c-9087-4d95-89aa-9acd2071d3fb](http://dx.doi.org/10.15124/3de4e00c-9087-4d95-89aa-9acd2071d3fb). This work is funded by EPSRC grants EP/K504178/1 and EP/M011372/1. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053 (project reference CfP-AWP17-IFE-CCFE-01). The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. This work was supported by Royal Society award IE140365. SNB in the hydrodynamic limit ============================= For long wavelength perturbations the diffusion term in equation \[eq:SNB\] can be ignored and thus the distribution function and nonlocal heat flow easily computed in this limit. An outline of the derivation is given here, note that using the BGK collision operator and $\bm{g}_1^{(\textrm{mb})}$ in the source term gives the same $\delta f_0$ as when using the AWBS operator with $\bm{f}_1^{(\textrm{mb})}$ in the source term if $r=2$. The different consequences of choosing each source term $\bm{g}_1^{(\textrm{mb})}$, $\bm{f}_1^{(\textrm{mb})}$ are distinguished by the terms, on the left and right respectively, inside the curly brackets $\left\{\cdot,\cdot\right\}$. that integration by parts is employed for the AWBS calculation and a change of variables to $u=v/\sqrt{2}v_\mathrm{T}$ is used. Additionally we define and use a further two dimensionless variables $X=\xi Z k^2\lambda_\mathrm{ei}(v)^2$ which is velocity-dependent and $X^{(\mathrm{B})}=\xi Z k^2\lambda_\mathrm{ei}^{\mathrm{(B)}2}$ which is independent of velocity. [$$\begin{aligned} &\qquad \textbf{BGK} & & \qquad \textbf{AWBS} \\ \delta f_0&=-\frac{\mathrm{i}Zk\lambda_\mathrm{ei}^{\mathrm(B)}}{r}\frac{\left\{\bm{g}_1^{(\textrm{mb})},\bm{f}_1^{(\textrm{mb})}\right\}}{3} &\delta f_0 &= \int_{\mathrlap{\infty}}^{\mathrlap{v}} \mathrm{d}v \frac{ \mathrm{i}Zk\lambda_\mathrm{ei}(v)}{v}\frac{\left\{\bm{g}_1^{(\textrm{mb})},\bm{f}_1^{(\textrm{mb})}\right\}}{3} \\ \delta \bm{Q} &= -\frac{2\uppi m_\mathrm{e}}{3}\int_{\mathrlap{0}}^{\mathrlap{\infty}}\mathrm{d}v \frac{X}{r} \frac{v^5\left\{\bm{g}_1^{(\textrm{mb})},\bm{f}_1^{(\textrm{mb})}\right\}}{3} & \delta \bm{Q} &= -\frac{2\uppi m_\mathrm{e}}{3}\int_{\mathrlap{0}}^{\mathrlap{\infty}} \mathrm{d}v \frac{X}{10} \frac{v^5\left\{\bm{g}_1^{(\textrm{mb})},\bm{f}_1^{(\textrm{mb})}\right\}}{3} \\ &= -\frac{32}{9\uppi}\int_{\mathrlap{0}}^{\mathrlap{\infty}} \mathrm{d} u \frac{u^{17}\{1,u^2-4\} \mathrm{e}^{-u^2}}{36} \frac{X^{\textrm{(B)}}}{r}\bm{Q}^\mathrm{(B)}. & &= -\frac{32}{9\uppi}\int_{\mathrlap{0}}^{\mathrlap{\infty}} \mathrm{d} u \frac{u^{17}\{1,u^2-4\} \mathrm{e}^{-u^2}}{360} X^{\textrm{(B)}}\bm{Q}^\mathrm{(B)}. \end{aligned}$$]{} Linearised SNB for arbitrary collisionality. ============================================ A similar analyis can be performed with slightly greater difficulty at arbitrary collisionality. Integration by parts must be used again for the AWBS derivation, along with some mathematical identities. Recall that the electric field correction made by the SNB model is a nonlinear correction and does not come into play if the amplitude of the perturbation is infinitesimal: [$$\begin{aligned} &\qquad \textbf{BGK} & & \qquad \textbf{AWBS} \\ \delta f_0&=-\frac{\mathrm{i}Zk\lambda_\mathrm{ei}^{\mathrm(B)}}{3r}\frac{\left\{\bm{g}_1^{(\textrm{mb})},\bm{f}_1^{(\textrm{mb})}\right\}}{1+X/3r} &\delta f_0 &= \mathrm{e}^{X/24}\int_{\mathrlap{\infty}}^{\mathrlap{v}} \mathrm{d}v \mathrm{e}^{-X/24}\frac{ \mathrm{i}Zk\lambda_\mathrm{ei}(v)}{v}\frac{\left\{\bm{g}_1^{(\textrm{mb})},\bm{f}_1^{(\textrm{mb})}\right\}}{3} \\ \delta \bm{Q} &= \frac{2\uppi m_\mathrm{e}}{3}\int_{\mathrlap{0}}^{\mathrlap{\infty}}\mathrm{d}v \frac{v^5\left\{\bm{g}_1^{(\textrm{mb})},\bm{f}_1^{(\textrm{mb})}\right\}}{1+X/3r} & \delta \bm{Q} &= -\frac{2\uppi m_\mathrm{e}}{3}\int_{\mathrlap{0}}^{\mathrlap{\infty}} \mathrm{d}v \frac{\gamma\left(\frac{5}{4},\frac{-X}{24}\right)\mathrm{e}^{-X/24}v^5\left\{\bm{g}_1^{(\textrm{mb})},\bm{f}_1^{(\textrm{mb})}\right\}}{(-X/24)^{1/4}} \\ \frac{ \bm{Q}}{\bm{Q}^\mathrm{(B)}}&= \int_{\mathrlap{0}}^{\mathrlap{\infty}} \mathrm{d}u \frac{u^{9}\{1,u^2-4\} \mathrm{e}^{-u^2}/12}{1+32X^{\textrm{(B)}}u^8/27\uppi r} . &\frac{ \bm{Q}}{\bm{Q}^\mathrm{(B)}} &= \int_{\mathrlap{0}}^{\mathrlap{\infty}}\mathrm{d}u \frac{\gamma\left(\frac{1}{4},\frac{-4X^{\textrm{(B)}}u^8}{27\uppi}\right)\mathrm{e}^{-4X^{\textrm{(B)}}u^8/27\uppi}u^9\{1,u^2-4\} \mathrm{e}^{-u^2}}{12(-4 X^{\textrm{(B)}}u^8/27\uppi)^{1/4}}, \end{aligned}$$]{} where $\gamma$ is the incomplete gamma function. Computing the definite integral numerically with Mathematica shows that the AWBS heat flow can become negative for $X^\mathrm{(B)} > 0.0154$, which corresponds to $k\lambda_\mathrm{ei}^{\textrm{(B)}} > 0.124/\sqrt{\xi Z}$. [80]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [**, ()](\doibase https://doi.org/10.1016/j.hedp.2011.03.008) [, ()](\doibase http://dx.doi.org/10.1016/j.cpc.2009.03.008) @noop [, ()]{} [, ()](http://stacks.iop.org/0741-3335/55/i=5/a=055009) [, ()](\doibase http://dx.doi.org/10.1016/j.jnucmat.2014.10.040) @noop [, ()]{} [, ()](\doibase 10.1063/1.4876617) [, ()](\doibase http://dx.doi.org/10.1016/j.jnucmat.2014.10.015), [, ()](\doibase 10.1063/1.1289512) [, ()](\doibase 10.1063/1.2179392) [, ()](\doibase 10.1063/1.4926824) [, ()](\doibase 10.1017/S0263034616000252) [, ()](\doibase 10.1063/1.4928445) “,” in @noop []{}, Vol. , (, , ) p. 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--- author: - 'Seyed Sadegh Mohseni Salehi[^1]' - Deniz Erdogmus - Ali Gholipour - '-, -, -' title: Tversky loss function for image segmentation using 3D fully convolutional deep networks --- Supplementary Material {#sec:intro} ====================== Layer Input Size Output Size Layer Input Size Output Size ------------- ---------------- ---------------- ------- ---------------- ---------------- C1 128,224,256,3 128,224,256,16 E12 128,224,256,48 128,224,256,2 C2 128,224,256,16 128,224,256,16 E11 128,224,256,48 128,224,256,48 C3-Pooling 128,224,256,16 64,112,128,16 E10 64,112,128,32 128,224,256,48 C4 64,112,128,16 64,112,128,32 E9 64,112,128,32 64,112,128,32 C5 64,112,128,32 64,112,128,32 E8 64,112,128,96 64,112,128,32 C6-Pooling 64,112,128,32 32,56,64,32 E7 32,56,64,64 64,112,128,96 C7 32,56,64,32 32,56,64,64 E6 32,56,64,64 32,56,64,64 C8 32,56,64,64 32,56,64,64 E5 32,56,64,192 32,56,64,64 C9-Pooling 32,56,64,64 16,28,32,64 E4 16,28,32,128 32,56,64,192 C10 16,28,32,64 16,28,32,128 E3 16,28,32,128 16,28,32,128 C11 16,28,32,128 16,28,32,128 E2 16,28,32,384 16,28,32,128 C12-Pooling 16,28,32,128 8,14,16,128 E1 8,14,16,256 16,28,32,384 C13 8,14,16,128 8,14,16,256 C14 8,14,16,256 8,14,16,256 [^1]: Corresponding author: S.S.M.Salehi (email: ssalehi@ece.neu.edu).
--- abstract: 'Using an algorithm for computing the symmetric function Kronecker product of D-finite symmetric functions we find some new Kronecker product identities. The identities give closed form formulas for trace-like values of the Kronecker product.' address: \| [Dept. Mathematics,]{}\ [ Simon Fraser University,]{}\ [ Burnaby, Canada]{}\ [ V5A 1S6]{} author: - \| Marni Mishna\ Dept. Mathematics, Simon Fraser University title: 'Kronecker product identities from D-finite symmetric functions' --- Introduction {#introduction .unnumbered} ============ In the process of showing how the scalar product of symmetric functions can be used for enumeration purposes, Gessel [@Gessel90], proved that this product, and the Kronecker product, [preserve D-finiteness]{}. Roughly, this means that if $F$ and $G$ are symmetric functions which both satisfy a particular kind of system of linear differential equations, then so will the scalar and Kronecker products of these functions. In an earlier work [@ChMiSa05], we give algorithms to calculate both of these systems of differential equations. In this short note we use this algorithm in a symbolic way to find explicit expressions for Kronecker products of pairs of several common series of symmetric functions, such as complete (${\mathsf{H}}=\sum_n h_n$), elementary (${\mathsf{E}}=\sum_n e_n$) and Schur (${\mathsf{S}}=\sum_n\sum_{\lambda \vdash n} s_\lambda$). Proposition 12 of [@ChMiSa05], is the following identity, $$\left(\sum_\lambda s_\lambda\right)\otimes\left(\sum_\lambda s_\lambda\right)={\exp\left(\sum_{n\geq 1}\frac{p_{2n-1}}{(2n-1)(1-p_{2n-1})}\right)}{\left({\prod_{n\geq 1}\left(1-p_n^2\right)}\right)^{-1/2}}.$$ This is the generating series of $\sum_n\left(\sum_{\lambda\vdash n,\mu\vdash n, \lambda<\mu} s_\lambda\otimes s_\mu\right)$. Here, we apply the same technique to give a table of new identities of the same flavour. Symmetric functions =================== We use notation as in Macdonald [@Macdonald95] for our symmetric functions. A [partition]{} of a positive integer $n$ is a decreasing sequence of integers $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_k)$ whose sum is $n$. This is denoted $\lambda\vdash n$. A partition is written in either vector or power notation, for example $(7,7,4,4,1)=[1\,4^2\,7^2]$ are both partitions of 23. A [symmetric function]{} is a sum of monomials in a some variable set, that is invariant under any permutation of that variable set. We can write any symmetric function as a sum of [monomial symmetric functions]{}, defined for the variable set $\{x_1, x_2, \ldots\}$ with respect to some partition $\lambda$ as $$m_\lambda:=\sum_{\sigma\in\cS_{{\mathbb N}\setminus\{0\}}} \left(r_1!\,r_2!\dotsm\right)^{-1} x_{\sigma(1)}^{\lambda_1}\dotsm x_{\sigma(k)}^{\lambda_k}.$$ For example, $m_{(3,2,2)}=x_1^3x_2^2x_3^2+x_3^3x_2^2x_1^2+x_4^3x_1^2x_3^2+\ldots$. We also have the [elementary symmetric functions]{}, $e_n=m_{\langle1^n\rangle}$, and $e_\lambda=e_{\lambda_1}\dotsm e_{\lambda_k}$; the [complete symmetric functions]{} $h_n=\sum_{\lambda\vdash n} m_\lambda$, and $h_\lambda=h_{\lambda_1}\dotsm h_{\lambda_k}$; and [power sum symmetric functions]{} $p_n=m_{(n)}=x_1^n+x_2^n+\ldots$, $p_\lambda=p_{\lambda_1}\dotsm p_{\lambda_k}$. We postpone the definition of the Schur symmetric functions to the next section, where we shall be better equipped. Any of the $h_\lambda$, $p_\lambda$, $e_\lambda$, or $s_\lambda$ can form a ${\mathbb Q}$-basis of the vector space $\Lambda$ of symmetric functions. We can also view $\Lambda$ as the ring ${\mathbb Q}[p_1, p_2 \ldots]$ and, finally, we also work in the ring $\hat\Lambda={\mathbb Q}[[p_1, p_2, \ldots]]$. The scalar product of symmetric functions ----------------------------------------- The ring of symmetric series is endowed with a scalar product defined as a symmetric bilinear form such that the bases $(h_\lambda)$ and $(m_\lambda)$ are dual to each other: $$\label{eq:scalhm} {\left\langlem_\lambda, h_\mu\right\rangle}=\delta_{\lambda\mu}.$$ It turns out that $$\langle p_\lambda,p_\mu\rangle=z_\lambda\delta_{\lambda,\mu},$$ with $z_\lambda=(1^{r_1}r_1!)(2^{r_2}r_2!)\dotsm$ when $\lambda=[1^{r_1}2^{r_2}\cdots]$. The Schur basis is an orthonormal symmetric function basis under this scalar product. In fact, Schur functions can be defined as the result of applying the Gram-Schmidt process for orthogonalizing a basis, applied to the monomial basis with the partitions ordered lexicographically[^1]. Plethysm of symmetric functions ------------------------------- To conclude this brief recollection of symmetric functions, we describe one type of composition that turns out to be quite useful here: [plethysm]{}, written $f[g]$. We can most easily define it using the power sum symmetric functions. It is defined by $p_n[\psi(p_1,p_2,\dots)]=\psi(p_n,p_{2n},\dots)$, along with $(\phi_1+c\phi_2)[\psi]=\phi_1[\psi]+c\phi_2[\psi]$ and $(\phi_1\cdot\phi_2)[\psi]=\phi_1[\psi]\cdot\phi_2[\psi]$. The Kronecker product of symmetric functions -------------------------------------------- In the ring of symmetric functions the usual polynomial multiplication serves as a product, but there is also a second product which arises from the connection between symmetric function and the characters of the symmetric group. This product has several names, including the [Kronecker product]{}, the tensor product and the internal product. Although we mostly follow the notation of Macdonald [@Macdonald95] for most matters relating to symmetric functions, we shall refer to it here as the Kronecker product, and denote it by ${}$. It was first described by Redfield as the [cap product]{} of symmetric functions and was rediscovered by Littlewood [@Littlewood56]. This product can be defined in representation theory pointwise product of characters, which corresponds to tensor products of representations, however here we use the following relation to the power sum symmetric functions, and extend linearly: $$\label{eq:iprod} p_\lambda\iprod p_\mu =\delta_{\lambda\mu}z_\lambda p_\lambda.$$ Calculating the connection coefficients $\gamma_{\lambda,\mu}^{(\rho)}$ for the Kronecker product in the Schur basis $$s_\lambda\iprod s_\mu = \sum_\rho \gamma_{\lambda,\mu}^{(\rho)} s_\rho$$ is also challenging, and quite interesting. There are some combinatorial interpretations of $\gamma_{\lambda,\mu}^{(\rho)}$ which have obtained results when $\lambda,\rho$ and $\mu$ are of a particular form, such as work of Goupil and Schaeffer [@GoSh98], Rosas [@Rosas01a], or Chauve and Goupil [@ChGo05]. The interest originates from the correspondence with irreducible representations, $$\chi({\mathcal{V}}_\lambda {}{\mathcal{V}}_\mu) = \sum_\rho \gamma_{\lambda,\mu}^{(\rho)}\, \chi({\mathcal{V}}_\rho).$$ When $\lambda$, $\mu$ and $\rho$ are all partitions of $n$, $\gamma_{\lambda,\mu}^{(\rho)}$ is the multiplicity of a character in the representation. For more details, the reader is pointed towards the text of Sagan [@Sagan01]. To compute $\gamma_{\lambda,\mu}^{(\rho)}$ using computer algebra systems, one typically expands the symmetric function into the power sum basis and then applies to a pairwise comparison of terms. (For example, in the SF package of Stembridge.) As we mentioned in the introduction, we introduced a generating function approach [@ChMiSa05]. The algorithm in [@ChMiSa05] that we use is called , and a Maple implementation on the author’s web page is available. A second approach, summarized in [@ScThWy93], uses a reduced notation that allows calculations with series of the form $\sum_n s_{(n,\lambda_2, \dots, \lambda_k)}z^n$, for fixed $\lambda_2, \dots, \lambda_k$. These computations are quite efficient; far more so than expanding the power-sum basis. Applications of D-finite symmetric series ========================================= The algorithms we use do not compute the products directly, rather they determine differential equations satisfied by the resulting function. The existence of such differential equations is a consequence of the D-finite closure properties of the scalar product. A function $\phi$ is said to be [D-finite]{} in $\bK[[x_1,\dots,x_r]]$ if and only if the partial derivatives $\partial_1^{\alpha_1}\dotsm\partial_r^{\alpha_r}\phi$ generate a finite dimensional vector space over $\bK(x_1,\dots,x_r)$. In this case, $\phi$ is determined by a system of linear differential equations. In order to treat symmetric functions, however, we must consider functions with an infinite number of variables. The function $\phi(x_1, x_2, \ldots)$ is D-finite in $K[[x_1, x_2, \ldots]]$ if for all $r$, $\phi(x_1, \ldots, x_r, 0, \cdots)$ is D-finite in $K[[x_1, x_2, \ldots, x_r]]$. This case does not enjoy all of the closure properties of the previous, nonetheless we have closure under $+$, $\times$, $\partial_i$, extension of coefficients, rational substitution, and exponentials of polynomials. We say that a symmetric function $\phi\in\hat\Lambda$ is D-finite if it is D-finite in $\bQ[[p_1,p_2,\dots]]$. For example, under this definition the two following famous symmetric function sums ${\mathsf{H}}$ and ${\mathsf{E}}$ which we introduced earlier satisfy the following relations, $${\mathsf{H}}=\exp\left(\sum_n\frac{p_n}{n}\right) \text{ and } {\mathsf{E}}=\exp\left(\sum_n(-1)^n\frac{p_n}{n}\right),$$ and thus are both D-finite. It was Gessel [@Gessel90] that first showed that the scalar product and the Kronecker product both preserve D-finiteness. The work [@ChMiSa05] makes this effective by transforming the system of differential equations satisfied by $F$ and $G$ in to one satisfied by $F\iprod G$, or $\langle F,G\rangle$. Kronecker product calculations ------------------------------ Many interesting problems which use the Kronecker product involve symmetric functions, which once they are expressed in the power sum basis, require an infinite number of $p_n$. Thus, at first glance they are seemingly unsuitable for direct application of our algorithms which, after all, require finite input! One approach is to apply these algorithms for several truncations of the symmetric functions and generate information upon which reasonable conjectures can be formulated. For each of these, we render the problem applicable by setting most $p_n$’s to 0. That is, we solve a sequence of problems involving an increasing number of $p_n$, and hope to identify a pattern. However, far more satisfying are the cases where there is sufficient form and structure which can be exploited to find exact results. We shall be more specific about precisely the “form and structure” we can exploit in a moment. First we remark that one important such class comes from symmetric series arising from plethysms. In this case, we can reduce the Kronecker product of functions each with an infinite number of $p_n$ variables to a finite number of symbolic calculations. For example, if two symmetric functions $F$ and $G$ can be expressed respectively in the form $$F(p_1, p_2, \ldots)=\prod_{n\ge 1} f_n(p_n) \qquad \text{ and } \qquad G(p_1, p_2, \ldots)=\prod_{n\ge 1} g_n(p_n),$$ then one can easily deduce that $$\label{eqn:prodform} F{}G=\prod_{n\ge 1} f_n(p_n){}g_n(p_n).$$ Essentially this follows from the fact that the Kronecker product of two power sum symmetric functions of differing order is 0. If, furthermore, the $f_n$ and $g_n$ are such that one can describe them in a finite way using D-finite functions, we can apply this method. Series which arise as plethysms of the form ${\mathsf{H}}[u]$ or ${\mathsf{E}}[u]$, where $u$ is a polynomial in the $p_i$, are precisely of this form. For example the sum of all Schur functions is of this type: $${\mathsf{S}}=\sum_\lambda s_\lambda={\mathsf{H}}[p_1+\frac{1}{2}p_1^{2}-\frac{1}{2}p_2] =\exp\left(\sum_n\frac{p_n^2}{2n}+\frac{p_{2n-1}}{2n-1}\right).$$ Thus, $${\mathsf{S}}=\left(\prod_{n\text { even}} \exp\left(\frac{p_n^2}{2n}\right)\right) \left(\prod_{n\text{ odd}} \exp\left(\frac{p_n^2}{2n}+\frac{p_n}{n}\right)\right)$$ We assign $f_n$ as follows $$f_{2n}=\exp\left(\frac{p_{2n}^2}{4n}\right)\quad \text{ and }\quad f_{2n-1}=\exp\left(\frac{p_{2n-1}^2}{2}+\frac{p_{2n-1}}{2n-1}\right).$$ Thus, to compute ${\mathsf{S}}\iprod {\mathsf{S}}$, we compute in turn $g_{2n}=f_{2n}{}f_{2n}$, and $g_{2n+1}=f_{2n+1}{}f_{2n+1}$. We find $g_{2n}$ by determining the differential equation that it satisfies, using adapted to handle a formal parameter. The adaptation amounts to performing a scalar product with adjunction formula $p^\diamond=n\partial$ for a formal parameter $n$. This gives $$(1-p_n^2)\frac{\partial g_n(p_n)}{\partial p_n}+p_ng_n(p_n)=0, \qquad\text{for even~$n$}.$$ We then solve for $g_n$. We do likewise for odd $n$, and then the identity in the introduction follows. Carbonara [et al.]{} [@CaCaRe03] are interested in the trace of $\sum_n\left(\sum_{\lambda\vdash n,\mu\vdash n, \lambda<\mu} s_\lambda{}s_\mu\right)$ given by $\sum_n\left(\sum_{\lambda\vdash n} s_\lambda{}s_\lambda\right)$. It is not immediately clear to me if our method could be adapted directly to this kind of calculation. A family of identities ---------------------- We now apply the above approach to create a number of different identities. The following table summarizes results. These formulas for ${\mathsf{H}}, {\mathsf{E}}, {\mathsf{S}}, {\mathsf{S}}{\mathsf{E}}^{-1}$ and ${\mathsf{S}}{\mathsf{H}}^{-1}$ are all derived in Macdonald [@Macdonald95]: $$\begin{array}{ll} {\mathsf{H}}=\sum h_nt^n=\exp\left(\sum_n \frac{p_n}{n}\right)\\ {\mathsf{E}}=\sum e_nt^n=\exp\left(\sum_n (-1)^{n+1}\frac{p_n}{n}\right)\\ {\mathsf{S}}=\sum_\lambda s_\lambda t^{\|\lambda\|}=\exp\left(\sum_n \frac{p^2_nt^{2n}}{2n}+\frac{p_{2n-1}t^{2n-1}}{2n-1}\right)\\ {\mathsf{S}}{\mathsf{E}}^{-1}=\sum_{\lambda\text{ all parts odd}} s_\lambda t^{\|\lambda\|}\\ {\mathsf{S}}{\mathsf{H}}^{-1}=\sum_{\lambda'\text{all parts even}} s_\lambda t^{\|\lambda\|} \end{array}$$ \[thm:table\] Given the above definitions for ${\mathsf{H}}, {\mathsf{E}}$ and ${\mathsf{S}}$. Then, there is the following multiplication table for the Kronecker product, . The products are expressed in terms of the following: $$\begin{array}{ll} {\mathsf{M}}_{odd (even)} = \exp\left(\sum_{n\;odd (even)}\frac{p_{n}}{n(1-p_n)}\right) \\ {\mathsf{N}} =\exp\left(\sum_{n} \frac{p_{n}^2}{2n(1-p_n^2)}\right)\\ {\mathsf{P}} = \exp\left(\sum_{n\;even} \frac{p_{n}}{n(1+p_n)}\right)\\ {\mathsf{G}} = \prod_{n\geq 1}\left(1-p_n^2\right)^{-1/2}. \end{array}$$ A Maple worksheet with the calculations behind the above table is available at the author’s website. We welcome all suggestions for other series of interest. It would equally easy to treat plethysms of the form ${\mathsf{H}}[\phi]$ for some symmetric polynomial $\phi$. A preliminary review of the work of Scharf, Thibon, and Wybourne, for example [@ScTh91; @ScThWy93] suggests that there may be more to do with series of the form $\sum_n s_{(n,\lambda_2, \dots, \lambda_k)}z^n$, for fixed $\lambda_2, \dots, \lambda_k$ if they can be shown to be D-finite. We are able to compute, with this method, expressions satified by some powers of ${\mathsf{S}}=\sum_\lambda s_\lambda$, with respect to the Kronecker product, for example ${\mathsf{S}}{}{\mathsf{S}}{}{\mathsf{S}}$, but these result in differential equations which we are presently unable to solve into explicit expressions. Conclusion {#conclusion .unnumbered} ========== The symbolic application of tensor product calculation yields, rather easily, families of Kronecker product identities. It is possible that these identities could be exploited for group theoretic gain, however, this remains to be investigated, as does finding connections between our formulas, and that of the trace co-characters. Acknowledgments {#acknowledgments .unnumbered} --------------- This work was initiated during a visit to Project Algorithms, in part from discussions with Frédéric Chyzak, and was funded in part by the NSERC (Canada). Thanks are due also to Rosa Orellana for an interesting discussion on the trace co-characters and an anonymous referee that suggested several interesting references. [1]{} J. O. Carbonara, L. Carini, and J. B. Remmel. Trace cocharacters and the [K]{}ronecker products of [S]{}chur functions. , 260(2):631–656, 2003. Frédéric Chyzak, Marni Mishna, and Bruno Salvy. Effective scalar products of [D]{}-finite symmetric series. , 112:1 – 43, 2005. Ira M. Gessel. Symmetric functions and [P]{}-recursiveness. , 53(2):257–285, 1990. Alain Goupil and Cedric Chauve. Combinatorial operators for kronecker powers of representations of $s_n$. , 54, 2006. Article B54j, 13 pages. Alain Goupil and Gilles Schaeffer. Factoring [$n$]{}-cycles and counting maps of given genus. , 19(7):819–834, 1998. D. E Littlewood. The kronecker product of symmetric group representations. , 31:89–93, 1956. Ian G. Macdonald. . The Clarendon Press Oxford University Press, New York, second edition, 1995. Mercedes H. Rosas. The [K]{}ronecker product of [S]{}chur functions indexed by two-row shapes or hook shapes. , 14(2):153–173, 2001. Bruce E. Sagan. , volume 203 of [Graduate Texts in Mathematics]{}. Springer-Verlag, New York, second edition, 2001. T. Scharf, and J.-Y. Thibon. A Hopf-algebra approach to inner plethysm 104:30–58 (1994). T. Scharf, J.-Y. Thibon and B.G. Wybourne. Reduced notation, inner plethysms and the symmetric group , 26(24):7461–7478, 1993. [^1]: In such an ordering, $1^n<1^{n-1}2<\cdots< n$.
--- abstract: 'We probe doubled geometry with dual fundamental branes, i.e. solitons. Restricting ourselves first to solitonic branes with more than two transverse directions we find that the doubled geometry requires an effective wrapping rule for the solitonic branes which is dual to the wrapping rule for fundamental branes. This dual wrapping rule can be understood by the presence of Kaluza-Klein monopoles. Extending our analysis to supersymmetric solitonic branes with less than or equal to two transverse directions we show that such solitons are precisely obtained by applying the same dual wrapping rule to these cases as well. This extended wrapping rule can not be explained by the standard Kaluza-Klein monopole alone. Instead, it suggests the existence of a class of generalized Kaluza-Klein monopoles in ten-dimensional string theory.' address: - \| Centre for Theoretical Physics, University of Groningen,\ Nijenborgh 4, 9747 AG Groningen, The Netherlands - \| INFN Sezione di Roma,\ Dipartimento di Fisica, Università di Roma “La Sapienza”\ Piazzale Aldo Moro 2, 00185 Roma, Italy author: - 'Eric A. Bergshoeff' - Fabio Riccioni title: Dual Doubled Geometry --- branes ,duality ,supersymmetry Introduction ============ Over the course of years supergravity has provided a number of key insights into string theory. Examples of such discoveries are the Green-Schwarz anomaly cancellation [@Green:1984sg] and the presence of branes in string theory such as the eleven-dimensional supermembrane [@Bergshoeff:1987cm]. A key signature for a $p$-brane, i.e. a brane with $p$ spatial directions, in string theory is the presence of a $(p+1)$-form potential in the corresponding supergravity theory. Usually, branes have more than two transverse directions and such branes couple to potentials that describe physical degrees of freedom. These are the so-called “standard” branes and they are well understood. Whenever a brane is standard the dual brane, i.e. the brane that couples to the dual potential, is also standard. The remaining “non-standard” branes are the branes with two transverse directions (“defect”-branes), one transverse direction (“domain-walls”) and no transverse direction at all (“space-filling branes”).[^1] The special thing about the defect-branes is that they couple to the duals of scalars parametrizing a coset manifold $G/H$. This leads to $(D-2)$-form potentials transforming in the adjoint of $G$ whose curvatures satisfy dim$H$ non-linear constraints that involve the scalars themselves. Due to this there is no one-to-one correspondence between a potential and a corresponding defect-brane. Domain-walls and space-filling branes are special in the sense that they couple to potentials that do not describe any physical degrees of freedom. In the case of domain walls the corresponding $(D-1)$-form potential is dual to an integration constant while the space-filling brane couples to a $D$-form potential that is not dual to anything at all. In recent years it has been realized that maximal supergravities can be extended with potentials of rank $D-1$ and $D$ occurring in specific U-duality representations [@Riccioni:2007au; @Bergshoeff:2007qi; @deWit:2008ta]. In recent work [@Bergshoeff:2010xc; @Bergshoeff:2011zk] we have developed a criterion to see which of the high-form potentials of rank $D-2, D-1$ and $D$ couple to supersymmetric defect-branes, domain walls or space-filling branes, respectively. Our requirement was that a gauge-invariant Wess-Zumino (WZ) term should exist that only involves worldvolume fields that fit into a supermultiplet. This is the minimum requirement for the construction of a kappa-symmetric worldvolume action. The result of our analysis was that not all potentials couple to supersymmetric branes. Furthermore, we found that the supersymmetric non-standard branes do not fill complete U-duality representations. A prime example of this phenomenon are the 8-forms of IIB supergravity. They transform as the ${\bf 3}$ of ${\rm SL}(2,\mathbb{R})$ S-duality but there is only a two-dimensional space of supersymmetric configurations, spanned by the D7-brane and its S-dual [@Bergshoeff:2002mb].[^2] In this letter we consider a specific class of branes suggested by supergravity, i.e. the supersymmetric solitons. These are branes whose tension scales with the inverse squared of the string coupling constant in the string frame. They are the duals of the fundamental branes whose tension is independent of the string coupling constant. It is well-known that a T-duality covariant formulation of the fundamental branes of toroidally compactified string theory with 32 supercharges requires a doubled geometry [@Hull:2004in]. In this letter we will probe this doubled geometry with the dual solitons. Before discussing doubled geometry and solitons we first review in the next section the relation between doubled geometry and fundamental branes and D-branes. For fundamental branes, doubled geometry requires an effective wrapping rule that gives rise to all the fundamental branes in a given dimension starting from ten dimensions. We will then discuss solitonic branes. We will first consider the standard solitons, i.e. those with more than two transverse directions. Probing the doubled geometry with such solitons requires an effective wrapping rule for the solitonic branes which is dual to the wrapping rule for fundamental branes. This dual wrapping rule can be understood by the presence of Kaluza-Klein (KK) monopoles. We next extend our analysis to the supersymmetric non-standard solitonic branes with less than or equal to two transverse directions whose existence is suggested by supergravity. These brane configurations have been classified in our previous work using the criterion mentioned above [@Bergshoeff:2011zk]. In this letter we will show that these non-standard solitons are precisely obtained by applying the same dual wrapping rule as in the case of the standard solitons. The fact that this wrapping rule works for the non-standard solitons as well cannot be explained by the presence of the standard KK monopole alone. Instead, it suggests the existence of a class of generalized KK monopoles in ten-dimensional string theory. Doubled Geometry ================ The only fundamental brane in ten dimensions is the fundamental string. It couples to the background metric $g_{\mu\nu}$ via a Nambu-Goto term and to the NS-NS two-form potential $B_{\mu\nu}$ via a WZ term. Schematically, we have $$\begin{aligned} {\cal L}^{\rm D=10}({\rm Fundamental\ String}) \ = \ T\,\sqrt{-g}\ +\ B_2\,,\end{aligned}$$ where $T$ is the string tension. The first term at the r.h.s. is the Nambu-Goto term containing the determinant of the pull-back of $g_{\mu\nu}$. The second term, where we have used form notation, is the WZ term containing the pull-back of $B_{\mu\nu}$. The special thing about fundamental branes is that their brane tension $T$ is independent of the string coupling constant $g_s = <e^\phi>$ with $\phi$ being the dilaton. In general the tension $T$ of a brane may scale like $T \ \sim \ e^{\alpha\,\phi}$ in terms of an integer number $\alpha \le 0$. This leads to a classification of branes according to $\alpha$:[^3] $$\begin{aligned} &&\alpha=0: \hskip .3truecm {\rm Fundamental\ Branes}\,, \nonumber \\ [.05truecm] &&\alpha=-1: \hskip .05truecm {\rm D-branes}\,, \\ [.05truecm] &&\alpha=-2: \hskip .1truecm {\rm Solitonic\ Branes}\,, \ \dots \nonumber \hskip 1truecm {\rm etc.}\end{aligned}$$ Another way of classifying branes is according to the number of transverse directions. As already mentioned in the introduction we will call branes with more than two transverse directions “standard” and branes with less than or equal to two transverse directions “non-standard”. Amongst the non-standard branes we will adapt the following nomenclature: $$\begin{aligned} && \#\ {\rm transverse\ directions} =2\,:\hskip .5truecm {\rm defect-branes}\,,\nonumber\\[.05truecm] && \#\ {\rm transverse\ directions} =1\,:\hskip .5truecm {\rm domain-walls}\,,\\[.05truecm] && \#\ {\rm transverse\ directions} =0\,:\hskip .5truecm {\rm space-filling\ branes}\,.\nonumber\end{aligned}$$ Restricting ourselves first to fundamental branes we not only have the fundamental string in $D < 10$ dimensions but also fundamental 0-branes, i.e. wrapped strings. They can be attached to the fundamental string and the corresponding WZ term gets accordingly modified with extra world-volume scalars that satisfy a self-duality condition [@Hull:2004in]: $$\label{WZstring} {\cal L}^{{\rm D}\le {\rm 10}}_{\rm WZ}{\rm (Fundamental\ String)} = B_2 + \eta^{AB} {\cal F}_{1, A} B_{1 ,B}\,.$$ Here $B_{1,A}$ are the NS-NS 1-forms and ${\cal F}_{1,A}=db_{0,A}+B_{1,A}$ are the 1-form world-volume curvatures of the extra scalars $b_{0,A}$. Both transform as a vector, indicated by the index $A$, under the T-duality group ${\rm SO}(d,d)$ with $d=10-D$. The number of extra scalars is twice the number of compactified dimensions in line with doubled geometry [@Hull:2004in]. Due to the self-duality condition that these scalars satisfy we obtain $(D-2) + 1/2 \cdot 2(10-D) = 8$ worldvolume degrees of freedom, where $D-2$ is the number of transverse scalars. This is the correct number that fits into a scalar supermultiplet. The WZ term for the fundamental 0-branes themselves does not contain extra scalars and is given by (omitting the explicit vector-index $A$) $$\label{WZ0branes} {\cal L}^{{\rm D} \le {\rm 10}}_{\rm {WZ}} ({\rm Fundamental\ 0-Branes}) = B_1\,.$$ In summary, in $D$ dimensions we have a T-duality vector of fundamental 0-branes and a singlet fundamental string. Alternatively, the above counting of branes is obtained by applying the following wrapping rule for fundamental branes: $$\begin{aligned} \label{fundamentalwrapping} & & {\rm wrapped} \ \ \ \ \rightarrow\ \ \ {\rm doubled}\,, \nonumber \\ & & {\rm unwrapped} \ \ \rightarrow \ \ {\rm undoubled}\,. \end{aligned}$$ The doubling of branes under wrapping is due to the fact that in each dimension there is an extra fundamental 0-brane resulting from the reduction of a pp-wave. This is precisely the manifestation of T-duality. Starting from a single fundamental string in ten dimensions (either IIA or IIB) one obtains the correct number of fundamental branes in each dimension by applying the fundamental wrapping rule (\[fundamentalwrapping\]) for each compactified dimension, see Table \[fundamental\]. [\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|]{} ------------------------------------------------------------------------ F$p$-brane &IIA/IIB& 9 & 8 & 7 & 6&5&4&3\ ------------------------------------------------------------------------ 0&&2&4&6&8&10&12&14\ ------------------------------------------------------------------------ 1&1/1&1&1&1&1&1&1&1\ We next consider the D-branes. In $D=10$ dimensions fundamental strings can end on D-branes and, accordingly, the WZ term gets deformed by an extra Born-Infeld worldvolume vector $b_1$, with 2-form curvature ${\cal F}_2=db_1+B_2$: $$\label{WZterm} {\cal L}^{\rm {D=10}}_{{\rm WZ}}{\rm (D-branes)} = e^{{\cal F}_2}C\,.$$ Here $C$ stands for the formal sum of all RR potentials which are of odd rank for IIA and of even rank for IIB string theory. In [@Bergshoeff:2010xc] we derived the T-duality-covariant expression of the D-brane WZ terms in $D < 10$ dimensions. Since now both wrapped and un-wrapped fundamental strings can end on the D-branes we get a further deformation by the extra worldvolume scalars $b_{0,A}$ [@Bergshoeff:2010xc]: $$\label{WZterm2} {\cal L}^{{\rm D}\le {\rm 10}}_{\rm {WZ}}{\rm (D-branes)} = e^{{\cal F}_2}e^{{\cal F}_{1,A}\Gamma^A}C\,,$$ where $\Gamma^A$ are the gamma-matrices of ${\rm SO}(d,d)$. The reason for the existence of the general expression (\[WZterm2\]) is that in any dimension the fundamental potentials transform as a singlet (2-form) and vector (1-form) under T-duality while the D-brane potentials transform as (chiral) spinor representations of the same duality group. We have omitted these spinor indices in eq. (\[WZterm2\]). At first sight the WZ term (\[WZterm2\]) does not seem to lead to the correct counting of worldvolume degrees of freedom. For any D$p$-brane the Born-Infeld vector corresponds to $p-1$ degrees of freedom. Considering also the $D-p-1$ embedding scalars one needs only $d$ extra scalars to fill the bosonic sector of a vector multiplet in $p+1$ dimensions. Instead, there are $2d$ scalar fields $b_{0,A}$, that is twice too many. Unlike in the case of the fundamental string one can this time not rescue the situation by imposing a self-duality condition on the extra scalars. Luckily, it turns out that the above counting is too naive. The expression (\[WZterm2\]) stands for the WZ term for a whole spinor representation of D-branes and it is enough to show that a single spinor component representing the WZ term of a particular D-brane contains only half of the $2d$ extra scalars. To show this, it is enough to expand (\[WZterm2\]) and consider only the first ${\cal F}_{1,A}$ term. For a given $p$ we obtain $$C_{p+1,\alpha} + {\cal F}_{1,A} (\Gamma^A)_\alpha{}^\beta C_{p, \beta} + ... \quad ,$$ where $\alpha$ is an ${\rm SO}(d,d)$ spinor index. Using an ${\rm SO}(d,d)$ lightcone basis, where the light-cone directions are denoted as $A=(1\pm$, $2\pm$,..., $d\pm)$, one can show that for a given value of the spinor index $\alpha$, for any fixed $n=1,...,d$, only one of the two matrices $(\Gamma^{n\pm})_\alpha{}^\beta$ gives a non-zero result when acting on a chiral spinor. The detailed proof can be found in [@Bergshoeff:2011zk]. This shows that for any given D$p$-brane only half of the $2d$ extra scalars $b_{0,A}$ actually occur, and this results in the correct number of degrees of freedom for a $(p+1)$-dimensional worldvolume vector multiplet. In summary, in each dimension $D<10$ we have a T-duality spinor of D-branes of dimension $2^{d-1}$. These D-branes can be obtained by applying the following D-brane wrapping rule: $$\begin{aligned} \label{Dbranewrapping} & & {\rm wrapped} \ \ \ \ \rightarrow\ \ \ {\rm undoubled}\,, \nonumber \\ & & {\rm unwrapped} \ \ \rightarrow \ \ {\rm undoubled}\,. \end{aligned}$$ Starting from the D-branes of ten-dimensional IIA or IIB string theory and using this wrapping rule one obtains the correct number of D-branes in $D<10$ dimensions, see Table \[Dbranetable\]. [\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|]{} ------------------------------------------------------------------------ D$p$-brane &IIA/IIB& 9 & 8 & 7 & 6&5&4&3\ ------------------------------------------------------------------------ 0&1/0&1&2&4&8&16&32&64\ ------------------------------------------------------------------------ 1&0/1&1&2&4&8&16&32&64\ ------------------------------------------------------------------------ 2&1/0&1&2&4&8&16&32&64\ ------------------------------------------------------------------------ $\vdots$&$\vdots$&$\vdots$&$\vdots$&$\vdots$&$\vdots$&$\vdots$&$\vdots$&\ ------------------------------------------------------------------------ 8&1/0&1&&&&&&\ ------------------------------------------------------------------------ 9&0/1&&&&&&&\ Unlike the fundamental wrapping rule the D-brane wrapping rule is self-contained, i.e. it does not need the assistance of gravitational solutions such as the pp-wave. The D-brane sector is also closed under duality in the sense that the dual of a D-brane is again a D-brane. This is not the case for fundamental branes which are dual to solitonic branes. Finally, we note that all potentials that couple to the supersymmetric fundamental branes and D-branes occur in the decomposition $$\label{decomposition} {\rm U-duality}\ \supset\ {\rm SO}(d,d)\times \mathbb{R}^+$$ of the U-duality representations according to which the potentials of maximal supergravity transform. The type of brane, i.e. the value of $\alpha$ in the tension $T=(g_s)^\alpha$, is determined by the weight of the potential under the $\mathbb{R^+}$-scaling symmetry. In particular, we find that the U-duality representations of the high-form potentials of rank $D-2, D-1$ and $D$ under the decomposition (\[decomposition\]) give rise to spinor representations of T-duality corresponding to D-branes with less than or equal to two transverse directions. This concludes our discussion of how fundamental branes and D-branes probe the doubled geometry structure. Requiring that for each brane the corresponding dual brane also belongs to string theory it is natural to include solitons in our discussion since they are dual to the fundamental branes. In the next section we will therefore extend our analysis to string solitons. Solitons and Dual Doubled Geometry ================================== We first restrict ourselves to solitonic branes with more than two transverse directions. Requiring that for every fundamental brane there is a dual solitonic brane one finds that the following dual wrapping rule must be introduced: $$\begin{aligned} \label{solitonicwrapping} & & {\rm wrapped} \ \ \ \ \rightarrow\ \ \ {\rm undoubled}\,, \nonumber \\ & & {\rm unwrapped} \ \ \rightarrow \ \ {\rm doubled}\,. \end{aligned}$$ The doubling of branes when unwrapped is due to the fact that in each dimension there is an extra solitonic $(D-4)$-brane resulting from the reduction of a KK monopole. Starting from the standard NS-NS five-brane of ten-dimensional IIA or IIB string theory and using the dual wrapping rule (\[solitonicwrapping\]) one obtains a singlet solitonic $(D-5)$-brane and a T-duality vector of solitonic $(D-4)$-branes in each dimension $D<10$, see Table \[sol1table\]. [\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|]{} ------------------------------------------------------------------------ S$p$-brane &IIA/IIB& 9 & 8 & 7 & 6&5&4&3\ ------------------------------------------------------------------------ 0&&&&&&1&12&\ ------------------------------------------------------------------------ 1&&&&&1&10&&\ ------------------------------------------------------------------------ 2&&&&1&8&&&\ ------------------------------------------------------------------------ 3&&&1&6&&&&\ ------------------------------------------------------------------------ 4&&1&4&&&&&\ ------------------------------------------------------------------------ 5&$1^\prime/1$&$1^\prime + 1$&&&&&&\ Unlike in the case of fundamental branes we find that the class of solitonic branes, like in the case of D-branes, extends to include non-standard branes, i.e. branes with less than or equal to two transverse directions, as well. Under the decomposition (\[decomposition\]) of the supergravity fields the string solitons (or S-branes) organize themselves as anti-symmetric tensor representations of the T-duality group, so that in $D$ dimensions an S$p$-brane has $(p+d-5)$ indices, see Table \[sol2table\] [@Bergshoeff:2011zk]. In each dimension it is understood that the number $m$ of anti-symmetric indices runs from $m=0$ to $m=d$, which means that the highest possible value of $p$ is 5. Furthermore, the representation with the maximum number of indices - that is $d$ indices - splits into a self-dual and an anti-selfdual representation of the T-duality group. The self-dual representation describes solitonic five-branes with a worldvolume vector multiplet while the anti-selfdual representation involves solitonic five-branes with a worldvolume tensor multiplet. [\|c\|]{} ------------------------------------------------------------------------ S$(D-5)$-brane\ ------------------------------------------------------------------------ \[S$(D-4)$-brane\]$_A$\ ------------------------------------------------------------------------ \[S$(D-3)$-brane\]$_{AB}$\ ------------------------------------------------------------------------ \[S$(D-2)$-brane\]$_{ABC}$\ ------------------------------------------------------------------------ \[S$(D-1)$-brane\]$_{ABCD}$\ Unlike the singlet and vector solitons, it turns out that not all components of the higher-rank antisymmetric tensor representations correspond to [supersymmetric]{} solitons. To understand this it is enough to consider only the leading and subleading term in the WZ term of a solitonic S$p$-brane which takes the schematic form ($m = p+d-5$) [@Bergshoeff:2011zk] $$D_{A_1\dots A_m} + {\overline {{\cal G}(c)}}\,\Gamma_{A_1\dots A_m}\,C\ + \dots \,.$$ Here $D$ is the solitonic target space potential, $C$ is the formal sum of RR target space gauge fields and ${\cal G}(c)$ is the formal sum of worldvolume RR $n$-form gauge fields $c$. Each $n$-form represents a possible D-brane ending on the soliton. Note that both the target space potentials $C$ and the worldvolume potentials $c$ transform as spinors under T-duality. In general there are too many worldvolume potentials to fit a supermultiplet. We therefore need to limit the number of worldvolume gauge fields as much as possible such that the independent ones do fit into a supermultiplet. One way to restrict this number is by imposing worldvolume duality conditions. It turns out that this is not enough [@Bergshoeff:2011zk]. To obtain a supersymmetric soliton one also needs to restrict the number of T-duality directions in the anti-symmetric tensor representation such that the $\Gamma_{A_1\dots A_m}$ matrix projects out the correct number of worldvolume gauge fields. Using a lightcone basis $A=(1\pm\,, 2\pm\,, \dots ,d\pm)$ we found that only the components $$\label{countingrule} [ABC \dots ]= [m\pm n\pm p\pm \dots]\ \ \ {\rm with}\ \ \ m\ne n \ne p \dots$$ are supersymmetric. For more details, we refer to [@Bergshoeff:2011zk]. This leads to a precise prediction of the number of supersymmetric solitons in each dimension. For instance, in $D=6$ dimensions there are solitonic domain-walls transforming in the three-index anti-symmetric tensor representation ${\bf 56}$ of the ${\rm SO(4,4)}$ T-duality group. This representation occurs in the decomposition (\[decomposition\]) of a 5-form gauge potential that transforms in the ${\bf 144}$ of the ${\rm SO}(5,5)$ U-duality group. According to the counting rule (\[countingrule\]) only the directions $1\pm 2\pm 3\pm\,, 1\pm 2\pm 4\pm\,, 1\pm 3\pm 4\pm$ and $2\pm3\pm4\pm$ correspond to supersymmetric solitons. We therefore find that only 32 out of the ${\bf 56}$ configurations are supersymmetric. Remarkably, precisely the same numbers of supersymmetric solitons are obtained by simply extending the dual wrapping rule (\[solitonicwrapping\]) to the non-standard solitons as well, see Table \[sol3table\]. This includes the 32 supersymmetric solitonic domain-walls in $D=6$ dimensions mentioned in the example above. This extension is non-trivial in the sense that not only the singlet soliton doubles when unwrapped but the other solitons double as well when unwrapped. Whereas the doubling of the singlet soliton can be understood by the presence of a singlet KK monopole in each dimension, a similar explanation for the doubling of the other solitons is not available. We will discuss a possible interpretation of this result in the next section. [\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|]{} ------------------------------------------------------------------------ S$p$-brane &IIA/IIB& 9 & 8 & 7 & 6&5&4&3\ ------------------------------------------------------------------------ 0&&&&&&1&12&84\ ------------------------------------------------------------------------ 1&&&&&1&10&60&280\ ------------------------------------------------------------------------ 2&&&&1&8&40&160&560\ ------------------------------------------------------------------------ 3&&&1&6&24&80&240&\ ------------------------------------------------------------------------ 4&&1&4&12&32&80&&\ ------------------------------------------------------------------------ 5&$1^\prime/1$&$1^\prime + 1$&$2^\prime + 2$&$4^\prime + 4$&$8^\prime + 8$&&&\ Generalized Kaluza-Klein monopoles ================================== To understand the wrapping rule for standard solitons it is enough to consider the standard KK monopole only. In $D=10$ dimensions the monopole solution is characterized by 6 worldvolume, one isometry and three transverse directions. To obtain a brane one must reduce over the isometry direction which leads to a solitonic S5-brane. The KK monopole is magnetically charged with respect to the KK vector which is represented by off-diagonal components of the metric. Formally, one may therefore say that the KK monopole is electrically charged with respect to the dual graviton. Although a dual graviton mixed-symmetry tensor $D_{7,1}$ can only be defined at the linearized level, see e.g. [@Bergshoeff:2008vc], for the present purposes it is convenient to introduce such a potential as an organizing principle. Upon reduction to $D=9$ dimensions a mixed-symmetry field $D_{7,1}$ gives rise to both a 7-form and a 6-form potential. Only the 6-form potential is dual to the KK vector and corresponds to the KK monopole. The 7-form potential is dual to the KK scalar and does not correspond to a supersymmetric soliton. We therefore need to restrict the possible reductions of $D_{7,1}$ with the condition that when the index after the comma in $7,1$ is internal also one of the indices before the comma has to be internal. The reduction to $D=9$ dimensions of the NS-NS solitonic 5-brane together with the $D=10$ KK monopole, represented by the mixed-symmetry tensor $D_{7,1}$, then leads to the desired dual wrapping rule (\[solitonicwrapping\]) (here $\sharp$ denotes the internal direction) $$\begin{aligned} & & D_6 \rightarrow D_{5 \sharp}\,, \quad D_6\,, \nonumber\\ & & D_{7,1} \rightarrow D_{6 \sharp, \sharp}\,. \end{aligned}$$ This yields a singlet S4-brane and an SO(1,1) vector of S5-branes, one with a vector and one with a tensor multiplet. This works for any dimension. For instance, reducing to $D=7$ dimensions we obtain $$\begin{aligned} & & D_6 \rightarrow D_{3 ijk} (1)\quad D_{4 ij} (3)\,, \nonumber\\ & & D_{7,1} \rightarrow D_{4 ijk, i}\,, (3)\,, \end{aligned}$$ where $i=1,2,3$ is a $\rm{GL}(3)$ index. The number between brackets indicates the number of potentials. Again we obtain a singlet S2-brane and an SO(3,3) T-duality vector of S3-branes. We next extend the analysis to include the non-standard solitons. For the dual wrapping rule to work in this case as well, we need an extra inflow of branes from objects which we call generalised KK monopoles. We represent these extra objects by mixed-symmetry fields, that a priori can be of the generic form $D_{m,n,p,...}$ for $m\ge n \ge p $ non-negative integers that denote the number of separately antisymmetric indices. Surprisingly, the following set of fields suffices: $$\label{mixedsymmetry} D_{6+n,n} \quad \quad ,\quad \quad n=0,1,2,3,4 \quad ,$$ where $6+n,n$ refers to the symmetries corresponding to a Young tableau with two columns, one with $6+n$ entries and a second one with $n$ entries. This includes the fields $D_6$ (the dual NS-NS 2-form), $D_{7,1}$ (the dual graviton), $D_{8,2}$ (which is another dual to the NS-NS 2-form) together with the higher-rank fields $D_{9,3}$ and $D_{10,4}$. The rule for all fields is that they give rise to supersymmetric branes only when the $n$ indices on the right of the comma in $D_{6+n,n}$ are compactified along directions on which also $n$ of the $6+n$ indices on the left of the comma are compactified. This reduction rule implies that there are no solitonic branes with a worldvolume dimension higher than 6. The restricted reduction rule for the mixed-symmetry fields (\[mixedsymmetry\]) yields exactly the right number of additional solitons such that the dual wrappping rule (\[solitonicwrapping\]) works. For instance, the reduction to $D=6$ dimensions yields the following potentials ($i=1,2,3,4$ is a GL(4) index): $$\begin{aligned} & & D_6 \rightarrow D_{2 ijkl} (1)\quad D_{3 ijk} (4) \quad D_{4ij} (6) \quad D_{5i} (4) \quad D_6 (1) \nonumber\\ & & D_{7,1} \rightarrow D_{3 ijkl, i} (4)\quad D_{4ijk,i} (12) \quad D_{5ij,i}(12) \quad D_{6i,i} (4)\nonumber \\ & & D_{8,2} \rightarrow D_{4ijkl,ij} (6) \quad D_{5ijk,ij} (12) \quad D_{6ij,ij} (6) \nonumber \\ & & D_{9,3} \rightarrow D_{5ijkl,ijk} (4) \quad D_{6ijk,ijk}(4) \nonumber \\ && D_{10,4} \rightarrow D_{6ijkl,ijkl} (1) \quad . \end{aligned}$$ This yields precisely the sequence of 1,8,24,32,16 potentials that can be found in the $D=6$ column of Table \[sol3table\]. The other dimensions work in the same way. It is not clear what the precise status of the mixed-symmetry fields (\[mixedsymmetry\]) for $n=2,3,4$ is. One point of view is to consider these fields as a formal framework to get a handle on the properties of the non-standard solitons after reduction. According to this point of view they should not themselves be associated with objects in ten-dimensional string theory. A more exciting possibility is that the mixed-symmetry potentials $D_{6+n,n}$ for $n=2,3,4$ can be associated with non-standard KK monopoles in string theory in the same way that the mixed-symmetry tensor $D_{7,1}$ encodes information about the standard KK monopole. It is suggestive to conjecture that the mixed-symmetry potentials $D_{6+n,n}$ represent supersymmetric solutions with 6 worldvolume directions, $n$ isometry directions and $4-n$ transverse directions. For $n=0$ this is the NS-NS 5-brane and for $n=1$ this is the standard KK monopole. The $n=2,3,4$ cases can probably be represented as supersymmetric single brane solutions by uplifting the lower-dimensional solitons. However, since they have less than or equal to 2 transverse directions they will not be well-defined by themselves. For instance, in the case of 2 transverse directions it is likely that one should consider multiple brane configurations to obtain finite energy solutions. This requires a further investigation. Conclusions =========== In this letter we first mentioned that the wrapping rule for fundamental branes, to be consistent with T-duality, requires extra 0-branes originating from the reduction of the pp-wave. The D-branes are consistent by themselves and their wrapping rule does not require any additional object. The fundamental branes are mapped under duality to the standard solitons. Accordingly, the wrapping rule for these standard solitons requires the dual of the pp-wave which is the KK-monopole. We next extended the analysis to the supersymmetric non-standard solitons which have been classified using supergravity input. Remarkably, these non-standard solitons result from the same dual wrapping rule that leads to the standard solitons. This rule can, however, not be explained by another use of the KK monopole. Other objects are needed and we showed that the corresponding fields are a limited number of mixed-symmetry tensors given in (\[mixedsymmetry\]). Remarkably, these solitonic mixed-symmetry fields are all contained in the solitonic sector of the spectrum of the very-extended Kac-Moody algebra ${\rm E}_{11}$ [@West:2001as]. It will be interesting to see whether this ${\rm E}_{11}$ algebra can play a guiding role in understanding the organizing principle here. One may extend the present analysis by considering branes whose tension scales as $(g_s)^\alpha$ with $\alpha \le -3$. The first objects to consider are the ones with $\alpha=-3$. In ten-dimensional string theory there is only one such object and that is the S-dual of the D7-brane. We will call these exceptional branes E-branes. It turns out that E-branes occur in any dimension as tensor-spinor representations of the T-duality group [@inpreparation]. The supersymmetric ones can be classified by considering their WZ term. To obtain the lower-dimensional supersymmetric E-branes from ten dimensions one needs the following exotic wrapping rule [@inpreparation] $$\begin{aligned} \label{exoticwrapping} & & {\rm wrapped} \ \ \ \ \rightarrow\ \ \ {\rm doubled}\,, \nonumber \\ & & {\rm unwrapped} \ \ \rightarrow \ \ {\rm doubled}\,. \end{aligned}$$ To realize this wrapping rule one needs a further inflow of branes resulting from other objects in string theory. A similar wrapping rule explaining the occurrence of branes in lower dimensions with $\alpha \le -4$ does not seem to exist [@inpreparation]. In summary, the classification of supersymmetric branes in lower dimensions with $0 \le \alpha \le -3$, suggested by supergravity, can be reproduced by a set of simple wrapping rules. These rules suggest the existence of a set of additional objects in ten-dimensional string theory. It remains to be seen wether these objects indeed have a meaning within string theory. Acknowledgements {#acknowledgements .unnumbered} ================ E.B. wishes to thank the hospitality of King’s College London, where part of this work was done. F.R. wishes to thank the University of Groningen for hospitality. [00]{} M. B. Green, J. H. Schwarz, “Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory,” Phys. Lett. [B149 ]{} (1984) 117-122. E. Bergshoeff, E. Sezgin, P. K. Townsend, “Supermembranes and Eleven-Dimensional Supergravity,” Phys. Lett. [B189 ]{} (1987) 75-78; M. J. Duff, K. S. Stelle, “Multimembrane solutions of D = 11 supergravity,” Phys. Lett. [B253 ]{} (1991) 113-118. F. Riccioni and P. C. West, “The E(11) origin of all maximal supergravities,” JHEP [0707]{} (2007) 063 \[arXiv:0705.0752 \[hep-th\]\]. E. A. Bergshoeff, I. De Baetselier and T. A. Nutma, “E(11) and the embedding tensor,” JHEP [0709]{} (2007) 047 \[arXiv:0705.1304 \[hep-th\]\]. B. de Wit, H. Nicolai and H. Samtleben, “Gauged Supergravities, Tensor Hierarchies, and M-Theory,” JHEP [0802]{} (2008) 044 \[arXiv:0801.1294 \[hep-th\]\]. E. A. Bergshoeff, F. Riccioni, “D-Brane Wess-Zumino Terms and U-Duality,” JHEP [1011 ]{} (2010) 139. \[arXiv:1009.4657 \[hep-th\]\]. E. A. Bergshoeff, F. Riccioni, “String Solitons and T-duality,” \[arXiv:1102.0934 \[hep-th\]\], to appear in JHEP. E. Bergshoeff, U. Gran and D. Roest, “Type IIB seven-brane solutions from nine-dimensional domain walls,” Class. Quant. Grav. [19]{} (2002) 4207 \[arXiv:hep-th/0203202\]; E. A. Bergshoeff, M. de Roo, S. F. Kerstan, T. Ortin and F. Riccioni, “IIB nine-branes,” JHEP [0606]{} (2006) 006 \[arXiv:hep-th/0601128\]; E. A. Bergshoeff, M. de Roo, S. F. Kerstan, T. Ortin and F. Riccioni, “SL(2,R)-invariant IIB Brane Actions,” JHEP [0702]{} (2007) 007 \[arXiv:hep-th/0611036\]. C. M. Hull, “A geometry for non-geometric string backgrounds,” JHEP [0510]{} (2005) 065 \[arXiv:hep-th/0406102\]; [ibidem]{}, “Doubled geometry and T-folds,” JHEP [0707]{} (2007) 080 \[arXiv:hep-th/0605149\]; C. M. Hull and R. A. Reid-Edwards, “Gauge Symmetry, T-Duality and Doubled Geometry,” JHEP [0808]{} (2008) 043 \[arXiv:0711.4818 \[hep-th\]\]. E. A. Bergshoeff, M. de Roo, S. F. Kerstan, A. Kleinschmidt, F. Riccioni, “Dual Gravity and Matter,” Gen. Rel. Grav. [41 ]{} (2009) 39-48. \[arXiv:0803.1963 \[hep-th\]\]. P. C. West, “${\rm E}_{11}$ and M theory,” Class. Quant. Grav. [18]{} (2001) 4443 \[arXiv:hep-th/0104081\]. E. A. Bergshoeff and F. Riccioni, in preparation. [^1]: It is well-known that these non-standard branes are not well-defined when considered as single branes. We will not discuss these issues here and instead only consider whether or not the single brane case is consistent with basic requirents such as gauge invariance and supersymmetry. [^2]: Note that there is a single constraint on the 9-form curvatures. This constraint is needed for the correct counting of physical degrees of freedom in IIB supergravity but not for the correct counting of supersymmetric branes. [^3]: We do not consider instantons.
--- abstract: 'Let $A,B\subset M$ be inclusions of $\sigma$-finite von Neumann algebras such that $A$ and $B$ are images of faithful normal conditional expectations. In this article, we investigate Popa’s intertwining condition $A\preceq_MB$ using their modular actions. In the main theorem, we prove that if $A\preceq_MB$ holds, then an intertwining element for $A\preceq_MB$ also intertwines some modular flows of $A$ and $B$. As a result, we deduce a new characterization of $A\preceq_MB$ in terms of their continuous cores. Using this new characterization, we prove the first W$^$-superrigidity type result for group actions on amenable factors. As another application, we characterize stable strong solidity for free product factors in terms of their free product components.' author: - 'Yusuke Isono[^1]' title: 'Unitary conjugacy for type III subfactors and W$^$-superrigidity*' --- Introduction {#Introduction} ============ In [@Po01], Sorin Popa obtained the first uniqueness result for certain Cartan subalgebras in non-amenable type II$_1$ factors up to unitary conjugacy. He used this result to compute some invariants of von Neumann algebras and succeeded to give the first examples of type II$_1$ factors which have trivial fundamental groups, solving a long standing open problem in von Neumann algebra theory. This breakthrough work led to great progress in the classification of non-amenable von Neumann algebras over the last years, which is now called Popa’s deformation/rigidity theory* (see the surveys [@Po06b; @Va10; @Io17]). An important technical ingredient in his theory is the intertwining-by-bimodules technique [@Po01; @Po03]. Let $M$ be a finite von Neumann algebra and $A,B \subset M$ von Neumann subalgebras. The intertwining condition, which will be written as $A \preceq_M B$, is defined as a weaker notion of unitary conjugacy from $A$ into $B$ (see Definition \[def corner intertwining\]). Popa proved that this condition is equivalent to an analytic condition: non-existence of a net of unitaries in $A$ with a certain convergence condition. This equivalence provides a very powerful tool to obtain a unitary conjugacy between certain subalgebras, and it is now regarded as a fundamental tool to study relations between general subalgebras in a von Neumann algebra. The proof of this analytic characterization relies on the bimodule structure via GNS representations of traces. The finiteness assumption of $M$ is hence crucial in this context. However since there are many natural questions for non-tracial von Neumann algebras (more specifically, for type III factors) which should be studied in deformation/rigidity theory, there have been many attempts to generalize the intertwining machinery to type III von Neumann algebras. In a joint work with C. Houdayer [@HI15], we succeeded to prove the aforementioned analytic characterization in the case when $A$ is finite (and $B \subset M$ can be general), but the general case is still open. See also [@CH08; @HR10; @HV12; @Ue12; @Is14; @Ue16; @BH16] for other partial generalizations of this technique. In the present article, we focus on this problem. We will investigate Popa’s intertwining condition $A\preceq_MB$ for general inclusions of von Neumann algebras. Before proceeding, we prepare some terminology. For a (possibly non-unital) inclusion of von Neumann algebras $A\subset M$, we say that $A\subset M$ is with expectation if there is a faithful normal conditional expectation $E_A \colon 1_AM1_A \to A$, where $1_A$ is the unit of $A$. For any such expectation $E_A$, we say that a faithful normal positive functional $\varphi\in M_$ is preserved by $E_A$* if it satisfies $\varphi = \varphi(1_A \cdot 1_A)+ \varphi(1_A^\perp \cdot 1_A^\perp)$ and $\varphi\circ E_A=\varphi$ on $1_AM1_A$, where $1_A^\perp:=1_M-1_A$. Now we introduce the main theorem in this article. The theorem shows that the intertwining condition $A\preceq_MB$ is equivalent to the same condition but together with additional conditions on Tomita–Takesaki’s modular actions. More precisely, an intertwining element, which manages a weak unitary conjugacy for $A\preceq_MB$, also intertwines some modular flows for $A$ and $B$. As a result, the condition $A\preceq_MB$ is equivalent to a condition on their continuous cores (see item (3) below). This provides new perspective for the intertwining machinery in type III von Neumann algebra theory. In the theorem below, $\sigma^\varphi$ is the modular action and $C_\varphi(M)$ is the continuous core of $M$ (with respect to $\varphi\in M_^+$), see Section \[Preliminaries\]. Recall that a factor $N$ is a type $\rm III_1$ factor* if its continuous core is a factor. See Definition \[def corner intertwining with modular actions\] and \[def corner intertwining with expectation\] for intertwining conditions with modular actions and with conditional expectations. \[thmA\] Let $M$ be $\sigma$-finite von Neumann algebra and $A,B\subset M$ (possibly non-unital) von Neumann subalgebras with expectations. We fix any faithful normal conditional expectation $E_B\colon 1_BM1_B \to B$, any faithful state $\varphi\in M_$ which is preserved by $E_B$. Then the following two conditions are equivalent. - We have $A\preceq_MB$. - We have $(A,\sigma^\psi)\preceq_M(B,\sigma^\varphi)$ for some faithful state $\psi\in M_$ such that $\sigma_t^\psi(A)=A$ for all $t\in {\mathbb{R}}$ (or equivalently, such that $\psi$ is preserved by some conditional expectation onto $A$). Moreover, for any fixed faithful normal conditional expectation $E_{A} \colon 1_AM1_A \to A$, any faithful state $\psi\in M_$ which is preserved by $E_A$, and any $\sigma$-finite type $\rm III_1$ factor $N$ equipped with a faithful state $\omega\in N_$, the following conditions are equivalent. - We have $(A,\sigma^\psi)\preceq_M(B,\sigma^\varphi)$. - We have $(A,E_A)\preceq_M(B,E_B)$. - We have $\Pi(C_{\psi\otimes\omega}(A{\mathbin{\overline{\otimes}}}N)) \preceq_{C_{\varphi\otimes\omega}(M{\mathbin{\overline{\otimes}}}N)}C_{\varphi\otimes \omega}(B{\mathbin{\overline{\otimes}}}N)$, where $\Pi\colon C_{\psi\otimes\omega}(M{\mathbin{\overline{\otimes}}}N) \to C_{\varphi\otimes\omega}(M{\mathbin{\overline{\otimes}}}N)$ is the canonical $\ast$-isomorphism given by the Connes cocycle. The following immediate corollary gives a new characterization of $A\preceq_MB$ in terms of their continuous cores. Since all continuous cores are semifinite, up to cutting down by a finite projection, one can use the analytic characterization of the intertwining condition at the level of continuous cores. Keep the setting as in Theorem \[thmA\] and fix a type $\rm III_1$ factor $N$ and a faithful state $\omega\in N_$. Then $A\preceq_MB$ holds if and only if item (3) in Theorem \[thmA\] holds for some $E_A$ and $\psi$. We emphasize that this corollary fails* if we do not take tensor products with a type III$_1$ factor. In fact, there is an inclusion $B\subset M=A$ such that $M\not\preceq _MB$ but $C_\varphi(M) \preceq_{C_\varphi(M)}C_\varphi(B)$ (see [@HI17 Theorem 4.9]). Hence the type III$_1$ factor $N$ is necessary. Here we explain the idea behind Theorem \[thmA\]. In [@Po04; @Po05a], Popa proved his celebrated cocycle superrigidity theorem. He developed a way of using his intertwining machinery to study cocycles of actions. If two discrete group actions $\Gamma \curvearrowright^\alpha M$ and $\Gamma \curvearrowright^\beta M$ on a finite von Neumann algebra $M$ are cocycle conjugate (so that $M\rtimes_\beta \Gamma = M\rtimes_\alpha \Gamma$), then the intertwining condition ${\mathbb{C}}1_M \rtimes_{\beta}\Gamma \preceq_{M\rtimes_\alpha\Gamma} {\mathbb{C}}1_M \rtimes_{\alpha} \Gamma$ is equivalent to a weak conjugacy condition for $\alpha$ and $\beta$ (see Definition \[def unital intertwining with actions\]). In [@HSV16], by assuming the subalgebra $A$ is trivial (but $B\subset M$ can be general), Houdayer, Shlyakhtenko, and Vaes applied this idea to the case of modular actions. They combined it with Connes cocycles and deduced a new characterization of intertwining conditions, in terms of their states. This new characterization enabled them to identify specific states on von Neumann algebras, and they applied it to the classification of free Araki–Woods factors. Our Theorem \[thmA\] is strongly motivated by these works. In fact, when the subalgebra $A$ is finite, Theorem \[thmA\] can be proved (without tensoring a type III$_1$ factor) by developing ideas in these works. Hence the main interest of Theorem \[thmA\] is the case that $A$ is of type III. It is technically more challenging, since both proofs of [@Po04; @Po05a] and [@HSV16] are no longer adapted. We will use another characterization of $A\preceq_MB$ which holds without the finiteness assumption (see Theorem \[thm corner intertwining\](2)). By taking tensor products with a type III$_1$ factor $N$ and by analyzing operator valued weights on basic constructions, we will connect this condition on $M$ to the one of $C_\varphi(M {\mathbin{\overline{\otimes}}}N)$. See Lemma \[III1 factor tensor lemma\] and \[key lemma\] for the use of type III$_1$ factors. Application: W$^$-superrigidity for actions on amenable factors {#application-w-superrigidity-for-actions-on-amenable-factors .unnumbered} ---------------------------------------------------------------- Our first application of Theorem \[thmA\] is on W$^$-superrigidity of group actions on amenable factors. For a group action $\Gamma \curvearrowright^\alpha B$ on a von Neumann algebra $B$, W$^$-superrigidity of $\alpha$ means that the isomorphism class of the action $\alpha$ can be recovered* from the one of the von Neumann algebra (or the W$^$-algebra) $B\rtimes_\alpha \Gamma$. To be precise, for any action $\Lambda\curvearrowright^\beta A$, if $B\rtimes_\alpha \Gamma\simeq A\rtimes_{\beta} \Lambda$ as von Neumann algebras, then one has $\alpha \simeq \beta$ as actions. Here for the action $\beta$, we only assume natural conditions in the framework (e.g. free and ergodic action) and do not impose any technical assumptions. The first example of W$^$-superrigid actions was discovered by Popa and Vaes [@PV09]. They proved that for a large class of amalgamated free groups, any free ergodic probability measure preserving action is W$^$-superrigid. After this breakthrough work, many examples have been obtained, see [@Pe09; @Io10; @HPV10; @PV11; @PV12; @Bo12; @Io12; @Va13; @CIK13]. All these works are on actions on probability spaces, namely, actions on commutative von Neumann algebras. In the present article, we investigate actions on amenable factors. Recall that a von Neumann algebra $M$ (with separable predual) is amenable* if it is generated by an increasing union of (countably many) finite dimensional von Neumann algebras. The amenable von Neumann algebras is the easiest class of von Neumann algebras and contains all commutative von Neumann algebras. Hence it is a natural question to ask if a W$^$-superrigidity phenomena occurs for actions on non-commutative amenable von Neumann algebras. However, because of the technical difficulties coming from non-commutativity, none of W$^$-superrigidity type results for such actions is known so far (even for type II$_1$ factors). We prepare some terminology. We say that a countable discrete group $\Gamma$ is in the class $\mathcal{C}$ [@VV14] if it is non-amenable and for any trace preserving cocycle action $\Gamma\curvearrowright B$ on a finite von Neumann algebra $B$, the following condition holds: - any projection $p \in B\rtimes \Gamma=:M$ and any amenable von Neumann subalgebra $A \subset pMp$, if $A' \cap pMp\subset A$ and if $\mathcal{N}_{pMp}(A)''\subset pMp$ is essentially finite index, then we have $A\preceq_M B$. The class $\mathcal C$ contains all weakly amenable group $\Gamma$ with $\beta_1^{(2)}(\Gamma)>0$ [@PV11], all non-amenable hyperbolic groups [@PV12] and all non-amenable free product groups [@Io12; @Va13]. Recall that a faithful normal state $\varphi$ on a von Neumann algebra $M$ is weakly mixing if the fixed point algebra of the modular action of $\varphi$ is trivial. In this case $M$ must be a type III$_1$ factor, and the unique amenable type III$_1$ factor admits such a state. The following theorem is the main application of Theorem \[thmA\]. This is the first W$^$-superrigidity type result for actions on amenable factors. As we will explain below, the proof of this theorem uses the modular theory in a crucial way, and hence cannot be adapted to type II$_1$ factors. \[thmB\] Let $\Gamma$ be an ICC countable discrete group in the class $\mathcal C$, $B_0$ a type $\rm III_1$ amenable factor with separable predual, and $\varphi_0$ a faithful normal state on $B_0$ which is weakly mixing. Then the Bernoulli shift action $\Gamma \curvearrowright^\alpha \bigotimes_{\Gamma}(B_0,\varphi_0)(=:(B,\varphi))$ is [W$^$-superrigid]{} in the following sense. Let $\Lambda\curvearrowright^{\beta} (A,\psi)$ be any state preserving outer action of a discrete group $\Lambda$ on an amenable factor $A$ with a faithful normal state $\psi$. If $B\rtimes_\alpha \Gamma \simeq A\rtimes_\beta \Lambda$, then there exist - a finite normal subgroup $\Lambda_0 \leq \Lambda$, so that one has a cocycle action $\Lambda/\Lambda_0 \curvearrowright^{\beta^{\Lambda/\Lambda_0}}(A\rtimes_{\beta}\Lambda_0,\psi')$ by a fixed section $s \colon \Lambda/\Lambda_0 \to \Lambda$, where $\psi'$ is the canonical extension of $\psi$ on $A\rtimes_\beta \Lambda_0$; - a state preserving cocycle action $({\operatorname{Ad}}(u_g))_{g\in \Gamma}$ of $\Gamma$ on a type $\rm I$ factor $({\mathbb{B}},\omega)$ equipped with a faithful normal state; such that two actions $\Lambda/\Lambda_0 \curvearrowright^{\beta^{\Lambda/\Lambda_0}}(A\rtimes_{\beta}\Lambda_0,\psi')$ and $\Gamma \curvearrowright^{\alpha\otimes {\operatorname{Ad}}(u)}(B{\mathbin{\overline{\otimes}}}{\mathbb{B}},\varphi\otimes \omega)$ are conjugate via a state preserving isomorphism. The Bernoulli action in this theorem was intensively studied in [@VV14; @Ve15]. They obtained similar conclusions if the action $\Lambda\curvearrowright^{\beta} (A,\psi)$ is also a Bernoulli action of a group in the class $\mathcal C$. Now thanks to our Theorem \[thmB\], we can put arbitrary actions as $\Lambda\curvearrowright^{\beta} (A,\psi)$. The conclusion of Theorem \[thmB\] is optimal. Indeed, subgroups and type I factors in the theorem can appear always, since the amenable type III$_1$ factor $B$ has decompositions such as $B=A\rtimes\Lambda_0$ and $B = B {\mathbin{\overline{\otimes}}}{\mathbb{B}}$. Note also that the cocycle action $\Lambda/\Lambda_0 \curvearrowright^{\beta^{\Lambda/\Lambda_0}}(A\rtimes_{\beta}\Lambda_0,\psi')$ above depends on the choice of the section $s$, but this dependence affects the cocycle action ${\operatorname{Ad}}(u)$ on a type I factor only. The proof of Theorem \[thmB\] splits into two steps. Firstly, we prove a unique crossed product decomposition theorem: we identify the base algebra $B$ from the von Neumann algebra $B\rtimes_\alpha \Gamma$, so that two actions are cocycle conjugate. Secondly, we prove a cocycle superrigidity type theorem: the corresponding cocycle is cohomologous to a coboundary, so that two actions are conjugate. The next theorem treats the first step. Such a unique crossed product decomposition theorem has been intensively studied during the last decade for actions on finite von Neumann algebras, see [@OP07; @CS11; @PV12; @HV12] (and see aforementioned works for W$^$-superrigidity). Thanks to our Theorem \[thmA\], we can put type III factors as base algebras $B$. \[thmC\] Let $\Gamma$ be an ICC countable discrete group in the class $\mathcal C$, $B$ a $\sigma$-finite, amenable, diffuse factor, and $\Gamma \curvearrowright^{\alpha} B$ an outer action. Assume that $B\rtimes_\alpha \Gamma \simeq A\rtimes_\beta \Lambda$ for some outer action $\Lambda\curvearrowright^{\beta} A$ of a countable discrete group $\Lambda$ on a $\sigma$-finite, amenable, diffuse factor $A$. Then there is an amenable normal subgroup $\Lambda_0 \leq \Lambda$ such that the induced cocycle action $\Lambda/\Lambda_0\curvearrowright^{\beta^{\Lambda/\Lambda_0}} A\rtimes_\beta\Lambda_0$ is cocycle conjugate to $\alpha$. In particular if $\Lambda$ has no amenable normal subgroups, then $\alpha$ and $\beta$ are cocycle conjugate. The following immediate corollary generalizes [@PV11 Theorem 1.10]. \[corD\] Let $\Gamma \curvearrowright^{\alpha} B$ and $\Lambda\curvearrowright^{\beta} A$ be outer actions of countable discrete ICC groups on $\sigma$-finite, amenable, diffuse factors such that $B\rtimes_\alpha \Gamma \simeq A\rtimes_\beta \Lambda$. If $\Gamma$ and $\Lambda$ are in the class $\mathcal{C}$, then $\alpha$ and $\beta$ are cocycle conjugate. We next need a cocycle superrigidity type theorem for the second step. Appropriate adaptations of techniques in [@Po05a; @Po05b] (see also [@VV14; @Ma16]) to our setting easily provides the following proposition. This proposition is however not* useful in our study, as we explain soon below. \[thmE\] Let $\Gamma$ be a non-amenable countable discrete group, $(B_0,\varphi_0)$ an amenable factor with separable predual and with a faithful normal state, and $\Gamma \curvearrowright^\alpha \bigotimes_{\Gamma}(B_0,\varphi_0)=:(B,\varphi)$ the Bernoulli shift action. Assume either that $\Gamma$ is a direct product of two infinite groups or has a normal subgroup with relative property (T). Assume that $\alpha$ is cocycle conjugate to some state preserving outer action $\Lambda\curvearrowright^{\beta} (A,\psi)$ of a countable discrete group $\Lambda$ on an amenable factor $A$ with a faithful normal state $\psi$. Then there exists an inner action $({\operatorname{Ad}}(u_g))_{g\in \Gamma}$ of $\Gamma$ on a type $\rm I$ factor ${\mathbb{B}}$ such that two actions $\beta$ and $\alpha\otimes {\operatorname{Ad}}(u)$ are conjugate. Idea of the proof of Theorem \[thmB\] {#idea-of-the-proof-of-theorem-thmb .unnumbered} ------------------------------------- We briefly explain the idea of the proof of Theorem \[thmB\]. The proof uses the modular theory in a crucial way. Consider two actions $\alpha$ and $\beta$ as in Theorem \[thmB\]. Since the group $\Gamma$ is in the class $\mathcal C$, we can first apply Theorem \[thmC\]. Then an induced cocycle action $\beta^{\Lambda/\Lambda_0}$ is cocycle conjugate to $\alpha$. If this cocycle action is a genuine action, by assuming that $\Gamma$ is a direct product or has property (T), one can apply Proposition \[thmE\] and obtain a conjugacy result. However it is not clear when the cocycle action, which comes from a section $s\colon \Gamma \simeq \Lambda/\Lambda_0 \to \Lambda$, is a genuine action. In other words, we do not know when the exact sequence $1 \to \Lambda_0 \to \Lambda \to \Gamma \to 1$ splits, where $\Lambda_0$ is amenable and $\Gamma$ is in the class $\mathcal C$ satisfying the assumption of Proposition \[thmE\]. This is the main technical issue to prove the W$^$-superrigidity theorem in our setting, and this is why such a result is not known even for type II$_1$ factors. In the present article, to avoid this problem, we use modular actions. Since we assumed that $\alpha$ and $\beta$ are state preserving, there is an isomorphism $$B\rtimes_{\alpha\times \sigma^\varphi} (\Gamma \times {\mathbb{R}}) \simeq A\rtimes_{\beta\times \sigma^\psi} (\Lambda \times {\mathbb{R}})$$ such that the corresponding (possibly cocycle) actions are cocycle conjugate. By assuming that $\varphi_0$ is weakly mixing (which means $\sigma^\varphi$ is weakly mixing), and combining with some rigidity property of Bernoulli actions, one can apply the proof of Proposition \[thmE\] to the direct product group $\Gamma \times {\mathbb{R}}$. Here we note that ${\mathbb{R}}$-actions are always genuine actions, so no technical problems appear in this context. Thus the cocycle is cohomologous to a coboundary as ${\mathbb{R}}$-actions. Since ${\mathbb{R}}\leq \Gamma \times {\mathbb{R}}$ is normal and since $\sigma^\varphi$ is weakly mixing, the same conclusion actually holds as $\Gamma \times {\mathbb{R}}$-actions and we can finish the proof. This is the main idea of the proof of Theorem \[thmB\]. Application: stable strong solidity of free product factors {#application-stable-strong-solidity-of-free-product-factors .unnumbered} ----------------------------------------------------------- The next application is on the structure of amalgamated free product von Neumann algebras. We will generalize Ioana’s work [@Io12] to the type III setting. Recall that for any (possibly non-unital) inclusions $A,B\subset M$ with expectations and with $1_B=1_M$, we say that $A$ is injective relative to $B$ in $M$* [@OP07; @Is17] if there is a conditional expectation $E \colon 1_A \langle M,B\rangle 1_A \to A$ which is faithful and normal on $1_AM1_A$. Recall that for any von Neumann algebra $M$ with the decomposition $M=M_a \oplus M_d$, where $M_a$ is atomic and $M_d$ is diffuse, we say that $M$ is strongly solid (resp. stably strongly solid) [@OP07; @BHV15] if for any diffuse amenable von Neumann algebra $A \subset M_d$ with expectation, $\mathcal{N}_{M_d}(A)''$ (resp. $s\mathcal{N}_{M_d}(A)''$) remains amenable. Here $s\mathcal{N}_{M_d}(A)$ is the set of all elements $x\in M_d$ such that $x A x^* \subset A$ and $x^Ax \subset A$, and such elements are called stable normalizers. Then $\mathcal{N}_{M_d}(A)$ is given by $s\mathcal{N}_{M_d}(A) \cap \mathcal{U}(M_d)$ and its elements are called normalizers. Note that these two notions of strong solidity coincide if $M$ is properly infinite. By definition, a strongly solid non-amenable factor $M$ does not admit any crossed product decomposition $M=A\rtimes \Gamma$ (for amenable $A$), so strong solidity should be understood as a strong indecomposability* of $M$. The following theorem is a generalization of Ioana’s theorem [@Io12 Theorem 1.6] (see also [@Va13; @HU15; @BHV15]). As a corollary, we characterize stable strong solidity of free product factors, see [@Io12 Theorem 1.8] for the same characterization for type II$_1$ factors. \[thmF\] Let $B \subset M_i$ be inclusions of $\sigma$-finite von Neumann algebras with expectations $E_i$ for $i=1,2$. Let $M:=(M_1,E_1)_B(M_2 ,E_2)$ be the amalgamated free product von Neumann algebra, $p\in M$ a projection, and $A\subset pMp$ a von Neumann subalgebra with expectation. Assume that $A$ is injective relative to $B$ in $M$ and assume that $A'\cap pMp \subset A$. Then at least one of the following conditions holds true: 1. $A\preceq_M B$; 2. $s\mathcal{N}_{pMp}(A)''\preceq_M M_i$ for some $i\in \{1,2\}$; 3. $s\mathcal{N}_{pMp}(A)''$ is injective relative to $B$. \[corG\] Let $I$ be a set and $(M_i,\varphi_i)_{i\in I}$ a family of nontrivial von Neumann algebras with faithful normal states. Put $M:=_{i\in I}(M_i,\varphi_i)$. Then $M$ is stably strongly solid if and only if so are all $M_i$’s. Examples of stably strongly solid factors have been obtained in several articles [@BHV15; @BDV17; @Ma18; @HT18]. Also all amenable von Neumann algebras are stably strongly solid. Using these algebras, Corollary \[corG\] provides plenty of new examples of stably strongly solid factors. [Acknowledgement.]{} The author would like to thank Cyril Houdayer, Amine Marrakchi, and Stefaan Vaes for many useful comments on the first draft of this manuscript. He also would like to thank Yuki Arano and Toshihiko Masuda for fruitful conversations on group actions on factors. He was supported by JSPS, Research Fellow of the Japan Society for the Promotion of Science. Preliminaries {#Preliminaries} ============= Tomita–Takesaki theory {#tomitatakesaki-theory .unnumbered} ---------------------- Let $M$ be a von Neumann algebra and $\varphi$ a faithful normal semifinite weight on $M$. Throughout the paper, for objects in Tomita–Takesaki’s modular theory, we will use the following notation. The modular operator, conjugation, and action are denoted by $\Delta_\varphi$, $J_\varphi$, and $\sigma^\varphi$ respectively. The continuous core, which is the crossed product von Neumann algebra $M\rtimes_{\sigma^\varphi}{\mathbb{R}}$, is denoted by $C_\varphi(M)$, and ${\mathord{\text{\rm Tr}}}_\varphi$ and $L_\varphi {\mathbb{R}}$ mean the canonical trace on $C_\varphi(M)$ and the canonical copy of $L{\mathbb{R}}$ in $C_\varphi(M)$ respectively. The centralizer algebra $M_\varphi$ is a fixed point algebra of the modular action. The norm $\\|\, \cdot \, \\|_\infty$ is the operator norm of $M$, while $\\|\, \cdot \, \\|_{2,\varphi}$ (or $\\| \, \cdot \, \\|_{\varphi}$) is the $L^2$-norm by $\varphi$. See [@Ta03] for definitions of all these objects. For any continuous action $G \curvearrowright^\alpha M$ of a locally compact group $G$, in this article, we will use the following canonical embeddings for crossed products: $\pi_\alpha\colon M \to M\rtimes_\alpha G$ by $(\pi_\alpha(x)\xi)(g)=\alpha_{g^{-1}}(x)\xi(g)$ for all $\xi \in L^2(G,L^2(M))$ and $g\in G$; and $G \to M\rtimes_\alpha G$ by $g \mapsto 1_M\otimes \lambda_g$ for all $g\in G$. Via these embeddings, we often regard $M$ and $LG$ as subalgebras of $M\rtimes_\alpha G$. Connes cocycle {#connes-cocycle .unnumbered} -------------- Let $G$ be a locally compact group, $M$ a von Neumann algebra and $G\curvearrowright^\alpha M$ a continuous action (see [@Ta03 Definition X.1.1] for continuity). Let $p \in M$ be a nonzero projection. We say that a $\sigma$-strongly continuous map $u\colon G \to pM$ is a generalized cocycle for $\alpha$ (with support projection $p$) if - $u_{gh}=u_g \alpha_g(u_h)$ for all $g,h\in G$; - $u_gu_g^* = p$, $u_g^* u_g = \alpha_g(p)$ for all $g\in G$. In this case, by putting $\alpha^u_g(pxp):=u_g\alpha_g(pxp)u_g^$ for all $x\in M$ and $g\in G$, one has a continuous $G$-action on $pMp$. It holds that $p(M \rtimes_\alpha G)p \simeq pMp \rtimes_{\alpha^u}G $. When $p=1$, we simply say that $u$ is a cocycle. Let $N$ be another von Neumann algebra and consider continuous actions $G\curvearrowright^\alpha M$ and $G\curvearrowright^\beta N$. We say that they are $\alpha$ is cocycle conjugate to $\beta$ via a generalized cocycle* if there exist a projection $p\in M$, a $\ast$-isomorphism $\pi\colon pMp \to N$ and a generalized cocycle $u\colon G \to pM$ for $\alpha$ with support projection $p$ such that $$\pi^{-1}\circ\beta_g\circ \pi (a) = u_g\alpha_g(a) u_g^, \quad \text{for all }a\in pMp, \ g\in G.$$ In this case, by identifying $pMp=N$ by $\pi$, we can define a partial isometry $U \colon L^2(G,L^2(M)) \to L^2(G,L^2(M)) $ by $(U\xi)(g) =u_{g^{-1}} \xi(g) = p u_{g^{-1}} \alpha_{g^{-1}}(p) \, \xi(g)$ for $g\in G$. Note that $U^ U = \pi_\alpha(p)$ and $UU^* = p\otimes 1_{L^2(G)}$. One has a $\ast$-isomorphism $$\Pi_{\beta,\alpha}:={\operatorname{Ad}}(U) \colon p(M\rtimes_\alpha G)p \to pMp \rtimes_{\beta} G$$ satisfying $\Pi_{\beta,\alpha}(x)=x$ for $x\in pMp$ and $\Pi_{\beta,\alpha}(p\lambda^\alpha_gp) = pu_g\lambda_g^\beta p=u_g\lambda_g^\beta$ for $g\in G$. If one can choose $p=1$, so that $u$ is a cocycle, then we simply say that $\alpha$ and $\beta$ are cocycle conjugate. Let $M$ be a von Neumann algebra and $\varphi,\psi$ normal semifinite weights on $M$. Assume that $\varphi$ is faithful and let $s(\psi)$ be the support projection of $\psi$. Consider modular actions $\sigma^\varphi$ on $M$ and $\sigma^\psi$ on $s(\psi)Ms(\psi)$. The Connes cocycle $([D\psi,D\varphi]_t)_{t\in {\mathbb{R}}}$ [@Co72] is a generalized cocycle for $\sigma^\varphi$ with support projection $s(\psi)$ such that $\sigma^\varphi$ is cocycle conjugate to $\sigma^\psi$ via $([D\psi,D\varphi]_t)_{t\in {\mathbb{R}}}$. In particular, there is a canonical $\ast$-isomorphism $$\Pi_{\psi,\varphi} \colon pC_\varphi(M)p= p(M\rtimes_{\sigma^\varphi} G)p \to pMp \rtimes_{\sigma^\psi} G= C_\psi(pMp) .$$ See [@Ta03 V.III.3.19-20] for this non-faithful version of the Connes cocycle. In this article, we need the following important theorem. \[connes cocycle existence\] Let $M$ be a von Neumann algebra and $\varphi$ a faithful normal semifinite weight on $M$. Let $p\in M$ be a projection and $(u_t)_{t\in {\mathbb{R}}}$ is a generalized cocycle for $(\sigma_t^\varphi)_t$ with support projection $p$. Then there is a unique normal semifinite weight $\psi$ on $M$ such that $s(\psi)=p$ and $u_t = [D\psi:D\varphi]_t$ for all $t\in {\mathbb{R}}$. Below, we record an elementary lemma. We use the notation $x\varphi y = \varphi(y \, \cdot \, x)$. \[connes cocycle lemma\] Let $M$ be a von Neumann algebra and $\varphi,\psi\in M_$ faithful positive functionals. 1. For any projection $e\in M_\psi$, we have $$[De\psi e, D\psi]_t=e \quad \text{and} \quad e[D\psi , D\varphi]_t = [De\psi e, D\varphi]_t .$$ In particular we have a chain rule: $$[De\psi e, D\psi]_t \, [D\psi , D\varphi]_t = [De\psi e, D\varphi]_t .$$ 2. Let $v \in M$ be a partial isometry such that $e:=v v^\in M_\psi$ and $f:=v^v \in M_\varphi$. Assume that $v\varphi v^ = e\psi e$ on $M$ (equivalently $f\varphi f = v^\psi v$). Then we have $$v\sigma_t^\varphi(v^xv)v^=\sigma_t^\psi(exe), \quad v^[D\psi , D\varphi]_t = \sigma_t^\varphi(v^), \quad x\in M,\ t\in {\mathbb{R}}.$$ Cocycle actions {#cocycle-actions .unnumbered} --------------- A more general notion of a group action is a cocycle action. We say that a locally compact group $G$ acts on a von Neumann algebra $M$ as a cocycle action* if there exist continuous maps $\alpha\colon G \to {\operatorname{Aut}}(M)$ and $v\colon G \times G \to \mathcal{U}(M)$ such that $$\begin{aligned} & \alpha_e ={\text{\rm id}}, \quad \alpha_g\circ \alpha_h = {\operatorname{Ad}}(v(g,h))\circ \alpha_{gh}, \quad v(g,h)v(gh,k) = \alpha_{g}(v(h,k))v(g,hk)\end{aligned}$$ for all $g,h,k \in G$, where $e$ is the neutral element. The map $v$ is called a 2-cocycle. Two cocycle actions $G\curvearrowright^{(\alpha,v)}M$ and $G\curvearrowright^{(\beta,w)}N$ are said to be cocycle conjugate if there exist a $\ast$-isomorphism $\pi \colon M \to N$ and a continuous map $u \colon G \to \mathcal{U}(M)$ such that, for all $g,h\in G$, $$\begin{aligned} \pi^{-1}\circ \beta_g \circ \pi={\operatorname{Ad}}(u_g)\circ \alpha_g , \quad \pi^{-1}(w(g,h)) = u_g \alpha_g(u_h)v(g,h)u^_{gh}.\end{aligned}$$ In this article, cocycle actions appear in the following two contexts. Let $\Gamma\curvearrowright^\alpha B$ be an action of a discrete group on a von Neumann algebra $B$. Let $p\in B$ be a projection and assume that $\alpha_g(p)\sim p$ in $B$ for all $g\in G$. Take any partial isometries $w_g\in B$ such that $w_gw_g^=p$ and $w_g^w_g=\alpha_g(p)$ for all $g\in \Gamma$. Define $\alpha_g^p(x):=w_g\alpha_g(x)w_g^$ and $v^p(g,h):=w_g\alpha_g(w_h)w_{gh}^$ for all $x\in pBp$, $g,h\in \Gamma$. Then $(\alpha^p,v^p)$ is a cocycle action on $pBp$ satisfying $p(B\rtimes_\alpha \Gamma)p \simeq pBp \rtimes_{(\alpha^p,v^p)}\Gamma$. Let $\Gamma \curvearrowright^\alpha B$ be the same group action. Let $\Lambda \leq \Gamma$ be a normal subgroup and fix a section $s\colon \Gamma /\Lambda \to \Gamma$ such that $s(\Lambda)$ is the unit of $\Gamma$. Inside $B\rtimes_\alpha \Gamma$, for all $g,h \in \Gamma/\Lambda$, we define $${\alpha}^{\Gamma/\Lambda}_{g} := {\operatorname{Ad}}(\lambda^{\Gamma}_{s(g)}) \in \mathrm{Aut}(B\rtimes_\alpha \Lambda),\quad \text{and} \quad v(g,h):=\lambda^{\Gamma}_{ s(g)s(h)s(gh)^{-1} }\in L\Lambda.$$ It is easy to verify that $\alpha^{\Gamma/\Lambda}$ and $v$ define a cocycle action of $\Gamma/\Lambda$ on $B\rtimes_\alpha \Lambda$ satisfying $B\rtimes_\alpha \Gamma \simeq (B\rtimes_\alpha \Lambda)\rtimes_{(\alpha^{\Gamma/\Lambda},v)}\Gamma/\Lambda$. Basic constructions and operator valued weights {#basic-constructions-and-operator-valued-weights .unnumbered} ----------------------------------------------- For operator valued weights, we refer the reader to [@Ha77a; @Ha77b]. We will say that a unital inclusion $B\subset M$ of von Neumann algebras is with operator valued weight* if there is an operator valued weight $E_B \colon M \to B$. Let $B \subset M$ be a unital inclusion of $\sigma$-finite von Neumann algebras with expectation $E_B$. Fix a faithful normal state $\varphi$ on $M$ such that $\varphi=\varphi\circ E_B$. Put $L^2(M):=L^2(M,\varphi)$ and $J:=J_\varphi$, and consider $B\subset M \subset {\mathbb{B}}(L^2(M))$. The von Neumann algebra $ \langle M,B\rangle := (JBJ)' $ is called the basic construction, and is generated by $Me_B M$, where $e_B$ is the Jones projection for $E_B$. Using the inclusion $JBJ \subset JMJ$ with expectation $JE_BJ := {\operatorname{Ad}}(J) \circ E_B \circ {\operatorname{Ad}}(J)$, one can define a canonical operator valued weight $(JE_BJ)^{-1}\colon (JBJ)' \to (JMJ)=M$. We will write as $\widehat{E}_B:=(JE_BJ)^{-1}$. It satisfies that $\widehat{E}_B(b^* e_Ba) = b^a$ for all $a,b \in M$. See [@Ko85; @ILP96] for the general theory of $\widehat{E}_B$. Below we collect well known facts for basic constructions and operator valued weights, which we will need in this article. - For any faithful $\psi\in M_^+$, one can define a faithful normal semifinite weight $\widehat{\psi}:=\psi \circ \widehat{E}_B$ on $\langle M,B\rangle$. It holds that $$\sigma_t^{\widehat{\psi}}\|_M = \sigma_t^\psi \quad \text{and} \quad [D\widehat{\psi}:D\widehat{\varphi}]_t = [D\psi:D\varphi]_t \quad \text{ for all }t\in {\mathbb{R}}.$$ - Let $E_{C_\varphi(B)}\colon C_\varphi(M) \to C_\varphi(B)$ be the canonical conditional expectation such that $E_{C_\varphi(B)}\|_M=E_B$ and $E_{C_\varphi(B)}\|_{L_\varphi {\mathbb{R}}} = {\text{\rm id}}$. Using $\sigma^{{\varphi}}_t\circ \widehat{E}_B = \widehat{E}_B \circ \sigma^{\widehat{\varphi}}_{t}$ for all $t\in {\mathbb{R}}$, one can define an operator valued weight from $\langle M,B\rangle \rtimes_{\sigma^{\widehat{\varphi}}} {\mathbb{R}}$ to $M \rtimes_{\sigma^\varphi} {\mathbb{R}}$ whose restriction on $\langle M,B\rangle^+$ coincides with $\widehat{E}_B$. We will denote it by $\widehat{E}_B \rtimes {\mathbb{R}}$. - We canonically have $$\langle C_\varphi(M),C_\varphi(B) \rangle = C_{\widehat{\varphi}}(\langle M,B\rangle).$$ The left hand side has a canonical operator valued weight $\widehat{E}_{C_\varphi(B)}$ onto $C_\varphi(M)$, and the right hand side has $\widehat{E}_B \rtimes {\mathbb{R}}$. Since constructions are canonical, these two operator valued weights coincide. Here we prove a lemma for type III$_1$ factors. \[III1 factor tensor lemma\] Let $A\subset M$ be a unital inclusion of von Neumann algebras with an operator valued weight $E_A$. Fix a faithful $\psi_A\in A_^+$, and put ${\psi}:=\psi_A \circ E_A$. Let $N$ be a type $\rm III_1$ factor with a faithful normal semifinite weight $\omega$. Then the following equation holds true: $$C_{\psi \otimes \omega}(A{\mathbin{\overline{\otimes}}}N)' \cap C_{\psi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N) = \left( A'\cap M_\psi \right) \otimes {\mathbb{C}}1_N \otimes {\mathbb{C}}1_{L^2({\mathbb{R}})}.$$ Since $N$ is a type III$_1$ factor, there is a faithful normal semifinite weight $\omega'$ such that $(N_{\omega'})' \cap N={\mathbb{C}}$ (see [@Ta03 Theorem XII.1.7]). Thanks to the Connes cocycle, there is a canonical isomorphism from $C_{\psi\otimes \omega'}(M{\mathbin{\overline{\otimes}}}N)$ to $C_{\psi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N)$ which sends $C_{\psi\otimes \omega'}(A{\mathbin{\overline{\otimes}}}N)$ onto $C_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}N)$ and which is the identity on $M{\mathbin{\overline{\otimes}}}N$. Hence to prove this lemma, by exchanging $\omega'$ with $\omega$, we may assume that $N_{\omega}' \cap N={\mathbb{C}}$. For simplicity we write as $ L_{\psi\otimes\omega}{\mathbb{R}}=L{\mathbb{R}}$. Observe that (e.g. [@HR10 Proposition 2.4]) $$\begin{aligned} C_{\psi\otimes \omega}({\mathbb{C}}1_{{A}}\otimes {\mathbb{C}}1_N)' \cap C_{\psi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N) \subset & \ (M{\mathbin{\overline{\otimes}}}N)_{{\psi\otimes \omega}} {\mathbin{\overline{\otimes}}}L{\mathbb{R}}.\end{aligned}$$ Since $({\mathbb{C}}1_A \otimes N_\omega )' \cap (M{\mathbin{\overline{\otimes}}}N)_{{\psi\otimes \omega}}= M_{{\psi}}\otimes {\mathbb{C}}1_N$, we have $$\begin{aligned} C_{\psi\otimes\omega}({\mathbb{C}}1_A\otimes N_\omega)' \cap C_{\psi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N) \subset & \ M_{{\psi}}{\mathbin{\overline{\otimes}}}{\mathbb{C}}1_N {\mathbin{\overline{\otimes}}}L{\mathbb{R}}.\end{aligned}$$ Since $C_\omega(N)$ is a factor, it holds that $\pi_\omega(N)' \cap ({\mathbb{C}}1_N\otimes L_{\omega} {\mathbb{R}}) ={\mathbb{C}}1_N \otimes {\mathbb{C}}1_{L^2({\mathbb{R}})}$, where $\pi_\omega(N)$ is the canonical image of $N$ in $C_\omega(N)$. This implies that $$\begin{aligned} C_{\psi\otimes\omega}({\mathbb{C}}1_A\otimes N)' \cap C_{\psi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N) \subset & \ M_{\psi}{\mathbin{\overline{\otimes}}}\left[ \pi_\omega(N)' \cap ({\mathbb{C}}1_N\otimes L {\mathbb{R}})\right]\\ = & \ M_{\psi}{\mathbin{\overline{\otimes}}}{\mathbb{C}}1_N {\mathbin{\overline{\otimes}}}{\mathbb{C}}1_{L^2({\mathbb{R}})}. \end{aligned}$$ Using the canonical embedding $\pi_{\psi \otimes \omega}$, the last term coincides with $\pi_{\psi \otimes \omega}(M_\psi \otimes {\mathbb{C}}1_N)$, hence $$\begin{aligned} C_{\psi \otimes \omega}(A{\mathbin{\overline{\otimes}}}N)' \cap C_{\psi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N) &=\pi_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}{\mathbb{C}}1_N)'\cap \pi_{\psi\otimes \omega}(M_\psi \otimes {\mathbb{C}}1_N)\\ &=\pi_{{\psi\otimes \omega}}(\left(A' \cap M_{{\psi}}\right) \otimes {\mathbb{C}}1_N) \\ &= \left( A'\cap M_{{\psi}} \right) \otimes {\mathbb{C}}1_N \otimes {\mathbb{C}}1_{L^2({\mathbb{R}})}. \end{aligned}$$ This is the conclusion. Popa’s intertwining theory {#popas-intertwining-theory .unnumbered} -------------------------- As explained in Section \[Introduction\], we refer the reader to [@Po01; @Po03] for the origin of intertwining theory. Here we give a definition introduced in [@HI15]. \[def corner intertwining\] Let $M$ be a $\sigma$-finite von Neumann algebra and $A,B\subset M$ (possibly non-unital) von Neumann subalgebras with expectation. We will say that a corner of $A$ embeds with expectation into $B$ inside $M$* and write $A \preceq_M B$ if there exist projections $e\in A$, $f\in B$, a partial isometry $v\in eMf$ and a unital normal $\ast$-homomorphism $\theta\colon eAe \to fBf$ such that - $\theta(eAe) \subset fBf$ is with expectation; - $v\theta(a)= av$ for all $a\in eAe$. In this case, we will say that $(e,f,\theta,v)$ witnesses $A\preceq_MB$. We recall known characterizations of the intertwining condition $A\preceq_MB$. For this, we borrow notation from [@HI15]. We refer the reader to [@HI15 Section 4] for items here. The same notation will be used in Section \[Popa’s intertwining techniques with conditional expectations\]. Let $M$ be a $\sigma$-finite von Neumann algebra and $A,B\subset M$ (possibly non-unital) von Neumann subalgebras with expectations. Fix a faithful normal conditional expectation $E_B$ for $B\subset 1_BM1_B$. Put $\widetilde{B}:=B\oplus {\mathbb{C}}(1_M-1_B)$ and let $E_{\widetilde{B}}\colon M\to \widetilde{B}$ be a faithful normal conditional expectation which extends $E_B$. Let $B = B_1 \oplus B_2 $ be the unique decomposition such that $B_1$ is finite and $B_2$ is properly infinite. Fix a faithful normal trace $\tau_{B_1}$ on $B_1$ and choose a faithful normal state $\varphi\in M_$ such that $\varphi$ is preserved by $E_B$ and $E_{\widetilde{B}}$ and that $\varphi\|_{B_1} = \tau_{B_1}$ (up to scalar multiples). Fix a standard representation $L^2(M):=L^2(M,\varphi)$ and its modular conjugation $J:=J_\varphi$. We write as $e_{\widetilde{B}}$ and $e_B$ corresponding Jones projections (note that $e_{\widetilde{B}}1_B=e_{\widetilde{B}}J1_BJ =e_B$), and as $\widehat{E}_{\widetilde{B}}$ the canonical operator valued weight from $\langle M,\widetilde{B}\rangle$ to $M$ given by $\widehat{E}_{\widetilde{B}}(xe_{\widetilde{B}}x^)=xx^$ for all $x\in M$. Denote by ${\mathord{\text{\rm Tr}}}$ the unique trace on $\langle M,\widetilde{B}\rangle J1_{B_1}J$ satisfying ${\mathord{\text{\rm Tr}}}( (x^e_{\widetilde B}x) J1_{B_1}J) = \tau_{B_1}( E_B(1_{B_1}xx^1_{B_1}))$ for all $x \in M$. Since $\mathcal{Z}(\langle M,\widetilde{B}\rangle J1_{B_1}J ) = J\mathcal{Z}(B_1)J$, there is a unique operator valued weight $\mathrm{ctr}\colon \langle M,\widetilde{B}\rangle J1_{B_1}J \to J\mathcal{Z}(B_1)J$ such that ${\mathord{\text{\rm Tr}}}= \overline{\tau_{B_1}(J\, \cdot \, J)} \circ \mathrm{ctr}$. Since ${\mathord{\text{\rm Tr}}}$ is a trace, $\mathrm{ctr}$ is an extended center valued trace. Let $\mathrm{ctr}_{B_1}$ be the center valued trace for $B_1$ and recall that $\tau_{B_1} \circ \mathrm{ctr}_{B_1} = \tau_{B_1}$. It holds that $$\mathrm{ctr}((x^e_{\widetilde B}x) J1_{B_1}J) = J\mathrm{ctr}_{B_1}\circ E_B(1_{B_1}xx^1_{B_1})J, \quad \text{for all }x\in M.$$ We mention that the decomposition $B=B_1\oplus B_2$ here is slightly different from the one in [@HI15], and that $\mathrm{ctr}$ was not used in [@HI15]. However the proof of [@HI15 Theorem 4.3] works without any change if we use $\mathrm{ctr}$ and our decomposition for $B$. Our items introduced here are more appropriate in the context of intertwining conditions with actions, which will be discussed in the next section. Now we introduce Popa’s intertwining theorem. We refer the reader to [@HI15 Theorem 4.3] and [@BH16 Theorem 2] for the proof of this version. \[thm corner intertwining\] The following conditions are equivalent. - We have $A \preceq_M B$. - There exists a nonzero positive element $d\in A' \cap 1_A\langle M,\widetilde{B}\rangle 1_A$ such that $$d=d J1_BJ \quad \text{and} \quad \widehat{E}_{\widetilde{B}}(d)\in M.$$ If $A$ is finite, then the following condition is also equivalent. - There is no net $(u_i)_i$ in $\mathcal{U}(A)$ such that $E_B(b^ u_ia) \to 0$ $\sigma$-strongly for all $a,b\in M1_B$. Using the next lemma, we can exchange the map $\theta$ for the condition $A\preceq_MB$ with a unital $\ast$-homomorphism on $A$. \[relation for unital and corner\] The following assertions hold true. - The condition $A\preceq_MB$ is equivalent to the following condition: there exist a separable Hilbert space $H$, a projection $f\in B{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)$, a partial isometry $w\in (1_A\otimes e_{1,1})(M{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H))f$, where $e_{1,1}$ is a minimal projection, and a unital normal $\ast$-homomorphism $\pi\colon A \to f(B{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H))f$ such that - $\pi(A) \subset f(B {\mathbin{\overline{\otimes}}}{\mathbb{B}}(H))f$ is with expectation; - $w\pi(a)= (a\otimes e_{1,1}) w$ for all $a\in A$. In this case, (to distinguish $A\preceq_MB$,) we will say that [$(H,f,\pi,w)$ witnesses $A\preceq^{\rm uni}_MB$]{}. - Assume either one of the following conditions holds: - $A$ does not have any direct summand which is semifinite and properly infinite; or - $B$ is properly infinite. If $A\preceq_MB$ holds, then the Hilbert space $H$ in item $(1)$ can be taken as finite dimensional. Since we will prove a very similar but a more complicated statement in Lemma \[relation for unital and corner of actions\], we omit the proof. Indeed, to prove this lemma, one can follow the proof of Lemma \[relation for unital and corner of actions\] by regarding actions are trivial (and by using [@HI15 Theorem 4.3 and Lemma 4.10]). Intertwining theory with modular actions {#Popa's intertwining techniques with conditional expectations} ======================================== In this section, we introduce several variants of Popa’s intertwining condition. We investigate these conditions as well as relations between them. At the end of this section, we prove Theorem \[thmA\]. Throughout this section, we always fix (possibly non-unital) inclusions $A,B \subset M$ of $\sigma$-finite von Neumann algebras with expectations $E_A,E_B$ respectively. Intertwining theory with group actions {#intertwining-theory-with-group-actions .unnumbered} -------------------------------------- We first consider the intertwining condition $A\preceq_MB$ when a locally compact group acts on them. This idea was first used in [@Po04; @Po05a] to study cocycle superrigidity for discrete group actions. Although our main interest is the case of modular actions, we first study this condition by assuming that a general locally compact group acts on $A,B \subset M$. We fix the following setting (which will be used in Definitions \[def unital intertwining with actions\] and Theorem \[thm unital intertwining with actions\]). We use notation introduced before Theorems \[thm corner intertwining\], so we use $A\subset 1_AM1_A$, $B \subset 1_BM1_B$, $B=B_1\oplus B_2$, $\widetilde{B}$, $E_B$, $E_{\widetilde{B}}$, $L^2(M)$, $\varphi$, $J$, $e_B$, $e_{\widetilde{B}}$, $\tau_{B_1}$, ${\mathord{\text{\rm Tr}}}$, $\widehat{E}_{\widetilde{B}}$, and $\mathrm{ctr}$. Let $G$ be a locally compact second countable group, and consider continuous actions $\alpha$ and $\beta$ of $G$ on $M$ such that - $\alpha_g(A) =A$ and $\beta_g(B) = B$ for all $g\in G$; - $\alpha_g\circ E_A = E_A \circ \alpha_g$ on $1_AM1_A$ and $\beta_g\circ E_B = E_B\circ \beta_g$ on $1_BM1_B$ for all $g\in G$; - $\alpha$ and $\beta$ are cocycle conjugate: there exists a $\beta$-cocycle $\omega \colon G \to M$ such that $\alpha_g={\operatorname{Ad}}(\omega_g) \circ \beta_g (=:\beta_g^\omega)$ for all $g\in G$. In this setting, based on the viewpoint of Lemma \[relation for unital and corner\](1), we define intertwining conditions with group actions as follows. \[def unital intertwining with actions\] Keep the setting. We say that $(A,\alpha)$ embeds with expectation into $(B,\beta)$ inside $M$ and write $(A,\alpha) \preceq^{\rm uni}_{M} (B,\beta)$ if there exist: $(H,f,\pi,w)$ which witnesses $A\preceq^{\rm uni}_MB$ (in the sense of Lemma \[relation for unital and corner\](1)), and a generalized cocycle $(u_g)_{g\in G}$ for $\beta\otimes {\text{\rm id}}_H$ with values in $B{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)$ and with support projection $f$ such that - $wu_g = (\omega_g\otimes 1_H )(\beta_g\otimes {\text{\rm id}}_H) (w)$ for all $g\in G$; - $u_g(\beta_g\otimes {\text{\rm id}}_H)(\pi(a))u_g^* = \pi(\alpha_g(a))$ for all $g\in G$ and $a\in A$. In this case, we will say that $(H,f,\pi,w)$ and $(u_g)_{g\in G}$ witness $(A,\alpha) \preceq^{\rm uni}_{M} (B,\beta)$. Before proceeding, we mention following remarks. - In the definition, using the polar decomposition, $w$ is not necessarily a partial isometry (e.g. [@HI15 Remark 4.2(1)]). - We can define a $\ast$-isomorphism $\Pi^{\omega}_{\beta,\alpha}\colon M\rtimes_{\alpha}G \to M\rtimes_\beta G$ such that $\Pi^{\omega}_{\beta,\alpha}(a)=a$ for $a\in M$ and $\Pi^{\omega}_{\beta,\alpha}(\lambda^\alpha_g) = \omega_g \lambda_g^\beta$ for $g\in G$. There exist unital inclusions $A\rtimes_\alpha G \subset 1_A(M\rtimes_\alpha G)1_A$ and $B\rtimes_\beta G \subset 1_B(M\rtimes_\beta G)1_B$. - Using compression maps by $e_B\otimes 1$ and $e_A \otimes 1$, faithful normal conditional expectations $E_{B\rtimes_\beta G}\colon 1_B(M\rtimes_\beta G)1_B \to B\rtimes_\beta G$ and $E_{A\rtimes_\alpha G}\colon 1_A(M\rtimes_\alpha G)1_A \to A\rtimes_\alpha G$ are defined. - For each $g\in G$, let $u_g^\beta \in \mathcal{U}(L^2(M))$ be the canonical implementing unitary for $\beta_g$. Then putting $\widehat{\beta}_g:={\operatorname{Ad}}(u_g^\beta)$, the action $\beta$ can be extended on $\langle M, \widetilde{B}\rangle$. - Putting $\widehat{\alpha}_g:={\operatorname{Ad}}(\omega_g u_g^\beta)={\operatorname{Ad}}(\omega_g)\circ \widehat{\beta}$ for $g\in G$, we can also extend $\alpha$ on $\langle M, \widetilde{B}\rangle$. Note that $\widehat{\alpha}_g(1_A) = 1_A$ and $\widehat{\alpha}_g(J1_BJ) = J1_{B}J$ for all $g\in G$. - For each $g\in G$, since $\beta_g $ commutes with $E_B$, it holds that $\widehat{E}_{\widetilde{B}}\circ \widehat{\beta}_g = \beta_g \circ \widehat{E}_{\widetilde{B}}$ on $(\langle M,\widetilde{B} \rangle J1_BJ)^+$. This implies that $\widehat{E}_{\widetilde{B}}\circ \widehat{\alpha}_g = \alpha_g \circ \widehat{E}_{\widetilde{B}}$ on $(\langle M,\widetilde{B} \rangle J1_BJ)^+$. Our first goal in this section is to prove the following theorem, which gives fundamental characterizations of the condition $(A,\alpha) \preceq_M (B,\beta)$. We mention the origins of these conditions can be found in [@Po04; @Po05a] (see also [@HSV16]). \[thm unital intertwining with actions\] Consider the following conditions. - We have $(A,\alpha) \preceq^{\rm uni}_M (B,\beta)$. - We have $\Pi^{\omega}_{\beta,\alpha}(A\rtimes_\alpha G) \preceq_{M\rtimes_\beta G} B\rtimes_\beta G$. - There exists no nets $(u_i)_{i}$ of unitaries in $\mathcal U(A)$ and $(g_i)_i$ in $G$ such that $$E_B(\beta_{g_i}(b^)\omega_{g_i}^ u_i a ) \rightarrow 0, \quad \text{$\sigma$-strongly for all $a,b\in M1_B$.}$$ - There exists a nonzero positive element $d\in A' \cap 1_A\langle M, \widetilde{B}\rangle^{\widehat{\alpha}}1_A$ such that $$d=dJ1_{B}J \quad \text{and} \quad \widehat{E}_{\widetilde{B}}(d)\in M.$$ Then we have $(4) \Leftrightarrow (1) \Rightarrow (2)$. Moreover the following assertion holds true. - Assume further that $A\rtimes_\alpha G$ is finite. Then we have $(2) \Leftrightarrow (3) \Rightarrow (4)$, hence all conditions are equivalent. In this case, we can choose a Hilbert space $H$ in item $(1)$ as finite dimensional. In the case $A={\mathbb{C}}$, combining with Theorem \[relation for expectations and actions\] below, this theorem generalizes [@HSV16 Theorem 3.1]. When $A$ is not finite, the theorem fails since there is a counterexample for the implication (2)$\Rightarrow$(1) by [@HI17 Theorem 4.9]. We will nevertheless use this theorem by taking tensor products with a type III$_1$ factor, see Lemma \[key lemma\]. Throughout the proof, we will write a tensor product with ${\mathbb{B}}(H)$ as with the symbol $H$ at the top, such as $M^H := M{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)$, $\alpha^H_g:=\alpha_g\otimes {\text{\rm id}}_{H}$, $\omega^H_g:=\omega_g\otimes 1_{H}$ etc. $(1) \Rightarrow (2)$ Fix $(H,f,\pi,w)$ and $(u_g)_{g\in G}$. The generalized cocycle $(u_g)_{g\in G}$ gives a $\ast$-isomorphism $$\Pi^u_{\beta^H, (\beta^H)^u}\colon f(M^H\rtimes_{(\beta^H)^u}G)f \to f(M^H\rtimes_{\beta^H} G)f$$ satisfying $\Pi^u_{\beta^H, (\beta^H)^u}(faf)=faf$ for $a\in M^H$ and $\Pi^u_{\beta^H, (\beta^H)^u}(f\lambda_g^{(\beta^H)^u}f) = fu_g \lambda_g^{\beta^H} f = u_g \lambda_g^{\beta^H}$ for $g\in G$. Note that this restricts to a $\ast$-isomorphism between $f(B^H\rtimes_{(\beta^H)^u}G)f$ and $f(B^H\rtimes_{\beta^H} G)f$. The equivariance property $(\beta^H)^u_g(\pi(a))=u_g\beta^H_g(\pi(a))u_g^* = \pi(\alpha_g(a))$ for $a\in A$ and $g\in G$ implies that there is a $\ast$-homomorphism $$A\rtimes_\alpha G \to \pi(A)\rtimes_{(\beta^H)^u} G\subset f (B^H\rtimes_{(\beta^H)^u} G) f.$$ Composing this map with $\Pi^u_{\beta^H,(\beta^H)^u}$, we get a $\ast$-homomorphism $$\widetilde{\pi}\colon A\rtimes_\alpha G \to f (B^H\rtimes_{\beta^H} G) f$$ such that $\widetilde{\pi}(a)=\pi(a)$ for $a\in A$ and $\widetilde{\pi}( \lambda_g^\alpha ) = u_g\lambda_g^{\beta^H}$ for $g\in G$. The partial isometry $w$ then satisfies that, inside $M^H\rtimes_{\beta^H} G$, for all $a\in A$ and $g\in G$, $$\Pi_{\beta^H,\alpha^H}^{\omega^H}(a\otimes e_{1,1}) w= w\widetilde{\pi}(a) \quad \text{and} \quad \Pi_{\beta^H,\alpha^H}^{\omega^H}(\lambda_g^{\alpha^H} )w = \omega_g^H \beta_g^H (w)\lambda_g^{\beta^H} =wu_g \lambda_g^{\beta^H} =w\widetilde{\pi}( \lambda_g^\alpha ).$$ Hence using the isomorphism $M^H\rtimes_{\beta^H} G = (M\rtimes_{\beta} G) {\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)$ and using $\Pi_{\beta^H,\alpha^H}^{\omega^H}=\Pi^{\omega}_{\beta,\alpha}\otimes {\text{\rm id}}_{H}$, $(H , \widetilde{\pi}, f, w)$ witnesses $\Pi^{\omega}_{\beta,\alpha}(A\rtimes_\alpha G) \preceq^{\rm uni}_{M\rtimes_{\beta} G} B\rtimes_{\beta} G$. This is equivalent to item (2) by Lemma \[relation for unital and corner\]. $(1)\Rightarrow (4)$ Take $(H,\pi,f,w)$ and $(u_g)_{g\in G}$ witnessing item (1). Write $w=\sum_j w_j \otimes e_{1,j}$, where $(e_{i,j})_{i,j}$ is a matrix unit of ${\mathbb{B}}(H)$, and put $W:=\sum_j w_je_{\widetilde{B}} \otimes e_{1,j}=w e_{\widetilde{B}}^H$ (where $e_{\widetilde{B}}^H :=e_{\widetilde{B}}\otimes 1_H $). Then it satisfies that for any $a\in A$, $$(a\otimes e_{1,1})WW^= (a\otimes e_{1,1})we_{\widetilde{B}}^Hw^ = w\pi(a)e_{\widetilde{B}}^Hw^* = WW^(a\otimes e_{1,1}) ,$$ so $WW^ \in (A\otimes {\mathbb{C}}e_{1,1})' \cap (1_A\otimes e_{1,1})\langle M^H , \widetilde{B}^H \rangle (1_A\otimes e_{1,1})= (A'\cap 1_A\langle M,\widetilde{B} \rangle1_A) \otimes {\mathbb{C}}e_{1,1}$. We also have that for any $g\in G$, $$\begin{aligned} \widehat{\alpha}_g^H(WW^) = \omega_g^H\widehat{\beta}_g^H(we_{\widetilde{B}}^Hw^)(\omega_g^H)^* = wu_ge_{\widetilde{B}}^Hu_g^w^ =WW^,\end{aligned}$$ so $WW^ \in (1_A\langle M,\widetilde{B} \rangle1_A)^{\widehat{\alpha}}\otimes {\mathbb{C}}e_{1,1}$. Using the equation $\widehat{E}_{\widetilde{B}{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)}=\widehat{E}_{\widetilde{B}}\otimes {\text{\rm id}}_H$, it holds that $$(\widehat{E}_{\widetilde{B}}\otimes {\text{\rm id}}_H)(WW^) = \widehat{E}_{\widetilde{B}{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)}(WW^) = ww^\in M\otimes {\mathbb{C}}e_{1,1}<\infty.$$ Thus by using the element $d$ such that $d\otimes e_{1,1} = WW^$, we get item (4). $(4)\Rightarrow (1)$ Take a nonzero spectral projection $p$ of $d$ such that $p\leq \lambda d$ for some $\lambda>0$. Then $p$ satisfies exactly the same assumption as the one of $d$. Fix a countably infinite dimensional Hilbert space $H$ (with a matrix unit $(e_{i,j})_{i,j}$ in ${\mathbb{B}}(H)$), and consider the inclusion $$A\otimes {\mathbb{C}}e_{1,1} \subset \langle M ,\widetilde{B}\rangle {\mathbin{\overline{\otimes}}}{\mathbb{B}}(H) = \langle M^H ,\widetilde{B}^H \rangle .$$ Then the projection $p\otimes e_{1,1}$ satisfies that $$\widehat{E}_{\widetilde{B}^H}(p\otimes e_{1,1}) = \widehat{E}_{\widetilde{B}}(p)\otimes e_{1,1}<\infty.$$ Since the projection $e_{\widetilde{B}}^H (1_B \otimes 1_H) = (e_{\widetilde{B}}1_B)\otimes 1_H$ is properly infinite, we can follow the proof of (6)$\Rightarrow$(2-b) of [@HI15 Theorem 4.3] (we do not need the finiteness of $A$). We can find a partial isometry $W \in \langle M^H ,\widetilde{B}^H\rangle$ (which is of the form $w e_{\widetilde{B}}^H= W$), a projection $f\in B^H$, a $\ast$-homomorphism $\pi\colon A \to fB^H f$ such that $\pi(a)e^H_{\widetilde{B}}=W^(a\otimes e_{1,1})W$ and $w\pi(a) = (a\otimes e_{1,1})w$ for all $a\in A$, and $WW^ = p \otimes e_{1,1} \in (1_A\langle M,\widetilde{B}\rangle1_A)^{\widehat{\alpha}} {\mathbin{\overline{\otimes}}}{\mathbb{B}}(H) $. Note that $(H,f,\pi,w)$ witnesses $A\preceq^{\rm uni}_MB$ (up to taking the polar decomposition of $w$). We next construct a generalized cocycle. For any $g\in G$, since $W^\omega^H_g \widehat{\beta}^H_g(W) \in 1_Be_{\widetilde{B}}^H \langle M,\widetilde{B}\rangle^H 1_Be_{\widetilde{B}}^H={B}^He_{\widetilde{B}}^H$, there is a unique $u_g\in {B}^H$ such that $u_ge_{\widetilde{B}}^H=W^\omega_g^H \widehat{\beta}_g^H(W)$. Since $g\mapsto \omega_g^H$ and $g \mapsto \widehat{\beta}_g^H(W)$ are $\ast$-strongly continuous, so is the map $G\ni g \mapsto u_g$. Observe that $$e_{\widetilde{B}}^Hu_gu_g^=W^\omega_g^H \widehat{\beta}_g^H(WW^)(\omega_g^H)^W = W^\widehat{\alpha}_g^H(WW^)W =fe_{\widetilde{B}}^H$$ and similarly $e_{\widetilde{B}}^Hu_g^u_g = {\beta}_g^H(f)e_{\widetilde{B}}^H$ for all $g\in G$. For $g,h \in G$, we compute that $$\begin{aligned} u_g{\beta}_g^H(u_h)e_{\widetilde{B}}^H &=& W^\omega_g^H\widehat{\beta}_g^H(W)\widehat{\beta}_g^H(W^\omega_h^H\widehat{\beta}_h^H(W)) \\ &=& W^\widehat{\alpha}_g^H(WW^)\omega_g^H\widehat{\beta}_g^H(\omega_h^H)\widehat{\beta}^H_{gh}(W) \\ &=& W^ \omega_{gh}^H\widehat{\beta}_{gh}^H(W) \\ &=& u_{gh} e_{\widetilde{B}}^H.\end{aligned}$$ Thus $(u_g)_{g\in G}$ is a generalized cocycle for $\beta^H$ with support projection $f$. Using the equation $(\omega_g^H)^W u_g = \widehat{\beta}_g^H(W)$, it holds that for any $a\in A$ and $g\in G$, $$\begin{aligned} \beta_g^H(\pi(a)) e_{\widetilde{B}}^H = \widehat{\beta}_g^H(W^(a\otimes e_{1,1})W) = u_g^W^{\alpha}^H_g(a\otimes e_{1,1}) W u_g = u_g^\pi({\alpha}_g(a))u_g e_{\widetilde{B}}^H. \end{aligned}$$ We get the equivariance property $ u_g\beta_g^H(\pi(a))u_g^ = \pi(\widehat{\alpha}_g(a))$ for all $a\in A$. Finally, since $W=w e_{\widetilde{B}}^H$, the equation $(\omega_g^H)^W u_g = \widehat{\beta}_g^H(W)$ for $g\in G$ implies $(\omega_g^H)^w u_g e_{\widetilde{B}}^H= \beta_g^H(w) e_{\widetilde{B}}^H$. We get $w u_g = \omega_g^H\beta_g^H(w)$ for all $g\in G$, and thus $(u_g)_{g\in G}$ is a desired cocycle. We get item (1). [From now on, we assume that $A\rtimes_\alpha G$ is finite.]{} $(2) \Leftrightarrow (3)$ Assume $A\rtimes_\alpha G$ is finite. Suppose first that item (3) does not hold, hence there exists a net $(u_i)_{i}$ of unitaries in $\mathcal U(A)$ and $(g_i)_{i}$ in $G$ such that $$E_B(\beta_{g_i}(b^)\omega_{g_i}^ u_i a ) \rightarrow 0, \quad \text{$\sigma$-strongly for all $a,b\in M1_B$.}$$ Then for any $a,b\in M1_B$ and $s,s'\in G$, we have $$\begin{aligned} & E_{B\rtimes_\beta G}(\lambda_s^\beta b^* \Pi^\omega_{\beta,\alpha}(\lambda_{g_i^{-1}}^\alpha)u_i a\lambda_{s'}^\beta) \\ =& \lambda_s^\beta E_{B\rtimes_\beta G}(b^* \lambda_{g_i^{-1}}^\beta \omega_{g_i}^* u_i a)\lambda_{s'}^\beta \\ =& \lambda_{sg_i^{-1}}^\beta E_{B}(\beta_{g_i}(b^* )\omega_{g_i}^* u_i a)\lambda_{s'}^\beta.\end{aligned}$$ The last term converges to 0 in the $\sigma$-strong topology for all $a,b\in M1_B$ and $s,s' \in G$. By Theorem \[thm corner intertwining\](3) (see also [@HI15 Theorem 4.3(5)]), this means $\Pi^{\omega}_{\beta,\alpha}(A\rtimes_\alpha G) \not\preceq_{M\rtimes_\beta G} B\rtimes_\beta G$. Conversely Suppose that $\Pi^{\omega}_{\beta,\alpha}(A\rtimes_\alpha G) \not\preceq_{M\rtimes_\beta G} B\rtimes_\beta G$. Then by Theorem \[thm corner intertwining\](3), there exist a net $(u_i)_{i}$ of unitaries in $\mathcal U(A)$ and $(g_i)_{i}$ in $G$ such that $$E_{B\rtimes_\beta G}(y^* \Pi^\omega_{\beta,\alpha}(\lambda_{g_i^{-1}}^\alpha)u_i x) \rightarrow 0, \quad \text{$\sigma$-strongly for all $x,y\in (M\rtimes_\beta G) 1_B$.}$$ Using the same computation as above, we get that item (3) does not hold. $(3)\Rightarrow (4)$ Assume that $A\rtimes_\alpha G$ is finite. Let $\psi$ be a faithful normal state on $M\rtimes_\alpha G$ which is preserved by $E_{A\rtimes_\alpha G}$ such that $\psi\|_{A\rtimes_\alpha G}$ is a trace. Observe that $\psi\|_{1_A M 1_A}$ is $\alpha$-preserving, since $1_A \lambda_g^\alpha \in (1_A M 1_A)_\psi$ for all $g\in G$. It then holds that $\widehat{\psi}\circ \widehat{\alpha}_g = \widehat{\psi}$ on $(1_A \langle M,\widetilde{B} \rangle1_A J1_BJ)^+$ for all $g\in G$. By assumption, there exist $\delta>0$ and a finite subset $\mathcal{F}\subset 1_AM1_B$ such that $$\sum_{a,b\in \mathcal{F}}\\|E_{B}(\beta_g(b^) w_{g}^ u a)\\|_{2,\varphi}^2>\delta, \quad \text{for all } u\in\mathcal{U}(A), \ g\in G.$$ Put $d_0:=\sum_{y\in \mathcal{F}}ye_{{\widetilde{B}}}y^\in (1_A\langle M,\widetilde{B} \rangle1_A)^+ $ and observe that $d_0=d_0 J1_BJ$, $\widehat{E}_{\widetilde{B}}(d_0) = \sum_{y \in \mathcal F} yy^ \in 1_AM1_A$ and $\mathrm{ctr}(d_0 \, J1_{B_1}J) = \sum_{y\in \mathcal F} J\mathrm{ctr}_{B_1}(E_B(1_{B_1}y^y1_{B_1})) J< +\infty$. Define $$\mathcal{K}:=\overline{\mathrm{co}}^{\rm weak}\left \{ u^ \widehat{\alpha}_g(d_0) u \mid u \in \mathcal U(A),\ g\in G \right \}\subset 1_A\langle M,\widetilde{B} \rangle 1_A.$$ Following the proof of (5)$\Rightarrow$(6) of [@HI15 Theorem 4.3], there exists a unique element $d \in \mathcal K$ of minimum $\\|\cdot\\|_{2,\widehat{\psi}}$-norm. Since $\widehat{\psi}$ is preserved by $\widehat{\alpha}$ and since $A$ is contained in the centralizer of $\widehat{\psi}$, we get that $d\in A'\cap (1_A\langle M,\widetilde{B}\rangle 1_A)^{\widehat{\alpha}}$. Note that $d = d J 1_{B} J$, since $d_0 = d_0 J1_BJ$. We prove that $d\neq0$. For all $u \in \mathcal{U}(A)$ and $g\in G$, we have $$\begin{aligned} \sum_{a\in\mathcal{F}}\langle u^\widehat{\alpha}_{g}(d_0)u \, \Lambda_\varphi(a),\Lambda_\varphi(a)\rangle_{\varphi} =&\sum_{a,b\in\mathcal{F}}\langle u^ \widehat{\alpha}_g(b e_{\widetilde{B}} b^)u \Lambda_\varphi(a), \Lambda_\varphi(a)\rangle_{\varphi}\\ =&\sum_{a,b\in\mathcal{F}}\langle u^ w_{g}\beta_{g}(b) e_{B} \beta_{g}(b^)w_{g}^u \Lambda_\varphi(a), \Lambda_\varphi(a)\rangle_{\varphi}\\ =&\sum_{a,b\in \mathcal{F}}\\|E_{B}(\beta_g(b^) w_{g}^ u a)\\|^2_{2,\varphi_B}>\delta .\end{aligned}$$ By taking convex combinations and a $\sigma$-weak limit, we obtain $\sum_{a\in\mathcal{F}}\langle d \Lambda_\varphi(a),\Lambda_\varphi(a)\rangle_{\varphi} \geq \delta$. This implies $d \neq 0$. We prove $\widehat{E}_{\widetilde{B}}(d)\in M$. Observe that for any $g\in G$, $$\begin{aligned} \widehat{E}_{\widetilde{B}}(u^\widehat{\alpha}_g(d_0)u) &=\sum_{y\in \mathcal{F}}\widehat{E}_{\widetilde{B}}(u^{\alpha}_g(y) \omega_ge_{\widetilde{B}}\omega_g^* {\alpha}_g(y^)u )\\ &=\sum_{y\in \mathcal{F}}u^{\alpha}_g(y) {\alpha}_g(y^)u \\ &=u^{\alpha}_g\left(\sum_{y\in \mathcal{F}}yy^* \right)u. \end{aligned}$$ Combined with the normality of $\widehat{E}_{\widetilde{B}}$, we conclude that $\\|\widehat{E}_{\widetilde{B}}(x)\\|_\infty \leq \\|\sum_{y\in \mathcal{F}}yy^\\|_\infty$ for all $x \in \mathcal{K}$, hence $\widehat{E}_{\widetilde{B}}(d)\in M$. We get item (4). Finally we prove that the Hilbert space $H$ in item (1) can be taken as finite dimensional. For this, we continue to use $d_0,d,\mathcal K$ and claim $\mathrm{ctr}(d J1_{B_1}J)<\infty$. Using the formula for $\mathrm{ctr}$ given in Section \[Preliminaries\] and using $\mathrm{ctr}_{B_1}\circ \beta_g = \beta_g \circ \mathrm{ctr}_{B_1}$ on $B_1$ for all $g\in G$, we compute that for any $g\in G$ and $u\in \mathcal{U}(A)$ $$\begin{aligned} \mathrm{ctr}(u^\widehat{\alpha}_g(d_0)u \, J1_{B_1}J) &= \sum_{y\in \mathcal{F}}\mathrm{ctr}([u^\omega_g{\beta}_g(y)] e_{\widetilde{B}} [{\beta}_g(y^)\omega_g^u] \, J1_{B_1}J)\\ &= \sum_{y\in \mathcal{F}}J\mathrm{ctr}_{B_1}\circ E_B(1_{B_1}[{\beta}_g(y^)\omega_g^u][{\beta}_g(y^)\omega_g^u]^1_{B_1})J\\ &= \sum_{y\in \mathcal{F}}J\mathrm{ctr}_{B_1}\circ E_B(1_{B_1}{\beta}_g(y^y)1_{B_1})J\\ &= J{\beta}_g\circ\mathrm{ctr}_{B_1}\circ E_B(\sum_{y\in \mathcal{F}} 1_{B_1}y^y1_{B_1})J.\end{aligned}$$ Combined with the normality of $\mathrm{ctr}$, we get $$\\|\mathrm{ctr}(x J1_{B_1}J)\\|_\infty \leq \\|\mathrm{ctr}_{B_1}(E_B(\sum_{y\in \mathcal{F}}1_{B_1}y^y 1_{B_1})\\|_\infty$$ for all $x\in \mathcal{K}$. Thus we get $\mathrm{ctr}(d J1_{B_1}J)<\infty$. We next follow the proof of (4)$\Rightarrow$(1) above. Take a nonzero spectral projection $p$ of $d$ such that $p\leq \lambda d$ for some $\lambda>0$, so that $\mathrm{ctr}(d J1_{B_1}J)<\infty$ and $\widehat{E}_{\widetilde{B}}(p)\in M$. We have either $p \, J1_{B_1}J\neq0$ or $p \, J1_{B_2}J\neq0$. Assume that $p \, J1_{B_2}J\neq0$. We may assume $p \, J1_{B_2}J = p$. Then since $B_2$ is properly infinite, we can follow the proof above (with $H={\mathbb{C}}$ and $B=B_2$), so we get item (1) with $H={\mathbb{C}}$. Assume that $p \, J1_{B_1}J\neq0$ and we may assume $p \, J1_{B_1}J = p$. Then using $\widehat{E}_{\widetilde{B}}(p)<\infty$ and $\mathrm{ctr}(p)<\infty$, there is a family $\{w_i\}_{i=1}^n \subset M1_{B_1}$ such that $W_i:=w_ie_{\widetilde{B}}$ are partial isometries for all $i$, $p = \sum_{i=1}^n w_i e_{\widetilde{B}} w_i^ = \sum_{i=1}^n W_iW_i^$, and $E_B(w_i^w_j) = \delta_{i,j}p_j$ for all $i,j$, where $p_j\in B_1$ are projections. (Indeed using $\widehat{E}_{\widetilde{B}}(p)<\infty$, one can first choose $\{p_i\}_{i \in I}$ as above but possibly $\|I\|=\infty$. Using a maximality argument, we can assume that the central support of $p_{i+1}$ in $B_1$ is smaller than $p_{i}$ for all $i$. Then using $\mathrm{ctr}(p)<\infty$, the family $\{p_i\}_{i}$ must be a finite set.) Consider a $\ast$-homomorphism $\pi\colon p\langle M,\widetilde{B} \rangle p \to B_1 {\mathbin{\overline{\otimes}}}{\mathbb{M}}_n$ given by $$\begin{aligned} pxp=\sum_{i,j=1}^nW_i(W_i^* x W_j)W_j^* \mapsto \sum_{i,j=1}^n E_B(w_i^xw_j) \otimes e_{i,j}, \quad (x\in \langle M,\widetilde{B}\rangle).\end{aligned}$$ Then using the identification $p\langle M,\widetilde{B} \rangle p \simeq p\langle M,\widetilde{B} \rangle p \otimes {\mathbb{C}}e_{1,1}$ and the partial isometry $W:=\sum_{j} W_j \otimes e_{1,j}$, the map $\pi$ satisfies $ \pi(x)(e_{\widetilde{B}} \otimes 1_n)=W^ (x\otimes e_{1,1})W$ for all $x\in p\langle M,\widetilde{B} \rangle p$. Define $f:=\pi(1_A)\in B_1\otimes {\mathbb{M}}_n$ and $w:=\sum_{j} w_j \otimes e_{1,j} \in M \otimes {\mathbb{M}}_n$, so that $W^W=f(e_{\widetilde{B}}\otimes 1_n)$ and $W=w(e_{\widetilde{B}}\otimes 1_n)$. By restricting $\pi$ to $Ap$ and composing with the map $A \to Ap$, we have a unital normal $\ast$-homomorphism $\pi \colon A \to f(B_1\otimes {\mathbb{M}}_n)f$ such that $ (a\otimes e_{1,1})W = W\pi(a)$ for all $a\in A$. Thus we are exactly in the same situation as in the proof of (4)$\Rightarrow$(1) but with $H={\mathbb{C}}^n$ and $B=B_1$. Following the same proof, we get item (1) with $H={\mathbb{C}}^n$ as desired. Intertwining theory with modular actions {#intertwining-theory-with-modular-actions .unnumbered} ---------------------------------------- We next focus on the case of modular actions. We continue to use $A,B \subset M$ and fix faithful normal conditional expectations $E_A,E_B$ for $A,B$ respectively. Let $\psi,\varphi \in M_$ be faithful normal positive functionals which are preserved by $E_A,E_{{B}}$ respectively. Then since $\sigma^\psi_t(A)=A$, $\sigma_t^\varphi(B)=B$ for all $t\in {\mathbb{R}}$, and $\sigma^\psi$ and $\sigma^\varphi$ are cocycle conjugate by $([D\psi:D\varphi]_t)_{t\in {\mathbb{R}}}$, one can think the condition $(A,\sigma^\psi) \preceq^{\rm uni}_M (B,\sigma^\varphi)$. In this setting, the extended actions of $\sigma^\psi$ and $\sigma^\varphi$ on $\langle M,\widetilde{B}\rangle$ are exactly the modular actions of $\widehat{\psi}:=\psi\circ \widehat{E}_{\widetilde{B}}$ and $\widehat{\varphi}:=\varphi\circ \widehat{E}_{\widetilde{B}}$ respectively. As in the usual intertwining condition, we introduce intertwining conditions with modular actions at a level of corners. \[def corner intertwining with modular actions\] Keep the setting. We will say that a corner of $(A,\sigma^\psi)$ embeds with expectation into $(B,\sigma^\varphi)$ inside $M$ and write $(A,\sigma^\psi) \preceq_M (B,\sigma^\varphi)$ if there exist $(e,f,\theta,v)$ which witnesses $A\preceq_M B$ with $e\in A_\psi$, and a generalized cocycle $(u_t)_{t\in {\mathbb{R}}}$ for $\sigma^\varphi$ with values in $B$ and with support projection $f$ such that, with $\omega_t:=[D\psi:D\varphi]_t$, - $vu_t = \omega_t \sigma^\varphi_t (v)$ for all $t\in {\mathbb{R}}$; - $u_t\sigma^\varphi_t(\theta(a))u_t^* = \theta(\sigma^\psi_t(a))$, for all $a\in eAe$ and $t\in {\mathbb{R}}$. In this case, we will say that $(e,f,\theta,u)$ and $(u_g)_{g\in G}$ witness $(A,\sigma^\psi) \preceq_M (B,\sigma^\varphi)$. Below we collect elementary lemmas. We omit proofs since they are straightforward. \[lemma for equivalence\] Assume $(A,\sigma^\psi)\preceq_M (B,\sigma^\varphi)$ and fix $(e,f,\theta,v)$ and $(u_t)_{t\in {\mathbb{R}}}$ which witness $(A,\sigma^\psi)\preceq_M (B,\sigma^\varphi)$ as in the sense of Definition \[def corner intertwining with modular actions\]. - For any projection $e_0 \in eA_\psi e$ with $e_0v=v\theta(e_0)\neq 0$, $(e_0,\theta(e_0),\theta\|_{e_0Ae_0}, e_0v)$ and $(\theta(e_0)u_t )_{t\in {\mathbb{R}}}$ witness $(A,\sigma^\psi)\preceq_M (B,\sigma^\varphi)$ (up to the polar decomposition of $e_0v$). - For any projection $z \in B \cap \theta(eAe)' \cap \{u_t\mid t\in {\mathbb{R}}\}'$ (e.g. $z\in \mathcal{Z}(B)$) with $vz\neq 0$, $(e,f z, \theta(\, \cdot\, ) z, vz)$ and $(u_t z)_{t\in {\mathbb{R}}}$ witness $(A,\sigma^\psi)\preceq_M (B,\sigma^\varphi)$ (up to the polar decomposition of $vz$). - Let $u\in A$ and $w\in B$ be partial isometries such that $e=u^u$ and $f=ww^$. Then $(uu^,w^w,{\operatorname{Ad}}(w^)\circ \theta \circ {\operatorname{Ad}}(u^),uvw)$ and the generalized cocycle $(w^u_t \sigma^\varphi_t(w))_{t\in {\mathbb{R}}}$ witness $(A,\sigma^{\psi'})\preceq_M (B,\sigma^{\varphi})$, where $\psi'\in M_^+$ is any faithful element which is preserved by $E_A$ such that $uu^* \psi' uu^* = u\psi u^$ and $uu^ \in A_{\psi'}$. - Let $\psi'$ and $\varphi'$ be any faithful normal positive functionals on $M$ which are preserved by $E_A$ and $E_B$ respectively such that $e\in A_{\psi'}$. Then $(e,f,\theta,v)$ and $(\theta(e[D\psi':D\psi]_te)u_t [D\varphi:D\varphi']_t)_{t}$ witness $(A,\sigma^{\psi'}) \preceq_M (B,\sigma^{\varphi'})$. Moreover all these statements hold if we consider $(H,f,\pi,w)$ and $(u_t)_{t\in {\mathbb{R}}}$ which witness $(A,\sigma^\psi)\preceq_M^{\rm uni} (B,\sigma^\varphi)$ as in the sense of Definition \[def unital intertwining with actions\]. (In this case, we use $\mathcal{Z}(A)$ and $B{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)$, instead of $A_\psi $ and $B$ in items $(1)$,$(2)$, and $(3)$, and item $(4)$ holds without the assumption $e\in A_{\psi'}$). The next lemma clarifies the relation between $\preceq$ and $\preceq^{\rm uni}$ for modular actions. It should be compared to Lemma \[relation for unital and corner\]. \[relation for unital and corner of actions\] The following assertions hold true. - We have that $(A,\sigma^\psi) \preceq_M (B,\sigma^\varphi)$ holds if and only if $(A,\sigma^\psi) \preceq^{\rm uni}_M (B,\sigma^\varphi)$ holds. In particular, these notions do not depend on the choice of $\psi$ and $\varphi$ (as long as they are preserved by $E_A$ and $E_B$ respectively). - Assume either one of the following conditions holds: - $A$ does not have any direct summand which is semifinite and properly infinite; or - $B$ is properly infinite. If $(A,\sigma^\psi) \preceq^{\rm uni}_M (B,\sigma^\varphi)$ holds, then the Hilbert space $H$ in Definition \[def unital intertwining with actions\] can be taken as finite dimensional. We decompose $A=A_1\oplus A_2\oplus A_3$ and $B=B_1\oplus B_2\oplus B_3$, where $A_1,B_1$ are finite, $A_2,B_2$ are semifinite and properly infinite, and $A_3,B_3$ are of type III. Then by Lemma \[lemma for equivalence\](1),(2) and [@HI15 Remark 4.2(2)], we have that $(A,\sigma^\psi) \preceq_M (B,\sigma^\varphi)$ holds if and only if $(A_i,\sigma^\psi) \preceq_M (B_j,\sigma^\varphi)$ holds for some $i,j$. Hence we can always assume that $A=A_i$ and $B=B_j$ for some $i,j$. The same thing is true for $(A,\sigma^\psi) \preceq^{\rm uni}_M (B,\sigma^\varphi)$. \(1) By Lemma \[lemma for equivalence\](4), the condition $(A,\sigma^{\psi}) \preceq^{\rm uni}_M (B,\sigma^\varphi)$ does not depend on the choice of $\psi,\varphi$. Hence if this statement is proven, then $(A,\sigma^{\psi}) \preceq_M (B,\sigma^\varphi)$ also does not depend on $\psi,\varphi$. Assume that $(A_i,\sigma^\psi) \preceq^{\rm uni}_M (B_j,\sigma^\varphi)$ holds for some $i,j$ and take $(H,f,\pi,w)$ and $(u_t)_t$ as in the definition. Let $z\in \mathcal{Z}(A)$ be a nonzero projection such that $Az \ni a \mapsto \pi(a)w^w$ is injective. Since $z\in A_\psi$, up to exchanging $Az$ by $A$, we may assume that $A\ni a \mapsto \pi(a)w^w$ is injective. In particular $w\pi(e) \neq 0$ for any nonzero projection $e\in A$. Assume that $B=B_2$ or $B=B_3$. Then since $1_B \otimes e_{1,1}$ is properly infinite, one has $f \prec 1_B \otimes e_{1,1}$. Up to equivalence of projections, using Lemma \[lemma for equivalence\](3), we may assume that $f$ is contained in $B\otimes {\mathbb{C}}e_{1,1}$. So using $M=M\otimes {\mathbb{C}}e_{1,1}$, we get $(A,\sigma^\psi) \preceq_M (B,\sigma^{\varphi})$. Assume that $B=B_1$. Then we must have that $A=A_1$ or $A_2$. If $A=A_2$, then by using $eA e$ for any fixed finite projection $e\in A_\psi$ (note that $A_\psi$ contains many finite projections, e.g. the first part of the proof of [@HU15 Lemma 2.1]) and using Lemma \[lemma for equivalence\](1), we may assume that $A$ is finite. By the last statement of Theorem \[thm unital intertwining with actions\], we may assume that $A$ is finite and $H$ is finite dimensional. We can still assume that $A\ni a \mapsto \pi(a)w^w$ is injective. Write $H={\mathbb{C}}^n$ for some $n\in {\mathbb{N}}$. As in the proof of [@BO08 Proposition F.10] or [@Ue12 Proposition 3.1 (ii)$\Rightarrow$(iii)], there is a projection $e\in A$ such that $\pi(e)$ is equivalent to a projection $f_0\otimes e_{1,1}$ for some $f_0 \in B$. By [@HU15 Lemma 2.1], $e$ is equivalent to a projection in $A_\psi$, so we may assume $e\in A_\psi$. Observe that, regarding $\pi$ as a map from $A\otimes {\mathbb{C}}e_{1,1}$, $(1_A\otimes e_{1,1},f,\pi,w)$ and $(u_t)_t$ witness $(A\otimes {\mathbb{C}}e_{1,1}, \sigma^\psi)\preceq_{M\otimes {\mathbb{M}}_n} (B\otimes {\mathbb{M}}_n, \sigma^{\varphi\otimes \mathrm{tr}_n})$. Since $\pi(e)w^w\neq 0$, by Lemma \[lemma for equivalence\](1), $(e\otimes e_{1,1}, \pi(e), \pi\|_{eAe\otimes e_{1,1}}, (e\otimes e_{1,1})w)$ witness $(A\otimes {\mathbb{C}}e_{1,1}, \sigma^\psi)\preceq_{M\otimes {\mathbb{M}}_n} (B\otimes {\mathbb{M}}_n, \sigma^{\varphi\otimes \mathrm{tr}_n})$ as well. We then apply Lemma \[lemma for equivalence\](3) for $\pi(e) \sim f_0\otimes e_{1,1}$, and obtain that $(e\otimes e_{1,1}, f_0\otimes e_{1,1}, \pi', w')$ and some generalized cocycle witness $(A\otimes {\mathbb{C}}e_{1,1}, \sigma^\psi)\preceq_{M\otimes {\mathbb{M}}_n} (B\otimes {\mathbb{M}}_n, \sigma^{\varphi\otimes \mathrm{tr}_n})$ for some $\pi'$ and $w'$. Finally since $f_0\otimes e_{1,1}$ and $w'$ are contained in $M\otimes {\mathbb{C}}e_{1,1}$, by identifying $M\otimes {\mathbb{C}}e_{1,1} = M$, we get $(A, \sigma^\psi)\preceq_{M} (B, \sigma^{\varphi})$. We next show the ‘only if’ direction. Assume that $(A,\sigma^\psi) \preceq_M (B,\sigma^\varphi)$ holds and take $(e,f,\theta,v)$ and $(u_t)_t$ as in the definition. As in the proof above, we can assume $eAe \ni a \mapsto v^v\theta(a)$ is injective and hence $v\theta(e_0) \neq 0$ for any nonzero projection $e_0\in eAe$. Let $z$ be the central support projection of $e$ in $A$, and take partial isometries $(w_i)_{i\in I}$ in $A$ such that $w_0=e$, $e_i:=w_i^w_i\leq e$ for all $i \in I$, and $\sum_{i \in I} w_iw_i^* =z$. Note that $I$ is a countable set, so we regard $I\subset {\mathbb{N}}$. We put $v_n:=w_n v $ for all $n \in I$ and $d=\sum_{n\in I} v_n e_{\widetilde{B}} v_n^$, and then it is easy to see that $d=dJ1_BJ$ and $\widehat{E}_{\widetilde{B}}(d)\in M .$ We note that $d\neq 0$, since each $v_n$ is nonzero by $w_n^v_n = w_n^w_n v=v \theta(w_n^w_n) \neq 0$. It is easy to compute that $ad=da$ for all $a\in A$, hence $d \in A' \cap 1_A \langle M,\widetilde{B}\rangle1_A$. Define a faithful normal positive functional $\psi'$ on $M$ by $$\psi' := \sum_{n \in I} \frac{1}{2^{n}} \, w_n \psi w_n^* + (1-z)\psi (1-z).$$ Note that $\psi'$ is preserved by $E_A$. By Lemma \[connes cocycle lemma\], the equation $e_n\psi' e_n = 2^{-n} w_n\psi w_n^$ implies $ \sigma_t^{\psi}(w_n) = 2^{-itn} [D\psi':D\psi]_t^ w_n$ for all $t\in {\mathbb{R}}$ and $n\in I$. An easy computation shows that $$\begin{aligned} \sigma_t^{\widehat{\psi}}(d) = [D\psi:D\varphi]_t\sigma_t^{\widehat\varphi}(d)[D\psi:D\varphi]_t^* = [D\psi':D\psi]_t^* \, d\, [D\psi':D\psi]_t, \quad \text{for all }t\in {\mathbb{R}}.\end{aligned}$$ We get that $\sigma_t^{\widehat{\psi}'}(d) = d$ for all $t\in {\mathbb{R}}$ and hence $d\in A' \cap (1_A \langle M,\widetilde{B}\rangle1_A)_{\widehat{\psi}'}$. By Theorem \[thm unital intertwining with actions\], this means $(A,\sigma^{\psi'}) \preceq^{\rm uni}_M (B,\sigma^\varphi)$. By Lemma \[lemma for equivalence\](4), this is equivalent to $(A,\sigma^{\psi}) \preceq^{\rm uni}_M (B,\sigma^\varphi)$. \(2) Assume that $(A_i,\sigma^\psi) \preceq^{\rm uni}_M (B_j,\sigma^\varphi)$ holds for some $i,j$. If $B=B_2$ or $B_3$, then the first half of the proof of item (1) shows that one can assume $H={\mathbb{C}}$. So we get the conclusion. If $A=A_3$, then we must have $B=B_3$, which we proved. Finally if $A=A_1$, then the last part of Theorem \[thm unital intertwining with actions\] gives the conclusion. Intertwining theory with conditional expectations {#intertwining-theory-with-conditional-expectations .unnumbered} ------------------------------------------------- In [@HSV16], a notion of intertwining conditions for states was introduced. Inspired from this, we introduce a notion of intertwining conditions for conditional expectations. We still fix $A,B \subset M$ with expectations $E_A,E_B$. \[def corner intertwining with expectation\] We say that a corner of $(A,E_A)$ embeds with expectation into $(B,E_B)$ inside $M$ and write $(A,E_A) \preceq_M (B,E_B)$ if there exist $(e,f,\theta,v)$ which witnesses $A\preceq_M B$, and faithful normal positive functionals $\psi,\varphi \in M_$ which are preserved by $E_A,E_B$ respectively such that $$vv^ \in (1_AM1_A)_\psi, \quad v^v\in (1_BM1_B)_{\varphi}, \quad \text{and} \quad vv^ \psi vv^* = v\varphi v^.$$ In this case, we say that $(e,f,\theta,v)$ and $\psi,\varphi$ witness* $(A,E_A) \preceq_M (B,E_B)$. The next lemma clarifies relations between $A\preceq_MB$ and $(A,E_{A})\preceq_M(B,E_B)$. Note that, as in the statement of Theorem \[thmA\], one can actually take $q=1_A$ in the next lemma (which will be proved later). \[intertwining implies expectation\] The condition $A\preceq_MB$ holds if and only if there is a nonzero projection $q\in A'\cap 1_AM1_A$ and a faithful normal conditional expectation $E_{Aq}\colon qMq \to Aq$ such that $(Aq,E_{Aq})\preceq_M(B,E_B)$. The ‘if’ direction is trivial, so we see the ‘only if’ direction. Take $(e,f,\theta,v)$ which witnesses the condition $A\preceq_M B$. By [@HI15 Remark 4.2(2),(3)], we may assume that $A$ is finite or of type III, and that $eAe\ni a \mapsto \theta(a)v^v$ is injective. Up to exchanging $e$ with a small one if necessary, we may assume that there exist finitely many orthogonal and equivalent projections $(e_{i})_{i=1}^n$ in $A$ such that $\sum_{i=1}^ne_i =:z_A(e)\in \mathcal{Z}(A)$. Fix a faithful normal conditional expectation $E_\theta$ for the inclusion $\theta(eAe) \subset fBf$, and take a faithful normal state $\varphi_B$ on $B$ such that $\varphi_B \circ E_{\theta} = \varphi_B$ on $fBf$. Put $\varphi:= \varphi_B \circ E_B$ on $1_BM1_B$ and observe that the modular action of $\varphi$ globally preserves $\theta(eAe)$ and $fBf$. In particular it also preserves $\theta(eAe)'\cap fMf$, so using [@HU15 Lemma 2.1], there is a partial isometry $w\in \theta(eAe)'\cap fMf$ such that $w^w = v^v$ and $ww^ \in (\theta(eAe)'\cap fMf)^{\sigma^{\varphi}}$. Up to exchanging $vw^$ by $v$, we may assume that $v^v$ is contained in $(fMf)^{\sigma^{\varphi}}$. We put $e_0:=vv^\in (eAe)' \cap eMe$ and $f_0:=v^v \in (\theta(eAe)'\cap fMf)^{\sigma^{\varphi}}$. Since $ \theta(eAe) f_0 \subset f_0Mf_0$ is globally preserved by $\sigma^\varphi$, it is with expectation, say $E\colon f_0Mf_0 \to \theta(eAe) f_0 $, which satisfies $\varphi\circ E = \varphi$ on $f_0Mf_0$. Observe that ${\operatorname{Ad}}(v)$ gives a spacial isomorphism from $\theta(eAe) f_0$ onto $(eAe)e_0$. Hence we can define a conditional expectation by $$E_A':={\operatorname{Ad}}(v)\circ E \circ {\operatorname{Ad}}(v^)\colon e_0Me_0 \to (eAe)e_0.$$ Define a positive functional $\psi_A':=v \varphi v^$ on $(eAe)e_0$ and put $\psi':=\psi_A'\circ E_A'$ on $e_0Me_0$. It holds that $v^v =f_0\in (1_BM1_B)_\varphi$ and $vv^ = e_0 \in (e_0Me_0)_{\psi'}$. By using $\psi_A'=v \varphi v^$ on $(eAe)e_0$ and $\varphi\circ E=\varphi$ on $f_0Mf_0$, we compute that, for any $x\in M$ $$\begin{aligned} vv^\psi'(x)vv^* &= \psi_A'\circ E_A'(vv^xvv^) \\ &= (v\varphi v^)( vE(v^ vv^xvv^ v) v^)\\ &= \varphi ( f_0E(v^xv) f_0)\\ &= \varphi \circ E(v^xv)\\ &= \varphi (v^xv).\end{aligned}$$ We get $vv^\psi' vv^=v\varphi v^$. Since they satisfy $\varphi=\varphi\circ E_B$ on $1_BM1_B$ and $\psi'=\psi' \circ E_A'$ on $e_0Me_0$, we can extend $\varphi$ and $\psi'$ to ones on $M$ which are preserved by $E_B$ and $E_A'$ respectively. In this case, we still have that $f_0 \in M_\varphi$, $e_0 \in M_{\psi'}$, and $vv^\psi' vv^=v\varphi v^$. We claim $((eAe)e_0, E_A') \preceq_M (B,E_B)$. Let $z\in \mathcal{Z}(eAe)$ be the central support projection of $e_0$ in $(eAe)'$ and observe that $(eAe)e_0 \simeq eAe z$. Since we assumed $eAe\ni a \mapsto v^v\theta(a)=v^ a v$ is injective, the map $eAe\ni a \mapsto {\operatorname{Ad}}(v)(v^v\theta(a))=ae_0$ is also injective. In particular we get $z=e$ and $(eAe)e_0 \simeq eAe$. Consider $\theta_0\colon (eAe)e_0\simeq eAe \to^{\theta} fBf$ given by $\theta_0(a e_0) := \theta(a)$ for $a\in eAe$. Then $(ee_0,f,\theta_0,v)$ witnesses $(eAe)e_0\preceq_MB$. Combined with $\varphi$ and $\psi'$ together, we obtain $((eAe)e_0, E_A') \preceq_M (B,E_B)$. Since $e_0 \in (eAe)' \cap (eMe) = (A'\cap 1_AM1_A)e$, there is a projection $q \in A'\cap 1_AM1_A$ such that $qe = e_0$ and $q=z_A(e)q$. Using projections $(e_i)_{i=1}^n$ which we fixed at the first paragraph, we have an identification $qMq \simeq e_0Me_0\otimes {\mathbb{M}}_n$ which restricts $Aq \simeq eAeq\otimes {\mathbb{M}}_n$. In particular, there is a faithful normal conditional expectation $E_{Aq}\colon qMq \to Aq$ such that $E_{Aq}\|_{e_0Me_0}=E_A'$. Since we chose $\psi'$ as any extension of $\psi'\|_{e_0Me_0}$ which is preserved by $E_A'$, we can particularly choose $\psi'$ as the one which is preserved by $E_A'$ and $E_{Aq}$. Then it is easy to see that the same $(ee_0,f,\theta_0,v)$ as above and $\psi',\varphi$ witness $(Aq, E_{Aq}) \preceq_M (B,E_B)$. The next theorem clarifies the relation between $(A,E_A)\preceq_M(B,E_B)$ and $(A,\sigma^{\psi})\preceq_M(B,\sigma^{\varphi})$. The proof uses Connes cocycles to construct a positive functional. Note that the case $A={\mathbb{C}}$ was proved in (the proof of) [@HSV16 Theorem 3.1]. \[relation for expectations and actions\] We have that $(A,E_A)\preceq_M(B,E_B)$ if and only if there exist faithful normal states $\psi,\varphi\in M_$ which are preserved by $E_A,E_B$ respectively such that $(A,\sigma^{\psi})\preceq_M(B,\sigma^{\varphi})$. Combined with Lemma \[relation for unital and corner of actions\](1), characterizations given in Theorem \[thm unital intertwining with actions\] can be adapted to $(A,E_A)\preceq_M(B,E_B)$ and $(A,\sigma^{\psi})\preceq_M(B,\sigma^{\varphi})$. Moreover $\psi$ and $\varphi$ for $(A,\sigma^{\psi})\preceq_M(B,\sigma^{\varphi})$ can be taken arbitrary as long as they are preserved by $E_A$ and $E_B$ respectively. Suppose $(A,E_A)\preceq_M(B,E_B)$ and take $(e,f,\theta,v)$ and $\psi,\varphi$. We put $d:=ve_{\widetilde{B}}v^$ and observe that $d\in (eAe)'\cap (e\langle M,\widetilde{B}\rangle e)$, $d=dJ1_BJ$, and $\widehat{E}_{\widetilde{B}}(d)<\infty$. By Lemma \[connes cocycle lemma\], the equation $vv^ \psi vv^* = v\varphi v^$ implies $[D\psi : D\varphi]_t\sigma_t^\varphi(v) = v $ for all $t\in {\mathbb{R}}$. It then holds that $\sigma_t^{\widehat{\psi}}(d)= d$ for any $t\in {\mathbb{R}}$, hence $d\in A' \cap (1_A \langle M,\widetilde{B}\rangle1_A)_{\widehat{\psi}}$. We get that $(eAe,\sigma^\psi)\preceq^{\rm uni}_M (B,\sigma^\varphi)$ by Theorem \[thm unital intertwining with actions\]. This implies $(eAe,\sigma^\psi)\preceq_M (B,\sigma^\varphi)$ by Lemma \[relation for unital and corner of actions\], and hence $(A,\sigma^\psi)\preceq_M (B,\sigma^\varphi)$. Suppose $(A,\sigma^{\psi})\preceq_M(B,\sigma^{\varphi})$ and take $(e,f,\theta,v)$ and $(u_t)_{t\in {\mathbb{R}}}$. Then since $(u_t)_{t\in{\mathbb{R}}}$ is a generalized cocycle for $\sigma^\varphi$ with support projection $f$, by Theorem \[connes cocycle existence\], there is a unique faithful normal semifinite weight $\mu_B$ on $fBf$ such that $[D\mu_B,D\varphi_B]_t = u_t$ for all $t\in {\mathbb{R}}$. Put $\mu:=\mu_B\circ E_B$ on $fMf$ and observe $[D\mu,D\varphi]_t = u_t$ for all $t\in {\mathbb{R}}$. For any $t\in {\mathbb{R}}$ and $a\in eAe$, using the equation $v u_t = \omega_t \sigma_t^\varphi(v)$ where $\omega_t=[D\psi:D\varphi]_t$, it is easy to compute that $$\begin{aligned} \sigma_t^\psi(vv^) =vv^, \quad \sigma_t^\mu(v^v) = v^v, \quad \text{and}\quad \sigma_t^\mu(\theta(a)) = \theta(\sigma_t^\psi(a)).\end{aligned}$$ We get that $vv^ \in eM_\psi e$ and $v^v\in (fMf)_\mu$. We extend $\mu$ by $f\mu f + (1-f)\varphi(1-f)$ and still denote by $\mu$. It satisfies that $\mu = \mu \circ E_B$ on $1_BM1_B$ and $1_B,f \in M_\mu$. We put $e_0:=vv^ \in eM_\psi e$ and $f_0:=v^v \in fM_\mu f$. For any $t\in {\mathbb{R}}$, using Lemma \[connes cocycle lemma\], we have $$\begin{aligned} [D(v\mu v^):D\varphi]_t &=& [D(v\mu v^):D\mu]_t[D\mu:D\varphi]_t\\ &=& v \sigma_t^\mu(v^) [D\mu:D\varphi]_t\\ &=& v[D\mu:D\varphi]_t \sigma_t^\varphi(v^) \\ &=& vu_t \sigma_t^\varphi(v^) \\ &=& \omega_t \sigma_t^\varphi(vv^) \\ &=& \sigma_t^\psi(v v^) \omega_t \\ &=& v v^* \omega_t \\ &=& [D(e_0\psi e_0): D\varphi]_t.\end{aligned}$$ We get $e_0\psi e_0= v\mu v^$. Hence $(e,f,\theta,v)$ and $\psi,\mu$ witness $(A,E_A) \preceq_M(B,E_B)$, but $\mu$ is not necessarily bounded. So we have to exchange $\mu$ by a bounded one. Since $e_0\psi e_0= v\mu v^$, it holds that $\mu_B( E_B (f_0)) = \mu(v^v) = \psi(e_0) <\infty$. Since $\sigma_t^{\mu_B}(E_B(f_0)) = E_B(\sigma_t^\mu(f_0))=E_B(f_0)$ for all $t\in {\mathbb{R}}$, and since $f_0=v^v\in \theta(eAe)'$, $E_B(f_0)$ is contained in $(fBf)_{\mu_B} \cap \theta(eAe)'$. Combined with the fact that $v^vE_B(f_0) \neq 0$ (because $E_B(v^vE_B(f_0)) = E_B(f_0)^2 \neq 0$), there is a nonzero spectral projection $f'\in (fBf)_{\mu_B}\cap \theta(eAe)'$ of $E_B(f_0)$ such that $vf'\neq 0$ and $\mu_B(f') <\infty$. Put $v':=vf'$, $\theta'(a):=\theta(a)f'$ for $a\in eAe$ and $u_t':= f'u_t$ for $t\in {\mathbb{R}}$. We claim that, up to the polar decomposition of $v'$, $(e,f',\theta',v')$ and $(u_t')_{t\in {\mathbb{R}}}$ witness $(A,\sigma^{\psi})\preceq _M(B,\sigma^{\varphi})$. It is easy to see that $v'\theta'(a) = a v'$ for all $a\in eAe$, hence $(e,f',\theta',v')$ witnesses $A\preceq_MB$. For any $t\in {\mathbb{R}}$, since $f'=\sigma_t^{\mu}(f')$, one has $$(u_t')^u'_t= u_t^ f'u_t=u_t^* \sigma_t^{\mu}(f')u_t= \sigma_t^\varphi(f').$$ This means $u_t'=f'u_t =u_t \sigma_t^\varphi(f') $ for all $t\in {\mathbb{R}}$. Using this, for any $a\in eAe$ and $t,s\in {\mathbb{R}}$, it is easy to compute that $$\begin{aligned} u'_{t+s} = u_t' \sigma_t^\varphi(u_s'), \quad v' u'_t = \omega_t \sigma_t^\varphi(v') \quad \text{and} \quad u_t'\sigma^\varphi_t(\theta'(a))(u_t')^* =\theta'(\sigma^\psi_t(a)).\end{aligned}$$ Thus $(e,f',\theta',v')$ and $(u_t')_{t\in {\mathbb{R}}}$ witness $(A,\sigma^{\psi})\preceq _M(B,\sigma^{\varphi})$. We exchange $v'$ with its polar part. Then by using $(e,f',\theta',v')$ and $(u_t')_{t\in {\mathbb{R}}}$, and by following the same construction as we did for $\mu$, we again construct a faithful normal semifinite weight $\mu'$ on $M$ such that $u_t' = [Df'\mu' f': D\varphi]_t$ for all $t\in {\mathbb{R}}$, and $e_0'\psi e_0'= v'\mu' \, v'^$, where $e_0':=v'v'^$. Since $$[Df'\mu' f': D\varphi]_t = u_t' = f' u_t = f'[Df\mu f: D\varphi]_t=[Df'\mu f': D\varphi]_t$$ for all $t\in {\mathbb{R}}$, it holds that $f'\mu' f' = f'\mu f'$. In particular, since $\mu(f')<\infty$, $f'\mu' f'$ is bounded. By construction, $\mu'$ is bounded on $M$ and hence $(e,f',\theta',v')$ and $\psi,\mu'$ witness $(A,E_A)\preceq _M(B,E_B)$. We record the following permanence property. \[intertwining for unital subalgebra\] Let $D \subset A$ be a unital von Neumann subalgebra with expectation $E_D$. - If $(A,\sigma^\psi) \preceq_M (B,\sigma^\varphi)$, then we have $(D,\sigma^{\psi'}) \preceq_M (B,\sigma^\varphi)$ for any faithful $\psi' \in M_^+$ which is preserved by $E_D \circ E_A$. - If $(A,E_A) \preceq_M (B,E_B)$, then we have $(D,E_D\circ E_A) \preceq_M (B,E_B)$. They are immediate by Lemma \[relation for unital and corner of actions\](1) and Theorem \[relation for expectations and actions\]. Proof of Theorem \[thmA\] {#proof-of-theorem-thma .unnumbered} ------------------------- Now we prove Theorem \[thmA\]. We continue to use $A,B \subset M$ with expectations, and we only fix $E_B$. We also fix a type III$_1$ factor $(N,\omega)$ as in the statement of Theorem \[thmA\]. The next lemma is the key observation to prove Theorem \[thmA\]. \[key lemma\] Let $E_A\colon 1_AM1_A\to A$ be a faithful normal conditional expectation, $\psi,\varphi\in M_$ be faithful states which are preserved by $E_A,E_B$ respectively. The following conditions are equivalent. 1. We have that $(A, E_A)\preceq_{M} (B, E_B)$. 2. We have that $(A{\mathbin{\overline{\otimes}}}N, E_A\otimes {\text{\rm id}}_N)\preceq_{M{\mathbin{\overline{\otimes}}}N} (B{\mathbin{\overline{\otimes}}}N, E_B\otimes {\text{\rm id}}_N)$. 3. We have that $\Pi_{\varphi\otimes\omega,\psi\otimes \omega}(C_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}N))\preceq_{C_{\varphi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N)}C_{\varphi\otimes \omega}(B{\mathbin{\overline{\otimes}}}N)$. (1)$\Rightarrow$(2) This is trivial (one only needs to take tensor products with $1_N$ or ${\text{\rm id}}_N$). \(2) $\Rightarrow$ (3) By Theorem \[relation for expectations and actions\] and Lemma \[relation for unital and corner of actions\](1), item (2) is equivalent to $(A{\mathbin{\overline{\otimes}}}N, \sigma^{\psi\otimes \omega})\preceq^{\rm uni}_{M{\mathbin{\overline{\otimes}}}N} (B{\mathbin{\overline{\otimes}}}N, \sigma^{\varphi\otimes \omega})$. By Theorem \[thm unital intertwining with actions\], we get item (3). \(3) $\Rightarrow$ (1) We first recall the following general facts (some of which were mentioned in Section \[Preliminaries\]). Since $\langle C_\varphi(M), C_\varphi(\widetilde{B}) \rangle$ is generated by $\langle M,\widetilde{B} \rangle$ and $L_{\varphi}{\mathbb{R}}$, and since $\sigma^{\widehat{\varphi}}_t = {\operatorname{Ad}}(\Delta_{\varphi}^{it})$, where $\widehat{\varphi}=\varphi\circ \widehat{E}_{\widetilde{B}}$, $\langle C_\varphi(M), C_\varphi(\widetilde{B}) \rangle$ is canonically identified as $C_{\widehat{\varphi}}(\langle M,\widetilde{B}\rangle)$. Put $\widehat{\psi}:=\psi\circ \widehat{E}_{\widetilde{B}}$. Since it satisfies $[D\widehat{\psi}:D\widehat{\varphi}]_t = [D\psi:D\varphi]_t$ for all $t\in {\mathbb{R}}$, the map $\Pi_{\widehat{\varphi},\widehat{\psi}}\colon C_{\widehat{\psi}}(\langle M,\widetilde{B}\rangle) \to C_{\widehat{\varphi}}(\langle M,\widetilde{B}\rangle)$ restricts to $\Pi_{{\varphi},{\psi}}\colon C_{\psi}(M)\to C_\varphi(M)$. Since $1_B=\pi_{\sigma^\varphi}(1_B)$ is the unit of $C_\varphi(B)$, for the modular conjugation $J_{C_\varphi(M)}$ on $L^2(C_\varphi(M))=L^2(M)\otimes L^2({\mathbb{R}})$ (with respect to the dual weight of $\varphi$), it holds that $$J_{C_\varphi(M)} 1_{C_\varphi(B)}J_{C_\varphi(M)}=J_{C_\varphi(M)} 1_{B}J_{C_\varphi(M)} = J 1_B J \otimes 1_{L^2({\mathbb{R}})}.$$ We note that the unitization of $C_\varphi(B)$ is contained in $C_\varphi(\widetilde{B})$, but they are different in general. We will use these observations for $A{\mathbin{\overline{\otimes}}}N, B{\mathbin{\overline{\otimes}}}N\subset M{\mathbin{\overline{\otimes}}}N$. Now we start the proof. We put $\mathcal{B}:=C_{\varphi\otimes \omega}(B{\mathbin{\overline{\otimes}}}N)$, $\mathcal{B}_1:=C_{\varphi\otimes \omega}(\widetilde{B}{\mathbin{\overline{\otimes}}}N)$, $\mathcal{M}:=C_{\varphi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N)$, $\mathcal{A}:=C_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}N)$, and $\Pi:=\Pi_{\widehat{\varphi\otimes \omega}, \widehat{\psi\otimes \omega}}$, so that our assumption is written as $\Pi(\mathcal A) \preceq_{\mathcal M} \mathcal B$. Note that the unitization of $\mathcal{B}$ is contained in $\mathcal{B}_1$. Take $(e,f,\theta,v)$ which witnesses $\Pi(\mathcal A) \preceq_{\mathcal M} \mathcal B$. Let $w_i\in \mathcal A$ be partial isometries such that $w_i^* w_i \leq e$ and $\sum_{i}w_iw_i^* = z_{\mathcal A}(e)$, where $z_{\mathcal A}(e)$ is the central support of $e$ in $\mathcal A$. Put $d:=\sum_i \Pi(w_i) v e_{\mathcal{B}_1} v^* \Pi(w_i^)$ and observe that $$d\in \Pi(\mathcal{A})' \cap 1_{\Pi(\mathcal{A})}\langle \mathcal M, \mathcal{B}_1\rangle 1_{\Pi(\mathcal{A})}, \quad d=d \mathcal J 1_{\mathcal{B}}\mathcal J, \quad \text{and} \quad \widehat{E}_{\mathcal{B}_1}(d) < \infty,$$ where $\mathcal J$ is the modular conjugation for $L^2(\mathcal {M})$. Note that $\mathcal{J}1_{\mathcal{B}}\mathcal{J} = J1_BJ\otimes 1_N\otimes 1_{L^2({\mathbb{R}})}$ as we have explained. The element $d$ is contained in $$\left[ A'\cap 1_A\langle M,\widetilde{B} \rangle J1_{B}J1_A\right]_{\widehat{\psi}} \otimes {\mathbb{C}}1_N \otimes {\mathbb{C}}1_{L^2({\mathbb{R}})}.$$ Observe that $$\Pi^{-1}(d)\in \mathcal{A}'\cap 1_{\mathcal{A}}\Pi^{-1}(\langle \mathcal{M},\mathcal{B}_1\rangle \mathcal{J}1_{\mathcal{B}}\mathcal{J})1_{\mathcal{A}}.$$ Observe $\Pi^{-1}(\langle \mathcal{M},\mathcal{B}_1\rangle ) = C_{\widehat{\psi\otimes \omega}}(\langle M{\mathbin{\overline{\otimes}}}N, \widetilde{B}{\mathbin{\overline{\otimes}}}N\rangle) $ and $\widehat{\psi\otimes \omega}=(\psi\otimes \omega)\circ \widehat{E}_{\widetilde{B}{\mathbin{\overline{\otimes}}}N} = \widehat{\psi}\otimes \omega$. Then using $\widehat{\psi}= \psi \circ E_A\circ \widehat{E}_{\widetilde{B}}$ on $1_A\langle M,\widetilde{B} \rangle 1_A$, we can apply Lemma \[III1 factor tensor lemma\] (to the inclusion $A \subset 1_A\langle M,\widetilde{B} \rangle 1_A$ with the operator valued weight $E_A\circ \widehat{E}_{\widetilde{B}}$) and get that $$\mathcal{A}'\cap 1_{\mathcal{A}}\Pi^{-1}(\langle \mathcal{M},\mathcal{B}_1 \rangle )1_{\mathcal{A}} = \left[ A'\cap 1_A\langle M,\widetilde{B} \rangle 1_A\right]_{\widehat{\psi}} \otimes {\mathbb{C}}1_N \otimes {\mathbb{C}}1_{L^2({\mathbb{R}})}.$$ Since $\Pi$ is the identity on $\langle M{\mathbin{\overline{\otimes}}}N,\widetilde{B}{\mathbin{\overline{\otimes}}}N\rangle$, $d$ is also contained in this set. Finally by multiplying $\mathcal{J}1_{\mathcal{B}}\mathcal{J} = J1_BJ\otimes 1_N\otimes 1_{L^2({\mathbb{R}})}$, we get the conclusion of the claim. By the claim, we can regard that $d$ is contained in $\left[ A'\cap 1_A\langle M,\widetilde{B} \rangle J1_{B}J1_A\right]_{\widehat{\psi}}$. As we mentioned in Section \[Preliminaries\], $\widehat{E}_{\mathcal{B}_1}$ coincides with $\widehat{E}_{\widetilde{B}{\mathbin{\overline{\otimes}}}N}\rtimes {\mathbb{R}}$ (the natural crossed product extension of $\widehat{E}_{\widetilde{B}{\mathbin{\overline{\otimes}}}N}$), hence the restriction of $\widehat{E}_{\mathcal{B}_1}$ on $\langle M{\mathbin{\overline{\otimes}}}N, \widetilde{B}{\mathbin{\overline{\otimes}}}N \rangle$ coincides with $\widehat{E}_{\widetilde{B}{\mathbin{\overline{\otimes}}}N}$. It then holds that $$\infty > \widehat{E}_{\mathcal{B}_1}(d) =\widehat{E}_{\widetilde{B}{\mathbin{\overline{\otimes}}}N}(d) = (\widehat{E}_{\widetilde{B}}\otimes {\text{\rm id}}_N)(d)=\widehat{E}_{\widetilde{B}}(d).$$ Thus $d$ satisfies the condition in Theorem \[thm unital intertwining with actions\](4) and we get $(A,\sigma^{\psi})\preceq^{\rm uni}_{M} (B,\sigma^{\varphi})$. By Lemma \[relation for unital and corner of actions\](1) and Theorem \[relation for expectations and actions\], this is equivalent to item (1). We first prove the equivalence of the first two conditions. Assume that $A\preceq_MB$. By Lemma \[intertwining implies expectation\], there is a projection $q\in A' \cap 1_AM1_A$ and a faithful normal conditional expectation $E_{Aq}\colon qMq\to Aq$ such that $(Aq,E_{Aq})\preceq_M(B,E_B)$. Put $A^q := W^\{A,q\} = Aq \oplus Aq^{\perp}$, where $q^\perp := 1_A -q$. Observe that $Aq^\perp \subset q^\perp Mq^\perp$ is with expectation, say $E_{Aq^\perp}$. Then by definition, the condition $(Aq,E_{Aq})\preceq_M(B,E_B)$ implies $(A^{q},E_{Aq}\oplus E_{Aq^{\perp}})\preceq_M(B,E_B)$. Since $A \subset 1_AM1_A$ is with expectation, $A \subset A^q$ is also with expectation. By Lemma \[intertwining for unital subalgebra\], it holds that $(A,E_{A})\preceq_M(B,E_B)$ for some faithful normal conditional expectation $E_A \colon 1_AM1_A\to A$. By Theorem \[relation for expectations and actions\], we get that $(A,\sigma^\psi)\preceq_M(B,\sigma^\varphi)$ for any faithful $\psi\in M_^+$ which is preserved by $E_A$. This finishes the proof of the first part of the theorem. We next prove the equivalence of items (1), (2), and (3). The equivalence of items (1) and (2) is proved in Theorem \[relation for expectations and actions\]. Using Lemma \[key lemma\], item (3) is also equivalent. Crossed products with groups in the class $\mathcal{C}$ {#Crossed products on type III factors} ======================================================= In this section we prove Theorem \[thmC\]. Throughout this section, we will fix an outer action $\Gamma \curvearrowright^\alpha B$ of a discrete group $\Gamma$ on a $\sigma$-finite diffuse factor $B$. We put $M:=B\rtimes_\alpha \Gamma$. General facts on outer actions {#general-facts-on-outer-actions .unnumbered} ------------------------------ We first recall several well known facts on outer actions and associated crossed products. \[lemma for core of crossed product\] Let $\varphi$ be a faithful normal state on $M$ which is preserved by $E_B$. Then one can define a $\Gamma$-action $\widetilde{\alpha}$ on $C_{\varphi}(B)$ by, for all $g\in \Gamma$, $b\in B$, $t\in {\mathbb{R}}$, $$\begin{aligned} \widetilde{\alpha}_g(b)=\alpha_g(b) \quad \text{and} \quad \widetilde{\alpha}_g(\lambda_t^{\varphi})= [D (\varphi\circ \alpha_{g^{-1}}) : D \varphi]_t \lambda_t^{\varphi}.\end{aligned}$$ We have a canonical identification $$(B\rtimes_{\alpha} \Gamma)\rtimes_{\sigma^\varphi}{\mathbb{R}}\simeq (B\rtimes_{\sigma^{\varphi}}{\mathbb{R}})\rtimes_{\widetilde{\alpha}} \Gamma$$ which is the identity on $B$, $L\Gamma$, and $L_\varphi {\mathbb{R}}$. This follows by direct computations by using ${\operatorname{Ad}}(\Sigma)$, where $\Sigma$ is the flip map on $L^2(B)\otimes \ell^2(\Gamma) \otimes L^2({\mathbb{R}})$ for the second and the third components. \[lemma for Fourier decomposition\] Let $p\in B$ be a projection, $B_0\subset pBp$ an irreducible subfactor, and $\beta\colon B_0 \to B_0$ a $\ast$-homomorphism such that $\beta(B_0)' \cap pBp = {\mathbb{C}}p$. Let $x\in pMp$ be any element with the Fourier decomposition $x= \sum_{g\in \Gamma}x_g \lambda_g$. If $x y = \beta(y)x$ for all $y\in B_0$, then we have that - $x_g \lambda_g y = \beta(y)x_g \lambda_g$ and $x_g \alpha_g(y) = \beta(y)x_g$ for all $y\in B_0$ and $g\in \Gamma$; - $x_g x_g^ \in {\mathbb{C}}p$ and $x_g^* x_g \in {\mathbb{C}}\alpha_{g}(p)$; - if $x\in \mathcal{U}(pMp)$ and $B_0' \cap pMp={\mathbb{C}}p$, there is a unique $g\in \Gamma$ such that $x= x_g \lambda_g$. For all $y \in B_0$, we have $$\sum_{g\in \Gamma} x_g \lambda_gy= x y=\beta(y)x = \sum_{g\in \Gamma} \beta(y)x_g \lambda_g.$$ By comparing coeffients, one has $x_g \lambda_g y = \beta(y)x_g \lambda_g$ and $x_g \alpha_g(y) = \beta(y)x_g$ for all $y\in B_0$ and $g\in \Gamma$. It holds that $x_g x_g^* = x_g \lambda_g ( x_g \lambda_g)^* \in \beta(B_0)' \cap pBp = B_0' \cap pBp={\mathbb{C}}p$, and $\alpha_{g^{-1}}(x_g^* x_g) = ( x_g \lambda_g)^x_g \lambda_g \in B_0' \cap pBp={\mathbb{C}}p$ for all $g\in \Gamma$. Assume further that $x$ is a unitary in $pMp$ and $B_0' \cap pMp={\mathbb{C}}p$. Fix $g\in \Gamma$ such that $x_g\neq 0$. Then it holds that $$x_g \lambda_g y = \beta(y)x_g \lambda_g = xyx^ x_g \lambda_g,$$ hence $x^x_g \lambda_g \in B_0' \cap pMp = {\mathbb{C}}p$. We conclude that $x=x_g\lambda_g$. \[lemma for normal subgroup\] Let $\Lambda\curvearrowright^\beta A$ be any outer action of a discrete group on a factor. Assume that $M = A \rtimes_\beta\Lambda$ such that $A\subset B$. Then there is a surjective homomorphism $\pi \colon \Lambda \to \Gamma$ such that - for any $h\in \Lambda$ there is a unique $u_h\in \mathcal{U}(B)$ such that $\lambda^\Lambda_h=u_h \lambda_{\pi(h)}^\Gamma$; - $B=A \rtimes_\beta \ker(\pi)$. In particular, $\beta$ induces a cocycle action $\Lambda/\ker(\pi) \curvearrowright A\rtimes_\beta \ker(\pi)$, and it is cocycle conjugate to $\alpha$ via $A\rtimes_\beta\ker(\pi) = B$ and $\pi\colon \Lambda/\ker(\pi) \simeq \Gamma$. Since $A' \cap M ={\mathbb{C}}$, by Lemma \[lemma for Fourier decomposition\], any $\lambda_h^\Lambda$ for $h\in \Lambda$ can be uniquely written as $\lambda^\Lambda_h=u_h \lambda_{g}^\Gamma$ for some $g\in \Gamma $ and some $u_h\in \mathcal{U}(B)$. By the uniqueness, if we put $g=\pi(h)$, then $\pi \colon \Lambda \to \Gamma$ define a homomorphism. Since $A$ and $\lambda^\Lambda_h$ $(h\in \Lambda)$ generate $M$, $B$ and $\pi(\Gamma)$ generate $M$ as well. This implies that $\pi(\Lambda)=\Gamma$ and $\pi$ is surjective. Put $\Lambda_0:=\ker(\pi)$. By construction, $\lambda_h = u_h$ for all $h\in \Lambda_0$ and hence $B_0:=A\rtimes_\beta \Lambda_0 \subset B$. We have to show the opposite inclusion. Let $E_B\colon M \to B$ and $E_{B_0}\colon M \to B_0$ be canonical conditional expectations. Observe that $E_{B_0}\circ E_B = E_{B_0}$. Fix any faithful normal state $\varphi$ on $B_0$ and extend it by $\varphi\circ E_{B_0}$. Then $E_B$ and $E_{B_0}$ extend to Jones projections $e_B$ and $e_{B_0}$ on $L^2(M,\varphi)$. Let $x=\sum_{h\in \Lambda} x_h \lambda_h^\Lambda \in A\rtimes_\beta \Lambda$ be any element with the Fourier decomposition. Then we have that $$e_{B}\Lambda_\varphi(x) = \sum_{h\in \Lambda} e_B\Lambda_\varphi(x_h \lambda_h^\Lambda) = \sum_{h\in \Lambda} e_B\Lambda_\varphi(x_h u_h\lambda_{\pi(h)}^\Gamma)= \sum_{h\in \Lambda_0} \Lambda_\varphi(x_h u_h) = \sum_{h\in \Lambda_0} \Lambda_\varphi(x_h \lambda_h^\Lambda) .$$ Since the last element is contained in $A\rtimes_\beta \Lambda_0$, we get that $B\subset A\rtimes_\beta\Lambda_0$. Put $\widetilde{\Lambda}:=\Lambda/\Lambda_0$ and $\widetilde{A}:=A\rtimes_\beta\Lambda_0$, and fix any section $s\colon \widetilde{\Lambda} \to \Lambda$ such that $s(\Lambda)=e$. For any $g,h \in \widetilde{\Lambda}$, we define $\lambda^{\widetilde{\Lambda}}_{g}:=\lambda^{\Lambda}_{s(g)}$, $\widetilde{\beta}_{g} := {\operatorname{Ad}}(\lambda^\Lambda_{s(g)}) \in \mathrm{Aut}(\widetilde{A})$, $\widetilde{u}_g:=u_{s(g)}$, and $c(g,h):=\lambda^{\Lambda}_{ s(g)s(h)s(gh)^{-1} }\in L\Lambda_0$. Then it is easy to check that $(\widetilde{\beta},c)$ defines a cocycle action of $\widetilde{\Lambda}$ on $\widetilde{A}$, and that $\widetilde{\beta}_{g} = {\operatorname{Ad}}(\widetilde{u}_{s(g)})\circ \alpha_{\pi(g)}$ and $ 1 = \widetilde{u}_{g}^ \widetilde{\beta}_{g}(\widetilde{u}_{h}^) c(g,h) \widetilde{u}_{gh}$ for all $g,h\in \widetilde{\Lambda}$. Thus using $\widetilde{A}=B$ and $\pi \colon \widetilde{\Lambda} \simeq \Gamma$, $(\widetilde{u}_{g})_{g\in \widetilde{\Lambda}}$ gives a cocycle conjugacy between $\widetilde{\Lambda}\curvearrowright^{(\widetilde{\beta},c)} \widetilde{A}$ and $\Gamma \curvearrowright^\alpha B$. Actions of groups in the class $\mathcal C$ {#actions-of-groups-in-the-class-mathcal-c .unnumbered} ------------------------------------------- We continue to use the outer action $\Gamma\curvearrowright^\alpha B$ on a $\sigma$-finite diffuse factor and $M=B\rtimes\Gamma$. The next proposition is a generalization of [@IPP05 Lemma 8.4]. \[intertwining for crossed products\] Let $p\in B$ be a projection and $A\subset pMp$ be a subfactor with expectation such that $A' \cap pMp ={\mathbb{C}}p$ and $\mathcal{N}_{pMp}(A)''=pMp$. 1. If $A\preceq_M B$, then there exist $(e,f,\theta,v)$ witnessing $A\preceq_M B$ and a finite normal subgroup $K \leq \Gamma$ such that $$\theta(eAe)' \cap fBf = {\mathbb{C}}f, \quad vv^=e, \quad v^v \in \theta(eAe)' \cap f(B\rtimes K)f .$$ Assume further that $\Gamma$ has no finite normal subgroups, and that either both of $A,B$ are of type II$_1$ or both are properly infinite. Then we can choose $e=f=p$ and $v\in \mathcal{U}(pMp)$. 2. Assume that $p=1$ and that $A$ has a decomposition $M=A\rtimes \Lambda$ for some outer action of a discrete group $\Lambda$ on $A$. Assume that $\Gamma$ and $\Lambda$ are ICC. If $A\preceq_M B$ and $B\preceq_M A$, then $A$ and $B$ are unitarily conjugate in $M$. \(1) Since $B$ is a factor, using [@HI15 Remark 4.5], we may assume that $A\preceq_{M} pBp$. We first show that, using the assumption $A' \cap pMp ={\mathbb{C}}p$, there is $(e,f,\theta,v)$ which witnesses $A\preceq_M pBp$ such that $\theta(eAe) \subset fBf$ is irreducible. Since $vv^ \in (eAe)' \cap eMe={\mathbb{C}}e$, one has $vv^=e$ and moreover $v^v$ is a minimal projection in $\theta(eAe)' \cap fMf$. Indeed, for any projection $r \leq v^v$ in $\theta(eAe)' \cap fMf$, $vrv^\in (eAe)' \cap eMe={\mathbb{C}}e$ is again $e$, hence $r=vv^$. We may assume that the support projection of $E_B(v^v)$, which is contained in $\theta(eAe)' \cap fBf$, coincides with $f$. Let $z$ be the central support projection of $v^v$ in $\theta(eAe)' \cap fMf$. Then since $v^v$ is minimal, $(\theta(eAe)' \cap fMf)z$ is a type I factor. Since $\theta(eAe) \subset fBf$ is with expectation, so is the inclusion $\theta(eAe)' \cap fBf \subset \theta(eAe) ' \cap fMf$. In particular, $(\theta(eAe)' \cap fBf)z$ is an atomic von Neumann algebra. Since $z$ commutes with $\theta(eAe)' \cap fBf$, there is a unique projection $w \in \mathcal{Z}(\theta(eAe)' \cap fBf)$ such that $(\theta(eAe)' \cap fBf)w \ni aw \mapsto az \in (\theta(eAe)' \cap fBf)z$ is isomorphic. Thus there is a minimal projection $q$ in $\theta(eAe)' \cap fBf$. Since $q \leq f$, $q$ is smaller than the support of $E_B(v^v)$, hence $vq \neq 0$. Now $(e,q,\theta(\cdot)q,vq)$ witness $A\preceq_M pBp$ (up to the polar decomposition of $vq$) and satisfies that $\theta(eAe)q \subset qBq$ is an irreducible inclusion. Thus we can start the proof by assuming $\theta(eAe)' \cap fBf={\mathbb{C}}f$. Put $B_0:=\theta(eAe) \subset fBf$ and note that $B_0 ' \cap fBf={\mathbb{C}}f$. Consider the Fourier decomposition $q:=v^v = \sum_{g\in \Gamma} x_g \lambda_g \in B\rtimes \Gamma$. Since $q \in B_0' \cap fMf$, by Lemma \[lemma for Fourier decomposition\], it holds that $x_g \lambda_g \in B_0' \cap fMf$, $x_gx_g^* = {\mathbb{C}}f$, and $x_g^x_g \in {\mathbb{C}}\alpha_{g}(f)$. Define subgroups $K, \Gamma_0 \leq \Gamma$ by $$\begin{aligned} K:=& \{ g\in \Gamma \mid {\operatorname{Ad}}(w_g)\circ \alpha_g\|_{B_0} = {\text{\rm id}}_{B_0} \text{ for some } w_g\in B \text{ s.t.\ } w_gw_g^ =f, \ w_g^w_g = \alpha_g(f) \};\\ \Gamma_0:=& \{ g\in \Gamma \mid {\operatorname{Ad}}(w_g)\circ \alpha_g(B_0) = B_0 \text{ for some } w_g\in B \text{ s.t.\ } w_gw_g^ =f, \ w_g^w_g = \alpha_g(f)\}.\end{aligned}$$ By definition, $q$ is contained in $B\rtimes K$ and $K$ is a normal subgroup of $\Gamma_0$. We will prove that $\|K\|<\infty$ and $\Gamma_0=\Gamma$. We claim that $K$ is a finite group. Fix $(w_g)_{g\in K}$ which appeared in the definition of $K$ such that $w_e=1$. For all $g,h\in K$, define $$\alpha^w_g :={\operatorname{Ad}}(w_g)\circ \alpha_g \quad \text{and} \quad \mu_{g,h}:= w_{g}\alpha_g(w_h)w_{gh}^ \in \mathcal{U}(fBf)$$ and observe that $(\alpha^w,\mu)$ gives a cocycle action of $K$ on $fBf$, so that $f(B\rtimes_\alpha K)f = fBf \rtimes_{(\alpha^w,\mu)}K$. The condition $\alpha^w\|_{B_0} = {\text{\rm id}}_{B_0}$ implies that $\mu_{g,h} \in {\mathbb{C}}f$ for all $g,h\in K$, hence we can regard $\mu$ as a scalar 2-cocycle. In particular $fBf \rtimes_{(\alpha^w,\mu)}K$ contains a finite von Neumann algebra $({\mathbb{C}}f)\rtimes_{(\alpha^w,\mu)}K$. Since $B_0 ' \cap fBf = {\mathbb{C}}f$ and $\alpha^w\|_{B_0} = {\text{\rm id}}_{B_0}$, using Fourier decompositions, it is easy to see that $$B_0' \cap [fBf \rtimes_{(\alpha^w,\mu)}K] = ({\mathbb{C}}f) \rtimes_{(\alpha^w,\mu)}K.$$ The left hand side contains the minimal projection $q$, and hence so does the right hand side. This implies that $K$ is a finite group. (Indeed if infinite, one has a sequence of unitaries which converges weakly to 0, but it is impossible in a finite von Neumann algebra with a minimal projection.) We next claim that $\Gamma=\Gamma_0$. Observe that $eAe\subset e(B\rtimes \Gamma)e$ is regular and $eAe$ is a diffuse factor. Since ${\operatorname{Ad}}(v^)$ is an isomorphism between $eAe\subset e(B\rtimes \Gamma)e$ and $B_0q \subset q(B\rtimes \Gamma)q$, it holds that $B_0q \subset q(B\rtimes \Gamma)q$ is regular. Fix $u\in \mathcal{N}_{q (B\rtimes \Gamma)q}(B_0q)$ and consider the Fourier decomposition $u = \sum_{g\in \Gamma} x_g \lambda_g \in B \rtimes \Gamma$. Since ${\operatorname{Ad}}(u)$ is an isomorphism on $B_0q$, using $B_0q \simeq B_0$, we can define $\beta^u \in \mathrm{Aut}(B_0)$ by $\beta^u(y)q=uyu^$ for all $y\in B_0$. By Lemma \[lemma for Fourier decomposition\], we get that for all $y\in B_0$ and $g\in \Gamma$, $$x_g \lambda_g y = \beta^u(y)x_g \lambda_g, \quad x_g x_g^* \in {\mathbb{C}}f, \quad \text{and} \quad x_g^* x_g \in {\mathbb{C}}\alpha_g(f).$$ So each $x_g \in fB\alpha_g(f)$ is a scalar multiple of a partial isomrtry. Observe that ${\operatorname{Ad}}(x_g\lambda_g)(y) = \beta^u(y)x_g x_g^* \in \beta^u(B_0)=B_0$ for all $y\in B_0$, so ${\operatorname{Ad}}(x_g\lambda_g)$ preserves $B_0$. By definition, this means that if $x_g\neq 0$, then $g \in \Gamma_0$. Hence it holds that $u\in q(B\rtimes \Gamma_0)q$. Since $Bq \subset q(B\rtimes \Gamma)q$ is regular, we conclude that $q(B\rtimes \Gamma)q=q(B\rtimes \Gamma_0)q$. Since $q \in B\rtimes \Gamma_0$ and since $B\rtimes \Gamma_0$ is a diffuse factor, we indeed have that $B\rtimes \Gamma= B\rtimes \Gamma_0$. This means that $\Gamma= \Gamma_0$. Finally assume that $\Gamma$ has no finite normal subgroups. Then $K$ must be trivial, so $v^v \in B$ and we may assume $f=v^v$. We have that there is a partial isometry $v\in pMp$ such that $vv^=e\in A$, $v^v =f\in pBp$, and $v^Av \subset fBf$. If both of $A,B$ are II$_1$ factors or if both of $A,B$ are properly infinite, then (up to exchanging $e,f$ by smaller ones if necessarily,) we can apply a usual patching method, and obtain that $e=f=p$ and $v\in \mathcal{U}(pMp)$. This is the conclusion. \(2) Observe that, since $A\rtimes \Lambda=M = B\rtimes \Gamma$, $A$ is a $\rm II_1$ factor if and only if so is $B$. Hence using item (1) of this proposition, we can find $v,w\in \mathcal{U}(M)$ such that $vAv^ \subset B$ and $wBw^* \subset A$. Put $u:=vw$ and observe that $uBu^* \subset B$ and $(uBu^)' \cap B \subset (uBu^)' \cap M = u(B' \cap M)u^* ={\mathbb{C}}$. By Lemma \[lemma for Fourier decomposition\], we can write $u=x_g \lambda_g$ for some $g\in \Gamma $ and $x_g \in \mathcal{U}(B)$. In particular we have $B = uBu^* = vw B w^* v^* \subset v A v^* \subset B$. We conclude that $vAv^=B$. The next lemma explains how we use the property of the class $\mathcal C$ for actions on type III factors. This uses our Theorem \[thmA\]. \[lemma for class C\] Let $p\in M$ be a projection, and $A \subset pMp$ be a subfactor with expectation $E_A$. Assume that $\Gamma$ is in the class $\mathcal{C}$, $A' \cap pMp = {\mathbb{C}}$, $A$ is amenable, and $\mathcal{N}_{pMp}(A)'' \subset pMp$ has finite index. Then we have $A \preceq_M B$. Put $P:=\mathcal{N}_{pMp}(A)''$ and let $N$ be the hyperfinite type $\rm III_1$ factor and $\omega$ a faithful normal state such that $N_\omega' \cap N ={\mathbb{C}}$. Let $E_A,E_P$ be any faithful normal conditional expectations for $A,P$ respectively. Observe that the condition $A' \cap pMp\subset A$ implies that normal expectations onto $A$ and $P$ are unique, hence $E_A\circ E_P=E_A$. Using this uniqueness and using Theorem \[thmA\], there exist $\psi,\varphi$, which are preserved by $E_A,E_B$ respectively such that $$\Pi_{\varphi\otimes \omega,\psi\otimes \omega}(C_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}N)) \not \preceq_{C_{\varphi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N)}C_{\varphi\otimes \omega}(B{\mathbin{\overline{\otimes}}}N).$$ There is a canonical inclusion $C_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}N) \subset C_{\psi\otimes \omega}(P{\mathbin{\overline{\otimes}}}N)$, which is regular by [@BHV15 Lemma 4.1]. For notation simplicity, we omit $\Pi_{\varphi\otimes \omega,\psi\otimes \omega}$ and write as $\mathcal{M}:=C_{\varphi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N)$, $\mathcal{B}:=C_{\varphi\otimes \omega}(B{\mathbin{\overline{\otimes}}}N)$, $\mathcal{A}:=C_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}N)$, and $\mathcal{P}:=C_{\psi\otimes \omega}(P{\mathbin{\overline{\otimes}}}N)$. Observe that $\mathcal{A}$ is amenable and $\mathcal{P} \subset \mathcal{M}$ has finite index. By Lemma \[lemma for core of crossed product\], there is an identification $\mathcal{M} = \mathcal B \rtimes_{\widetilde{\alpha}}\Gamma$. Let $r\in L_{\varphi\otimes \omega}{\mathbb{R}}$ be any projection such that ${\mathord{\text{\rm Tr}}}_{\varphi\otimes \omega}(r)<\infty$. Then since $\mathcal{B}$ is a type II$_\infty $ factor and since $\widetilde{\alpha}$ preserves the canonical trace on $\mathcal B$, $r\mathcal {M}r$ is realized as a cocycle crossed product $r\mathcal{B}r \rtimes_{(\widetilde{\alpha}^r,u)} \Gamma$ for some 2-cocycle $u\colon \Gamma \times \Gamma \to r\mathcal{B}r$. Since $\mathcal{M}$ is a II$_\infty$ factor, $p$ is infinite, and $r$ is finite, there is $v\in \mathcal{M}$ such that $vv^ = r$ and $p_0:=v^v \in p\mathcal{A}p$. Put $\mathcal A^v:= v\mathcal A v^$. Observe that $\mathcal{A}^v$ is amenable and that $(\mathcal{A}^v)' \cap r\mathcal{M}r={\mathbb{C}}r$ (use Lemma \[III1 factor tensor lemma\]). Since $\mathcal{A}$ is a II$_\infty$ factor, it holds that $p_0\mathcal{N}_{p\mathcal Mp}(\mathcal A)''p_0 = \mathcal{N}_{p_0\mathcal Mp_0}(p_0\mathcal Ap_0)''$. In particular $\mathcal{N}_{r\mathcal{M}r}(\mathcal{A}^v)'' \subset r\mathcal{M}r$ has finite index. Hence by the definition of the class $\mathcal C$, we have $\mathcal{A}^v\preceq_{r\mathcal{M}r}r\mathcal{B}r$. This implies $\mathcal{A}\preceq_{\mathcal{M}}\mathcal{B}$ and hence by Theorem \[thmA\], we obtain $A \preceq_M B$. By Lemma \[lemma for class C\], we have $A\preceq_M B$. Observe that, $A$ is a type $\rm II_1$ factor if and only if so is $B$. Hence we can apply Proposition \[intertwining for crossed products\], and find a unitary $u\in \mathcal{U}(M)$ such that $uAu^* \subset B$. Thus we may assume that $A \subset B$. We then apply Lemma \[lemma for normal subgroup\] and get the conclusion. Note that $\ker(\pi)$ is amenable since $A\rtimes \ker(\pi)$ is amenable and $A$ is a factor. Rigidity of Bernoulli shift actions =================================== In this section, we will study Bernoulli shift actions with type III base algebras. We particularly prove Theorem \[thmB\] and Proposition \[thmE\]. Popa’s criterion for cocycle superrigidity {#popas-criterion-for-cocycle-superrigidity .unnumbered} ------------------------------------------ The next proposition is a variant of Popa’s theorem which was used to prove cocycle superrigidity [@Po04; @Po05a; @Po05b]. See also [@VV14 Theorem 7.1]. \[prop for cocycle superrigidity\] Let $G$ be a locally compact second countable group, $G_1\leq G$ a closed normal subgroup, $(P,\varphi)$ a von Neumann algebra with a faithful normal state. Let $G \curvearrowright^\alpha (P,\varphi)$ be a state preserving continuous action. Let $\omega\colon G \to \mathcal{U}(P)$ be a $\sigma$-strongly continuous map such that $\beta_g:={\operatorname{Ad}}(\omega_g)\circ \alpha_g$ and $v(g,h):=\omega_g \alpha_g(\omega_h) \omega_{gh}^$ for $g,h\in G$ define a cocycle action of $G$. Assume that - $v(g,h)=1=v(h,g)$ for all $g\in G_1$ and $h\in G$ (hence $\beta\|_{G_1}$ is a genuine action); - there is a faithful state $\psi\in P_$ which is preserved by $\beta\|_{G_1}$; - $({\mathbb{C}}p , \beta\|_{G_1}) \preceq^{\rm uni}_{P} ({\mathbb{C}}1_P ,\alpha\|_{G_1})$ for all projections $p\in P^\beta$; - $\alpha\|_{G_1}$ is weakly mixing. Then there exist a separable Hilbert space $H$, a projection $f\in {\mathbb{B}}(H)$, a $\sigma$-strongly continuous map $u\colon G \to \mathcal{U}(f{\mathbb{B}}(H)f)$, a partial isometry $w\in P{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)$ such that $$w^w = f, \quad ww^ = 1\otimes e_{1,1}, \quad \text{and} \quad wu_g = (w_g\otimes 1_H )(\alpha_g\otimes {\text{\rm id}}_H) (w) \quad \text{for all }g\in G,$$ where $e_{1,1} $ is a minimal projection in ${\mathbb{B}}(H)$. In particular, $({\operatorname{Ad}}(u_g))_{g\in G}$ and $(u_gu_h u_{gh}^)_{g,h\in G}$ define a cocycle action on $f{\mathbb{B}}(H)f$, and $\beta$ is conjugate to the cocycle action $(\alpha_g\otimes {\operatorname{Ad}}(u_g))_{g\in G}$ by $w$: $$\beta_g(wxw^)=\alpha_g^\omega( wxw^) = w (\alpha_g\otimes {\operatorname{Ad}}(u_g))(x) w^, \quad \text{for all }x\in P{\mathbin{\overline{\otimes}}}f{\mathbb{B}}(H)f.$$ Since most of proofs are straightforward adaptations of [@VV14 Theorem 7.1], we give only a sketch of the proof. Take $(H,f,\pi,w)$ and $(u_g)_{g\in G_1}$ which witness $({\mathbb{C}}p , \beta\|_{G_1})\preceq_{P}({\mathbb{C}}1_P, \alpha\|_{G_1})$ (and $H$ can be finite dimensional). Observe that $w^w \in (P{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H))^{\alpha\otimes {\operatorname{Ad}}(u)\|_{G_1}} = {\mathbb{C}}1_P {\mathbin{\overline{\otimes}}}{\mathbb{B}}(H) $ (because $\alpha\|_{G_1}$ is weakly mixing), hence up to exchanging $f$ by $w^w$, we may assume that $w^w = f$. Thus the condition $({\mathbb{C}}p , \beta\|_{G_1})\preceq_{P}({\mathbb{C}}1_P, \alpha\|_{G_1})$ means that there exist $(n,f,w,u)$: a projection $f\in {\mathbb{M}}_n$, a continuous homomorphism $u \colon G_1 \to \mathcal{U}(f{\mathbb{M}}_nf)$, and a partial isometry $w\in (p\otimes e_{1,1})(P\otimes {\mathbb{M}}_n)f$ such that $wu_g = (\omega_g\otimes 1_n )(\alpha_g\otimes {\text{\rm id}}_n) (w)$ for all $g\in G_1$. There exist a separable Hilbert space $H$, a projection $f\in {\mathbb{B}}(H)$, a partial isometry $w\in P{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)$, and a continuous homomorphism $u\colon G_1 \to \mathcal{U}(f{\mathbb{B}}(H)f)$ such that - $wu_g = (\omega_g\otimes 1_H )(\alpha_g\otimes {\text{\rm id}}_H) (w)$ for all $g\in G_1$; - $w^w = f$ and $ww^* \in pP^\beta p{\mathbin{\overline{\otimes}}}{\mathbb{C}}e_{1,1}$, where $e_{1,1}$ is a fixed minimal projection; - there exist finite rank projections $(P_k)_{k\in {\mathbb{N}}}$ in ${\mathbb{B}}(H)$ such that $P_k \to 1_H$ as $k\to \infty$ and that each $P_k$ commutes with $u_g$ for all $g\in G_1$. Let $\mathcal E$ denote the set of all nonzero projections $e\in P (=P\otimes {\mathbb{C}}e_{1,1})$ such that there exist $(n,f,w,u)$ which witnesses $({\mathbb{C}}p , \beta\|_{G_1})\preceq_{P}({\mathbb{C}}1_P, \alpha\|_{G_1})$ with $e = ww^$. Then it is straightforward to check that $\mathcal E$ is closed under the following operations: $\alpha_h(e)\in\mathcal E$ for all $h\in G$ and for all $e\in \mathcal{E}$; $e\vee f\in\mathcal E$ for all $e,f \in \mathcal{E}$; and $e_0 \in \mathcal E$ for all projections $e_0 \in eP^{\beta\|_{G_1}}e$ and $e\in \mathcal{E}$. Fix any countable dense subset $X \subset G$. Observe that $\sup_{h\in X} \alpha_h(e) \in pP^{\beta}p$ is realized as a (countably) infinite direct sum of projections in $\mathcal E$, that is, there is a family $(n_i,f_i,w_i,u^i)_{i\in I}$ such that $\sum_{i\in I}w_iw_i^ = \sup_{h\in X} \alpha_h(e)$, where $I$ is a countable set. By defining $H:= \bigoplus_{i\in I} {\mathbb{C}}^{n_i}$, $f:=\bigoplus_{i\in I} f_i$, $w=[w_i]_{i\in I} \in (p\otimes e_{1,1})(B{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H))f$, and $u:=\bigoplus_{i\in I}u^i$, we get the conclusion. Now we define $\mathcal F$ as the set of all nonzero projections $e\in P^\beta (= P^\beta \otimes {\mathbb{C}}e_{1,1})$ such that there exists $(H,f,w,u)$ which witnesses the conclusion of the claim above with $e=ww^$. Now using the assumption $({\mathbb{C}}p , \beta\|_{G_1})\preceq_P ({\mathbb{C}}1_P , \alpha\|_{G_1})$ for all $p \in P^\beta$ and applying a maximality argument, there is a family $(H_i,f_i,w_i,u^i)_{i\in I}$ such that $\sum_{i\in I}w_iw_i^ = 1_P (=1_P \otimes e_{1,1})$, where $I$ is a countable set. Define $(H,f,w,u)$ as a direct sum of all $(H_i,f_i,w_i,u^i)_{i\in I}$ (with $w=[w_i]_{i\in I} \in (1\otimes e_{1,1})(B{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H))$), and then it satisfies all the conditions in the claim above with $ww^* =1 \otimes e_{1,1}$. Hence $(H,f,w,u)$ satisfies the conclusion of this theorem but only for $G_1$. We have to extend the conditions on $G_1$ to that on $G$, using the weak mixingness of $\alpha\|_{G_1}$. Put $\omega_g^H:=\omega_g\otimes 1_H$, $\alpha_g^H:=\alpha_g \otimes {\text{\rm id}}_H$, $\beta_g^H:=\beta_g \otimes {\text{\rm id}}_H$, and $v^H(g,h):=v(g,h)\otimes 1_H$ for all $g,h \in G$. Extend the map $u$ to the one on $G$ by $$u_g := w^* \omega_g^H\alpha_g^H (w), \quad \text{for all } g\in G.$$ It is easy to compute that for any $g,h\in G$, $$u_gu_g^* = f = u_g^u_g \quad \text{and}\quad u_{g}\alpha^H_g(u_h)=w^ v^H(g,h)w u_{gh}.$$ In particular, $u\colon G \to \mathcal{U}(P{\mathbin{\overline{\otimes}}}f{\mathbb{B}}(H)f)$ is a cocycle for $\alpha^H$ with a 2-cocycle $w^v^H(\cdot ,\cdot ) w$. To finish the proof, we have only to show that $u$ is a map into $f{\mathbb{B}}(H)f$, so that $\alpha_g^H(u_h) = u_h$ and $u_gu_h u_{gh}^=w^v^H(g,h)w \in f{\mathbb{B}}(H)f$ for all $g,h \in G$. Fix $g\in G$ and $k\in {\mathbb{N}}$. Put $H_k:=P_kH$ and $u_h^k:= P_k u_hP_k$ for all $h\in G$, where $(P_n)_{n\in {\mathbb{N}}}$ is a family of finite rank projections as in the claim (and we regard $P_k = 1_P \otimes P_k$). Then since $P_k$ commutes with $u_h$ for all $h\in G_1$, putting $\alpha_h^u:={\operatorname{Ad}}(u_h)\circ \alpha_h$, it holds that $$\alpha_h^u(u_g^k)= P_k\alpha_h^u(u_g)P_k = u_{g}^k u^k_{g^{-1}hg}(u_h^k)^ \in u_g^k {\mathbb{B}}(H_k) , \quad \text{for all }h\in G_1.$$ Observe that $\alpha_h^u$ is of the form that $\alpha_h\otimes {\operatorname{Ad}}(u_h)$ for all $h\in G_1$. Then combining the weak mixingness of $\alpha\|_{G_1}$ with $(\alpha_h\otimes {\operatorname{Ad}}(u_h^k))(u_g^k) \in u_g^k {\mathbb{B}}(H_k)$ for all $h\in G_1$, it holds that $u_g^k \in {\mathbb{B}}(H_k)$. Since $k$ is arbitrary, we obtain that $u_g \in {\mathbb{B}}(H)$ as required. Rigidity of Bernoulli shifts for cocycle actions {#rigidity-of-bernoulli-shifts-for-cocycle-actions .unnumbered} ------------------------------------------------ Let $\Gamma$ be a countable discrete group, $B_0$ an amenable von Neumann algebra with separable predual, $\varphi_0$ a faithful normal state on $B_0$, and $\Gamma \curvearrowright^\alpha \bigotimes_{\Gamma}(B_0,\varphi_0)=:(B,\varphi)$ the Bernoulli shift action. Put $M:=B\rtimes_\alpha \Gamma$. Here we recall the following fact. \[Bernoulli theorem2\] Let $p \in M$ be a projection and $A \subset p Mp$ a von Neumann subalgebra with expectation $E_A$. Fix a faithful $\psi \in M_$ which is preserved by $E_A$, and ${P}:={A}' \cap p {M}_\psi p$. If $C_\psi({A}) \not\preceq _{C_\varphi(M)} C_\varphi(L \Gamma)$, then $P$ has an amenable direct summand. This can be proved by applying arguments in [@CPS11 Theorem 4.1], which is based on the arguments in [@Po03; @Po04; @Po06a] (together with the deformation given in [@Io06]). Actually one has to modify the spectral gap argument [@Po06a] as follows. Put $\widetilde B:= \bigotimes_{\Gamma}(B_0L{\mathbb{Z}},\varphi_0\tau_{L{\mathbb{Z}}})$ and extend $\varphi$ and $\alpha$ on $\widetilde{B}$, so that there are canonical inclusions $M \subset \widetilde{B}\rtimes_\alpha \Gamma =:\widetilde{M}$ and $C_\varphi(M) \subset C_\varphi(\widetilde{M})$. Then we can prove the following weak containment: $${}_M L^2(C_\varphi(\widetilde{M}))\ominus L^2(C_\varphi(M))_{C_\varphi(M)} \prec {}_M L^2(C_\varphi(M))\otimes L^2(C_\varphi(M))_{C_\varphi(M)}$$ (e.g. see the proof of [@Ma16 Theorem 5.2]). Then using the spectral gap argument given in [@Ma16 Lemma 4.1], we can follow the proof of [@CPS11 Theorem 4.1]. Put $M:=B\rtimes_\alpha \Gamma = A\rtimes_\beta \Lambda$. Using Lemma \[lemma for class C\] and Proposition \[intertwining for crossed products\], we may assume $A \subset B$. Then by Lemma \[lemma for normal subgroup\], there is a surjective homomorphism $\pi \colon \Lambda \to \Gamma$ such that $A\rtimes_\beta \Lambda_0 = B$, where $\Lambda_0:=\ker \pi$, and for any $h\in \Lambda$, there is a unique $u_h \in \mathcal{U}(B)$ such that $\lambda_h^\Lambda = u_h \lambda_{\pi(h)}^\Gamma$. Put $\widetilde{A}:=A\rtimes_\beta \Lambda_0$ and $\widetilde{\Lambda}:=\Lambda/\Lambda_0$. Using a fixed section $s\colon \widetilde{\Lambda} \to \Lambda$ such that $s(\Lambda_0)$ is the unit, we will use the following notation: for all $g,h\in \widetilde{\Lambda}$, $\widetilde{\beta}_g:={\operatorname{Ad}}(\lambda^{\Lambda}_{s(g)}) \in {\operatorname{Aut}}(\widetilde{A})$, $c(g,h):=\lambda^{\Lambda}_{ s(g)s(h)s(gh)^{-1} }$, $\lambda_g^{\widetilde{\Lambda}} :=\lambda_{s(g)}^\Lambda$, and $u_g:=u_{s(g)}$. We have a cocycle action $\widetilde{\Lambda}\curvearrowright^{(\widetilde{\beta},c)} \widetilde{A}$ with relations $$\lambda_h^{\widetilde{\Lambda}}= u_g \lambda_{\pi(h)}^\Gamma, \quad {\operatorname{Ad}}(u_g)\circ \alpha_{\pi(g)} = \widetilde{\beta}_g, \quad c(g,h) =\widetilde{u}_{g}\alpha_{g}(\widetilde{u}_{h}) \widetilde{u}_{gh}^ \quad \text{for all }g,h\in \widetilde{\Lambda}.$$ For simplicity we identify $C_\psi(M)=C_\varphi(M)$. Then using Lemma \[lemma for core of crossed product\], there is an inclusion $$L_\psi {\mathbb{R}}\subset C_\psi(\widetilde{A}\rtimes_{\widetilde{\beta}}\widetilde{\Lambda}) = C_\varphi(M) = C_\varphi(B)\rtimes_\alpha \Gamma.$$ Observe that, since $\widetilde{\beta}$ is $\psi$-preserving, $(L_\psi{\mathbb{R}})' \cap C_\varphi(M)$ contains a copy of $L\widetilde{\Lambda}$ with expectation, hence $(L_\psi{\mathbb{R}})' \cap C_\varphi(M)$ has no amenable direct summand. We have $({\mathbb{C}}p, \sigma^\psi) p \preceq_{B} ({\mathbb{C}}1_B , \sigma^\varphi)$ for all projections $p\in B_{\psi}^{\widetilde{\beta}} $. Fix any projection $p\in B_{\psi}^{\widetilde{\beta}} $. Since $L\widetilde{\Lambda} p$ has no amenable summand, by applying Theorem \[Bernoulli theorem2\] to $L_\psi{\mathbb{R}}p$, we obtain that $L_\psi {\mathbb{R}}p \preceq_{C_\varphi(M)}C_\varphi(L\Gamma)$. By Theorem \[thm unital intertwining with actions\], to prove this claim, we have only to show that $L_\psi{\mathbb{R}}p \preceq_{C_\varphi(B)} L_\varphi {\mathbb{R}}$. Suppose by contradiction that $L_\psi{\mathbb{R}}p\not\preceq_{C_\varphi(B)} L_\varphi {\mathbb{R}}$. Take a net $(u_i)_i$ in $\mathcal{U}(L_\psi {\mathbb{R}})$ such that $$E_{L_\varphi {\mathbb{R}}} (b^* u_i p a) \to 0 , \quad \text{for all }a,b \in C_\varphi(B).$$ Observe that for all $h\in \widetilde{\Lambda}$ and $u_i\in L_\psi{\mathbb{R}}$, since $u_i$ commutes with $\lambda_h^{\widetilde{\Lambda}}$, $$\lambda^\Gamma_{\pi(h)} u_i p (\lambda^\Gamma_{\pi(h)})^={u}_{h}^\lambda_h^{\widetilde{\Lambda}} u_i p(\lambda^{\widetilde{\Lambda}}_{h})^{u}_{h}={u}_{h}^ u_i p {u}_{h}.$$ It holds that for all $a,b \in C_\varphi(B)$ and $g,h\in \widetilde{\Lambda}$, $$\begin{aligned} E_{C_\varphi(L\Gamma)} ( b \lambda_{\pi(h)}^\Gamma u_ip a\lambda_{\pi(g)}^\Gamma) &= E_{C_\varphi(L\Gamma)} ( b \left[\lambda_{\pi(h)}^\Gamma u_ip (\lambda_{\pi(h)}^\Gamma)^* \right] \alpha_{\pi(h)}(a)\lambda_{\pi(hg)}^\Gamma)\\ &= E_{C_\varphi(L\Gamma)} ( b \left[{u}_{h}^* u_i p {u}_{h} \right] \alpha_{\pi(h)}(a)\lambda_{\pi(hg)}^\Gamma)\\ &= E_{L_\varphi {\mathbb{R}}} ( b {u}_{h}^* u_i p {u}_{h} \alpha_{\pi(h)}(a)) \lambda_{\pi(hg)}^\Gamma \to 0.\end{aligned}$$ By [@HI15 Theorem 4.3(5)], we get $L_\psi{\mathbb{R}}p\not\preceq_{C_\varphi(M)} C_\varphi(L\Gamma)$, a contradiction. Put $G:=\Gamma \times {\mathbb{R}}$. Since $\alpha$ and $\sigma^\varphi$ commute, we can define a continuous action $G\curvearrowright^{\alpha^\varphi} (B,\varphi)$ by $$\alpha^\varphi_{(g,t)}:= \alpha_g\circ\sigma_t^\varphi = \sigma_t^\varphi \circ \alpha_g, \quad \text{for all }(g,t)\in G$$ The condition $B_\varphi={\mathbb{C}}$ then means that $\alpha^\varphi\|_{{\mathbb{R}}}$ is weakly mixing. In the same say, we can define a continuous cocycle action $\widetilde{\Lambda}\times {\mathbb{R}}\curvearrowright^{\widetilde{\beta}^\psi} (\widetilde{A},\psi)$ with the 2-cocycle $c^\psi((g,t), (h,s)) := c(g,h)$ for all $(g,t), (h,s)\in \widetilde{\Lambda}\times {\mathbb{R}}$. Identify $\widetilde{\Lambda} = \Gamma$ and $\widetilde{A} =B$. Define a $\sigma$-strongly continuous map $\omega\colon G \to \mathcal{U}(B)$ by $$\omega_{(g,t)}:=[D\psi:D\varphi]_t\sigma_t^{\varphi}(u_g)=\sigma_t^{\psi}(u_g) [D\psi:D\varphi]_t , \quad g\in \Gamma, \ t\in {\mathbb{R}}.$$ Then $\omega$ gives a cocycle conjugacy between $\alpha^\varphi$ and $\widetilde{\beta}^\psi$: for all $(g,t),(h,s)\in G$, $${\operatorname{Ad}}(\omega_{(g,t)})\circ \alpha_{(g,t)}^\varphi = \widetilde{\beta}_{(g,t)}^\psi \quad \text{and} \quad \omega_{(g,t)} \alpha^\varphi_{(g,t)} (\omega_{(h,s)}) = c^\psi((g,t),(h,s))\omega_{(gh,t+s)}.$$ Observe that for any $(g,t)\in G$, since $\lambda_t^\varphi$ and $\lambda_g^\alpha$ commute in $C_\varphi(M)$, $$\begin{aligned} &\lambda_g^\alpha \lambda_t^\varphi = u_g^* \lambda_g^{\widetilde{\beta}} [D\varphi: D\psi]_t \lambda_t^\psi = u_g^* \widetilde{\beta}_g([D\varphi: D\psi]_t) \lambda_g^{\widetilde{\beta}} \lambda_t^\psi \\ = \ & \lambda_t^\varphi \lambda_g^\alpha = [D\varphi: D\psi]_t \lambda_t^\psi u_g^* \lambda_g^\beta = [D\varphi: D\psi]_t \sigma_t^\psi(u_g^) \lambda_t^\psi \lambda_g^{\widetilde{\beta}} .\end{aligned}$$ Since $\lambda_t^\psi \lambda_g^{\widetilde{\beta}} = \lambda_g^{\widetilde{\beta}} \lambda_t^\psi $, using $[D\varphi: D\psi]_t^=[D\psi: D\varphi]_t$, we get that $$\omega_{(g,t)}=\sigma_t^\psi(u_g)[D\psi: D\varphi]_t = \widetilde{\beta}_g([D\psi: D\varphi]_t)u_g =u_g \alpha_g([D\psi: D\varphi]_t).$$ Recall that we have cocycle relations: $$\begin{aligned} &c(g,h) = {u}_{g}\alpha_{g}({u}_{h}) {u}_{gh}^, \quad \text{for all }g,h\in \Gamma; \\ & [D\psi: D\varphi]_{t+s} = [D\psi: D\varphi]_t\sigma_t^\varphi([D\psi:D\varphi]_s), \quad \text{for all }t,s\in {\mathbb{R}}.\end{aligned}$$ We then compute that for any $(g,t),(h,s)\in G$, $$\begin{aligned} \omega_{(g,t)} \alpha^\varphi_{(g,t)} (\omega_{(h,s)}) &= u_g \alpha_g([D\psi: D\varphi]_t)\alpha_g\circ\sigma^\varphi_{t} ([D\psi:D\varphi]_s\sigma_s^{\varphi}(u_h))\\ &= u_g \alpha_g([D\psi:D\varphi]_{t+s}\sigma^\varphi_{t+s}(u_h))\\ &= u_g \alpha_g(w_{(h,t+s)})\\ &= u_g \alpha_g(u_h \alpha_h([D\psi:D\varphi]_{t+s}))\\ &= c(g,h)u_{gh} \alpha_{gh}([D\psi:D\varphi]_{t+s})\\ &= c^\psi((g,t),(h,s))\omega_{(gh,t+s)},\end{aligned}$$ and similarly ${\operatorname{Ad}}(\omega_{(g,t)})\circ \alpha_{(g,t)}^\varphi = \widetilde{\beta}_{(g,t)}^\psi$. Now we put $G_1 := {\mathbb{R}}\leq G$. Then since we already have $({\mathbb{C}}p, \sigma^\psi)\preceq_B ({\mathbb{C}},\sigma^\varphi)$ for all projections $p\in B_\psi^{\widetilde{\beta}}=B^{\widetilde{\beta}^\psi}$, we can apply Proposition \[prop for cocycle superrigidity\]. Thus there exist a separable Hilbert space $H$, a projection $f\in {\mathbb{B}}(H)$, a $\sigma$-strongly continuous map $v\colon G=\Gamma \times {\mathbb{R}}\to \mathcal{U}(f{\mathbb{B}}(H)f)$, a partial isometry $w\in B{\mathbin{\overline{\otimes}}}{\mathbb{B}}(H)$ such that, - $wv_g = (\omega_g\otimes 1_H )(\alpha_g^\varphi\otimes {\text{\rm id}}_H) (w)$ for all $g\in G$; - $w^w = f$ and $ww^* = 1\otimes e_{1,1}$, where $e_{1,1}\in {\mathbb{B}}(H)$ is a minimal projection; - $({\operatorname{Ad}}(v_g))_{g\in G}$ and $(v_gv_h v_{gh}^)_{g,h\in G}$ define a cocycle action on $f{\mathbb{B}}(H)f$; - $\widetilde{\beta}^\psi_g( wxw^) = w (\alpha^\varphi_g\otimes {\operatorname{Ad}}(v_g))(x) w^$ for all $x\in B{\mathbin{\overline{\otimes}}}f{\mathbb{B}}(H)f.$ As in the proof of Proposition \[prop for cocycle superrigidity\], the first equation implies $v_{t+s}=v_tv_s$ for all $t,s \in {\mathbb{R}}$, hence $(v_t)_{t\in {\mathbb{R}}}$ is a continuous homomorphism. By Stone’s theorem, there is a unique analytic generator $h$ on $fH$, so that $[{\mathord{\text{\rm Tr}}}_H(h\, \cdot \, ),f{\mathord{\text{\rm Tr}}}_H f]_t = h^{it} = v_t$ for all $t\in {\mathbb{R}}$, where ${\mathord{\text{\rm Tr}}}_H$ is a fixed semifinite trace on ${\mathbb{B}}(H)$ (with ${\mathord{\text{\rm Tr}}}_H(e_{1,1})=1$). We then compute that for all $t\in {\mathbb{R}}$, with $\varphi^H:= \varphi \otimes {\mathord{\text{\rm Tr}}}_H$, $\psi^H:=\psi \otimes {\mathord{\text{\rm Tr}}}_H$ and $h=1_B\otimes h$, using Lemma \[connes cocycle lemma\], $$\begin{aligned} &[Df\varphi^H(h\, \cdot \, ) f: D\psi^H \circ {\operatorname{Ad}}(w)]_t\\ =\ & [Df\varphi^H(h\, \cdot \, )f: Df\varphi^Hf]_t [Df\varphi^Hf: D\psi^H\circ {\operatorname{Ad}}(w)]_t \\ =\ & v_t [Df\varphi^H f: Df\psi^Hf]_t [Df\psi^Hf: D\psi^H\circ {\operatorname{Ad}}(w)]_t \\ =\ & v_t ([D\varphi:D\psi]_t \otimes 1_H) (\sigma_t^\psi\otimes {\text{\rm id}}_H)(w^) w \\ =\ & v_t (\sigma_t^\varphi\otimes {\text{\rm id}}_H)(w^) ([D\varphi:D\psi]_t \otimes 1_H) w\\ =\ & w^([D\psi,D\varphi]_t\otimes 1_H) ([D\varphi:D\psi]_t \otimes 1_H) w\\ =\ & f.\end{aligned}$$ We get that $\varphi^H(h\, \cdot \, ) = \psi^H \circ {\operatorname{Ad}}(w)$. In particular, putting $\mu:={\mathord{\text{\rm Tr}}}_H(h \, \cdot \, )$, $${\operatorname{Ad}}(w^)\colon B = B \otimes {\mathbb{C}}e_{1,1} \to B{\mathbin{\overline{\otimes}}}f{\mathbb{B}}(H)f$$ satisfies $\psi=(\varphi\otimes \mu)\circ {\operatorname{Ad}}(w^)$. Since ${\operatorname{Ad}}(w^)$ gives a conjugacy between $\alpha^\varphi\otimes {\operatorname{Ad}}(u)$ and $\widetilde{\beta}^\psi$, by restriction, it gives a state preserving conjugacy between $\alpha\otimes {\operatorname{Ad}}(u)$ and $\widetilde{\beta}$. Finally we show that $\Lambda_0 $ is a finite group. Observe that ${\mathord{\text{\rm Tr}}}_H(h) = \psi(1) <\infty$, so $h$ is a compact operator on $fH$. It holds that $$A_\psi\rtimes_\beta \Lambda_0= (A\rtimes_\beta \Lambda_0)_\psi \simeq (B{\mathbin{\overline{\otimes}}}f{\mathbb{B}}(H)f)_{\varphi\otimes \mu}.$$ Since $h$ is a compact operator, there exist finite rank projections $r_n$ on $fH$ which commutes with $h$ such that $r_n \to f$. Then since $\sigma^\varphi$ is weakly mixing, one has $r_n(B{\mathbin{\overline{\otimes}}}f{\mathbb{B}}(H)f)_{\varphi\otimes \mu}r_n = {\mathbb{C}}\otimes (r_n{\mathbb{B}}(H)r_n)_\mu$ for all $n$. In particular $(B{\mathbin{\overline{\otimes}}}f{\mathbb{B}}(H)f)_{\varphi \otimes \mu}$ is an atomic von Neumann algebra, so that $A_\psi\rtimes_\beta \Lambda_0$ as well. This implies that $\Lambda_0$ is a finite group (and $A_\psi$ is atomic). Rigidity of Bernoulli shifts for genuine actions {#rigidity-of-bernoulli-shifts-for-genuine-actions .unnumbered} ------------------------------------------------ We continue to use the Bernoulli shift action $\Gamma \curvearrowright^\alpha \bigotimes_{\Gamma}(B_0,\varphi_0)=(B,\varphi)$ and $M=B\rtimes_\alpha \Gamma$, assuming that $B_0$ is amenable. We recall the following fact. \[Bernoulli theorem\] Let $p \in M$ be a projection, $A \subset pMp$ a finite von Neumann subalgebra with expectation. - If $A \not\preceq_M L\Gamma$, then $A' \cap pMp$ has an amenable direct summand. - If $A$ has relative property (T) in $pMp$, then $A \preceq_M L\Gamma$. By assumption, there are isomorphisms $\Gamma \simeq \Lambda$, $A\simeq B$, and there is a cocycle $\omega\colon \Gamma \to \mathcal{U}(B)$ such that $\beta=\alpha^\omega$. Assume that $\Gamma$ has a normal subgroup $\Gamma_1 \leq \Gamma$ with relative property (T). Put $\Lambda_1\leq \Lambda$ as the image of $\Gamma_1$. For any projection $q\in L\Lambda_1' \cap B$, we apply Theorem \[Bernoulli theorem\](2) to $L\Lambda_1 q$ and get that $L\Lambda_1 q \preceq_M L\Gamma$. Assume that $\Gamma$ is a direct product $\Gamma=\Gamma_1\times \Gamma_2$ with $\Gamma_2$ non-amenable. We put $\Lambda_i\leq \Lambda$ as images of $\Gamma_i$ for $i=1,2$. For any projection $q\in L\Lambda_1' \cap B$, we apply Theorem \[Bernoulli theorem\](1) to $L\Lambda_1 q$. We get that $L\Lambda_1 q \preceq_M L\Gamma$. Thus in both cases, one has $L\Lambda_1 q \preceq_M L\Gamma$ for any projection $q\in L\Lambda_1' \cap B$. Fix such $q\in L\Lambda_1' \cap B$ and we claim that $({\mathbb{C}}q , \beta\|_{\Lambda_1})\preceq_{B}({\mathbb{C}}, \alpha\|_{\Gamma_1})$. Indeed, suppose by contradiction that there is $(g_i)_{i\in I}$ in $\Lambda_1$ such that $$\varphi(\alpha_{g_i}(b^)\omega_{g_i}^q a ) \rightarrow 0, \quad \text{$\sigma$-strongly for all $a,b\in B$.}$$ Then for any $a,b\in B$ and $s,s'\in \Gamma$, we have $$\begin{aligned} E_{L\Gamma}(\lambda_s^{\alpha} b^ \Pi^\omega_{\alpha,\beta}(\lambda_{g_i^{-1}}^\beta)q a\lambda_{s'}^\alpha) = \lambda_s^\alpha E_{L\Gamma}(b^* \lambda_{g_i^{-1}}^\alpha \omega_{g_i}^* q a)\lambda_{s'}^\alpha = \lambda_{sg_i^{-1}}^\alpha \varphi(\alpha_{g_i}(b^* )\omega_{g_i}^q a)\lambda_{s'}^\alpha.\end{aligned}$$ The last term converges to 0, hence we get $L\Lambda_1 q \not\preceq_M L\Gamma$, a contradiction. Finally since $\Lambda_1 \leq \Lambda$ is normal, we can apply Proposition \[prop for cocycle superrigidity\] and get a cocycle action $({\operatorname{Ad}}(u_g))_{g\in \Gamma}$ on a factor ${\mathbb{B}}$. By construction, this cocycle action is a genuine action and we finish the proof. Strong solidity of free product factors {#Strong solidity of free product factors} ======================================= For amalgamated free product von Neumann algebras and their modular theory, we refer the reader to [@VDN92; @Ue98]. Throughout this section we fix the following setting. Let $I$ be a set, $(M_i)_{i\in I}$ a family of $\sigma$-finite von Neumann algebras, $B \subset M_i$ a common unital von Neumann subalgebra with expectations $E_i$ for all $i \in I$. Denote by $M:=_{B} (M_i,E_i)_{i\in I}$ the amalgamated free product von Neumann algebra, and by $E_B\colon M \to B$ the canonical conditional expectation. For any subset $\mathcal F \subset I$, we denote by $M_{\mathcal F}:=_{B} (M_i,E_i)_{i\in \mathcal F}$, and $E_{\mathcal{F}}\colon M \to M_\mathcal{F}$ is the canonical conditional expectation. To prove Theorem \[thmF\], we first prove the following special case. This is a variant of Ioana’s theorem [@Io12 Theorem 1.6] (see also [@Va13; @HU15]), and the proof uses a theorem in [@BHV15]. \[semifinite AFP lemma\] Let $I=\{1,2\}$. Assume that there is a semifinite trace ${\mathord{\text{\rm Tr}}}_B$ on $B$ such that ${\mathord{\text{\rm Tr}}}_B\circ E_i$ are tracial for all $i\in I$. Then the conclusion of Theorem \[thmF\] holds for any $p\in M$ and $A\subset pMp$ as in the statement, provided that ${\mathord{\text{\rm Tr}}}_B\circ E_B(p)<\infty$. Recall that for any semifinite von Neumann algerbas, relative injectivity and relative semidiscreteness are the same conditions (see [@Is17 Theorem A.6]). To prove this lemma, we follow the argument in the paragraph just before [@HU15 Theorem A.4]. In this argument, we can apply [@BHV15 Theorem 3.11], instead of [@PV11 Theorem 1.6]. Then all other proofs work if we exchange the normalizer algebra with the stable normalizer algebra. Thus the conclusion of [@HU15 Theorem A.4] holds for the stable normalizer von Neumann algebra and the lemma is proven. Suppose that $A \not\preceq_M B$ and $s\mathcal{N}_{pMp}(A)'' \not\preceq_M M_i$ for $i=1,2$. We will prove that $P:=s\mathcal{N}_{pMp}(A)''$ is injective relative to $B$ in $M$. Let $E_A$ and $E_P$ be faithful normal conditional expectations for $A$ and $P$ respectively, $N$ the hyperfinite type $\rm III_1$ factor, and $\omega$ a faithful normal state such that $N_\omega' \cap N ={\mathbb{C}}$. Observe that $A' \cap pMp \subset A$ implies that $E_A$ and $E_P$ are unique normal expectations, hence it holds that $E_A\circ E_P=E_A$. Using this uniqueness and using Theorem \[thmA\], there exist $\psi$ which is preserved by $E_A,E_P$, and $\varphi$ which is preserved by $E_B,E_{M_i}$ for $i=1,2$, such that $$\begin{aligned} &\Pi_{\varphi\otimes \omega,\psi\otimes \omega}(C_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}N)) \not \preceq_{C_{\varphi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N)}C_{\varphi\otimes \omega}(B{\mathbin{\overline{\otimes}}}N), \\ &\Pi_{\varphi\otimes \omega,\psi\otimes \omega}(C_{\psi\otimes \omega}(P{\mathbin{\overline{\otimes}}}N)) \not \preceq_{C_{\varphi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N)}C_{\varphi\otimes \omega}(M_i{\mathbin{\overline{\otimes}}}N), \quad \text{for }i=1,2.\end{aligned}$$ Observe that, since $A{\mathbin{\overline{\otimes}}}N$ is properly infinite, by [@FSW10 Lemma 2.4] $$A{\mathbin{\overline{\otimes}}}N \subset P {\mathbin{\overline{\otimes}}}N \subset s\mathcal{N}_{pMp {\mathbin{\overline{\otimes}}}N}(A{\mathbin{\overline{\otimes}}}N)'' = \mathcal{N}_{pMp {\mathbin{\overline{\otimes}}}N}(A{\mathbin{\overline{\otimes}}}N)''.$$ In particular the inclusion $A{\mathbin{\overline{\otimes}}}N \subset P{\mathbin{\overline{\otimes}}}N$ is regular, and hence by [@BHV15 Lemma 4.1], the inclusion $C_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}N)\subset C_{\psi\otimes \omega}(P{\mathbin{\overline{\otimes}}}N)$ is regular as well. For notation simplicity, we omit $\Pi_{\varphi\otimes \omega,\psi\otimes \omega}$ and write as $\mathcal{M}:=C_{\varphi\otimes \omega}(M{\mathbin{\overline{\otimes}}}N)$, $\mathcal{M}_{i}:=C_{\varphi\otimes \omega}(M_{i}{\mathbin{\overline{\otimes}}}N)$ for $i=1,2$, $\mathcal{B}:=C_{\varphi\otimes \omega}(B{\mathbin{\overline{\otimes}}}N)$, and $\mathcal{A}:=C_{\psi\otimes \omega}(A{\mathbin{\overline{\otimes}}}N)$. Let $\mathcal{E}_i\colon \mathcal{M}_i \to \mathcal{B}$ be the faithful normal conditional expectation such that $\mathcal{E}_i\|_{M_i{\mathbin{\overline{\otimes}}}N}=E_{i}\otimes {\text{\rm id}}_N$ and $\mathcal{E}\|_{L{\mathbb{R}}_{\varphi}}={\text{\rm id}}_{L{\mathbb{R}}_\varphi}$ and note that $\mathcal{M}$ has an amalgamated free product structure $$\mathcal{M}=(\mathcal{M}_1,\mathcal{E}_1)_{\mathcal B}(\mathcal{M}_2,\mathcal{E}_2).$$ In this setting, our assumptions are translated to that, $\mathcal A \not \preceq_{\mathcal M} \mathcal{B}$, $\mathcal{N}_{p\mathcal M p}(\mathcal{A})'' \not \preceq_{\mathcal M} \mathcal{M}_i$ for all $i=1,2$, and $\mathcal {A}$ is injective relative to $\mathcal B$ in $\mathcal M$ (use [@Is17 Corollary 3.6 and Theorem 3.2]). Fix any projection $r\in L_{\psi\otimes \omega}{\mathbb{R}}$ such that ${\mathord{\text{\rm Tr}}}_{\psi\otimes \omega}(r)<\infty$, and observe that one has $r\mathcal A r \not \preceq_{\mathcal M} \mathcal{B}$ and $r\mathcal{N}_{p\mathcal M p}(\mathcal{A})''r \not \preceq_{\mathcal M} \mathcal{M}_i$ for all $i=1,2$. Using the inclusion $r\mathcal{N}_{p\mathcal M p}(\mathcal A)''r \subset s\mathcal{N}_{pr\mathcal M pr}(r\mathcal A r)''$ (e.g. [@FSW10 Proposition 2.10]), by applying Lemma \[semifinite AFP lemma\] to $r\mathcal{A}r \subset rp\mathcal{M}rp$, we get that $r\mathcal{N}_{p\mathcal M p}(\mathcal A)''r$ is injective relative to $\mathcal B$. Since $r$ is arbitrary, by [@HI17 Lemma 3.3(v)], we conclude that $\mathcal{N}_{p\mathcal M p}(\mathcal A)''$ is injective relative to $\mathcal B$ in $\mathcal M$. Since $\mathcal{N}_{p\mathcal M p}(\mathcal A)''$ contains $C_{\psi\otimes \omega}(P{\mathbin{\overline{\otimes}}}N)$ with expectation, by [@Is17 Theorem 3.2], it holds that $P{\mathbin{\overline{\otimes}}}N$ is injective relative to $B{\mathbin{\overline{\otimes}}}N$ in $M{\mathbin{\overline{\otimes}}}N$. Finally it is easy to see that $P$ is injective relative to $B$ in $M$. This is the conclusion. If $M$ is stably strongly solid, then since all $M_i$’s are von Neumann subalgebras with expectation, all $M_i$’s are stably strongly solid. We have to show the converse. Let $p\in M$ be a projection and $A \subset pMp$ a diffuse amenable von Neumann subalgebra with expectation. We have to show that $P:=s\mathcal{N}_{pMp}(A)''$ is amenable. Since $pMp$ is solid by [@HU15 Theorem 6.1], $A'\cap pMp$ is amenable. Then as in the proof of [@BHV15 Main theorem], up to exchanging $A\vee (A'\cap pMp)$ by $A$, we may assume that $A'\cap pMp \subset A$. Let $z\in P$ be the unique projection such that $P(p-z)$ is amenable and $Pz$ has no amenable direct summand. We will deduce a contradiction by assuming that $z\neq 0$. In this case, using $Pz \subset s\mathcal{N}_{zMz}(Az)''$, up to exchanging $z$ by $p$, we may assume that $P$ has no amenable direct summand. Define $M^\infty := M{\mathbin{\overline{\otimes}}}{\mathbb{B}}(\ell^2)$, $M^\infty_i := M_i{\mathbin{\overline{\otimes}}}{\mathbb{B}}(\ell^2)$, $A^\infty := A{\mathbin{\overline{\otimes}}}{\mathbb{B}}(\ell^2)$, and $E^\infty_i:=E_i \otimes {\text{\rm id}}_{{\mathbb{B}}(\ell^2)}$, and observe that $M^\infty = _{{\mathbb{B}}(\ell^2)}(M_i^\infty, E_i^\infty)_{i\in I}$ and $s\mathcal{N}_{pM^\infty p}(A^\infty)'' = \mathcal{N}_{pM^\infty p}(A^\infty)''$ (since $A^\infty$ is properly infinite). Since $A^\infty$ is diffuse, we have $A^\infty \not \preceq_{M^\infty} {\mathbb{B}}(\ell^2)$. Suppose first that $I=\{1,2\}$. We can apply Theorem \[thmF\] to $A^\infty \subset pM^\infty p$, and get that (ii) $\mathcal{N}_{pM^\infty p}(A^\infty)''\preceq_{M^\infty} M_i^\infty$ for some $i\in \{1,2\}$ or (iii) $\mathcal{N}_{pM^\infty p}(A^\infty)''$ is amenable. If (iii) holds, then since $P{\mathbin{\overline{\otimes}}}{\mathbb{B}}(\ell^2) \subset \mathcal{N}_{pM^\infty p}(A^\infty)''$ is with expectation, we get that $P$ is amenable, a contradiction. Hence one has the condition (ii). Fix $i$ such that $\mathcal{N}_{pM^\infty p}(A^\infty)''\preceq_{M^\infty} M_i^\infty$, and take $(H,f,\pi,w)$ witnessing this condition. Observe that $\pi(A^\infty) \subset f(M_i^\infty \otimes {\mathbb{M}}_n)f$ is a diffuse amenable von Neumann subalgebra with expectation and that $\pi(P{\mathbin{\overline{\otimes}}}{\mathbb{B}}(\ell^2)) \subset \mathcal{N}_{f(M_i^\infty\otimes {\mathbb{M}}_n)f}(\pi(A^\infty))''$ is with expectation. Since $M_i$ is assumed to be stably strongly solid, $M_i^\infty \otimes {\mathbb{M}}_n$ is strongly solid by [@BHV15 Corollary 5.2]. We thus get that $\pi(P{\mathbin{\overline{\otimes}}}{\mathbb{B}}(\ell^2))$ is amenable. Since $\pi$ is a normal $\ast$-homomorphism, $P$ has an amenable direct summand, a contradiction. We have thus proved this theorem in the case $I=\{1,2\}$. Now we prove the general case. Let $I$ be a general set and we put $M_{\mathcal{F}}:=_{i\in \mathcal F}(M_i,\varphi_i)$ for any subset $\mathcal F \subset I$. We fix any finite subset $\mathcal {F}\subset I$ and observe that $M_{\mathcal F}$ is stably strongly solid by the result in the last paragraph. we apply the same argument as in the case $I=\{1,2\}$ to $A\subset pMp$ using the decomposition $M = M_{\mathcal F} * M_{\mathcal F^c}$. Then since $M_{\mathcal F}$ is stably strongly solid, the only possible condition is that $\mathcal{N}_{pM^\infty p}(A^\infty)''\preceq_{M^\infty} M_{\mathcal F^c}^\infty$. By assuming that this condition holds for all finite subsets $\mathcal F \subset I$, we will deduce a contradiction. Since $P {\mathbin{\overline{\otimes}}}{\mathbb{B}}(\ell^2) \subset \mathcal{N}_{pM^\infty p}(A^\infty)''$, using [@HI15 Lemma 4.8], we indeed have that $P{\mathbin{\overline{\otimes}}}{\mathbb{B}}(\ell^2)\preceq_{M^\infty} M_{\mathcal F^c}^\infty$ for all finite subsets $\mathcal F \subset I$. Then as in the proof of Theorem \[thmF\], by applying Theorem \[thmA\] (and using $N\simeq N{\mathbin{\overline{\otimes}}}{\mathbb{B}}(\ell^2)$), one has $\mathcal{P} \preceq_{\mathcal {M}} \mathcal{M}_{\mathcal{F}^c}$ for all finite subsets $\mathcal F \subset I$, where we used similar notations to ones in the proof of Theorem \[thmF\], such as $\mathcal{P}:=C_{\psi\otimes \omega}(P{\mathbin{\overline{\otimes}}}N)$, $\mathcal{M}_{\mathcal F^c}:=C_{\varphi\otimes \omega}(M_{\mathcal F^c}{\mathbin{\overline{\otimes}}}N)$ for appropriate $E_P,\psi,\varphi$. Fix any projection $r\in L_{\psi\otimes \omega}{\mathbb{R}}$ such that ${\mathord{\text{\rm Tr}}}_{\psi\otimes \omega}(r)<\infty$. Fix any projection $z\in \mathcal P ' \cap p\mathcal Mp = (P' \cap pMp)_{\psi} = \mathcal{Z}(P)$ (e.g. Lemma \[III1 factor tensor lemma\]). We will prove that $r\mathcal{P}rz \preceq_{\mathcal {M}} \mathcal{M}_{\mathcal{F}^c}$ for all finite subsets $\mathcal F \subset I$. Then using [@HU15 Proposition 4.2], this will imply the amenability of $r\mathcal Pr$ and hence the one of $\mathcal P$, a contradiction. To prove this condition, fix $\mathcal F$, $r$ and $z$. Observe that $Pz \subset s\mathcal{N}_{zMz}(Az)''$. Then since $Pz$ has no amenable direct summand, we can apply the same argument to $Az \subset Pz$ (as we applied to $A\subset P$), and get that $\mathcal{P}z\preceq_{\mathcal M} \mathcal{M}_{\mathcal F^c}$. Since the central support of $rz$ in $\mathcal P z$ is $z$, by [@HI15 Remark 4.2(3)], we get $r\mathcal{P}rz \preceq_{\mathcal {M}} \mathcal{M}_{\mathcal{F}^c}$. This is the desired condition. [^1]: Research Institute for Mathematical Sciences, Kyoto University, 606-8502, Kyoto, Japan\ E-mail: `isono@kurims.kyoto-u.ac.jp`
--- address: \| Bogoliubov Laboratory of Theoretical Physics,\ JINR, Dubna, Moscow Reg., 141 980, Russia\ E-mail: filippov@thsun1.jinr.ru, isaevap@thsun1.jinr.ru author: - 'A. T. FILIPPOV , A. P. ISAEV' title: 'PATH-INTEGRAL FOR QUADRATIC HAMILTONIAN SYSTEMS AND BOUNDARY CONDITIONS' --- =cmr8 1.5pt \#1\#2\#3\#4[[\#1]{} [\#2]{}, \#3 (\#4)]{} Boundary conditions for path-integral. ====================================== Let us consider a Hamiltonian system in the phase space ${\cal M}$ with coordinates $Z_{A} = \{q_a, \, p^a, \, \psi_\alpha, \, \overline{\psi}^{\alpha} \}$ where $q_a$ and $p^a$ are bosonic coordinates and momenta $(a=1, \dots , N)$ while $\psi_\alpha$ and $\overline{\psi}^{\alpha}$ $(\alpha = 1, \dots , K)$ are fermionic degrees of freedom. The action for this system can be written in the form \[1\] S = \_[0]{}\^[T]{} dt , where $t \in [0,T]$ is an evolution parameter and ${\cal H}$ is the Hamiltonian. The variational principle, $\delta S =0$, gives the equations of motion, $\dot{Z}_A = \Omega_{AB} \, \partial^B \, {\cal H}$. $(\partial^{B} \equiv \vec{\partial}^{B})$. Here $\Omega_{AB} = -(-)^{AB} \, \Omega_{BA} = \{ Z_A , \, Z_B \}$ is inverse to the matrix $$\Omega^{AB} = (\partial^A \, P^B - (-)^{A + B + AB } \, \partial^B \, P^A ) = -(-)^{A +B +AB} \, \Omega^{BA} \; .$$ Note that the symplectic 2-form $\omega$ on the superspace ${\cal M}$ is defined by a slightly different supermatrix: $\omega = (\partial^A \, P^B - (-)^{AB } \, \partial^B \, P^A ) \, dZ_A \wedge dZ_B$. Below we concentrate on the case of a constant supermatrix $\Omega^{AB}$. Then we have $P^{A} = \frac{1}{2} \, Z_B \, \Omega^{BA}$ and the action (\[1\]) can be represented in the form S = \_[0]{}\^[T]{} dt . We will investigate the path-integral representation [@FaSl] for the HK of the evolution operator $U$: < Z\^f \| U(T,0) \| Z\^i > = d { Z\_A } exp {i (S + [B]{}) } where $d \{ Z_A \}$ is a measure over the space of trajectories in ${\cal M}$. The boundary terms ${\cal B}$ depending on the initial and final points, $Z^i$, $Z^f$, are determined by relevant boundary conditions.\ [Proposition 1.]{} [Boundary conditions can be specified by the boundary terms in the path-integral (\[3\]): = , where $Z^{f,i}_A$ are fixed supervectors in ${\cal M}$ and matrices $P_{f,i}$ are constant projectors ($P^2 =P$) in ${\cal M}$ of the rank $(N+K)$. These projectors should satisfy the conditions $P^{\T} \, \Omega = \Omega \, (1-P)$, where $\T$ denotes a super-transposition [@Ber; @Wi] $(P^{\T})_{AB} = (-)^{AB + A} \, P_{BA}$. Substituting the boundary terms (\[4\]) into (\[3\]) is equivalent to fixing the initial and final states of the system by the conditions: $(1-P_i ) \, Z(0) = Z^i$, $(1-P_f ) \, Z(T) = Z^f$. ]{}\ [Hint for proof.]{} Cosider the equation $\delta (S +{\cal B}) = 0$ at the points $t=0,T$.\ [Note.]{} Different choices of the projectors $P_{f,i}$ lead to different choices of the boundary states in (\[3\]). In this way one can write down the HK (\[3\]) in coordinate, momentum or holomorphic representations in unified form. Path-integral (\[3\]) for quadratic Hamiltonians. ================================================= Now, consider systems with general quadratic Hamiltonians (Z) = Z\_A Z\_D , where the supermatrix ${\cal A}$ depends on the evolution parameter $t$ and is independent of $Z_A$. The hermiticity of (5) requires that ${\cal A}^{\T} \Omega + \Omega {\cal A} =0$, i.e. ${\cal A} \in osp(2K\|2N)$. With the Hamiltonian (\[5h\]) the action (\[2\]) acquires the form S = \_[0]{}\^[T]{} dt Z Z .\ [Proposition 2.]{} [In the case of the Hamiltonian system with the action (\[5\]) and boundary terms (\[4\]), the path-integral (\[3\]) gives the following representation for the evolution operator $U$: < Z\^f \| U(T,0) \| Z\^i > Ber\^[1/2]{} ( ) { - i S\_[eff]{} (Z\^f, Z\^i) } , $$S_{eff} = \frac{1}{2} \, \left[ Z^i \Omega \frac{1}{V^{+-}} V^{++} Z^i - 2 Z^i \Omega \frac{1}{V^{+-}} Z^f + Z^f \Omega V^{--} \frac{1}{V^{+-}} Z^f \right]$$ where $V=T \, exp \{ \int^{T}_{0} dt A(t) \}$, $V^{\alpha \beta} = P^{\alpha}_f V P^{\beta}_i$ and $P^{-}_{f,i} \equiv P_{f,i}$, $P^{+}_{f,i} \equiv 1-P_{f,i}$. ]{}\ [Proof.]{} Direct computation. Heat kernel for the evolution operator of “Discrete Strings”. ============================================================= Finally, consider the system with the action (\[5\]) and interpret the Hamiltonian (\[5h\]) as a linear combination of the first class constraints. In this case, the elements of the supermatrix $A$ play the role of Lagrange multipliers. Note that the fermionic string models can be described by the action of the same type. For this reason we call such systems as “Discrete Strings” (for details see [@FI1; @FI2; @FI3; @FI4]). The Action with boundary terms, $S'=S+{\cal B}$ (where S and ${\cal B}$ are defined in (\[5\]) and (\[4\])), is invariant under the gauge transformations $$Z \rightarrow F(t)Z \;\; , \;\;\; A \equiv A_M e^M \rightarrow F A F^{-1} + \dot{F}F^{-1} \Longleftrightarrow V \rightarrow F(T) V F(0) \; ,$$ where $F^{\T} \Omega F = \Omega$ (i.e., the gauge group $G$ is a subgroup of $Osp(2K\|2N)$). Matrices $e^M$ form a basis in the algebra of $G$. The boundary terms in $S'$ fix the gauge group parameters at the final and initial points up to residual gauge transformations: $[ F(T,0), \, P_{f,i} ] = 0$. Let us denote the corresponding groups by $G_{f,i}$. Using the freedom of rotating the boundary states $Z^{f,i} \rightarrow F(T,0)Z^{f,i}$, one can show that the evolution operator (\[6\]) depends only on the coordinates of the moduli space $\{ V \} / G_f \otimes G_i \equiv \overline{V}$, which is an analogue of the Teichmueller space used in string theories. The standard procedure [@FaSl; @FI3; @FI5] for quantizing constrained Hamiltonian systems via path-integrals allows to obtain (after fixing a gauge and representing Faddeev-Popov determinant as an integral over ghosts $\{ B, \, C \}$) the expression for the propagator of “Discrete Strings”: K\_[f,i]{} = d { } < Z\^f \| U(T,0) \| Z\^i > . Here $d \{ \overline{V}\}$ is a measure over the moduli space and the HK for the ghost evolution operator $\widetilde{U}$ is represented by the path-integral: < B\^f, C\^f \| (T,0) \| B\^i , C\^i> = d { B } d { C } exp {i (S\_[gh]{} + [B]{}\_[gh]{}) } . The ghost action $S_{gh}$ and boundary terms ${\cal B}_{gh}$ have the form $$S_{gh} = \int^{T}_{0} dt \, B^N (\delta^M_N \partial_t - \widetilde{A}^M_N )C_M \;\; , \;\;\; {\cal B}_{gh} = B^i \widetilde{P}_i C(0) - B^f \widetilde{P}_f C(T) \; ,$$ where ghosts $B^{N}$, $C_{M}$ realize the adjoint representations of gauge group $G$ ($\widetilde{A}$ is the adjoint analogue of $A$) and their parity is opposite to the parity of the Lagrange multipliers $A_N$. Constant projectors $\widetilde{P}_{f,i}$ are defined by the relations $ (\widetilde{P}_{f,i})^{M}_{N} e^{N} = P^{+}_{f,i} e^{M} P^{-}_{f,i} + P^{-}_{f,i} e^{M} P^{+}_{f,i} $, which impose some restrictions on the choice of the boundary projectors $P_{f,i}$. As in Proposition 1, the boundary terms ${\cal B}_{gh}$ fix the boundary states for ghosts: $B(T,0)\widetilde{P}_{f,i} = B^{f,i}$, $(1-\widetilde{P}_{f,i}) C(T,0) = C^{f,i}$.\ [Proposition 3.]{} [The path-integral (\[8\]) for the ghost HK is equal to < B\^f, C\^f \| (T,0) \| B\^i , C\^i> (\^[+-]{}) { i \_[eff]{} } , $$\widetilde{S}_{eff} = \left[ (B^i - B^f \widetilde{V}^{--}) \frac{1}{\widetilde{V}^{+-}} (C^f - \widetilde{V}^{++} C^i) - B^f \widetilde{V}^{-+} C^i \right] \; ,$$ where $\widetilde{V} = T \exp \{ \int^{T}_{0} dt \, \widetilde{A}(t) \}$, $\widetilde{V}^{\alpha \beta} = \widetilde{P}^{\alpha}_f \widetilde{V} \widetilde{P}^{\beta}_i$ and $\widetilde{P}^{-}_{f,i} \equiv \widetilde{P}_{f,i}$, $\widetilde{P}^{+}_{f,i} \equiv 1-\widetilde{P}_{f,i}$. ]{}\ [Proof.]{} Direct computation.\ [Note.]{} We have to stress the following: if the operator $\widetilde{V}^{+-}$ has zero eigenvalues, the right-hand side of (\[9\]) has to be modified. Namely, we have to remove the zero eigenvalues from the superdeterminant $\widetilde{Ber}$ and also multiply the final expression by the product of delta-functions of corresponding zero-modes of the ghost variables $B,C$ (to remove infinities from $\widetilde{S}_{eff}$). This work is supported in part by the RFBR grant No. 97-01-01041 and by INTAS grant 93-127-ext. We are grateful to CERN TH (A.P.I.) and INFN Torino (A.T.F.) for kind hospitality and support. References {#references .unnumbered} ========== [99]{} , Gauge Fields: Introduction to Quantum Theory, [(Frontiers in Physics [83]{}, Benjamin-Cummings, London, 1991)]{}. , Introduction to Superanalysis, [(Dordrecht Reidel, 1987)]{}. , Supermanifolds, [(Cambridge University Press, 1992)]{}. , Mod. Phys. Lett. A [4]{} [2167 (1989)]{}. A.T. Filippov and A.P. Isaev, [Mod. Phys. Lett. A]{}[ 5]{} 325 (1990). A.T. Filippov, D. Gangopadhyay and A.P. Isaev, [Int. J. Mod. Phys. A]{}[ 7]{} 2487, 3639 (1992). A.T. Filippov, [Acc. Sc. Torino - Memorie Sc. Fis.]{} [19-20]{} 129 (1995). A.P. Isaev, [Discrete Systems with Chiral Structure and Quantum Symmetries]{}, unpublished, (1997).
--- abstract: \| In this paper, we are interested in the properties of solution of the nonlocal equation $$\begin{cases} u_t+(-\Delta)^su=f(u),\quad t>0, \ x\in\mathbb{R}\\ u(0,x)=u_0(x),\quad x\in\mathbb{R} \end{cases}$$ where $0\le u_0<1$ is a Heaviside type function, $\Delta^s$ stands for the fractional Laplacian with $s\in (0,1)$, and $f\in C([0,1],\R^+)$ is a non negative nonlinearity such that $f(0)=f(1)=0$ and $f'(1)<0$. In this context, it is known that the solution $u(t,s)$ converges locally uniformly to 1 and our aim here is to understand how fast this invasion process occur. When $f$ is a Fisher-KPP type nonlinearity and $s \in (0,1)$, it is known that the level set of the solution $u(t,x)$ moves at an exponential speed whereas when $f$ is of ignition type and $s\in \left(\frac{1}{2},1\right)$ then the level set of the solution moves at a constant speed.\ In this article, for general monostable nonlinearities $f$ and any $s\in (0,1)$ we derive generic estimates on the position of the level sets of the solution $u(t,x)$ which then enable us to describe more precisely the behaviour of this invasion process. In particular, we obtain a algebraic generic upper bound on the “speed” of level set highlighting the delicate interplay of $s$ and $f$ in the existence of an exponential acceleration process. When $s\in\left (0,\frac{1}{2}\right]$ and $f$ is of ignition type, we also complete the known description of the behaviour of $u$ and give a precise asymptotic of the speed of the level set in this context. Notably, we prove that the level sets accelerate when $s\in\left(0,\frac{1}{2}\right)$ and that in the critical case $s=\frac{1}{2}$ although no travelling front can exist, the level sets still move asymptotically at a constant speed. These new results are in sharp contrast with the bistable situation where no such acceleration may occur, hightligting therefore the qualitative difference between the two type of nonlinearities. author: - 'Jérôme Coville[^1] , Changfeng Gui [^2] , Mingfeng Zhao [^3]' bibliography: - 'biblio-asym.bib' title: Propagation acceleration in reaction diffusion equations with anomalous diffusions --- Introduction ============ The study of propagation phenomena is a classical topic in analysis as its provide a robust way to understand some pattern formations that arises in a wide range of context ranging from population dynamics in ecology [@Fisher1937; @Kolmogorov1937], to combustion [@Kanel1961] and phase transition [@Aronson1978]. Concretely, this often leads to analyse the asymptotic properties of the solution $u(t,x)$ of the parabolic problem used to model the phenomenon considered. When this model is a reaction diffusion equation, this lead then to the study of the properties of the solutions of $$\label{rd} \begin{cases} \partial_t u(t,x)=\Delta u(t,x) +f(u(t,x)) \quad \text{ for } t>0, x\in \R^N\\ u(0,x)=u_0(x) \end{cases}$$ with respect to the nonlinearity $f$ and the initial data $u_0$. In this particular situation, when $f$ is a smooth bistable, ignition or monostable nonlinearity, say $f$ Lipschitz such that $f(0)=f(1)=0$ , $f'(1)<0$, it is known that the solution of the equation can exhibit some phase transition behaviour. More precisely, for a Heaviside type initial datum $u_0$ i.e. $u_0(x)=\mathds{1}_{ H_e}(x)$ where $H_e$ denotes a half-space $\{x\in \R^N\,\|\, x\cdot e<0\}$, then the solution $u(t,x)$ of converges locally uniformly as $t\to +\infty$ to $1 $ and the “invasion process” resulting of this initial datum can be characterised by particular solutions of called planar front $\varphi(x\cdot e-ct)$ [@Aronson1978; @Fife1977; @Kolmogorov1937; @Weinberger1982], where $(\varphi, c)$ solves here the following equations $$\begin{cases} \varphi''(z)+c\varphi'(z)+f(\varphi(z))=0 \quad \text{ for } z\in \R,\\ \ds{\lim_{z\to-\infty }\varphi(z)=1}, \\ \ds{\lim_{z\to+\infty }\varphi(z)= 0.} \end{cases}$$ In particular, for any $\lambda \in (0,1)$ the superlevel set $E_{\lambda}(t):=\{x\in \R^N\|u(t,x)\ge \lambda\}$ grows at a constant speed. That is there exists $x^+(\lambda),x^-(\lambda)$ in $\R^N$ and a family of Half-space $H^+(t)$ defined by $$H^+(t):=\{x\in \R^N\,\|\, x\cdot e -ct \le 0\}$$ such that $E_\lambda$ satisfies $$x^-(\lambda)+H^+(t)\subset E_\lambda (t)\subset x^+(\lambda)+H^+(t).$$ Thanks to the comparison principle satisfied by such semi-linear equations , clearly this particular phase transition behaviour appears also for other type of initial data $u_0\ge \mathds{1}_{H_e}$ that have some decay as $x\cdot e \to -\infty$. For those initial data, we may then wonder if the above description of the behaviour of superlevel set $E_\lambda$ still holds true and if not how can we characterise it. As shown in [@Alfaro2017a; @Hamel2010; @Kay2001; @Needham1999; @Sherratt2005], the above characterisation may not hold in general and in some situation an accelerated transition may occur. Indeed when $N=1$ and for a monostable $f$ of KPP type, that is $f\in C([0,1])$ such that $f(0)=f(1)=0$, $f>0$, $f'(0)>0,f'(1)<0$, and such that $f(s)\le f'(0)s$, then for $u_0(x)>0$ F. Hamel and L. Roques have obtained in [@Hamel2010] a sharp description of the speed of the level line of the solution of the corresponding Cauchy problem. In particular, they show that when $u_0$ is such that $u_0(x)\sim \frac{1}{x^\alpha} $, as $x\to +\infty$, then the level lines of the solution move exponentially fast. That is, for any $\lambda \in (0,1)$ there exists points $x(t)\in E_\lambda(t)$ such that $x(t)\sim e^{f'(0)t}$. More generally, they prove that Let $u_0$ be for regular ($C^2$) nonincreasing initial data $u_0$ on some semi-infinite interval $[\xi_0,+\infty)$ and such that $$\partial_{xx}u_0 (x) = o(u_0 (x))\quad \text{ as }\quad x \to +\infty.$$ Then, for any $\lambda \in (0, 1)$, $\eps \in (0, f'(0))$, $\mu > 0$ and $\nu > 0$, there exists $T_{\lambda,\eps,\mu,\nu}\ge t_\lambda$ such that $$\Gamma_\lambda(t) \subset u_0^{-1}([ \mu e^{-(f'(0)+\eps)t}, \nu e^{-(f'(0)-\eps)t}]),$$ where $\Gamma_\lambda$ denotes $$\Gamma_\lambda(t):=\{x\in \R\, \|\, u(t,x)=\lambda\}$$ From this result, we can see the clear dependence of the speed of the level sets of the solution $u(t,x)$ with respect to the decay behaviour of $u_0$. Similar sharp descriptions of the speed of the level sets have been obtain for more general monostable type of nonlinearity, see for example [@Alfaro2017a; @Kay2001; @Needham1999; @Sherratt2005]. On the other hand, thanks to the work of Fife and Mc Leod [@Fife1977], and Alfaro [@Alfaro2017a] we can see that accelerated transitions will never occur when the non-linearity considered is bistable or of ignition type. In this spirit, in this paper we are interested in propagation acceleration phenomena that are caused by anomalous diffusions such as super diffusions, which plays important roles in various physical, chemical, biological and geological processes. (See, e.g., [@Volpert13] for a brief summary and references therein.) A typical feature of such anomalous diffusions is related to Lévy stochastic processes which may possesses discontinuous “jumps” in their paths and have long range dispersal, while the standard diffusion is related to the Brownian motion. Analytically, certain Lévy processes ($\alpha$ stable) may be modeled by their infinitesimal generators which are fractional Laplace operators $(-\Delta )^{s} u $ with $0<s<1$, whose Fourier transformation $\widehat {(-\Delta )^{s} u} $ is $ (2\pi\|\xi\|)^{2s} \widehat{u}$. (See [@Metzler].) More precisely, we consider the following one-dimensional reaction-diffusion equation involving the fractional Laplacian: $$\begin{aligned} \label{eqn:1-1} \left\{ \begin{array}{l} u_t+(-\Delta)^su=f(u),\quad t>0, \ x\in\mathbb{R}\\ u(0,x)=u_0(x),\quad x\in\mathbb{R} \end{array} \right.\end{aligned}$$ where - $(-\Delta)^s$ ($0<s<1$) denotes the fractional Laplacian operator: $$(-\Delta)^s u(x)=C_{1,s}\textnormal{P.V. }\int_{\mathbb{R}} \frac{u(x)-u(y)}{\|x-y\|^{1+2s}} dy,$$ where $C_{1,s}$ is a positive normalization constant in the sense that $\widehat{ (-\Delta)^s u}(\xi)=\|\xi\|^{2s}\widehat{u}(\xi)$. For simplicity, in the whole article, let’s assume that $C_{1,s}=1$ after a suitable normalization. - $f$ is a nice function on $[0,1]$. - $u_0(x)$ is the initial condition. The precise assumptions on $f$ and $u_0$ will be given later on. Along with other types of nonlocal models (integrodifferential or integrodifference) such nonlocal fractional reaction diffusion model has received a lot of attention lately. Contrary to the standard reaction diffusion equation , accelerated transitions can be observed for Heaviside type initial data [@Nec08; @Mancinelli; @Cabre2013; @Engler; @Gui2015] in the anomalous reaction diffusion systems. The mechanism that triggers the acceleration in this situation is then intrinsically different from that in the classical reaction diffusion and seems governed by subtle interplay between the long range jumps in the diffusion processes and the strength of the pushes and pullings in the reaction part, mathematically, i.e., the tails of the kernel and the properties of nonlinearity $f$ considered. Namely, when $f$ is of bistable type then planar wave exists for all $s \in (0,1)$ [@Gui2015a; @Palatucci2013] and the solution to with a reasonable Heaviside initial data $u_0$ will converge to a planar front, see [@Achleitner2015]. On the other hand, for the same initial data but for a $KPP$ type nonlinearity, the solution will accelerate exponentially fast [@Cabre2013; @Engler], that is, for $x(t)\in \Gamma(t)$ we have $x(t)\sim e^{f'(0)t}$. For more general monostable nonlinearities $f$, including those of ignition type, the picture is less clear and only results on the existence/ non-existence of planar front have been obtained. More precisely, when $f$ is an ignition nonlinearity then a planar front can only exist in the range $s\in \left(\frac{1}{2},1\right)$ see [@Gui2015; @Mellet2009]. Whereas for a general monostable nonlinearity $f$(i.e. $f(t)\sim t^p(1-t))$) the existence of a planar front only occurs when $p>2$ and in the range $s \in (\frac{p}{2(p-1)},1)$ see [@Gui2015]. In the later case, this suggest that as in the KPP case, an accelerated transition will then occur for any $s \in (0,1)$ when $1<p<2$. A natural question is then, as in the KPP case, does the level sets move with an exponential speed when $1<p<2$? One objective of this paper is to answer to this question and give a more detailed characterisation of the speed of the level set for general monostable nonlinearities $f$. Main Results ------------ Let us now describe more precisely the assumptions we made and state our main results. \[ass:1-1\] The nonlinearity $f: [0,1]\longrightarrow [0,\Vert f\Vert_\infty]$ is of class $C^1$, and is of the monostable type, in the sense that $$\begin{aligned} \label{eqn:1-2} f(0)=f(1)=0,\qquad f(u)>0,\quad\textnormal{for all $u\in(0,1)$}. \end{aligned}$$ The steady state $0$ is degenerate, in the sense that, there exist some constants $r>0$ and $\beta>1$ such that $$\begin{aligned} \label{eqn:1-3} f(u)\leq ru^\beta,\quad\textnormal{for all $u\in[0,1]$}. \end{aligned}$$ The steady state $1$ is stable, in the sense that $$\begin{aligned} \label{eqn:1-4} f'(1)<0. \end{aligned}$$ \[ass:1-2\] The initial data $u_0:\mathbb{R}\longrightarrow[0,1]$ is of class $C^1$ and satisfies - $0\leq u_0(x)\leq1$ for all $x\in\mathbb{R}$. - $\displaystyle \liminf_{x\rightarrow-\infty} u_0(x)>0$. - $u_0(x)\equiv0$ on $[a,+\infty)$ for some $a$. Under this two assumptions, we first prove that \[thm:1-1\] For any $0<s<1$, assume that the nonlinearity $f$ satisfies [Assumption \[ass:1-1\]]{}, and the initial data $u_0(x)$ satisfies [Assumption \[ass:1-2\]]{}. Let $u(t,x)$ be the solution to the problem with the initial data $u_0(x)$, consider the superlevel set $E_\lambda(t)=\{ x\in\mathbb{R}\| u(t,x)>\lambda \}$ of the solution $u(t,x)$, and define $$x_\lambda(t)=\sup E_\lambda(t).$$ If further assume that $\frac{\beta}{2s(\beta-1)}>1$, then for any $\lambda\in(0,1)$, there exist some constants $T_\lambda>0$ and $C(\lambda)>0$ such $$\begin{aligned} E_\lambda(t)\subseteq (-\infty, x_\lambda(t)),\quad \textnormal{and}\quad x_\lambda(t)\leq C(\lambda) t^{\frac{\beta}{2s(\beta-1)} },\qquad \forall t>T_\lambda. \end{aligned}$$ When $\beta>2$ and $\frac{\beta}{2s(\beta-1)}\leq1$, the existence of the traveling wave to the problem provided $\frac{\beta}{2s(\beta-1)}\leq1$ was proved by Gui and Huan in [@Gui2015] meaning that for the solution $u(t,x)$ to with some front-like data, if we look at the level set of $u(t,x)$, then the spatial variable $x$ may linearly depend on the time variable $t$. In this sense, our condition $\frac{\beta}{2s(\beta-1)}>1$ is sharp. In addition, we can observe from our results that when $\beta>1$ then the level set of the solution $u(t,x)$ to the equation moves at most at a polynomial rate i.e $\ds{x_\lambda(t)\sim t^\gamma}$ with $\ds{\gamma:=\sup\left\{1;\frac{1}{2s}+\frac{1}{\beta -1}\right\}}$. These results are in sharp contrasts with the results of Cabre et al. [@Cabre2013] for the KPP case. In particular, they highlight the fact that the exponential acceleration is strongly thigh to the non-degeneracy of the nonlinearity $f$ and only occur when $f$ is such that $f'(0)>0$, a situation that allows an exponential growth at low density. (-0.1,0)–(21,0) node\[right\][$s$]{}; (0,-0.1)–(0,11) node\[right\][$\beta$]{}; plot (,[(-2\(/20)/(1-2\(/20))) ]{}); (0,1)–(21,1) node\[right\]; (0,2)–(21,2) node\[right\]; (20,0)–(20,11); (10,0)–(10,11); at (-0.5,-0.5)[$0$]{}; at (-0.5,1)[$1$]{}; at (-0.5,2)[$2$]{}; at (20,-0.5)[$1$]{}; at (10,-0.5)[$\frac{1}{2}$]{}; at (5,0.5)[$x_\lambda(t) \sim e^{\rho t}$]{}; at (5,1.5); at (5,5.5); at (15,0.5)[$x_\lambda(t) \sim e^{\rho t}$]{}; at (15,1.5); at (12,3.25); at (15,7.5)[$x_\lambda(t) \sim t$]{}; Next, we prove a first lower bound of the speed of the level set. Namely, we show that \[thm:1-2\] For any $0<s<1$, assume that the nonlinearity $f$ satisfies $f(u)\geq0$ for all $u\in[0,1]$, and the initial data $u_0(x)$ satisfies [Assumption \[ass:1-2\]]{}. Let $u(t,x)$ be the solution to the problem with the initial data $u_0(x)$, consider the superlevel set $E_\lambda(t)=\{ x\in\mathbb{R}\| u(t,x)>\lambda \}$ of the solution $u(t,x)$, and define $$x_\lambda(t)=\sup E_\lambda(t).$$ Then for any $\lambda\in(0,1)$, there exists some constants $T_\lambda'>0$ and $C'(\lambda)>0$ such $$\begin{aligned} x_\lambda(t)\geq C'(\lambda) t^{\frac{1}{2s} },\qquad\forall t>T_\lambda' \end{aligned}$$ Combining the later with the upper bound obtained in Theorem \[thm:1-2\], as a immediate corollary we then get \[cor:1-1\] For any $0<s<1$, assume that the nonlinearity $f$ satisfies [Assumption \[ass:1-1\]]{} and $f(u)\geq0$ for all $u\in[0,1]$, and the initial data $u_0(x)$ satisfies [Assumption \[ass:1-2\]]{}. Let $u(t,x)$ be the solution to the problem with the initial data $u_0(x)$, consider the superlevel set $E_\lambda(t)=\{ x\in\mathbb{R}\| u(t,x)>\lambda \}$ of the solution $u(t,x)$, and define $$x_\lambda(t)=\sup E_\lambda(t).$$ If further assume that $\frac{\beta}{2s(\beta-1)}>1$, then for any $\lambda\in(0,1)$, there exists some constants $T_\lambda>0$, $C(\lambda)>0$ and $C'(\lambda)>0$ such $$\begin{aligned} C'(\lambda) t^{\frac{1}{2s}}\leq x_\lambda(t)\leq C(\lambda) t^{\frac{\beta}{2s(\beta-1)} },\qquad \forall t>T_\lambda. \end{aligned}$$ Although these first estimates on the speed seem rather crude this are still quite interesting, in particular in the case $0<s<\frac{1}{2}$, as they give a very simple way of showing the non-existence of the traveling wave solution to the problem with any general non negative function $f$ and in particular for the Fisher-KPP nonlinearity. These results also highlight a fundamental difference between nonlocal model versus local model when considering an ignition type nonlinearity. Indeed, when the nonlinearity $f$ is of ignition type, we can easily deduce from the work of Alfaro [@Alfaro2017a] that accelerated transitions never occur in the classical reaction diffusion model whereas they do in the nonlocal reaction diffusion when $s\in(0,1/2)$. This is also a clear evidence that in the nonlocal setting, unlike in the local setting () the two types of nonlinearities: bistable and ignition type are not alike in the sense that the dynamic obtained are completely different. In this nonlocal setting, a condition on the decay of the tail of the kernel appears then of crucial importance in order to guarantee the existence of traveling front. Namely, from our results we can see that when $f$ is non negative the kernel must satisfy some first moment integrability condition to expect to observe traveling front solutions. This finite first moment condition suggests that a similar condition should hold true as well for convolution type nonlocal models studied in [@Coville2007d] as the these two models shares many similarities. That is, in such convolution type models, for a traveling front to exist the kernel need to satisfy a first moment condition. (-0.1,0)–(21,0) node\[right\][$s$]{}; (0,-0.1)–(0,16) node\[right\][$\beta$]{}; plot (,[(-2\(/20)/(1-2\(/20))) ]{}); (0,1)–(21,1) node\[right\]; (0,2)–(21,2) node\[right\]; (20,0)–(20,16); (10,0)–(10,16); at (-0.5,-0.5)[$0$]{}; at (-0.5,1)[$1$]{}; at (-0.5,2)[$2$]{}; at (20,-0.5)[$1$]{}; at (10,-0.5)[$\frac{1}{2}$]{}; at (5,0.5)[$x_\lambda(t) \sim e^{\rho t}$]{}; at (5,1.5); at (5,15); at (15,0.5)[$x_\lambda(t) \sim e^{\rho t}$]{}; at (15,1.5); at (7.7,6.25); at (15,7.5)[$x_\lambda(t) \sim t$]{}; Let us look now more deeply at the consequences of these first estimates on the speed for the combustion model and for supercritical fractional Laplacians (that is, $0<s<\frac{1}{2}$). In this situation, from the above estimate we can in fact derive a sharp estimate on the speed of propagation. Namely, we show \[cor:1-2\] For any $0<s\leq \frac{1}{2}$, assume that the initial data $u_0(x)$ satisfies [Assumption \[ass:1-2\]]{}, and the nonlinearity $f$ is a combustion type nonlinearity, in the sense that there exists some $\theta\in(0,1)$ such that $$\begin{aligned} \label{eqn:1-5} f(1)=0=f(u),\quad\textnormal{for all $u\in[0,\theta]$},\qquad\textnormal{and}\qquad f(u)>0\quad\textnormal{for all $u\in(\theta,1)$} \end{aligned}$$ Let $u(t,x)$ be the solution to the problem with the initial data $u_0(x)$, consider the superlevel set $E_\lambda(t)=\{ x\in\mathbb{R}\| u(t,x)>\lambda \}$ of the solution $u(t,x)$, and define $$x_\lambda(t)=\sup E_\lambda(t).$$ Then for any $\eps>0$, and for any $\lambda\in(0,1)$, there exists some constants $T_{\lambda,\eps}>0$, $C(\lambda,\eps)>0$ and $C'(\lambda)>0$ such $$\begin{aligned} C'(\lambda) t^{\frac{1}{2s}}\leq x_\lambda(t)\leq C(\lambda,\eps) t^{\frac{1}{2s}+\eps },\qquad \forall t>T_{\lambda,\eps}. \end{aligned}$$ The proof of this corollary is quite straightforward. Indeed, the combustion model can be thought as some limit case of the degenerated monostable situation (i.e., $f(u)$ monostable with $f^{(k)}(0)=0$ for all $k\in \N$). In particular, for any combustion nonlinearity $f$ we may find a constant $C_0>0$ such that for all $ \beta>1$ we have $$f(u)\le f_\beta(u):= C_0u^\beta(1-u).$$ Recall that since we assume that the fractional Laplacian is either super-critical or critical (i.e. $s\in (0,\frac{1}{2}]$) then we can check that for all $ \beta>1$ the condition below is satisfied $$\frac{\beta}{2s(\beta -1)}=\frac{1}{2s}+\frac{1}{2s(\beta-1)}>1$$ and then using a standard comparison argument and [Corollary \[cor:1-1\]]{} we may deduce that for any $\beta>1$ there exists $C(\beta)$ and $T_\beta$ such that for all $t\ge T_\beta$ $$x_\lambda(t)\le C(\lambda,\beta)t^{\frac{1}{2s}+\frac{1}{2s(\beta-1)}}$$ The results of [Corollary \[cor:1-2\]]{} follows then by picking $\beta$ so large that we have $\frac{1}{2s(\beta-1)}\le \eps$. Note that this estimate is sharp in the sense it gives the right asymptotic for the speed of the level set i.e. we get $x_\lambda(t)\sim t^{\frac{1}{2s}}$ as $t\to \infty$. It also provides a useful information for the critical case $s=\frac{1}{2}$, where we see that the level set moves asymptotically with a constant speed although there is no existence of a traveling front in this situation. Lastly, in the spirit of [@Alfaro2017], let us obtain a finer lower bound on the speed for general degenerate monostable nonlinearities $f$, i.e. $\exists \, \beta\in (1,+\infty), \text{ such that } \lim_{u\to 0}\frac{f(u)}{u^\beta}=l>0$. \[thm:1-3\] For any $0<s<1$, assume that the nonlinearity $f$ satisfies [Assumption \[ass:1-1\]]{} and $f(u)\geq r_1u^\beta$ as $u\rightarrow 0^+$ for some small $r_1>0$, and the initial data $u_0(x)$ satisfies [Assumption \[ass:1-2\]]{}. Let $u(t,x)$ be the solution to the problem with the initial data $u_0(x)$, consider the superlevel set $E_\lambda(t)=\{ x\in\mathbb{R}\| u(t,x)>\lambda \}$ of the solution $u(t,x)$, and define $$x_\lambda(t)=\sup E_\lambda(t).$$ If further assume that $\frac{1}{2s(\beta-1)}>1$, then for any $\lambda\in(0,1)$, there exists some constants $T_\lambda>0$, $C(\lambda)>0$ and $C'(\lambda)>0$ such $$\begin{aligned} C'(\lambda) t^{\frac{1}{2s(\beta-1)}}\leq x_\lambda(t)\leq C(\lambda) t^{\frac{\beta}{2s(\beta-1)} },\qquad \forall t>T_\lambda. \end{aligned}$$ Notice that $\frac{1}{2s}<\frac{1}{2s(\beta-1)}$ if and only if $1<\beta<2$. Hence when $\frac{1}{2}<s<1$ and $1<\beta<2$, the lower bound in [Theorem \[thm:1-3\]]{} is better than the one in [Theorem \[thm:1-2\]]{}. From these estimates we can then deduce the following generic estimate: \[thm:1-4\] For any $0<s<1$, assume that the nonlinearity $f$ satisfies [Assumption \[ass:1-1\]]{} and $f(u)\geq r_1u^\beta$ as $u\rightarrow 0^+$ for some small $r_1>0$, and the initial data $u_0(x)$ satisfies [Assumption \[ass:1-2\]]{}. Let $u(t,x)$ be the solution to the problem with the initial data $u_0(x)$, consider the superlevel set $E_\lambda(t)=\{ x\in\mathbb{R}\| u(t,x)>\lambda \}$ of the solution $u(t,x)$, and define $$x_\lambda(t)=\sup E_\lambda(t).$$ Then for any $\lambda\in(0,1)$, there exists some constants $T_\lambda>0$, $C(\lambda)>0$ and $C'(\lambda)>0$ such $$\begin{aligned} C'(\lambda) t^{\sup\left\{\frac{1}{2s(\beta-1)};\frac{1}{2s}\right\}}\leq x_\lambda(t)\leq C(\lambda) t^{\frac{1}{2s(\beta-1)} +\frac{1}{2s} },\qquad \forall t>T_\lambda. \end{aligned}$$ This last results clearly indicate that the speed of the level sets is the result of a fine interplay between the diffusion process intimately linked to the quantity $t^1/2s$ and the reaction term $f$ which, as we will see in the proof, is strongly linked to the quantity $t^{\frac{1}{2s(\beta-1)}}$. (-0.1,0)–(21,0) node\[right\][$s$]{}; (0,-0.1)–(0,16) node\[right\][$\beta$]{}; plot (,[4\(1- (1/(1-2\(/20)))) ]{}); plot (,[4\(1+1/(2\(/20))) ]{}); plot (,[4\(1+1/(2\(/20))) ]{}); (0,4)–(21,4) node\[right\]; (0,8)–(10,8) node\[right\]; (10,8)–(21,8) node\[right\]; (20,0)–(20,16); (10,0)–(10,16); at (-0.5,-0.5)[$0$]{}; at (-0.5,4)[$1$]{}; at (-0.5,8)[$2$]{}; at (20,-0.5)[$1$]{}; at (10,-0.5)[$\frac{1}{2}$]{}; at (5,2)[$x_\lambda(t) \sim e^{\rho t}$]{}; at (5,5.5); at (15,2)[$x_\lambda(t) \sim e^{\rho t}$]{}; at (15,5.5); at (9.1,11.25); at (15,16)[$x_\lambda(t) \sim t$]{}; Further comments ---------------- Before going to the proofs of our results, we would like to make some further comments. First, we would like to emphasize that similar results were previously obtained in [@Alfaro2017] in the context of integrodifferential equation $$\label{noloc} \begin{cases} \partial_tu(t,x)=J\star u(t,x) -u(t,x) + f(u(t,x)) \quad \text{ for }\quad t>0, x\in \R\\ u(0,x)=u_0(x) \end{cases}$$ where $J\star u$ stands for the standard convolution and $J$ is a positive probability density with a finite first moment i.e $J\in L^1(\R)$ such that $J\ge 0,\int_{\R}J(z)\,dz=1,\int_{\R}J(z)\|z\|\,dz<+\infty$. The two equations and shares some similarities, and in particular the equation may be viewed as a reformulation of the equation but with a non integrable singular kernel. However, the treatment of the singularity is of crucial importance here and induces some tricky technical difficulties, which the ideas developed to analyse seem not able to overcome. Indeed, the challenge of singularity here is intrinsic and related to the physical nature of the fractional Laplacian. The approach here is hence not just an adaptations of the proofs given in [@Alfaro2017], and we have to deal with the singularity carefully. In particular, we go a step further in our understanding of the mechanism triggering acceleration by describing the situation for $s \in (0,\frac{1}{2})$, a situation which is not treated in [@Alfaro2017] at all. We believe that some of the techniques developed here will be also useful to apprehend propagation phenomena in the equation for kernels that do not satisfy this first moment condition. In particular, the analysis presented here should provide the ground for a deeper understanding of nonlocal combustion problems modeled by the equation studied in [@Coville2007d], by ensuring that the existence of traveling front is conditioned to a first moment property satisfied by the kernel. Works in this direction are currently underway. We also want to stress that although our results gives some good insights on the speed of the level sets, apart from situations involving combustion nonlinearities where a precise asymptotic is known, there is still a gap in our estimates and the right behaviour,that we believe is $t^{\frac{\beta}{2s(\beta-1)}}$ has not been capture yet. Using new approaches, recent progress have been made on the understanding of acceleration phenomena in various situations , namely for semilinear equation like with a nonlinear diffusion instead of the classical diffusion [@Alfaro2017b; @Alfaro2019; @Garnier2017] as well as for the equation with Fisher -KPP type nonlinearity [@Bouin2018; @Garnier2011]. The different approaches developed in these works may be of some help in this task. Works in this direction are also under consideration. The paper is organised as follows. In Section \[sec1\], we prove Theorem \[thm:1-1\] and obtain the upper bound on the speed of the level set.Then in Section \[sec-l\], we obtain the generic lower bound on this speed, Theorem \[thm:1-2\]. Finally, in the last section, Section \[sec3\], we prove the a refine estimate of this speed when a degenerate monostable nonlinearity $f$ is considered, Theorem \[thm:1-3\]. Upper bound on the speed of the super level sets {#sec1} ================================================ Construction of a supersolution: For some constant $p>0$ which will be determined later, let’s define $$\begin{aligned} v_0(x)=\left\{ \begin{array}{ll} 1, &\textnormal{if $x\leq1$},\\ \displaystyle \frac{1}{x^p}, &\textnormal{if $x>1$}. \end{array} \right.\end{aligned}$$ (-1.5,0)–(3,0) node\[right\][$x$]{}; (0,-0.2)–(0,1.5) node\[right\][$y$]{}; plot (,[1/() ]{}); (-1.5,1)–(1,1) node\[right\]; (1,0)–(1,1); at (-0.1,-0.1)[$0$]{}; at (-0.1,1.1)[$1$]{}; at (1,-0.1)[$1$]{}; at (2,0.4)[$v_0(x)$]{}; For any $\gamma>0$, let $w(t,x)$ be the solution to the following initial-value problem: $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{dw(t,x)}{dt}=\gamma[w(t,x)]^\beta,\\ w(0,x)=v_0(x) \end{array} \right.\end{aligned}$$ Since $\beta>1$, it’s easy to solve the above problem and obtain $$\begin{aligned} w(t,x)&=&\frac{1}{\left[[v_0(x)]^{1-\beta}-\gamma(\beta-1)t \right]^{\frac{1}{\beta-1}}}. \end{aligned}$$ By the definition of $v_0(x)$, it’s easy to see that $w(t,x)$ is well defined for $t\in\left[0,\frac{1}{\gamma(\beta-1)}\right)$ if $x\leq1$; and $w(t,x)$ is well defined for $t\in\left[0, \frac{x^{p(\beta-1)}}{\gamma(\beta-1)}\right)$ if $x>1$. When $t$ is fixed, the function $w(t,x)$ is decreasing with respect to $x$. Let $x_0(t)=\left[ 1+\gamma(\beta-1)t \right]^{\frac{1}{p(\beta-1)}}$ for all $t>0$, it’s easy to know that $x_0(t)>1$, $w(t,x_0(t))=1$, $w(t,x)>1$ for all $x<x_0(t)$, and $w(t,x)<1$ for all $x>x_0(t)$. Let’s consider the function $$\begin{aligned} m(t,x)=\left\{ \begin{array}{ll} 1, &\textnormal{if $x\leq x_0(t)$},\\ w(t,x),& \textnormal{if $x>x_0(t)$}. \end{array} \right.\end{aligned}$$ It’s easy to see that $m(t,x)$ is well defined for all $t\geq0$ and all $x\in\mathbb{R}$, and $0<m(t,x)\leq 1$ for all $t\geq0$ and all $x\in\mathbb{R}$. \[claim:2-1\] If $p+1\geq p\beta$, then there exists some constant $C(p,\beta)>0$ such that $$\|\partial_x m(t,x)\|+\|\partial^2_{xx} m(t,x)\|\leq C(p,\beta),\qquad\forall t>0,\ \forall x\in\mathbb{R}.$$ It’s easy to see that $\partial_xm(t,x)=\partial^2_{xx}m(t,x)=0$ for all $(t,x)$ such that $x\leq x_0(t)$. Now for any $x>x_0(t)$, then $$\begin{aligned} m(t,x)=w(t,x)=[ [v_0(x)]^{1-\beta}-\gamma(\beta-1)t ]^{\frac{1}{1-\beta} } \end{aligned}$$ Since $x_0(t)>1$, then $v_0(x)=\frac{1}{x^p}$. A direct computation shows that $$\begin{aligned} \partial_xm(t,x)&=&\partial_xw(t,x)\\ &=&\frac{1}{1-\beta}\cdot [ [v_0(x)]^{1-\beta}-\gamma(\beta-1)t ]^{\frac{1}{1-\beta} -1}\cdot (1-\beta)[v_0(x)]^{-\beta}\cdot v_0'(x)\\ &=&-p[m(t,x)]^{\beta}\cdot x^{p\beta-p-1}. \end{aligned}$$ Since $\beta>1$, $p+1\geq p\beta$, $0<m(t,x)\leq 1$, and $x>x_0(t)>1$, then we have $$\|\partial_xm(t,x)\|\leq p.$$ On the other hand, we have $$\begin{aligned} \partial^2_{xx}m(t,x)&=&-p\partial_x\left[ [m(t,x)]^{\beta}\cdot x^{p\beta-p-1} \right]\\ &=&-p\left[\beta[m(t,x)]^{\beta-1}\cdot \partial_xw(t,x)\cdot x^{p\beta-p-1}+[m(t,x)]^\beta\cdot (p\beta-p-1)\cdot x^{p\beta-p-2} \right]\\ &=&-p\left[\beta[m(t,x)]^{\beta-1}\cdot \left[ -p[m(t,x)]^{\beta}\cdot x^{p\beta-p-1} \right]\cdot x^{p\beta-p-1}+[m(t,x)]^\beta\cdot (p\beta-p-1)\cdot x^{p\beta-p-2} \right]\\ &=&p^2\beta[m(t,x)]^{2\beta-1}x^{2(p\beta-p-1)}+p\beta(p+1-p\beta)[m(t,x)]^{\beta}x^{p\beta-p-2}. \end{aligned}$$ Since $\beta>1$, $p+2>p+1\geq p\beta$, $0<m(t,x)\leq 1$, and $x>x_0(t)>1$, then we have $$\|\partial^2_{xx}m(t,x)\|\leq p^2\beta+p\beta(p+1-p\beta).$$ \[claim:2-2\] If $p+1\geq p\beta$, then there exists some constant $C(s,p,\beta)>1$ such that $$\| (-\Delta)^sm(t,x)\|\leq C(s,p,\beta),\qquad\forall t>0,\ \forall x\in\mathbb{R}.$$ By the definition of $(-\Delta)^sm(t,x)$, it’s easy to see that $$\begin{aligned} (-\Delta)^sm(t,x)&=&\frac{1}{2}\int_{\mathbb{R}} \frac{ m(t,x+h)+m(t,x-h)-2m(t,x) }{\|h\|^{1+2s}} dh\\ &=&\frac{1}{2}\int_{\|h\|\geq1 } \frac{ m(t,x+h)+m(t,x-h)-2m(t,x) }{\|h\|^{1+2s}} dh+\frac{1}{2}\int_{\|h\|<1} \frac{ m(t,x+h)+m(t,x-h)-2m(t,x) }{\|h\|^{1+2s}} dh \end{aligned}$$ Since $0<m(t',x')\leq1$ for all $t'>0$ and all $x'\in\mathbb{R}$, by [Claim \[claim:2-1\]]{}, then there exists some $C_1(s,p,\beta)>0$ such that $$\begin{aligned} \|(-\Delta)^sm(t,x)\|&\leq& \frac{1}{2}\int_{\|h\|\geq1} \frac{4}{\|h\|^{1+2s}} dh+\frac{1}{2}\int_{\|h\|<1} \frac{ C_1\|h\|^{2} }{\|h\|^{1+2s}} dh\\ &=&C(s,p,\beta). \end{aligned}$$ \[claim:2-3\] For any $(t,x)$ such that $x\leq x_0(t)$, we have $$\partial_tm(t,x)+(-\Delta)^sm(t,x)-f(m(t,x))>0.$$ In fact, since $x\leq x_0(t)$, then $m(t,x)=1$. By [Assumption \[ass:1-1\]]{}, then $f(m(t,x))=f(1)=0$. By the definition of $m(t',x')$, it’s easy to see that $\partial_tm(t,x)=0$. Since $m(t,x)=1$, $0<m(t',x')\leq1$ for all $t'>0$ and all $x'\in\mathbb{R}$, by the definition of $(-\Delta)^sm(t,x)$, it’s easy to obtain that $(-\Delta)^sm(t,x)>0$. In summary, we obtain that $$\partial_tm(t,x)+(-\Delta)^sm(t,x)-f(m(t,x))>0.$$ \[claim:2-4\] If $p+1\geq p\beta$, let $C_1$ be the positive constant defined in [Claim \[claim:2-2\]]{}, for any fixed $\gamma>\gamma_0:=r+2C_1$, let’s define $$x_\gamma(t)=\left[\left(\frac{\gamma-r}{C_1}\right)^{\frac{\beta-1}{\beta} }+\gamma(\beta-1)t \right]^{\frac{1}{p(\beta-1)}}.$$ Then $x_0(t)<x_\gamma(t)$ for all $t>0$, and for any $(t,x)$ such that $x_0(t)<x\leq x_\gamma(t)$, we have $$\partial_tm(t,x)+(-\Delta)^sm(t,x)-f(m(t,x))\geq0.$$ Since $\gamma-r\geq 2C_1>2$, since $\beta>1$, then $x_\gamma(t)>[1+\gamma(\beta-1)t]^{\frac{1}{p(\beta-1)}}=x_0(t)$. Now for any $(t,x)$ such that $x_0(t)<x\leq x_\gamma(t)$, by [Assumption \[ass:1-1\]]{} and [Claim \[claim:2-2\]]{}, then we have $$\begin{aligned} \partial_tm(t,x)+(-\Delta)^sm(t,x)-f(m(t,x))&=&\partial_tw(t,x)+(-\Delta)^sm(t,x)-f(w(t,x))\\ &=&\gamma[w(t,x)]^\beta-f(w(t,x))+(-\Delta)^sm(t,x)\\ &\geq&\gamma[w(t,x)]^\beta- r[w(t,x)]^\beta-C_1\\ &=& (\gamma-r)[w(t,x)]^\beta-C_1. \end{aligned}$$ Since $w(t,x')$ is decreasing with respect to $x'$, since $x_0(t)<x\leq x_\gamma(t)$, then we obtain $$\begin{aligned} \partial_tm(t,x)+(-\Delta)^sm(t,x)-f(m(t,x))\geq (\gamma-r)[w(t,x_\gamma(t))]^\beta-C_1=0. \end{aligned}$$ In the following, let’s verify the supersolution inequality for $(t,x)$ with $x>x_\gamma(t)$. First, we introduce notations $q:=p(\beta-1)$ and $\sigma:=\gamma(\beta-1)t$. For some constant $K>2$ which will be determined later, let’s write $$\begin{aligned} -(-\Delta)^sm(t,x)&=&{\int_{-\infty}^{\frac{x_0(t)-x}{K} } \frac{m(t,x+z)-m(t,x)}{\|z\|^{1+2s}} dz }+{\textnormal{P.V. }\int^{+\infty}_{\frac{x_0(t)-x}{K} } \frac{m(t,x+z)-m(t,x)}{\|z\|^{1+2s}} dz }\\ &=:&I_1+I_2.\end{aligned}$$ Since $\beta>1$, by the definitions of $x_\gamma(t)$ and $x_0(t)$, we can find some large $\gamma_1$ which may depend on $K$ such that $\gamma_1>\gamma_0$ (where $\gamma_0$ is defined in [Claim \[claim:2-4\]]{}), and $x_0(t)-x<-K$ for all $\gamma\geq\gamma_1$ and all $x>x_\gamma(t)$. Since $0<m(t',x')\leq1$ for all $t'>0$ and $x'\in\mathbb{R}$, then $$\begin{aligned} I_1 \leq \int_{-\infty}^{\frac{x_0(t)-x}{K} } \frac{1}{\|z\|^{1+2s}} dz=\frac{1}{2s}\cdot \left[ \frac{K}{x-x_0(t)}\right]^{2s}.\end{aligned}$$ By choosing $q<1$ (that is, $p(\beta-1)<1$), then $x^q\leq [x_0(t)]^q+(x-x_0(t))^q$, that is, $[x^q-[x_0(t)]^q ]^{\frac{1}{q}}\leq x-x_0(t)$, which implies that $$\begin{aligned} \frac{1}{[x-x_0(t)]^{2s }}\leq \frac{1}{[x^q-[x_0(t)]^q]^{\frac{2s}{q}} }.\end{aligned}$$ Since $x>x_0(t)+K>2$, then $(x^q-1)^{\frac{1}{q}}>1$. So we get $\left[ v_0((x^q-1)^{\frac{1}{q}}) \right]^{1-\beta}=x^q-1$, which implies that $$\begin{aligned} w(t,(x^q-1)^{\frac{1}{q}}) &=&\frac{ 1 }{\left[ \left[ v_0((x^q-1)^{\frac{1}{q}}) \right]^{1-\beta}-\gamma(\beta-1)t \right]^{\frac{1}{\beta-1} }}\\ &=&\frac{1}{\left[ x^q-1-\gamma(\beta-1)t \right]^{\frac{1}{\beta-1} }}\\ &=&\frac{1}{\left[ x^q-[x_0(t)]^q \right]^{\frac{1}{\beta-1} }}.\end{aligned}$$ Since $\frac{2s}{q}\cdot (\beta-1)=\frac{2s}{p}$ and $x-x_0(t)>K$, then we can obtain $$\begin{aligned} I_1\leq \frac{1}{2s}\cdot K^{2s}\cdot \left[ w(t,(x^q-1)^{\frac{1}{q}}) \right]^{\frac{2s}{p}}.\end{aligned}$$ Since $x>1$ and $q=p(\beta-1)$, then we have $$\begin{aligned} \frac{w(t,(x^q-1)^{\frac{1}{q}}) }{w(t,x)}&=&\frac{1}{w(t,x)}\cdot \frac{1}{\left[ x^q-1-\gamma(\beta-1)t \right]^{\frac{1}{\beta-1} }}\\ &=&\frac{1}{w(t,x)}\cdot\frac{1}{\left[ x^q-1- [v_0(x)]^{1-\beta}+[w(t,x)]^{1-\beta}) \right]^{\frac{1}{\beta-1} }}\\ &=&\frac{1}{w(t,x)}\cdot\frac{1}{\left[ x^q-1- x^{-p(1-\beta)}+[w(t,x)]^{1-\beta}) \right]^{\frac{1}{\beta-1} }}\\ &=&\frac{1}{w(t,x)}\cdot\frac{1}{\left[ -1+[w(t,x)]^{1-\beta}) \right]^{\frac{1}{\beta-1} }}\\ &=&\frac{ 1 }{\left[ 1-[w(t,x) ]^{\beta-1} \right]^{ \frac{1}{\beta-1 }}}.\end{aligned}$$ Since $x>x_\gamma(t)$, and $w(t',x')$ is decreasing with respect to $x'$, then $$\begin{aligned} w(t,x)<w(t,x_\gamma(t))=\left[ \frac{C_1}{\gamma-r} \right]^{\frac{1}{\beta}}.\end{aligned}$$ So we can find some large $\gamma_2>\gamma_1$ such that for all $\gamma>\gamma_2$, we have $\left[ \frac{C_1}{\gamma-r} \right]^{\frac{1}{\beta}}\leq 1$, which implies that $$\begin{aligned} I_1\leq C_2 K^{2s}\cdot [w(t,x)]^{\frac{2s}{p}}.\end{aligned}$$ For $I_2$, since $x_0(t)-x<-K$, then we have $$\begin{aligned} I_2&=&{\int_{\frac{x_0(t)-x}{K} }^{-1} \frac{m(t,x+z)-m(t,x)}{\|z\|^{1+2s}} dz } +{\textnormal{P.V. }\int_{-1}^1 \frac{m(t,x+z)-m(t,x)}{\|z\|^{1+2s}} dz }+{ \int_1^{+\infty} \frac{m(t,x+z)-m(t,x)}{\|z\|^{1+2s}} dz }\\ &=:&I_3+I_4+I_5.\end{aligned}$$ For $I_3$, for all $\frac{x_0(t)-x}{K}\leq z\leq -1$, since $x_0(t)>1$ and $K>2$, then we have $x+z>\frac{x}{2}>1$. By changing variables, $z=xu$, then we have $$\begin{aligned} I_3&=&\int_{\frac{x_0(t)-x}{K} }^{-1} \frac{w(t,x+z)-w(t,x)}{\|z\|^{1+2s}} dz\\ &=&\int_{\frac{x_0(t)-x}{Kx} }^{-\frac{1}{x}} \frac{w(t,x+xu)-w(t,x)}{\|xu\|^{1+2s}} \cdot xdu\\ &=&xw(t,x)\int_{\frac{x_0(t)-x}{Kx} }^{-\frac{1}{x}} \frac{1}{\|xu\|^{1+2s}}\cdot\left[\frac{w(t,x+xu)}{w(t,x)}-1 \right]du\\ &=&xw(t,x)\int_{\frac{x_0(t)-x}{Kx} }^{-\frac{1}{x}} \frac{1}{\|xu\|^{1+2s}}\cdot\left[ \frac{[(x+xu)^{p(\beta-1)}-(\beta-1)\gamma t]^{\frac{1}{1-\beta}} }{[x^{p(\beta-1)}-(\beta-1)\gamma t]^{\frac{1}{1-\beta}}} -1 \right]du\\ &=&xw(t,x)\int_{\frac{x_0(t)-x}{Kx} }^{-\frac{1}{x}}\frac{1}{\|xu\|^{1+2s}}\cdot\left[ \frac{1}{ \left[ \frac{(1+u)^q-1}{1-\frac{\sigma}{x^q}} +1 \right]^{\frac{p}{q}} } -1\right] du.\end{aligned}$$ \[claim:2-5\] For $q<1$, there exists some $K(q)>0$ such that for all $t>0$, all $K\geq K(q)$, all $x>x_0(t)$, and all $u\in\left[\frac{x_0(t)-x}{Kx} ,0 \right]$, we have $$\frac{(1+u)^q-1}{1-\frac{\sigma}{x^q}}-1\geq -\frac{1}{2}.$$ The proof goes identically with the one in [@Alfaro2017]. By [Claim \[claim:2-5\]]{} and Lagrange’s Mean Value Theorem, then there exists some constant $C_3>0$ such that $$\begin{aligned} \frac{1}{ \left[ \frac{(1+u)^q-1}{1-\frac{\sigma}{x^q}} +1 \right]^{\frac{p}{q}} } -1&\leq&-\frac{p}{q}[1+C_3]\cdot \frac{(1+u)^q-1}{1-\frac{\sigma}{x^q}}\\ &=&\frac{p}{q}[1+C_3]\cdot x^q\cdot \frac{1-(1+u)^q}{x^q-\sigma}.\end{aligned}$$ Since $[w(t,x)]^{1-\beta}=x^q-\sigma$ and $0<q<1$, then $$\begin{aligned} \frac{1}{ \left[ \frac{(1+u)^q-1}{1-\frac{\sigma}{x^q}} +1 \right]^{\frac{p}{q}} } -1&\leq&\frac{p}{q}[1+C_3]\cdot x^q [w(t,x)]^{\beta-1}[1-(1+u)^q] \\ &\leq&\frac{p}{q}[1+C_3]\cdot x^q [w(t,x)]^{\beta-1}\cdot \|u\|^q\\ &=&C_4\cdot x^q [w(t,x)]^{\beta-1}\cdot \|u\|^q. \end{aligned}$$ So we have $$\begin{aligned} I_3&\leq& xw(t,x)\int_{\frac{x_0(t)-x}{Kx} }^{-\frac{1}{x}}\frac{1}{\|xu\|^{1+2s}}\cdot C_4\cdot x^q [w(t,x)]^{\beta-1}\cdot \|u\|^qdu\\ &=&C_4[w(t,x)]^\beta \int_{\frac{x_0(t)-x}{Kx} }^{-\frac{1}{x}}\frac{1}{\|xu\|^{1+2s}}\cdot \|xu\|^q\cdot x du\\ &=&C_4[w(t,x)]^\beta\int_{\frac{x_0(t)-x}{K}}^{-1} \frac{1}{\|z\|^{1+2s-q}} dz.\end{aligned}$$ Let’s take $q$ such that $2s-q>0$, that is, $2s>q=p(\beta-1)$, then we have $$\begin{aligned} I_3\leq C_4[w(t,x)]^\beta\int_{-\infty}^{-1} \frac{1}{\|z\|^{1+2s-q}} dz=C_5[w(t,x)]^\beta.\end{aligned}$$ For $I_4$, it’s easy to see that $$I_4=\frac{1}{2}\int_{-1}^1 \frac{m(t,x+z)+m(t,x-z)-2m(t,x) }{\|z\|^{1+2s}} dz.$$ By the proof of [Claim \[claim:2-1\]]{}, since $p+1\geqq p\beta$ and $x>1$, then we know that $$\begin{aligned} \|\partial^2_{xx}m(t,x)\|&\leq& p^2\beta[m(t,x)]^{2\beta-1}+(p+1-p\beta)[m(t,x)]^{\beta}.\end{aligned}$$ For any $z\in [-1,1]$, since $x>2$, then $x+z>1$, which implies that $$\begin{aligned} \|\partial^2_{xx}m(t,x)\|&\leq& p^2\beta[w(t,x)]^{2\beta-1}+(p+1-p\beta)[w(t,x)]^{\beta}.\end{aligned}$$ Since $\beta>1$ and $0<w(t,x)\leq 1$, then there exists some $C_6>0$ such that $$\begin{aligned} \|\partial^2_{xx}m(t,x)\|&\leq& C_6[w(t,x)]^{\beta}.\end{aligned}$$ So we know that $$\begin{aligned} I_4\leq\frac{1}{2}\int_{-1}^1 \frac{C_6[w(t,x)]^\beta \|z\|^2 }{\|z\|^{1+2s}} dz=C_7[w(t,x)]^\beta.\end{aligned}$$ For $I_5$, for all $z\geq0$, since $m(t',x')$ is decreasing with respect to $x'$, then $$I_5\leq 0.$$ So we have $$\begin{aligned} \partial_tm(t,x)+(-\Delta)^sm(t,x)-f(m(t,x))&=&\partial_tw(t,x)+(-\Delta)^sm(t,x)-f(w(t,x))\\ &=&\gamma[w(t,x)]^\beta-f(w(t,x))+(-\Delta)^sm(t,x)\\ &\geq&\gamma[w(t,x)]^\beta- r[w(t,x)]^\beta-I_1-I_2\\ &=& (\gamma-r)[w(t,x)]^\beta-I_1-I_3-I_4-I_5\\ &\geq&[w(t,x)]^\beta[\gamma-r C_2 K^{2s}\cdot [w(t,x)]^{\frac{2s}{p}-\beta}- C_5-C_7 ].\end{aligned}$$ Let’s take $p=\frac{2s}{\beta}$ (in this case, we have $p+1\geq p\beta$ and $2s>p(\beta-1 )$), when $\gamma$ is large enough, we have $$\partial_tm(t,x)+(-\Delta)^sm(t,x)-f(m(t,x))\geq0.$$ In summary, we can conclude that $m(t,x)$ is a supersolution to the problem . By the comparision principle, then $$u(t,x)\leq m(t,x),\qquad\forall t>0,\ \forall x\in\mathbb{R}.$$ So for any $\lambda\in(0,1)$ and any $x\in\Gamma_\lambda(t)$, then $$\lambda\leq u(t,x) \leq m(t,x)=\frac{1}{\left[x^{\frac{2s(\beta-1)}{\beta} }-\gamma(\beta-1)t \right]^{\frac{1}{\beta-1}}}.$$ Since $\beta>1$, hence we get $$x\leq \left[ \left(\frac{1}{\lambda} \right)^{\beta-1}+\gamma(\beta-1)t \right]^{\frac{\beta}{2s(\beta-1)} }.$$ So when $T_\lambda\gg1$, we have $$\begin{aligned} x_\lambda(t)\leq C(\lambda)\cdot t^{\frac{\beta}{2s(\beta-1)}}.\end{aligned}$$ Lower bound on the speed of the super level sets {#sec-l} ================================================ By [Assumption \[ass:1-2\]]{} on the initial data $u_0(x)$, we can construct a non-increasing function $\widetilde{u}_0(x)$ such that $\widetilde{u}_0(x)\leq u_0(x)$ for all $x\in\mathbb{R}$ and $$\begin{aligned} \widehat{u}_0(x)=\left\{ \begin{array}{ll} c_0, &\textnormal{if $x\leq-R_0-1$},\\ 0, &\textnormal{if $x\geq -R_0$}, \end{array} \right. \end{aligned}$$ for some small $0<c_0\ll1$ and some large $R_0\gg1$. Let $v(t,x)$ be the solution of the following problem: $$\begin{aligned} \left\{ \begin{array}{l} v_t+(-\Delta)^sv=0,\quad t>0, \ x\in\mathbb{R},\\ v(0,x)=\widehat{u}_0(x),\quad x\in\mathbb{R}. \end{array} \right. \end{aligned}$$ Since $f(u)\geq0$ for all $u\in[0,1]$, it’s easy to see that $v(t,x)$ is a subsolution to the problem . By the comparison principle, we have $$v(t,x)\leq u(t,x),\qquad\forall t>0,\ x\in\mathbb{R}.$$ Let $p_s(t,x)$ be the heat kernel for $(-\Delta)^s$, then we have $$\begin{aligned} v(t,x)&=&\int_{\mathbb{R}} \widehat{u}_0(x-y)p_s(t,y) dy,\qquad\forall t>0,\ x\in\mathbb{R}. \end{aligned}$$ For the heat kernel for $(-\Delta)^s$, it’s well known that there exists some constant $1>C_1>0$ such that $$\begin{aligned} \frac{C_1}{t^{\frac{1}{2s}}[1+\|t^{-\frac{1}{2s}} x\|^{1+2s} ] }\leq p_s(t,x)\leq \frac{C_1^{-1}}{t^{\frac{1}{2s}}[1+\|t^{-\frac{1}{2s}} x\|^{1+2s} ] },\qquad\forall t>0,\ x\in\mathbb{R}. \end{aligned}$$ So we get $$\begin{aligned} v(t,x)&\geq&\int_{\mathbb{R}} \widehat{u}_0(x-y)\cdot \frac{C_1}{t^{\frac{1}{2s}}[1+\|t^{-\frac{1}{2s}} y\|^{1+2s} ] } dy \nonumber \\ &\geq&\int_{x+R_0+1 }^{+\infty} \frac{ c_0\cdot C_1 }{t^{\frac{1}{2s}}[1+\|t^{-\frac{1}{2s}} y\|^{1+2s} ] } dy\nonumber \\ &=&\int_{t^{-\frac{1}{2s}}(x+R_0+1) }^{+\infty} \frac{ c_0\cdot C_1 }{1+\|z\|^{1+2s} } dz \label{esti-diff} \end{aligned}$$ In particular, we have $$u(1,x)\ge \int_{x+R_0+1}^{+\infty} \frac{ c_0\cdot C_1 }{1+\|z\|^{1+2s} } dz.$$ Thus for $x>1$, we may find a constant $C_2>0$ such that $$u(1,x)\ge \frac{C_2}{x^{2s}}.$$ As a result, we can find a small enough $d>0$ such that $$\label{ac:def-v0-1} u(1,x)\geq v(1,x) \ge \tilde{u}_0(x):= \begin{cases} d &\text{ for } x \le 1\\ \frac{d}{x^{2s}} &\text{ for } x \ge 1. \end{cases}$$ Hence, from the comparison principle and up to a shift in time, we only need to get the lower estimate for the case that $u(t,x)$ is the solution to the problem with the initial data $ \tilde{u}_0$. Since $\tilde{u}_0(x)$ is decreasing, it’s easy to see that $u(t,x)$ is decreasing with respect to $x$. Let $\displaystyle \lambda_0:=\int_{\frac{1}{2} }^{+\infty} \frac{ c_0\cdot C_1 }{1+\|z\|^{1+2s} } dz$, $\displaystyle x_B(t):=\frac{t^{\frac{1}{2s} }}{4}$, and $x_{\lambda_0}(t)$ be such that $u(t,x_{\lambda_0}(t))=\lambda_0$. From , then there exist $T_{\lambda_0}\gg1$ such that for all $t\ge T_{\lambda_0}$ we have, $$v(t,x_B(t))\ge \int_{\frac{1}{4}+t^{-\frac{1}{2s}}(R_0+1) }^{+\infty} \frac{ c_0\cdot C_1 }{1+\|z\|^{1+2s} } dz\ge \int_{\frac{1}{2} }^{+\infty} \frac{ c_0\cdot C_1 }{1+\|z\|^{1+2s} } dz =\lambda_0.$$ Since $\widehat{u}_0(x)$ is decreasing, it’s easy to see that $v(t,x)$ is decreasing with respect to $x$. Since $u(t,x)\geq v(t,x)$ for all $t\geq0$ and all $x\geq0$, and $u(t,x)$ and $v(t,x)$ are decreasing with respect to $x$, then we can get $\frac{t^{\frac{1}{2s} }}{4}=x_B(t)\le x_{\lambda_0}(t)$. The above argument holds as well for any $0<\lambda \leq \lambda_0$ which provides the lower estimate. It remains to obtain a similar bound for a given $\lambda_0<\lambda <1$. To obtain the bound we can argue as in [@Alfaro2017]. So first let us prove an invasion lemma on the solution of the Cauchy problem . Namely, \[prop:level-set\] For any $0<s<1$, assume that the nonlinearity $f$ satisfies [Assumption \[ass:1-1\]]{}, and the initial data $u_0(x)$ satisfies [Assumption \[ass:1-2\]]{}. Assume $\beta>1$ and $\frac{\beta}{2s(\beta-1)}>1$, and let $u(t,x)$ be the solution to the problem with the initial data $u_0(x)$. Then, for any $A \in \R$, $$\label{invasion} \lim_{t\to \infty}u(t,x)=1 \quad \text{ uniformly in } (-\infty,A],$$ and, for any $\lambda \in (0,1)$, $$\label{acc-prop} \lim_{t\to\infty}\frac{x_{\lambda}(t)}{t}=+\infty.$$ Let us postpone for a moment the proof of [Proposition \[prop:level-set\]]{} and finish the proof of [Theorem \[thm:1-2\]]{}. Let us denote by $w(t,x)$ the solution of starting from a nonincreasing $w_0$ such that $$\label{initial-data-w} w_0(x)= \begin{cases} \lambda_0 &\text{ if } x\leq -1 \\ 0 &\text{ if }x\geq 0. \end{cases}$$ It follows from [Proposition \[prop:level-set\]]{} that there is a time $\tau_{\lambda}>0$ such that $$\label{w-grand} w(\tau_{\lambda},x)>\lambda, \quad \forall x\leq 0.$$ On the other hand, since $\widehat{u}_0(x)$ is decreasing, it’s easy to see that $v(t,x)$ is also decreasing with respect to $x$. Since $u(t,x_B(t))\geq\lambda_0$, then $$\begin{aligned} u(t,x)\geq v(t,x)\geq \lambda_0,\qquad\forall x\leq x_B(t). \end{aligned}$$ So it follows from that $$u(T,x)\ge w_0(x-x_{\lambda_0}(T)),\quad \forall T\geq 0, \forall x\in \R.$$ So the comparison principle yields $$u(T+\tau,x)\geq w(\tau,x-x_{\lambda_0}(T)),\quad \forall T\geq 0, \forall \tau \geq 0, \forall x\in \R.$$ In view of , this implies that $$u(T+\tau_{\lambda},x)>\lambda,\quad \forall T\geq 0, \forall x\leq x_{\lambda_0} (T).$$ Hence, for $t\geq \tau _\lambda$, the above implies $$x_\lambda(t)\geq x_{\lambda_0}(t-\tau _\lambda)=\frac{(t-\tau _\lambda)^{\frac{1}{2s} }}{4} \geq \underline C t^{\frac{1}{2s}},$$ provided $t\geq T_\lambda '$, with $T_\lambda '>\tau _\lambda$ large enough. This concludes the proof of the lower estimate. In summary, we can conclude that for any $\lambda\in(0,1)$, then there exist some constants $T_\lambda'>0$ and $C'(\lambda)>0$ such $$\begin{aligned} x_\lambda(t)\geq C'(\lambda) t^{\frac{1}{2s} },\qquad\forall t>T_\lambda'. \end{aligned}$$ In order to prove [Proposition \[prop:level-set\]]{}, let us first establish the following result. \[prop:cinfinie\] For any $0<s<1$, assume that the nonlinearity $f$ satisfies [Assumption \[ass:1-1\]]{}, and the initial data $u_0(x)$ satisfies [Assumption \[ass:1-2\]]{}. Assume $\beta>1$ and $\frac{\beta}{2s(\beta-1)}>1$. Let $(g_{n})=(g_{\theta _n })$ be a sequence of bistable nonlinearities such that $g_n\le g_{n+1}\le f$ and $g_n\to f$. Let $(c_n,U_n)$ be the associated sequence of traveling waves. Then $$\lim_{n\to \infty }c_n=+\infty.$$ Since $g_{n+1}\geq g_n$ it follows from standard sliding techniques [@Berestycki1991a; @Chen1997; @Coville2005; @Coville2006; @Coville2007] that $c_{n+1}\geq c_n$. Assume now by contradiction that $c_n \nearrow \bar c$ for some $\bar c\in \R$. Observe that since $g_n\to f$ and $\int_0^1f(s)\,ds>0$, we have $c_n\ge c_0>0$ for $n$ large enough, says, $n\ge n_0$. As a consequence, for all $n\ge n_0$, $U_n$ is smooth and since any translation of $U_n$ is a still a solution, without loss of generality, we can assume the normalization $U_n(0)=1/2$. Now, thanks to Helly’s Theorem [@Brunk1956] and up to extraction, $U_n$ converges to a monotone function $\bar U$ such that $\bar U(0)=\frac{1}{2}$. Also, since $c_n<\bar c$, from the equation we get an uniform bound on $U^{'}_n, U^{''}_n$ and up to extraction, $U_n$ also converges in $C^2_{loc}(\R)$, and the limit has to be $\bar U'$. As a result, $\bar U$ is monotone and solves $$\begin{cases} (-\Delta^{s})\bar U+\bar c\bar U'+f(\bar U)=0 \quad \text{ on } \R,\\ \bar U(-\infty)=1,\quad \bar U(0)=\frac{1}{2}, \quad \bar U(\infty)=0. \end{cases}$$ In other words, we have constructed a monostable traveling wave under assumption that $\beta>1$ and $\frac{\beta}{2s(\beta-1)}>1$ , which is a contradiction with the result in [@Gui2015]. Equipped with this technical result we can establish [Proposition \[prop:level-set\]]{}. First, we prove for the particular case where the initial datum $u_0$ is a smooth decreasing function such that $$\label{d0} u_0(x)=\begin{cases} d_0 &\text{ for } x \le -1\\ 0 &\text{ for } x \ge 0, \end{cases}$$ for an arbitrary $0<d_0<1$. Since $u_0$ is nonincreasing, we deduce from the comparison principle that, for all $t>0$, the function $u(t,x)$ is still decreasing in $x$. Let us now extend $f$ by $0$ outside the interval $[0,1]$. From [@Achleitner2015] and [Proposition \[prop:cinfinie\]]{}, there exists $0<\theta<d_0$ and a Lipschitz bistable function $g\le f$ — i.e. $g(0)=g(\theta)=g(1)=0$, $g(s)<0$ in $(0,\theta)$, $g(s)>0$ in $(\theta,1)$, and $g'(0)<0$, $g'(1)<0$, $g'(\theta)>0$— such that there exists a smooth decreasing function $U_\theta$ and $c_\theta>0$ verifying $$\begin{aligned} &(-\Delta)^{s}U_\theta +c_{\theta} U_\theta'+g(U_\theta)=0 \quad \text{ on } \R, \\ &U_\theta(-\infty)=1, \qquad U_\theta(\infty)=0.\end{aligned}$$ Let us now consider $v(t,x)$ the solution of the Cauchy problem $$\begin{aligned} &\partial_t v(t,x) =-(-\Delta)^s v(t,x) + g(v(t,x)) \quad \text{ for } t>0, x\in \R, \\ &v(0,x)=u_0(x).\end{aligned}$$ Since $g\le f$, $v$ is a subsolution of the Cauchy problem and by the comparison principle, $v(t,x)\le u(t,x)$ for all $t>0$ and $x \in \R$. Now, thanks to the global asymptotic stability result [@Achleitner2015 Theorem 3.1], since $d_0>\theta$, then we know that there exists $\xi\in\R, C_0>0$ and $\kappa>0$ such that for all $t\ge 0$ $$\\|v(t,\cdot)-U_{\theta}(\cdot-c_\theta t+\xi)\\|_{L^\infty} \le C_0 e^{-\kappa t}.$$ Therefore, for all $t>0$ and $x \in \R$, we have $$u(t,x)\ge v(t,x)\ge U_{\theta}(x-c_\theta t+\xi)-C_0e^{-\kappa t}.$$ Since $c_\theta>0$, by sending $t \to \infty$, we get $1\ge \liminf_{t\to \infty}u(t,x)\ge \lim_{t\to \infty}[U_{\theta}(x-c_\theta t+\xi)-C_0e^{-\kappa t}]= 1$. As a result, for all $x\in\mathbb{R}$, we have $1\ge \limsup_{t\to \infty} u(t,x)\ge \liminf_{t\to \infty} u(t,x)= 1$, which implies that $u(t,x)\to 1$ as $t \to \infty$. Since $u(t,x)$ is decreasing in $x$, then the convergence is uniform on any set $(-\infty,A]$. This concludes the proof of for our particular initial datum. For a generic initial data satisfying [Assumption \[ass:1-2\]]{}, we can always, up to a shift in space, construct a smooth decreasing $\tilde{u_0}$ satisfying and $\tilde{u_0}\leq u_0$. Since the solution $\tilde u(t,x)$ of the Cauchy problem starting from $\tilde{u_0}$ satisfies , so does $u(t,x)$ thanks to the comparison principle. Another better lower bound on the speed of the super level sets {#sec3} =============================================================== Here we prove another lower bound on the speed of $x_\lambda(t)$ when $1<\beta<2$ and$\frac{1}{2s(\beta-1)}>1$ (notice that $\frac{1}{2s(\beta-1)}>\frac{1}{2s}$ if and only if $1<\beta<2$). In the whole of this section, let’s assume the conditions in [Theorem \[thm:1-3\]]{} hold. As above to measure the acceleration, we use a subsolution that fills the space with a superlinear speed. The construction of this subsolution is an adaptation of the one proposed by Alfaro and Coville [@Alfaro2017] for a nonlocal diffusion with an integrable kernel. It essentially contains three steps. [Step one.]{} It consists in using the diffusion to gain an algebraic tail at time $t=1$. From the proof of [Theorem \[thm:1-2\]]{}, we have known that we can find a small enough $d>0$ such that $$\label{ac:def-v0} u(1,x)\geq v(1,x) \ge v_0(x):= \begin{cases} d &\text{ for } x \le 1\\ \frac{d}{x^{2s}} &\text{ for } x \ge 1. \end{cases}$$ Hence, from the comparison principle and up to a shift in time, it is enough to prove the lower estimate for $u(t,x)$ which is the solution starting from the initial data $v_0$, which we do below. [Step two.]{} Here we construct explicitly the subsolution that we are considering. Following Alfaro-Coville [@Alfaro2017] let us consider the function $g(y):=y(1-By)$, with $B>\frac{1}{2d}$, it’s easy to see that $g(y)\le 0$ if and only if $0\leq y\leq \frac{1}{B} $, and $g(y)\le g(\frac{1}{2B})=\frac{1}{4B}<d$ for all $y\in\mathbb{R}$. As in the previous subsection, for any $\gamma>0$, let $w(\cdot,x)$ denote the solution to the Cauchy problem $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{dw}{dt}(t,x)=\gamma [w(t,x)]^{\beta},\\ w(0,x)=v_0(x). \end{array} \right.\end{aligned}$$ That is $$w(t,x)=\frac{1}{\left[ [v_0(x)]^{1-\beta} -\gamma(\beta -1) t\right]^{\frac{1}{\beta -1}}},$$ where $v_0$ is defined in . Notice that $w(t,x)$ is not defined for all times. When $x\le 1$, $w(t,x)$ is defined for $t\in[0,\frac{1}{d^{\beta -1}\gamma(\beta-1)})$, whereas for $x>1$, $w(t,x)$ is defined for $t\in\left[0,T(x):=\frac {x^{2s(\beta -1)}}{d^{\beta -1}\gamma(\beta-1)}\right)$. Let us define $$\label{ac-def-xB} x_{B}(t):=d^{\frac{1}{2s}} \left[\left(2B \right)^{\beta-1}+\gamma (\beta -1)t \right]^{\frac{1}{2s(\beta-1)}}.$$ Since $B>\frac{1}{2d}$ and $\beta>1$, then $x_B(t)>1$ and $w(t,x_B(t))=\frac{1}{2B}$. For $x<1$ and $0<t<\frac{1}{d^{\beta-1}\gamma(\beta-1)}$, since $v_0(x')=d$ for all $x'\leq 1$, then we have $\partial_xw(t,x)=\partial_{xx} w(t,x)=0$. For $x>1$ and $0< t< T(x)$, we compute $$\begin{aligned} \partial_xw(t,x)&=&\frac{1}{1-\beta}\cdot [ [v_0(x)]^{1-\beta}-\gamma(\beta-1)t ]^{\frac{1}{1-\beta} -1}\cdot (1-\beta)[v_0(x)]^{-\beta}\cdot v_0'(x)\\ &=&-2d^{1-\beta}s[w(t,x)]^\beta\cdot x^{2s\beta-2s-1}\\ &<&0\\ \partial_{xx} w(t,x)&=&-2d^{1-\beta}s\cdot[ \beta [w(t,x)]^{\beta-1}\cdot \partial_x w(t,x) \cdot x^{2s\beta-2s-1}+[w(t,x)]^\beta\cdot (2s\beta-2s-1)x^{2s\beta-2s-2} ]\\ &=&-2d^{1-\beta}s\cdot\left[ \beta [w(t,x)]^{\beta-1}\cdot \left(-2d^{1-\beta}s[w(t,x)]^{\beta}\cdot x^{2s\beta-2s-1}\right) \cdot x^{2s\beta-2s-1}\right.\\ &&\left.+[w(t,x)]^{\beta}\cdot (2s\beta-2s-1)x^{2s\beta-2s-2} \right]\\ &=&2d^{1-\beta}s [w(t,x)]^{\beta}\cdot x^{2s\beta-2s-2}\cdot[ 2d^{1-\beta}s\beta\cdot [w(t,x)]^{\beta-1}x^{2s\beta-2s}+2s+1-2s\beta ]\\ &=&2d^{1-\beta}s [w(t,x)]^{\beta}\cdot x^{2s\beta-2s-2}\cdot[ 2d^{1-\beta}s\beta\cdot [[v_0(x)]^{1-\beta}-\gamma(\beta-1)t]^{-1} x^{2s\beta-2s}+2s+1-2s\beta ]\\ &>&2d^{1-\beta}s [w(t,x)]^{\beta}\cdot x^{2s\beta-2s-2}\cdot[ 2d^{1-\beta}s\beta\cdot [v_0(x)]^{\beta-1} x^{2s\beta-2s}+2s+1-2s\beta ]\\ &=&2d^{1-\beta}s [w(t,x)]^{\beta}\cdot x^{2s\beta-2s-2}\cdot\left[ 2d^{1-\beta}s\beta\cdot \left(\frac{d}{x^{2s}} \right)^{\beta-1} x^{2s\beta-2s}+2s+1-2s\beta \right]\\ &=&2d^{1-\beta}s [w(t,x)]^{\beta}\cdot x^{2s\beta-2s-2}\cdot(2s+1)\\ &>&0.\end{aligned}$$ In the first inequality of the computation of $\partial_{xx}w(t,x)$, we used the condition $\beta>1$ and $\gamma>0$. Hence, for any $t>0$, the function $w(t,\cdot)$ is decreasing and convex with respect to the variable $x$. Let us now define the continuous function $$m(t,x):= \begin{cases} \frac{1}{4B} &\text{ for } x \le x_{B}(t)\\ g(w(t,x)) &\text{ for } x_B(t)<x. \end{cases}$$ Note that by the construction of $m(t,x)$ for all $t>0$, it’s easy to see that the function $m(t,x)$ is $C^{1,1}(\R)$ in $x$, and $$\partial_xm(t,x)=\partial_xw(t,x)(1-2Bw(t,x))^+,$$ which is a Lipschitz function. Observe that: when $x>x_B(0)=(2d B)^{\frac{1}{2s}}>1$, we have $m(0,x)=g(w(0,x))=g(v_0(x))\leq v_0(x)$; when $x<1$, we have $m(0,x)=\frac{1}{4B}< d=v_0(x)$; when $1\leq x\leq x_B(0)=(2dB)^{\frac{1}{2s}}$, we have $v_0(x)=\frac{d}{x^{2s}} \geq\frac{d}{2dB}=\frac{1}{2B}>\frac{1}{4B}=m(0,x)$. Hence $m(0,x)\leq v_0(x)$ for all $x\in \R$. Let us now show that $m(t,x)$ is a subsolution for some appropriate choices of $\gamma$ and $B$. By the definition of $m(t,x)$, we have $\partial_tm(t,x)=\gamma w^{\beta}(t,x)(1-2Bw(t,x))^+$, therefore we get $$\label{eq-dtm} \partial _t m(t,x)\le \begin{cases} 0 &\text{ for } x \le x_B(t)-1\\ \gamma w^\beta(t,x)&\text{ for } x_B(t)-1<x\\ \end{cases}$$ Since $f$ satisfies $f(u)\geq r_1u^{\beta}$ as $u\rightarrow0^+$, then there exists a small $r_2>0$ such that $f(u)\geq r_2 u^\beta (1-u)$ for all $0\leq u \leq 1$. When $x\leq x_B(t)$, then $m(t,x)=\frac{1}{4B}$. Since $w(t,x_B(t))=\frac{1}{2B}$, then $f(m(t,x))\geq r_2[m(t,x)]^{\beta}[1-m(t,x)]=r_2\left[ \frac{1}{2}w(t,x_B(t)) \right]^{\beta}\left(1-\frac{1}{4B}\right)=\frac{r_2}{2^{\beta}}\left( 1-\frac{1}{4B} \right) [w(t,x_B(t))]^{\beta}$. When $x>x_B(t)$, since $0\leq g(y)\leq \frac{1}{4B}$ and $w(t,x)\leq \frac{1}{2B}$, then $f(m(t,x))\geq r_2[w(t,x)(1-Bw(t,x)) ]^{\beta}[1-g(w(t,x))]\geq r[w(t,x)(1-B\cdot \frac{1}{2B}) ]^{\beta}[1-\frac{1}{4B}]=\frac{r}{2^{\beta}}\left( 1-\frac{1}{4B} \right) [w(t,x)]^{\beta}$. In summary, we have $$\label{ac-eq-f} f(m(t,x))\geq \begin{cases} C_0[w(t,x_B(t))]^\beta &\text{ for } x \le x_B(t)\\ C_0[w(t,x)]^{\beta} &\text{ for } x > x_B(t), \end{cases}$$ where $C_0:=\frac{r}{2^\beta}\left(1- \frac{1}{4B}\right)$. Let us now derive some estimate on the fractional diffusion term $ (-\Delta)^{s}m(t,x)$ on the three regions $x\leq x_B(t)-1$, $x_B(t)-1<x<x_B(t)+1$ and $x>x_B(t)+1$. For simplicity of the presentation, we dedicated a subsection to each region and let us start with the region $x\le x_B(t)-1$. ### : {#section .unnumbered} \ In this region of space, we claim that: \[subsolcla1\] 1. If $\frac{1}{2}< s<1$, then there exists $C_3>0$ such that for all $x\le x_B(t)-1$, we have $$(-\Delta)^{s}m(t,x)\le -C_3 v_0'(x_B(t)) [v_0(x_B(t))]^{-\beta} [w(t,x_B(t)) ]^{\beta}.$$ 2. If $0<s\le \frac{1}{2}$, for large enough $B\gg1$, then there exists $C_3>0$ such that for all $x\le x_B(t)-1$, we have $$(-\Delta)^{s}m(t,x)\le \frac{C_3}{B^2}.$$ Note that the singularity here play a major role and the estimate strongly depends on the value of $s$. For $x\le x_B(t)-1$, since $m(t,y)=\frac{1}{4B}=m(t,x_B(t))$ for all $y\leq x_B(t)$, then we have $$\begin{aligned} (-\Delta)^{s}m(t,x)= \int_{x_B(t)}^{+\infty}\frac{m(t,x_{B}(t))-m(t,y)}{\|x-y\|^{1+2s}}dy.\end{aligned}$$ We now treat separately the following two situations: $\frac{1}{2}<s<1$, $0<s\le \frac{1}{2}$. [Case I]{}: $\frac{1}{2}<s<1$. By using the Fundamental Theorem of Calculus, then we have $$\begin{aligned} -(-\Delta)^{s}m(t,x)&= \int_{x_B(t)}^{+\infty}\int_0^1\frac{(y-x_B(t))}{\|x-y\|^{1+2s}}\partial_xm(t,x_B(t)+\tau(y-x_B(t)))\,d\tau dy,\\ &=\int_{0}^{\infty}\int_0^1\frac{z}{\|x-x_B(t)-z\|^{1+2s}} \partial_xm(t,x_B(t)+\tau z) \,d\tau dz.\end{aligned}$$ Now, since $w(t,\cdot)$ is a positive, decreasing and convex function, for any $z,\tau>0$, then we have $$\begin{aligned} \partial_xm(t,x_B(t)+\tau z)&=&\partial_xw(t,x_B(t)+\tau z)\left(1 -2Bw(t,x_B(t)+\tau z)\right)\\ &\geq&\partial_xw(t,x_B(t)).\end{aligned}$$ So we can obtain that $$(-\Delta)^{s}m(t,x)\ge \partial_xw(t,x_B(t)) \int_{0}^{\infty}\frac{z}{\|x-x_B(t)-z\|^{1+2s}}dz.$$ For any $z>0$, since $x<x_B(t)-1 $, then $x-x_B(t)-z<-1-z<0$, which implies that $\|x-x_B(t)-z\|\geq \|1+z\|>0$. Since $\frac{1}{2}<s<1$, then we have $$\int_{0}^{\infty}\frac{z}{\|x-x_B(t)-z\|^{1+2s}}dz<C_3:=\int_{0}^{\infty}\frac{z}{\|1+z\|^{1+2s}}dz<+\infty.$$ As a result $$-(-\Delta)^{s}m(t,x)\ge C_3 \partial_x w(t,x_B(t))=-C_3 v_0'(x_B(t)) [v_0(x_B(t))]^{-\beta} [w(t,x_B(t)) ]^{\beta},\quad \forall x\leq x_B(t)-1,$$ [Case II]{}: $0<s\le \frac{1}{2} $. In this situation, the previous argumentation fails and we argue as follows. Since $w(t,y)\geq0$ for all $t$ and all $y$, for any constant $R>1$ which will be determined later, we have $$\begin{aligned} (-\Delta)^{s}m(t,x)&=& \int_{x_B(t)}^{+\infty}\frac{m(t,x_{B}(t))-m(t,y)}{\|x-y\|^{1+2s}}dy \\ &=&\int_{x_B(t)}^{+\infty}\frac{w(t,x_B(t))-B[w(t,x_B(t))]^2-w(t,y)+B[w(t,y)]^2 }{\|x-y\|^{1+2s}}dy \\ &=&\int_{x_B(t)}^{+\infty} \frac{[w(t,x_B(t)) -w(t,y) ][ 1-B[w(t,x_B(t))+w(t,y) ] ] }{\|x-y\|^{1+2s}} dy\\ &\leq&\int_{x_B(t)}^{+\infty} \frac{w(t,x_B(t)) -w(t,y) }{\|x-y\|^{1+2s}} dy\qquad\textnormal{Since $w(t,y)\geq0$ }\\ &\leq&\int_{x_B(t)}^{+\infty} \frac{w(t,x_B(t)) -w(t,y) }{\|y-x_B(t)+1\|^{1+2s}} dy\\ &=&\int_{0}^{+\infty} \frac{ w(t,x_B(t))-w(t, x_B(t)+z) }{\|1+z\|^{1+2s}} dz\\ &=&\int_{0}^{R} \frac{ w(t,x_B(t))-w(t, x_B(t)+z) }{\|1+z\|^{1+2s}} dz+\int_{R}^{+\infty} \frac{ w(t,x_B(t))-w(t, x_B(t)+z) }{\|1+z\|^{1+2s}} dz\\ &=&I_1+I_2. \end{aligned}$$ Let us now estimate $I_1$ and $I_2$. Since $w(t,y)\geq0$ for all $t$ and all $y$, for $I_2$ we have $$\begin{aligned} I_2&\leq&\int_{R}^{+\infty} \frac{ w(t,x_B(t)) }{\|1+z\|^{1+2s}} dz \nonumber \\ &\leq &\frac{1}{2B}\cdot \int_R^{+\infty} \frac{1}{z^{1+s} } dz \nonumber \\ &=&\frac{1}{2B}\cdot \frac{1}{s}\cdot \frac{1}{R^{s}}.\label{com1-1} \end{aligned}$$ On the other hand, by using the Fundamental Theorem of Calculus and the convexity of $w(t,y)$ is convex with respect to $y$, we get for $I_1$ $$\begin{aligned} I_1=\int_{0}^{R}\int_0^1 \frac{ -\partial_x w(t,x_B(t)+\tau z )\cdot z }{\|1+z\|^{1+2s}} d\tau dz, &\leq\int_{0}^{R}\int_0^1 \frac{ -\partial_x w(t,x_B(t) )\cdot z }{\|1+z\|^{1+2s}} d\tau dz \\ &\le -\partial_x w(t,x_B(t)) \int_0^R \frac{z}{ \|1+z\|^{1+2s} } dz. \\ \end{aligned}$$ Thus, by using the definition of $\partial_xw(t,x_B(t))$, $R>1$ and since $\|y\|^{2s}>\|y\|^s$ in $(1,R)$ we get $$\begin{aligned} I_1&\leq& -\partial_x w(t,x_B(t)) \int_0^R \frac{z}{ \|1+z\|^{1+2s} } dz. \\ &\leq& -\partial_x w(t,x_B(t))\int_1^{2R} \frac{1}{y^{2s}} dy\\ &\leq& -\partial_x w(t,x_B(t))\int_1^{2R} \frac{1}{y^{s}} dy\\ I_1&\le &2d^{1-\beta}s\left(\frac{1}{2B}\right)^\beta\cdot [x_B(t)]^{2s\beta-2s-1} \cdot \frac{1}{1-s}\cdot (2R)^{1-s},\end{aligned}$$ which using that $x_B(t)\geq d^{\frac{1}{2s}} (2B)^{\frac{1}{2s}}$ and $2s\beta-2s-1<0$ enforces that $$\begin{aligned} I_1&\leq& 2d^{1-\beta}s\left(\frac{1}{2B}\right)^\beta\cdot [d^{\frac{1}{2s}} (2B)^{\frac{1}{2s}}]^{2s\beta-2s-1} \cdot \frac{1}{1-s}\cdot (2R)^{1-s}\\ &=&C_{3,1} B^{-\left( 1+\frac{1}{2s} \right)} R^{1-s}. \end{aligned}$$ Combining the latter estimate with , then we have $$\begin{aligned} (-\Delta)^{s}m(t,x)\leq I_1+I_2\leq C_{3,1}B^{-\left( 1+\frac{1}{2s} \right)} R^{1-s}+\frac{1}{2B}\cdot \frac{1}{s}\cdot \frac{1}{R^{s}}. \end{aligned}$$ By taking $R$ such that $C_{3,1} B^{-\left( 1+\frac{1}{2s} \right)} R^{1-s}=\frac{1}{2B}\cdot \frac{1}{s}\cdot \frac{1}{R^{s}}$, that is, $R=\frac{1}{2s C_{3,1}} B^{\frac{1}{2s}}$, we then achieve $$\begin{aligned} (-\Delta)^{s}m(t,x)\leq \frac{C_3}{B^2}. \end{aligned}$$ Let us now obtain some estimate in the region $x\ge x_B(t)+1$. ### : {#section-1 .unnumbered} \ In this region, we claim that \[subsolcla2\] 1. If $\frac{1}{2}<s<1$, then there exists positive constant $C_4$ such that for all $x\ge x_B(t)+1$, we have $$(-\Delta)^{s}m(t,x)\le -C_4 v_0'(x) [v_0(x)]^{-\beta}[w(t,x)]^{\beta}.$$ 2. If $0<s\le\frac{1}{2}$, for large enough $B\gg1$, then there exists positive constant $C_4$ such that for all $x\ge x_B(t)+1$ , we have $$(-\Delta)^{s}m(t,x)\le -C_4\partial_xw(t,x)+C_4[w(t,x)]^{1-2s+2s\beta} x^{2s(2s\beta-2s-1)}.$$ First, we have $$\begin{aligned} (-\Delta)^{s}m(t,x)&=&\textnormal{P.V. }\int_{\mathbb{R}} \frac{m(t,x)-m(t,y)}{\|x-y\|^{1+2s}} dy\\ &=&\int_{-\infty}^{x_B(t)}\frac{m(t,x)-m(t,y)}{\|x-y\|^{1+2s}} dy+\textnormal{P.V. }\int_{x_B(t)}^{+\infty} \frac{m(t,x)-m(t,y)}{\|x-y\|^{1+2s}} dy\\ &=:&I_1+I_2 \end{aligned}$$ For $I_1$, since $\partial_xm(t,x)=\partial_xw(t,x)(1-2Bw(t,x))^+$, then $m(t,\cdot)$ is decreasing. Since $x\geq x_B(t)+1$, then $$\begin{aligned} I_1\leq0.\label{com2-1} \end{aligned}$$ For $I_2$, we have $$\begin{aligned} I_2& = \textnormal{P.V. } \int_{x_B(t)-x}^{\infty}\frac{m(t,x)-m(t,x+z) }{\|z\|^{1+2s}}dz \\ & = \int_{x_B(t)-x}^{-\frac{1}{2}}\frac{m(t,x)-m(t,x+z)}{\|z\|^{1+2s}}dz +\textnormal{P.V. } \int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{m(t,x)-m(t,x+z) }{\|z\|^{1+2s}}dz+\int_{\frac{1}{2}}^{+\infty}\frac{m(t,x)-m(t,x+z) }{\|z\|^{1+2s}}dz\\ &=: I_3+I_4+I_5. \end{aligned}$$ Since $m(t,\cdot)$ is decreasing and $x\ge x_B(t)+1$, it’s easy to see that $$\begin{aligned} I_3\leq0.\label{com2-2} \end{aligned}$$ For $I_4$, since the function $m(t,\cdot)$ is smooth (at least $C^2$) on $(x-\frac{1}{2},x+\frac{1}{2})$, then we have $$I_4=\frac{1}{2} \int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{m(t,x+z)+m(t,x-z)-2m(t,x)}{\|z\|^{1+2s}}dz \leq C_{4,1}\|\partial_{xx}m(t,x)\|$$ A direct computation of $\partial_{xx}m(t,x)$ shows that $\|\partial_{xx}m(t,x)\|\le -C_{4,2}\|\partial_xw(t,x)\|$, which implies that $$\begin{aligned} I_4\leq -C_{4,3}\partial_{x}w(t,x). \label{com2-3} \end{aligned}$$ To complete our proof we need to estimate $I_5$ and to do so we treat separately the following two situations: $\frac{1}{2}<s<1$, $0<s\le \frac{1}{2}$. [Case I]{}: $\frac{1}{2}<s<1$. In this situation, by using the Fundamental Theorem of Calculus, we obtain $$I_5=-\int_{\frac{1}{2}}^{+\infty}\int_0^1\frac{z\partial_x w(t,x+\tau z)\left(1 -2Bw(t,x+\tau z)\right)}{\|z\|^{1+2s}} \,dzd\tau ,$$ which by the convexity and the monotonicity of $w(t,\cdot)$ and since $\frac{1}{2}<s<1$, then yields $$\begin{aligned} I_5&\leq -\int_{\frac{1}{2}}^{\infty}\int_0^1\frac{z}{\|z\|^{1+2s}} \partial_x w(t,x+\tau z)\left(1 -2Bw(t,x+\tau z)\right) \,dzd\tau, \\ &\le -\partial_x w(t,x)\int_{\frac{1}{2}}^\infty\frac{z}{\|z\|^{1+2s}}\,dz,\\ &= -\partial_x w(t,x)\frac{1}{2^{1-2s}(2s-1)}.\end{aligned}$$ By combining the latter with , , and , we can therefore find a constant $C_4>0$ such that $$(-\Delta)^{s}m(t,x)\le- C_4\partial_xw(t,x)=-C_4 v_0'(x) v_0^{-\beta}(x)w^{\beta}(t,x).$$ [Case II :]{} $ 0<s\le\frac{1}{2} $. In this situation, again the previous argumentation fails and we argue as follows. For any $R>1$, let us rewrite $I_5$ as follows $$\begin{aligned} I_5&=&\int_{\frac{1}{2}}^{R}\frac{m(t,x)-m(t,x+z) }{\|z\|^{1+2s}}dz+\int_{R}^{+\infty}\frac{m(t,x)-m(t,x+z) }{\|z\|^{1+2s}}dz\\ &=:&I_6+I_7.\end{aligned}$$ An easy computation show that since $m(t,x)\leq w(t,x)$, and $R>1$ we get $$\begin{aligned} I_7\le \int_{R}^{+\infty}\frac{m(t,x) }{\|z\|^{1+2s}}dz&\leq& w(t,x)\cdot \int_R^{+\infty} \frac{1}{z^{1+2s}} dz\nonumber\\ &\le& w(t,x)\cdot \int_R^{+\infty} \frac{1}{z^{1+s}} dz= \frac{1}{s} w(t,x)\cdot \frac{1}{R^{s}}.\label{com2-4}\end{aligned}$$ On the other hand by using the Fundamental Theorem of Calculus, we obtain the following for $I_6$ $$\begin{aligned} I_6&=&-\int_{\frac{1}{2}}^{R}\int_0^1\frac{z\partial_x w(t,x+\tau z)\left(1 -2Bw(t,x+\tau z)\right)}{\|z\|^{1+2s}} \,dzd\tau,\end{aligned}$$ and by using the convexity and the monotonicity of $w(t,\cdot)$, we then achieve $$\begin{aligned} I_6&\leq & -\int_{\frac{1}{2}}^{R}\int_0^1\frac{z}{\|z\|^{1+2s}} \partial_x w(t,x+\tau z)\left(1 -2Bw(t,x+\tau z)\right) \,dzd\tau, \nonumber\\ &\le & -\partial_x w(t,x)\int_{\frac{1}{2}}^{R}\frac{z}{\|z\|^{1+2s}}\,dz, \nonumber\\ &\le & -\partial_x w(t,x)\left(\int_{\frac{1}{2}}^{1}\frac{1}{z^{2s}}\,dz+\int_{1}^{R}\frac{1}{z^{2s}}\,dz\right),\nonumber\\ &\le & -\partial_x w(t,x)\left(\int_{\frac{1}{2}}^{1}\frac{1}{z^{2s}}\,dz+\int_{1}^{R}\frac{1}{z^{s}}\,dz\right)\le -\partial_x w(t,x)\left(C(s)+\frac{1}{s}R^{1-s}\right).\label{com2-5}\end{aligned}$$ Combining with and using the definition of $\partial_xw(t,x)$ enforce that $$I_5\leq \frac{1}{s} w(t,x)\cdot \frac{1}{R^{s}}+C_{6,2}[w(t,x)]^\beta\cdot x^{2s\beta-2s-1}R^{1-s} +C(s)\partial_xw(t,x).$$ By taking $R$ such that $C_{6,2}[w(t,x)]^\beta\cdot x^{2s\beta-2s-1}R^{1-s}=\frac{1}{s} w(t,x)\cdot \frac{1}{R^{s}}$, that is, $R=\frac{1}{s C_{6,2} }\cdot [w(t,x)]^{1-\beta} \cdot x^{-2s\beta+2s+1}$, we then get $$I_5 \leq C_{7,2} [w(t,x)]^{1-2s+2s\beta}x^{2s(2s\beta-2s-1)}+ C(s)\partial_xw(t,x),$$ which combined with , and then yields $$(-\Delta)^{s}m(t,x)\le -C_4\partial_xw(t,x)+C_4[w(t,x)]^{1-2s+2s\beta} x^{2s(2s\beta-2s-1)}.$$ Lastly, let us estimate $-(-\Delta)^{s}m(t,x)$ in the region $x_B(t)-1<x<x_B(t)+1$. ### : {#section-2 .unnumbered} \ In this last region, we claim that: \[subsolcla3\] 1. If $\frac{1}{2}<s<1$, then there exists positive constant $C_5$ such that for all $ x_B(t)-1\le x\le x_B(t)+1$, we have $$(-\Delta)^{s}m(t,x)\le -C_5 v_0'(x) [v_0(x)]^{-\beta}[w(t,x)]^{\beta}.$$ 2. If $0<s\le \frac{1}{2}$, for large enough $B\gg1$, then there exists positive constant $C_5$ such that for all $ x_B(t)-1\le x\le x_B(t)+1$, , we have $$(-\Delta)^{s}m(t,x)\le -C_5\partial_xw(t,x)+C_5[w(t,x)]^{1-2s+2s\beta} x^{2s(2s\beta-2s-1)}.$$ Again let us rewrite the fractional Laplacian in the following way : $$\begin{aligned} (-\Delta)^{s}m(t,x)&=&\textnormal{P.V. }\int_{\mathbb{R}} \frac{m(t,x)-m(t,y)}{\|x-y\|^{1+2s}} dy\\ &=&\int_{-\infty}^{x-1} \frac{m(t,x)-m(t,y)}{\|x-y\|^{1+2s}} dy+\textnormal{P.V. }\int_{x-1}^{+\infty} \frac{m(t,x)-m(t,y)}{\|x-y\|^{1+2s}} dy\\ &=:&I_1+I_2 \end{aligned}$$ By using the monotone character of $m(t,\cdot)$, we have $$\begin{aligned} I_1\leq0.\label{com3-1} \end{aligned}$$ For $I_2$, we have $$\begin{aligned} I_2& = \textnormal{P.V. }\int_{-1}^{\infty}\frac{m(t,x)-m(t,x+z)}{\|z\|^{1+2s}}dz \\ & =- \frac{1}{2} \int_{-1}^{1}\frac{m(t,x+z)+m(t,x-z)-2m(t,x)}{\|z\|^{1+2s}}dz + \int_{1}^{\infty}\frac{m(t,x)-m(t,x+z)}{\|z\|^{1+2s}}dz \\ &= I_3+I_4. \end{aligned}$$ Observe that again to estimate $I_2$ we break the integral into two part and we can easily see that the contribution of $I_4$ can be estimated as in the proof of the previous claim so we won’t repeat it. If fact, here the only change with respect to the situation $x>x_B(t)+1$, is the contribution of $I_3$ since unlike the previous case, the function $m$ is not any more a $C^2$ smooth function on the domain of integration and we need then more precise estimate. So let us now look more closely at $I_3$. For $I_3$, thanks to the definition of $m$, we can get $$\begin{aligned} I_3&=-\frac{1}{2} \int_{-1}^1\int_{0}^{1}z\frac{\partial_xm(t,x+\tau z) -\partial_xm(t,x-\tau z)}{\|z\|^{1+2s}}d\tau dz\\ &=-\frac{1}{2}\int_{-1}^1\int_{0}^{1}\frac{z}{\|z\|^{1+2s}}[\partial_x w(t,x+\tau z)(1-2Bw(t,x+\tau z))^+ -\partial_x w(t,x-\tau z)(1-2Bw(t,x-\tau z))^+ ]\,d\tau dz \end{aligned}$$ Let us rewrite the bracket inside the integral as follows: $$\begin{aligned} && \partial_x w(t,x+\tau z)(1-2Bw(t,x+\tau z))^+ -\partial_x w(t,x-\tau z)(1-2Bw(t,x-\tau z))^+\\ &=& [\partial_x w(t,x+\tau z)- \partial_x w(t,x-\tau z)]\cdot (1-2Bw(t,x+\tau z))^+\\ && \quad +\, \partial_x w(t,x-\tau z)[(1-2Bw(t,x+\tau z))^+-(1-2Bw(t,x-\tau z))^+ ] \end{aligned}$$ Then we can decompose $I_3$ into two integrals $ I_3=I_5+I_6$ with $$\begin{aligned} &I_5:=\frac{1}{2}\int_{-1}^1\int_{0}^{1}\frac{z}{\|z\|^{1+2s}}[\partial_x w(t,x+\tau z)- \partial_x w(t,x-\tau z)] (1-2Bw(t,x+\tau z))^+ \,d\tau dz\\ &I_6:=\frac{1}{2} \int_{-1}^1\int_{0}^{1}\frac{z}{\|z\|^{1+2s}} \partial_x w(t,x-\tau z)[(1-2Bw(t,x+\tau z))^+-(1-2Bw(t,x-\tau z))^+ ]\,d\tau dz. \end{aligned}$$ Now since $w(t,x)$ is smooth, using the Fundamental Theorem of Calculus, we get $$\begin{aligned} I_5=-\frac{1}{2}\int_{-1}^1\int_{0}^{1}\int_{-1}^1\frac{2\tau z^2}{\|z\|^{1+2s}}\partial_{xx} w(t,x+\sigma \tau z) (1-2Bw(t,x+\tau z))^+ \,d\sigma d\tau dz. \end{aligned}$$ Since $w(t,\cdot)$ is convex, then $$\begin{aligned} I_5\leq0. \end{aligned}$$ For $I_6$, by using the convexity of $w(t,\cdot)$ and the uniform Lipschitz continuity of the function $(1-2Bw(t,x))^+$ for $x \in (x_B(t)-1,x_B(t)+1)$, then we have $$\begin{aligned} I_6&\le -\frac{1}{2} \partial_xw(t,x-1)\int_{-1}^1\int_{0}^{1}\frac{\|z\|}{\|z\|^{1+2s}} \|(1-2Bw(t,x+\tau z))^+-(1-2Bw(t,x-\tau z))^+ \|\,d\tau dz\\ &\le -C\partial_xw(t,x-1)\int_{-1}^{1}\frac{z^2}{\|z\|^{1+2s}}dz \\ &=-C_{5,2}\partial_x w(t,x-1). \end{aligned}$$ A direct computation can give us some $C_{5,1}>0$ such that $\partial_xw(t,x-1)\ge C_{5,1}\partial_x w(t,x)$, which implies that $$I_6\le -C_{5,3}\partial_x w(t,x).$$ Hence $$\begin{aligned} I_3\leq -C_{5,3}\partial_x w(t,x).\label{com3-2} \end{aligned}$$ By collecting , [Claim \[subsolcla1\]]{}, [Claim \[subsolcla2\]]{}, [Claim \[subsolcla3\]]{} and , now we can show that $m(t,x)$ is a subsolution for some appropriate choices of $B$, $\gamma$ and $\epsilon$. If $\frac{1}{2}<s<1$, by , [Claim \[subsolcla1\]]{}, [Claim \[subsolcla2\]]{}, [Claim \[subsolcla3\]]{} and , we have $$(\partial _t m +(-\Delta)^{s}m-f(m))(t,x) \leq \begin{cases} -[w(t,x_B(t))]^{\beta}\left[C_0+h(t,x_B(t))\right] &\text{ for } x \le x_B(t)-1\\ -[w(t,x)]^{\beta}\left[C_0+h(t,x) -\gamma\right] &\text{ for } x > x_B(t)-1, \end{cases}$$ where $h(t,x)= C_6v_0'(x)[v_0(x)]^{-\beta}$ with $C_6\ge \max\{C_3,C_4,C_5\}$. We now choose $\gamma\le \frac{C_0}{2}$. In the view of the above inequalities, to complete the construction of the subsolution $m(t,x)$, it suffices to find a condition on $B$ so that $h(t,x)\geq -\frac{C_0}{2}$ for all $t>0$ and all $x\in \R$. From the definitions of $h(t,x)$ and $v_0(x)$, it suffices to achieve $$x^{(\beta-1)2s-1}\le \frac{C_0d^{\beta-1}}{4s dC_6 }, \quad \text{ for all } t>0, x\geq x_B(t)-1.$$ Since $(\beta-1)2s<1$, this reduces to the following condition on $x_B(0)$: $$x_B(0)\ge \left(\frac{C_0d^{\beta-1}}{4s dC_6}\right)^{\frac{1}{1-2s(\beta-1)}}+1. \label{ac-x0t-ineq}$$ From we have $x_B(0)=(2Bd)^{\frac{1}{2s}}$. Hence, in view of the definition of $C_0$, the above inequality holds by selecting $B\geq B_0$, with $B_0>0$ large enough. If $0<s\le \frac{1}{2}$, when $x\leq x_B(t)-1$, by , [Claim \[subsolcla1\]]{}, and , we have $$\begin{aligned} (\partial _t m +(-\Delta)^{s}m-f(m))(t,x)&\leq&\frac{C_3}{B^2}-C_0[w(t,x_B(t))]^{\beta}\\ &=&\frac{C_3}{B^2}-C_0\left( \frac{1}{2B} \right)^\beta\\ &=&B^{-2}[C_3-2^{-\beta} C_0B^{2-\beta} ]\end{aligned}$$ Since $\beta<2$, then there exists some $B_1\gg1$ such that $C_3-2^{-\beta} C_0B^{2-\beta} <0$ for all $B\geq B_1$. Hence in this case, we have $$(\partial _t m +(-\Delta)^{s}m-f(m))(t,x)<0.$$ When $x>x_B(t)-1$, by , [Claim \[subsolcla2\]]{}, [Claim \[subsolcla3\]]{} and , we have $$\begin{aligned} &&(\partial _t m +(-\Delta)^{s}m-f(m))(t,x)\\ &\leq& \gamma [w(t,x)]^{\beta}-C_6\partial_xw(t,x)+C_6[w(t,x)]^{1-2s+2s\beta} x^{2s(2s\beta-2s-1)}-C_0w^{\beta}(t,x)\\ &=&\gamma [w(t,x)]^{\beta}+C_7[w(t,x)]^\beta\cdot x^{2s\beta-2s-1}+C_6[w(t,x)]^{1-2s+2s\beta} x^{2s(2s\beta-2s-1)}-C_0[w(t,x)]^{\beta}\\ &=&[w(t,x)]^{\beta}[\gamma+C_7 x^{2s\beta-2s-1}+C_6[w(t,x)]^{1-2s+2s\beta-\beta}x^{2s(2s\beta-2s-1)} -C_0 ]\end{aligned}$$ Take $\gamma=\frac{C_0}{3}$. It’s easy to see that $x_B(t)\geq d^{\frac{1}{2s}} (2B)^{\frac{1}{2s}}$. Since $2s\beta-2s-1<0$, so when $B$ is large enough, we have $$C_7 x^{2s\beta-2s-1}\leq \frac{C_0}{3}.$$ Note that since $\beta>1$ and $0<s\le \frac{1}{2}$, we have $1-2s+2s\beta-\beta\le 0$ and therefore since $w(t,x)\geq w(0,x)=v_0(x)=\frac{d}{x^{2s}}$, we have $$\begin{aligned} C_6[w(t,x)]^{1-2s+2s\beta-\beta}x^{2s(2s\beta-2s-1)} &\leq& C_8 x^{-2s(1-2s+2s\beta-\beta)}\cdot x^{2s(2s\beta-2s-1)}\\ &=&C_8x^{2s(\beta-2)}\end{aligned}$$ Using that $\beta<2$ and since $x\geq x_B(t)-1\geq d^{\frac{1}{2s}} (2B)^{\frac{1}{2s}}-1 $, so when $B$ is large enough, we have $C_8x^{2s(\beta-2)}\leq \frac{C_0}{3}$, which implies that $$C_6[w(t,x)]^{1-2s+2s\beta-\beta}x^{2s(2s\beta-2s-1)} \leq \frac{C_0}{3}.$$ So $$(\partial _t m +(-\Delta)^{s}m-f(m))(t,x)\leq0.$$ In summary, for any $0<s<1$, after some good choices of $\gamma$, $B$ and $\epsilon$, then the function $m(t,x)$ indeed is a subsolution. [Step three.]{} It consists in using the subsolution to prove the lower estimate in [Theorem \[thm:1-3\]]{}. Fix $\gamma>0$ and $B_0 >0$ as in the previous step so that $m(t,x)$ is a subsolution. From the comparison principle we get $m(t,x)\le u(t,x)$, for all $t>0$ and $x \in \R$. Recall that $m(t,x_{B_0}(t))=\frac{1}{4B_0}$ and that $u(t,\cdot)$ is nonincreasing (since initial datum $v_0$ is nonincreasing) so that $$\label{u-grand} u(t,x)\geq \frac{1}{4B_0}, \quad \forall x\leq x_{B_0}(t).$$ In particular, for any $0<\lambda \leq \frac{1}{4B_0}$, the largest” element $x_\lambda(t)$ of the super level set $\Gamma _\lambda (t)$ has to satisfy $$x_\lambda(t)\geq x_{B_0}(t)\geq d^{\frac{1}{\alpha -1}}[\gamma (\beta -1)t]^{\frac{1}{2s(\beta-1)}},$$ which provides the lower estimate. It now remains to obtain a similar bound for a given $\frac{1}{4B_0}<\lambda <1$. Such estimate can be obtained by redoing the argument in [Section \[sec-l\]]{}. Acknowledgement. J.Coville carried out this work in the framework of Archimède Labex (ANR-11-LABX-0033) and of the A\MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir" French Government program managed by the French National Research Agency (ANR). He has also received funding from the ANR DEFI project NONLOCAL (ANR-14-CE25-0013). C. Gui is supported by NSF grants DMS-1601885 and DMS-1901914 and Simons Foundation Award 617072. M. Zhao is supported by the National Natural Science Foundation of China No.11801404. [^1]: UR 546 Biostatistique et Processus Spatiaux, INRA, Domaine St Paul Site Agroparc, F-84000 Avignon, France, [email:* ]{}jerome.coville@inra.fr [^2]: One USA Circle, Department of Mathematics, University of Texas at San Antonio, USA [email: ]{}changfeng.gui@utsa.edu [^3]: Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China, [email: ]{}mingfeng.zhao@tju.edu.cn
--- abstract: 'The ground state properties of a frustrated spin-1/2 system is studied on a trellis ladder which is composed of two zigzag ladders interacting through rung interactions. The presence of rung interaction between the zigzag ladders induces a non-magnetic ground state, although, each of zigzag ladders has ferromagnetic order in weak anti-ferromagnetic leg interaction limit. The rung interaction also generates rung dimers and opens spin gap which increases rapidly with rung interaction strength. The correlation between spins decreases exponentially with the distance between them.' address: 'S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata - 700106, India' author: - Debasmita Maiti - Manoranjan Kumar title: Melting of Ferromagnetic Order on a Trellis Ladder --- Frustrated magnetic systems, Dimer phase, Density Matrix Renormalization Group Method \[sec:intro\]Introduction ========================= The interaction-driven quantum phase transition in low-dimensional frustrated systems like spin chain [@chubukov1991; @furukawa2012], ladder or any quasi-one dimensional system has been a frontier area of current research [@white1994; @verkholyak2012]. Many realizations of these systems like (N$_2$H$_5$)CuCl$_3$ [@Maeshima2003], LiCuSbO$_4$ [@dutton_prl], LiCuVO$_4$ [@Mourigal2012], Li$_2$CuZrO$_4$ [@Drechsler2007] are frustrated due to competing spin exchange interactions. $J_1-J_2$ Heisenberg spin-1/2 model with nearest neighbor (NN) ferromagnetic (FM) $J_1$ and next nearest neighbor (NNN) anti-ferromagnetic (AFM) $J_2$ interactions is used extensively to understand the ground state magnetic properties of many of these materials [@white96; @OKAMOTO1992; @Chitra1995; @mk2015]. This model can explain the gapless spin fluid, gapped dimer and incommensurate gapped spiral phases [@white96; @mk2010a; @mk2015]. On the other hand, the ground state of quasi-1D ladder like structure with AFM leg and rung interactions shows the presence of gapped short-range order in the system. This kind of phase is observed in SrCu$_2$O$_3$ [@sandvik1995], (VO)$_2$P$_2$O$_7$ [@dcjhon1987; @dagotto96], CaV$_2$O$_5$, MgV$_2$O$_5$ [@Korotin1999; @Tdasgupta2000] etc. The $J_1-J_2$ chain can also be considered as a two-chain lattice with diagonal or zigzag like couplings. Hence it can be alternatively called as zigzag ladder. The isolated ladders like zigzag [@Chitra1995; @white96] and normal ladder [@white96; @white1994] have been extensively studied. However, the theoretical study of the effect of inter-ladder coupling on the ladders is still an open field. One of the extended networks of the coupled ladders can form a trellis lattice like structure. The trellis lattice is composed of a number of normal ladders coupled through zigzag bonds; alternatively, it can be considered as coupled zigzag ladders through rung interactions as shown in Fig. \[fig:figure1\]. In this lattice, $J_2$ and $J_3$ are leg and rung couplings of a normal ladder, respectively and $J_1$ is inter-ladder coupling through zigzag bonds. ![\[fig:figure1\] Two coupled zigzag ladders form trellis ladder. The broken lines show the extension of trellis ladder to trellis lattice structure. The arrows show respective spin arrangements. The reference site is labeled by ’0’ and the distances of all other sites are shown with respect to it. The distance of the sites in the same zigzag ladder as the reference site are written in bold numbers, and the sites on the other ladder are written in normal numbers. ](fig1){width="3.0"} There are several theoretical studies for two coupled zigzag ladders, e.g. a two-leg honeycomb ladder is considered in Ref. [@Luo2018], where both $J_1$ and $J_2$ are AFM, but $J_3$ can be either FM or AFM. This system shows two types of Haldane phases for the FM $J_3$ and, columnar dimer and rung singlet phases for the AFM $J_3$. Normand et al. considered the similar coupled ladders with all three AFM $J_1$, $J_2$ and $J_3$ interactions. They find dimerized chains for large $J_2$ and small $J_3$ limit, spiral long range order for both large $J_2$ and $J_3$ limit, Néel long range order in the small $J_2 < 0.4$ and for all $J_3$ [@Normand1997]. Ronald et al. have shown the effect of inter-chain coupling on spiral ground state of $J_1-J_2$ model [@Zinke2009]. The effect of inter-ladder coupling on spin gap and magnon dispersion has been discussed in [@Miyahara1998] exploiting a theoretical model which has also been compared with the experimental data of SrCu$_2$O$_3$ and CaV$_2$O$_5$. The study of interladder coupling effect is important to explain the physical properties of some other materials like LaCuO$_{2.5}$ [@Troyer1997], Sr$_{14}$Cu$_{24}$O$_{41}$ [@Uehara1996], MgV$_2$O$_5$ [@Korotin1999], NaV$_2$O$_5$ [@smolinski1998; @Tanokura2010] etc. The weak interladder exchange interactions in these materials form an effective 2D trellis like structure. Recently Yamaguchi et al. showed magnetic field induced spin nematic phase in the verdazyl radical $\beta$-2,3,5-$Cl_3$-V [@Yamaguchi2018]. The $J_1$-$J_2$-$J_3$ spin system promises a zoo of exotic phases. In the current paper we consider a trellis ladder which is composed of two zigzag ladders with FM $J_1$ and AFM $J_2$ and coupled by AFM $J_3$ as shown in Fig. \[fig:figure1\]. The ground state of an isolated zigzag ladder in small $J_2$ limit is FM. Our main focus of this paper is to understand the effect of rung interaction $J_3$ on the FM ground state exhibited by a single zigzag ladder in the limit of $J_2 \le \frac{\|J_1\|}{4}$, and study the transition of ground state from the FM to the singlet dimer state. This paper is divided into four sections. In section \[sec2\], the model Hamiltonian and the numerical methods are explained. The numerical results are given in section \[sec3\]. All the results are discussed and summarized in section \[sec4\]. \[sec2\]Model Hamiltonian and Numerical Method ============================================== We consider four-leg ladder system where two zigzag ladders are coupled through an AFM Heisenberg interaction as shown in Fig. \[fig:figure1\]. The diagonal interactions $J_1$ in zigzag ladders are FM whereas $J_2$ bonds along the legs are AFM. Two zigzag ladders interact with each other through AFM rungs $J_3$. Thus we can write an isotropic Heisenberg spin-1/2 model Hamiltonian for the system as $$\begin{aligned} \label{eq:ham} H &= &\sum_{a = 1,2}\sum_{i=1}^{N/2} \ J_1 \, \vec{S}_{a,i} \cdot \vec{S}_{a,i+1} + J_2 \, \vec{S}_{a,i} \cdot \vec{S}_{a,i+2} \nonumber \\ & & \qquad \qquad + J_3 \, \vec{S}_{1,i} \cdot \vec{S}_{2,i},\end{aligned}$$ where $a=1,2$ are the zigzag ladder indices. $\vec{S}_{a,i}$ is the spin operator at reference site $i$ on zigzag ladder $a$. We set the FM $J_1=-1$ and treat the AFM $J_2$ and $J_3$ as variable quantities. We use periodic boundary condition along the rungs, whereas it is open along the legs of the system. We use the exact diagonalization (ED) method for small systems and density matrix renormalization group (DMRG) method to handle the large degrees of freedom for large systems. The DMRG is a state of art numerical technique for 1D or quasi-1D system, and it is based on the systematic truncation of irrelevant degrees of freedom at every step of growth of the chain [@white-prl92; @karen2006; @schollwock2005]. We use recently developed DMRG method where four new sites are added at every DMRG steps [@mk2010b]. For the renormalization of operators, we keep $m$ eigenvectors corresponding to largest eigenvalues of the density matrix of the system in the ground state of the Hamiltonian in Eq. (\[eq:ham\]). We have kept $m$ up to 300 to constraint the truncation error less than $10^{-10}$. We have used system sizes up to $N=400$ to minimize the finite size effect. \[sec3\]Results =============== It is well known that the zigzag spin-1/2 ladder has a FM ordered ground state for FM $J_1$ and AFM $J_2 \le \frac{\|J_1\|}{4}$ [@mk2015]. The AFM coupling, i.e., the rung interaction $J_3$ between two isolated zigzag spin-1/2 ladders retains the FM arrangement in each zigzag ladder; however, the spins on two zigzag ladders are in AFM arrangement to each other, as depicted in the schematic Fig. \[fig:figure1\]. In other words, the effective ground state of the whole system is in $S^z=0$ manifold, though the spins on each zigzag ladder are arranged ferromagnetically. The ground state forms singlet dimer along the rung in the large $J_3$ limit, and ground state wave function can be represented as the product of singlet dimers. However, to study the effect of $J_3$ on the ground state, we analyze longitudinal correlation function, longitudinal bond order and spin gap. ![\[fig:cor\]The longitudinal spin-spin correlation C(r) shown for N = 122, $J_2=0.15$ and different $J_3$ as indicated adjacent the respective curves. The solid lines represent respective exponential fits. The inset shows the correlation length $\xi$ vs. $J_3$ plots for $J_2=0.1$ and $0.15$ in log-log scale. ](fig2){width="3.5"} In this paper we focus $J_2/\|J_1\| < \frac{1}{4}$ limit where each zigzag ladder have the FM order in the ground state for $J_3=0$. For a finite $J_3$, we calculate the longitudinal spin-spin correlations $C(r)=<{S^z}_i {S^z}_{i+r}>$, where $S^z_i$ and $S_{i+r}^z$ are the z-component of the spin operators at the reference site $i$ and the site at a distance $r$ from $i$, respectively. In \[fig:figure1\], We have shown the distance $r$ along the same zigzag ladder with bold numerics, and otherwise in normal numerics, where the reference site is at the 0th position. We note that in $J_2/\|J_1\|< \frac{1}{4}$ limit all the spins are aligned in parallel on each zigzag ladder and have short range longitudinal correlation for finite $J_3$. As we increase the strength of $J_3$, $C(r)$ shows an exponential behavior as shown in the main Fig. \[fig:cor\] for $J_2 = 0.15$. We notice that each zigzag ladder shows FM arrangement as $C(r)>0$, but it decays exponentially with $r$, i.e. $ C(r) \propto exp(-r/\xi)$. The correlation length $\xi$ follows an algebraic decay with $J_3$ for a given $J_2$, as shown in the inset of Fig. \[fig:cor\]. $\xi$ for $J_3=0.1$ is approximately 28.5, however it decrease to 1.58 for $J_3=0.6$. $\xi$ becomes less than 1 for $J_3 > 0.9$ for $J_2=0.15$, and in this parameter regime the system is completely dimerized along the rung. ![\[fig:cor1\] Correlation function C of the reference site with its first neighbors along the rung ($C^R$, circle), diagonal direction ($C^D$, square), and the leg ($C^L$, diamond), are shown as function of $J_3$. We have drawn $\|C^R\|$ to take care of the AFM $J_3$ interaction. The lines represent respective exponential fits. ](fig3){width="3.3"} To study the effect of $J_3$ on the bonds along the rung, diagonal and leg directions, we calculate correlations $C^R$, $C^D$ and $C^L$, respectively as shown in Fig. \[fig:cor1\]. The calculations of these three correlations are confined to the first neighbor along the respective directions. $\|C^R\|$ increases exponentially with $J_3$ and follows $\|C^R\| = 0.25(1-0.60exp(-1.48J_3))$. $C^D$ and $C^L$ are represented by square and diamond symbols, respectively, and both these bond orders exponentially decrease with $J_3$. The exponent of the $C^D$ and $C^L$ are 0.42 and 1.50 respectively. $C^L$ decays faster than $C^D$, because $J_2$ allows the magnon to deconfine along the leg of the zigzag ladder; therefore weaker $J_2$ reduces $C^L$. The correlation function $C(r)$ of the system shows the short range spin order. Therefore, we explore the excitation energy or spin gap in the system. The rung interaction dominates other interactions; thus we expect the opening of the spin gap $\Delta$. We calculate $\Delta$ for various $J_3=0.1, 0.3, 0.4$ and 0.5. The main Fig. \[fig:spin\_gap\] shows the extrapolation of the spin gap $\Delta$, from which we obtain the spin gap $\Delta_\infty$ in the thermodynamic limit. $\Delta_\infty$ increases algebraically with $J_3$, as shown in the inset of Fig. \[fig:spin\_gap\] for $J_2=0.1$ and 0.15. The algebraic exponent $\gamma$ for $J_2=0.1$ and 0.15 are 3.33 and 3.13, respectively. We notice that $\gamma$ decreases with increasing $J_2$. This may be due to the delocalization of magnon along the leg of zigzag ladder. The large $J_2/\|J_1\|$ enhances the interaction of spin along each leg of zigzag ladder and the each leg of the system can have quasi-long range correlation like a normal Heisenberg chain, in $J_2/\|J_1\|>>1$ limit. In this limit, system behaves as four non-interacting spin-1/2 Heisenberg chains with gapless spectrum [@mk2016]. ![\[fig:spin\_gap\] The extrapolation of spin-gap $\Delta$ with respect to system size $N$ for different $J_3=0.1, 0.3, 0.4$ and 0.5 at $J_2=0.1$. The solid lines are the fitted curves. In the inset, the spin gap in thermodynamic limit, $\Delta_\infty$ vs. $J_3$ plots for $J_2=0.1$ and 0.15 are shown in log-log scale.](fig4){width="3.5"} \[sec4\]Discussion and Conclusions ================================== In this paper the effect of $J_3$ on FM order in each zigzag ladder of a trellis ladder with FM $J_1$ and AFM $J_2 < \frac{\|J_1\|}{4}$ is studied. We show that even a small $J_3$ induces spin gap in the system. The correlation between spins on a zigzag ladder decays exponentially with distance. It may be because of the confinement of magnon along the rungs. As shown in Fig. \[fig:spin\_gap\] the gap increases rapidly with $J_3$ for $J_2 = 0.1$. The correlation length of the system reduced to less than a unit lattice for $J_3 > 0.9$ at $ J_2 \approx 0.15$. This implies the setting of the dimerized state. This model can also be mapped to a two interacting $J_1-J_2$ Heisenberg spin-1/2 chains. This system is studied recently by Ronald et al. [@Zinke2009]. They have mostly studied the effect of inter-chain coupling on the spiral nature of the ground state in large $J_2$ and low $J_3$ limit. There are many compounds like CaV$_2$O$_5$ [@ONODA1996], SrCu$_2$O$_3$ [@Miyahara1998; @dagotto96] etc, which have both strong $J_3$ and $J_2$. However, our prediction are confined to the $J_2 < \frac{\|J_1\|}{4}$ and large $J_3$ limit. In summary, this model system goes from a FM ordered ground state along a zigzag ladder in the $J_3=0$ limit to a rung dimer state in large $J_3$ limit. 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--- abstract: 'We consider advection of a passive scalar $\theta(t,{\bbox r})$ by an incompressible large-scale turbulent flow. In the framework of the Kraichnan model the whole PDF’s (probability distribution functions) for the single-point statistics of $\theta$ and for the passive scalar difference $\theta({\bbox r}_1)-\theta({\bbox r}_2)$ (for separations ${\bbox r}_1-{\bbox r}_2$ lying in the convective interval) are found.' address: \| $^a$ Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia;\ $^b$ Landau Institute for Theoretical Physics, RAS,\ Kosygina 2, Moscow 117940, Russia;\ $^c$ Department of Physics of Complex Systems, Weizmann Institute of Science,\ Rehovot 76100, Israel;\ $^d$ Institute of Automation and Electrometry, RAS,\ Novosibirsk 630090, Russia;\ $^e$ Novosibirsk State University, Novosibirsk, 630090, Russia. author: - 'I. Kolokolov$^{a,e}$, V. Lebedev$^{b,c}$, and M. Stepanov$^{d,e}$.' title: \| Notes about Passive Scalar in\ Large-Scale Velocity Field --- Introduction {#introduction .unnumbered} ============ We treat advection of a passive scalar field $\theta(t,{\bbox r})$ by an incompressible turbulent flow, the role of the scalar can be played by temperature or by pollutants density. The velocity field is assumed to contain motions from some interval of scales restricted from below by $L_v$. A steady situation with a permanent random supply of the passive scalar is considered. We wish to establish statistics of the passive scalar $\theta$ for scales that are less than both the scale $L_v$ and the pumping scale $L$, and larger than the diffusion scale $r_{\rm dif}$ (for definiteness we assume that $L<L_v$). Such convective interval of scales exists if the Peclet number ${\rm Pe}=L/r_{\rm dif}$ is large enough, we will assume the condition. Since all scales from the convective interval are assumed to be smaller than $L_v$ we will say about advection by a large-scale turbulent flow. The problem is of physical interest for the dimensionalities $d=2,3$, but formally it can be treated for an arbitrary dimensionality $d$ of space. Below we will treat $d$ as a parameter. Particularly, all the expressions will be true for a space of large dimensionality $d$. Description of a small-scale statistics of a passive scalar advected by a large-scale solenoidal velocity field is a special problem in turbulence theory. This problem was treated consistently from the very beginning and some rigorous results have been obtained which is quite unusual for a turbulence problem. Batchelor [@Bat] examined the case of external velocity field being so slow that it does not change during the time of the spectral transfer of the scalar from the external scale to the diffusion scale. Then Kraichnan [@Kra] obtained plenty of results in the opposite limit of a velocity field delta-correlated in time. The pair correlation function of the passive scalar $\langle\theta({\bbox r})\theta({\bbox 0})\rangle$ was found to be proportional to the logarithm $\ln(L/r)$ and the pair correlation function of the passive scalar difference $\langle[\theta({\bbox r})-\theta({\bbox 0})]^2\rangle$ was found to be proportional to $\ln(r/r_{\rm dif})$ in both cases. The assertions are really correct for any temporal statistics of the velocity field [@94SS; @95CFKLa]. Thus we are dealing with the logarithmic case which is substantially simpler than cases with power-like correlation functions usually encountered in turbulence problems [@53Bat; @MY; @Frish]. Now about high-order correlation functions of the passive scalar. As long as all distances between the points are much less than $L$, the $2n$-point correlation functions of $\theta$ are given by their reducible parts (that is are expressed via products of the pair correlation function) until $n\sim\ln(L/r)$ where $r$ is either the smallest distance between the points or $r_{\rm dif}$ depending on what is larger [@95CFKLa]. The reason for such Wick decoupling is simply the fact that reducible parts contain more logarithmic factors (which are considered as large ones) than non-reducible parts do. Consistent calculations of the fourth-order correlation function of the passive scalar at $d=2$ [@95BCKL] confirm the assertion. Therefore e.g. the single-point PDF of $\theta$ has a Gaussian core (that describes the first moments with $n<\ln{\rm Pe}$) and a non-Gaussian tail (that describes moments with $n>\ln{\rm Pe}$). The tail appears to be exponential (see [@94SS; @95CFKLa]). The same is true for the passive scalar difference $\Delta\theta=\theta({\bbox r})-\theta({\bbox 0})$ where instead of $\ln{\rm Pe}$ we should take $\ln(r/r_{\rm dif})$. The tails do not depend on $\ln{\rm Pe}$ or on $\ln(r/r_{\rm dif})$ and contain only coefficients depending on the statistics of the advecting velocity. Correlation functions of the passive scalar can be written as averages of integrals of the pumping along Lagrangian trajectories (see e.g. [@94FL]). For example, the pair correlation function $\langle\theta({\bbox r})\theta({\bbox 0})\rangle$ is proportional to an average time needed for two points moving along Lagrangian trajectories to run from the distance $r$ to the distance $L$. Generally, correlation functions of the passive scalar are determined by the spectral transfer that is by an evolution of Lagrangian separations up to the scale $L$. For the large-scale velocity field Lagrangian dynamics is determined by the stretching matrix $\sigma_{\alpha\beta} =\nabla_\beta v_\alpha$ and, consequently, the statistics of the matrix determines correlation functions of the passive scalar. For example, the coefficient at the logarithm in the pair correlation function of the passive scalar is $P_2/\bar\lambda$ [@Bat; @Kra; @94SS; @95CFKLa] where $P_2$ is the pumping rate of $\theta^2$ and $\bar\lambda$ is Lyapunov exponent that is the average of the largest eigen value of the matrix $\hat\sigma$. The coefficients in the exponential tails are more sensitive to the statistics of $\hat\sigma$, particurlaly they depend on the dimensionless parameter $\bar\lambda\tau$ [@95CFKLa] where $\tau$ is the correlation time of $\hat\sigma$. The motion of the fluid particles in the random velocity field resembles in some respects random walks, but one should remember that correlation lengths of both the advecting velocity and of the pumping are much larger than scales from the convective interval we are interested in. Thus the situation is opposite to one usually occurring in solid state physics, where e.g. random potential is shortly correlated in space. Since really $\ln(L/r)$ is not very large it is of interest to find the whole PDF’s for the single-point statistics of $\theta$ and for the passive scalar difference $\Delta\theta$. It is possible to do for the Kraichnan short-correlated case $\bar\lambda\tau\ll1$ when the statistics of $\hat\sigma$ can be regarded to be Gaussian. An attempt to do this was made in [@94CGK; @96FKLM] in terms of the statistics of the main eigen value of the matrix $\hat\sigma$. Unfortunately the scheme works only for the dimensionality $d=2$ where the matrix $\hat\sigma$ has a single eigen value. This fact was noted in the work [@98BGK] where also the correct coefficient in the exponential tails for an arbitrary dimensionality of space $d$ was found. Here, we develop a scheme enabling to obtain the whole PDF’s for arbitrary $d$. The scheme is also interesting from the methodical point of view. For example, its modification enables one to calculate the statistics of local dissipation [@98GK]. The paper is organized as follows. In Section \[sec:gen\] we find a path integral representation for the simultaneous statistics of the passive scalar. In Section \[sec:fun\] we analyze the generating functional for correlation functions of the passive scalar in the convective interval of scales. Using different approaches we obtain the functional and establish the applicability conditions of our consideration. In Section \[sec:pdf\] we find explicit expressions for the single-point PDF and for the PDF of the passive scalar difference. In Conclusion we shortly discuss the obtained results. General Relations {#sec:gen} ================= The dynamics of the passive scalar $\theta$ advected by the velocity field ${\bbox v}$ is described by the equation $$\partial_t\theta+{\bbox v}\nabla\theta -\kappa\nabla^2\theta=\phi \,. \label{ga1}$$ Here, the term with the velocity ${\bbox v}$ describes the advection of the passive scalar, the next term is diffusive ($\kappa$ is the diffusion coefficient) and $\phi$ describes a pumping source of the passive scalar. Both ${\bbox v}(t,{\bbox r})$ and $\phi(t,{\bbox r})$ are supposed to be random functions of $t$ and ${\bbox r}$. We regard the statistics of the velocity and of the source to be independent. Therefore, all correlation functions of $\theta$ are to be treated as averages over both statistics. Simultaneous Statistics {#simsta} ----------------------- The source $\phi$ is believed to possess a Gaussian statistics and to be $\delta$-correlated in time. The statistics is entirely characterized by the pair correlation function $$\langle\phi(t_1,{\bbox r}_1)\phi(t_2,{\bbox r}_2)\rangle =\delta(t_1-t_2)\chi\left(\|{\bbox r}_1-{\bbox r}_1\|\right) \,, \label{gaa1}$$ where we assumed that the pumping is isotropic. The function $\chi(r)$ is accepted to have a characteristic scale $L$ which is the pumping length. We will be interested in the statistics of the passive scalar on scales much smaller than $L$. Simultaneous correlation functions of the passive scalar $\theta$ can be represented as coefficients of the expansion over $y$ of the generating functional $${\cal Z}(y)=\left\langle\exp\left\{{\rm i}y \int {\rm d}{\bbox r}\, \beta({\bbox r}) \theta(0,{\bbox r})\right\}\right\rangle \,, \label{gaa3}$$ where $\beta$ is a function of coordinates and angular brackets designate averaging over both the statistics of the pumping $\phi$ and over the statistics of the velocity ${\bbox v}$. The generating functional ${\cal Z}(y)$ contains the entire information about the simultaneous statistics of the passive scalar $\vartheta$. Particularly, knowing ${\cal Z}(y)$ one can reconstruct the simultaneous PDF of the passive scalar, the problem is discussed in Section \[sec:pdf\]. If characteristic scales of $\beta$ in (\[ga5\]) are much larger than the diffusion scale $r_{\rm dif}$ then it is possible to neglect diffusivity at treating the generating functional (\[gaa3\]). Then the left-hand side of the equation (\[ga1\]) describes the simple advection and it is reasonable to consider a solution of the equation in terms of Lagrangian trajectories ${\bbox\varrho}(t)$ introduced by the equation $$\partial_t{\bbox\varrho}= {\bbox v}(t,{\bbox\varrho}) \,. \label{ga2}$$ We will label the trajectories by ${\bbox r}$, those are positions of Lagrange particles at $t=0$: ${\bbox\varrho}(0,{\bbox r})={\bbox r}$. Next, introducing $\tilde\theta(t,{\bbox r})=\theta(t,{\bbox\varrho})$ we rewrite the equation (\[ga1\]) as $\partial_t\tilde\theta=\phi$, that leads us to $$\theta(0,{\bbox r})=\int_{-\infty}^0\!\! {\rm d}t\, \phi(t,{\bbox\varrho}) \,. \label{ga3}$$ Here we have taken into account that at $t=0$ the functions $\theta$ and $\tilde\theta$ coincide. Starting from (\[ga3\]) and using Gaussianity of the pumping statistics we can average explicitly the generating functional (\[gaa3\]) over the statistics. The result is $$\begin{aligned} && {\cal Z}(y)=\left\langle\exp\left[ -\frac{y^2}{2}\int_{-\infty}^0\!\! {\rm d}t\,U\right] \right\rangle \,, \label{gaa5} \\ && U=\int {\rm d}{\bbox r}_1 {\rm d}{\bbox r}_2\, \beta({\bbox r}_1)\beta({\bbox r}_2) \chi(\|{\bbox\varrho}_1-{\bbox\varrho}_2\|) \,, \label{ga5}\end{aligned}$$ where angular brackets mean averaging over the statistics of the velocity field only. Being interested in the single-point statistics of $\theta$ we should take $\beta({\bbox r})=\delta({\bbox r})$. But it is impossible since we have neglected diffusivity. We will take $\beta({\bbox r})=\delta_\Lambda({\bbox r})$ instead where the function $\delta_\Lambda({\bbox r})$ tends to zero at $\Lambda r>1$ fast enough and is normalized by the condition $$\begin{aligned} && \int d{\bbox r}\, \delta_\Lambda({\bbox r})=1 \,. \nonumber \end{aligned}$$ Then the generating functional (\[gaa5\]) will describe the statistics of an object $$\begin{aligned} && \theta_{\Lambda}=\int{\rm d}{\bbox r}\, \delta_\Lambda({\bbox r})\theta({\bbox r}) \,, \label{theta} \end{aligned}$$ smeared over a spot of the size $\Lambda^{-1}$. If $r_{\rm dif}\Lambda\ll1$ then the statistics of the object is not sensitive to diffusivity. From the other hand if $\Lambda L\gg1$, then knowing correlation functions of $\theta_{\Lambda}$, we can reconstruct single-point statistics due to the logarithmic character of the correlation functions. To obtain single-point correlation functions one should substitute simply $\Lambda\to r_{\rm dif}^{-1}$ into the correlation functions of $\theta_{\Lambda}$. The above inequalities $\Lambda r_{\rm dif}\ll1$ and $\Lambda L\gg1$ are compatible because of ${\rm Pe}\gg1$. If we are interested in the statistics of the passive scalar differences in points with a separation ${\bbox r}_0$ (where $r_0\gg r_{\rm dif}$) then instead of $\delta_\Lambda({\bbox r})$ we should take $$\begin{aligned} && \beta({\bbox r})= \delta_\Lambda({\bbox r}-{\bbox r}_0/2) -\delta_\Lambda({\bbox r}+{\bbox r}_0/2) \,. \label{difb} \end{aligned}$$ Then the generating functional (\[gaa5\]) will describe the statistics of an object $$\begin{aligned} && \Delta\theta_{\Lambda}=\theta_{\Lambda}({\bbox r}_0/2) -\theta_{\Lambda}(-{\bbox r}_0/2) \,. \label{diff} \end{aligned}$$ Again, correlation functions of the passive scalar differences can be found from correlation functions of $\Delta\theta_{\Lambda}$ after the substitution $\Lambda\to r_{\rm dif}^{-1}$. Path Integral ------------- Below, we treat advection of the passive scalar by a large-scale velocity field, that is we assume that the velocity correlation length $L_v$ is larger than the scales from the convective interval. Then for the scales one can expand the difference $$\begin{aligned} && v_\alpha({\bbox r}_1)- v_\alpha({\bbox r}_2) =\sigma_{\alpha\beta}(t)(r_{1\beta}-r_{2\beta}) \,, \quad \sigma_{\alpha\beta}=\nabla_\beta v_\alpha \,. \label{ga8} \end{aligned}$$ Here $\sigma_{\alpha\beta}(t)$ can be treated as an ${\bbox r}$-independent matrix field. Then the equation (\[ga2\]) leads to $$\partial_t({\varrho}_{1,\alpha}-{\varrho}_{2,\alpha})= \sigma_{\alpha\beta}(t)({\varrho}_{1,\beta}-{\varrho}_{2,\beta})\,. \label{gaa4}$$ A formal solution of the equation (\[gaa4\]) is $$\begin{aligned} && {\varrho}_{1,\alpha}-{\varrho}_{2,\alpha} =W_{\alpha\beta}(r_{1,\beta}-r_{2,\beta}) \,, \nonumber \\ && \partial_t \hat W =\hat\sigma\hat W \,, \quad \hat W={\cal T}\exp\left(-\int_t^0 \!\! {\rm d}t\,\hat\sigma\right) \,, \label{ga7} \end{aligned}$$ where ${\cal T}$ means an antichronological ordering. Note that ${\rm Det\,}{\hat W}=1$, those property is a consequence of ${\rm Tr}\,\hat\sigma=0$ and the initial condition $\hat W=1$ at $t=0$. The Lagrangian difference in (\[ga5\]) is now rewritten as $$\begin{aligned} && \|{\bbox\varrho}_1-{\bbox\varrho}_2\| =\sqrt{(r_{1\alpha}-r_{2\alpha}) B_{\alpha\beta}(r_{1\beta}-r_{2\beta})} \,, \quad \hat B=\hat W^T \hat W \,, \label{gaa7}\end{aligned}$$ where the subscript $T$ designates a transposed matrix. Note that ${\rm Det\,}{\hat B}=1$ since ${\rm Det\,}{\hat W}=1$. The generating functional ${\cal Z}(y)$ (\[gaa5\]) can be explicitly calculated in the Kraichnan case [@Kra] when the statistics of the velocity is $\delta$-correlated in time. Then the velocity statistics is Gaussian and is entirely determined by the pair correlation function which in the convective interval is written as $$\begin{aligned} && \left\langle v_\alpha(t_1,{\bbox r}_1) v_\beta(t_2,{\bbox r}_2)\right\rangle=\delta(t_1-t_2) \left[{\cal V}_0\delta_{\alpha\beta} -{\cal K}_{\alpha\beta}({\bbox r}_1-{\bbox r}_2)\right] \,, \label{vel} \\ && {\cal K}_{\alpha\beta}({\bbox r})= D(r^2\delta_{\alpha\beta}-r_\alpha r_\beta) +\frac{(d-1)D}{2}\delta_{\alpha\beta}\, r^2 \,. \label{vel1} \end{aligned}$$ Here ${\cal V}_0$ is a huge ${\bbox r}$-independent constant and $D$ is a parameter characterizing the amplitude of the strain fluctuations. The structure of (\[vel1\]) is determined by the isotropy and space homogeneouty accepted, and by the incompressibility condition ${\rm div}\,{\bbox v}=0$. Then the statistics of $\hat\sigma$ is Gaussian and is determined by the pair correlation function which can be found from (\[vel\],\[vel1\]): $$\left\langle\sigma_{\alpha\beta}(t_1)\sigma_{\mu\nu}(t_2)\right\rangle =D\left[(d+1)\delta_{\alpha\mu}\delta_{\beta\nu} -\delta_{\alpha\nu}\delta_{\beta\mu} -\delta_{\alpha\beta}\delta_{\mu\nu}\right]\delta(t_1-t_2) \,. \label{ga9}$$ Note that the correlation function (\[ga9\]) is ${\bbox r}$-independent, as it should be. We see from (\[ga9\]) that the parameter $D$ characterizes amplitudes of $\hat\sigma$ fluctuations. Averaging over the statistics of $\hat\sigma$ can be substituted by the path integral over unimodular matrices $\hat W(t)$ with a weight $\exp\left(i{\cal I}\right)$. The effective action ${\cal I}=\int{\rm d}t\,{\cal L}_0$ is determined by (\[ga9\]): $${\rm i}{\cal L}_0=-\frac{1}{2d(d+2)D} \left[(d+1){\rm Tr}(\hat\sigma^T\hat\sigma) +{\rm Tr}\hat\sigma^2\right] \,. \label{ga17}$$ Then the generating functional (\[ga5\]) can be rewritten as the following functional integral over unimodular matrices $$\begin{aligned} && {\cal Z}(y)=\int {\cal D}\hat W \, \exp\left[\int_{-\infty}^0 \!\! {\rm d}t\, \ \left( {\rm i}{\cal L}_0-\frac{y^2}{2}U\right)\right] \,, \label{ga16} \\ && U = \int {\rm d}{\bbox r}_1 {\rm d}{\bbox r}_2 \, \beta({\bbox r}_1)\beta({\bbox r}_2) \chi\left[\sqrt{(r_{1\alpha}-r_{2\alpha}) B_{\alpha\beta}(r_{1\beta}-r_{2\beta})}\right] \,. \label{ga18}\end{aligned}$$ Here, we should substitute $\hat\sigma=\partial_t \hat W(\hat W)^{-1}$ and remember about the boundary condition $\hat W=1$ at $t=0$. Some words about the ‘potential’ $U$ (\[ga5\]) figuring in (\[ga18\]). The characteristic value of ${\bbox r}_1-{\bbox r}_2$ in the integral (\[ga5\]) is of order $\Lambda^{-1}$ for $\beta({\bbox r})=\delta_\Lambda({\bbox r})$. Since we assume $\Lambda L\gg1$ then for a single-point statistics $U\approx P_2$, where $P_2=\chi(0)$, if $B$ is not very large. Particularly it is correct at moderate times $\|t\|$ since $\hat B=\hat 1$ at $t=0$. With increasing $\|t\|$ the argument of $\chi$ in (\[ga18\]) grows and $U$ tends to zero when the argument of $\chi$ becomes greater than $L$. For the passive scalar difference when $\beta$ is determined by (\[difb\]) the situation is a bit more complicated. Then $U$ is a difference of two contributions. The first contribution behaves like for a single-point statistics. The second contribution contains $\chi$ with the argument determined by ${\bbox r}_1-{\bbox r}_2\approx\pm{\bbox r}_0$. Then at $t=0$ the meaning of the second contribution is determined again by $P_2$ but it turns to zero at increasing $\|t\|$ earlier than the first contribution. Thus we reduced our problem to the path integral (\[ga16\]) that is to the quantum mechanics with $d^2-1$ degrees of freedom. Nevertheless to solve the problem we should perform an additional reduction of the degrees of freedom. The conventional way to do this is passing to eigen values, say, of the matrix $\hat B$ figuring in (\[ga18\]) (see e.g. [@mehta]) and excluding angular degrees of freedom. Just this way was used by Bernard, Gawedzki and Kupiainen in [@98BGK]. Then the authors using known facts about the quantum mechanics associated with the eigenvalues (see e.g. [@OlPe]) have found the coefficient in the exponential tail in the single-point PDF of $\theta$. Unfortunately this way is not very convenient to find the whole PDF. To do this we will use a special representation of the matrix $\hat W$ in the spirit of the nonlinear substitution introduced by Kolokolov [@Kol]. That is a subject of the next subsection. Choice of Parametrization ------------------------- To examine the generating functional ${\cal Z}(y)$ we will use a mixed rotational-triangle parametrization $$\hat W=\hat R \hat T \,, \quad \hat B=\hat T^T\hat T \,, \label{gy1}$$ where $\hat R$ is an orthogonal matrix and $\hat T$ is a triangular matrix: $T_{ij}=0$ at $i>j$. The parametrization (\[gy1\]) is the direct generalization of the $2d$ substitution suggested in [@98CFK]. Note that ${\rm Det}\,\hat T=1$ since ${\rm Det}\,\hat W=1$. Note also that the matrix $\hat B$ introduced by (\[gaa7\]) does not depend on $\hat R$ as is seen from (\[gy1\]). That is a motivation to exclude the matrix $\hat R$ from the consideration performing the integration over the corresponding degrees of freedom in the path integral (\[ga16\]). A jacobian appears at the integration. To avoid an explicit calculation of the jacobian (which needs a discretization over time and then an analysis of an infinite matrix [@94CGK]) we will use an alternative procedure described below. Let us examine the dynamics of the matrix $\hat T$. It is determined by the equation $$\partial_t T_{ij}=\Sigma_{ii}T_{ij} +\sum\limits_{i<k\le j}(\Sigma_{ik}+\Sigma_{ki})T_{kj} \,, \label{gy3}$$ following from (\[ga7\],\[gy1\]). Here we used the designation $${\hat\Sigma}=\hat R^T\hat\sigma\hat R \,. \label{ga12}$$ Next introducing the quantities $$T_{ii} = \exp(\rho_i)\,, \quad T_{ij} = \exp(\rho_i)\eta_{ij} \,, \quad {\rm if}\ i < j \,, \label{gy5}$$ we rewrite the equation (\[gy3\]) as $$\begin{aligned} && \partial_t \rho_i=\Sigma_{ii} \,, \label{gy6} \\ && \partial_t \eta_{ij}=(\Sigma_{ij}+\Sigma_{ji}) \exp(\rho_j-\rho_i) +\sum\limits_{i<k<j}(\Sigma_{ik}+\Sigma_{ki}) \exp(\rho_k-\rho_i)\eta_{kj} \,. \label{gy7} \end{aligned}$$ Comparing (\[ga7\]) with (\[gy1\]) one can find the following expression for $\hat A=\hat R^T\partial_t\hat R$ $$A_{ij}=\Sigma_{ij} \quad {\rm if}\ i > j \qquad A_{ij}=-\Sigma_{ji} \quad {\rm if}\ i < j \,. \label{gy2}$$ One can easily check that the irreducible pair correlation function of $\Sigma_{ij}$ has the same form as for $\sigma_{ij}$ that is determined by (\[ga9\]): $$\begin{aligned} && \langle\Sigma_{ij}(t_1)\Sigma_{mn}(t_2)\rangle =D[(d+1)\delta_{im}\delta_{jn} -\delta_{in}\delta_{jm} -\delta_{ij}\delta_{mn}]\delta(t_1-t_2) \,. \label{sig} \end{aligned}$$ Besides, the average value of $\Sigma_{ij}$ is nonzero [@94CGK]: $$\langle \Sigma_{ij}\rangle = - D\frac{d(d-2i+1)}2 \delta_{ij} \,. \label{gy4}$$ Nonzero averages of $\Sigma_{ij}$ are related to Lyapunov exponents (not only the first one) [@83Dor] (for our model see also [@KG]). To obtain (\[gy4\]) one should first solve the equation $\hat A=\hat R^T\partial_t\hat R$ for $\hat R$ on a small interval $\tau$: $$\begin{aligned} && \hat R(t+\tau)\approx\hat R(t)\left[1 +\int_t^{t+\tau}\!\!\!{\rm d}t'\,\hat A(t')\right] \,. \nonumber \end{aligned}$$ Then with the same accuracy we get from the expression (\[ga12\]) $$\begin{aligned} && \hat\Sigma(t+\tau)\approx \hat R^T(t)\hat\sigma(t+\tau)\hat R(t) +\left[\hat\Sigma(t+\tau), \int_t^{t+\tau}\!\!\!{\rm d}t'\,\hat A(t')\right] \,. \label{rox} \end{aligned}$$ The average value of $\hat\Sigma$ arises from the second term in the right-hand side of (\[rox\]). The explicit form of the average can be found using $$\begin{aligned} && \left\langle\Sigma_{ij}(t+\tau) \int_t^{t+\tau}\!\!\!{\rm d}t'\, \Sigma_{mn}(t')\right\rangle= \frac{D}{2}[(d+1)\delta_{im}\delta_{jn} -\delta_{in}\delta_{jm}-\delta_{ij}\delta_{mn}] \,. \label{rox1} \end{aligned}$$ Here we utilized Eq. (\[sig\]) and substituted the integral $$\begin{aligned} \int_t^{t+\tau}\!\!\!{\rm d}t'\,\delta(t+\tau-t') \nonumber \end{aligned}$$ by $1/2$. The reason is that really the correlation function of $\hat\sigma$ has a finite correlation time and therefore $\delta(t)$ (representing this correlation function) should be substituted by a narrow function symmetric under $t\to-t$. Then we will get $1/2$. Expressing in (\[rox\]) $\hat A$ via $\hat\Sigma$ from (\[gy2\]) and calculating its average using (\[rox1\]) we get the answer (\[gy4\]). The expressions (\[gy6\],\[gy7\],\[sig\],\[gy4\]) entirely determine the stochastic dynamics of $\rho_i$ and $\eta_{ij}$. Using the conventional approach [@61Wyl; @73MSR; @76Dom; @76Jan; @DP78] correlation functions of these degrees of freedom can be described in terms of a path integral over $\rho_i$, $\eta_{ij}$ and over auxiliary fields which we will designate as $m_i$ and $\mu_{in}$ ($i<n$). This integral should be taken with the weight $\exp({\rm i}\int {\rm d}t\, {\cal L})$ where the Lagrangian is $$\begin{aligned} && {\cal L}=\sum\limits_{a=1}^{d}m_a \left[\partial_t\rho_a+D\frac{d(d-2a+1)}2 \right] +\frac{{\rm i}D}{2}\left[d\sum\limits_a m_a^2 -\left(\sum\limits_a m_a\right)^2\right] \nonumber \\ && +{\rm i}Dd \sum\limits_{i<j}\exp(2\rho_j-2\rho_i)\mu_{ij}^2 +2{\rm i}Dd\sum\limits_{i<k<j}\mu_{ij}\mu_{ik} \exp(2\rho_k-2\rho_i)\eta_{kj} \nonumber \\ && +\sum\limits_{i<j}\mu_{ij}\partial_t\eta_{ij} +{\rm i}Dd\sum\limits_{i<k<m,n} \mu_{im}\mu_{in}\eta_{km}\eta_{kn} \exp(2\rho_k-2\rho_i) \,. \label{gy8} \end{aligned}$$ Since the matrix $\hat B$ in accordance with (\[gy1\]) does not depend on $\hat R$ it is enough to know the statistics of $\rho_a$ and $\eta_{ij}$ to determine the average (\[gaa5\]). Therefore, instead of (\[ga16\]) we get $$\begin{aligned} && {\cal Z}(y)=\int {\cal D}\rho{\cal D}\eta{\cal D}m{\cal D}\mu\, \exp\left[\int_{-\infty}^0 \!\! {\rm d}t\, \ \left( {\rm i}{\cal L}-\frac{y^2}{2}U\right)\right] \,. \label{gaa16}\end{aligned}$$ Here $U$ is determined by (\[ga18\]) where (\[gy1\],\[gy5\]) should be substituted. Thus we obtained the expression for the generating functional (\[gaa3\]) in terms of the functional (path) integral which is convenient for an analysis which is presented in the subsequent section. Generating Functional {#sec:fun} ===================== Here, we are going to calculate the generating functional (\[gaa3\]) for a single-point statistics of $\theta$ that is of the object (\[theta\]) corresponding to $\beta({\bbox r})=\delta_\Lambda ({\bbox r})$ and also the statistics of the difference that is of the object (\[diff\]) corresponding to (\[difb\]). The starting point for the subsequent consideration is the expression (\[gaa16\]). There are different ways to examine ${\cal Z}(y)$. We will describe two schemes leading to the same answer but carrying in some sense a complimentary information. We believe also that the consideration of the different schemes is useful from the methodical point of view. A modification of the second scheme is presented in Appendix. Saddle-Point Approach --------------------- The first way to obtain the answer for the generating functional (\[gaa3\]) is in using the saddle-point approximation for the path integral (\[gaa16\]). The inequalities justifying the approximation are $\Lambda L\gg1$ for the object (\[theta\]) and $\Lambda r\gg1$ for the object (\[diff\]). As we will see large values of the differences $\rho_i-\rho_k$ ($i<k$) will be relevant for us. At the condition fluctuations of $\eta$ and $\mu$ are suppressed and it is possible to neglect the fluctuations. Therefore we can omit the integration over $\eta$ and $\mu$ in (\[gaa16\]) substituting $\eta=\mu=0$ into (\[gy8\]). After that we obtain a reduced Lagrangian. Introducing an auxiliary field $s$ we can rewrite the reduced Lagrangian as $${\rm i}{\cal L}_{\rm r} = {\rm i} \sum_{a=1}^{d} m_a \left[\partial_t\rho_a + D\frac{d(d+1-2a)}2+s\right] -\frac{Dd}{2}\sum\limits_a m_a^2-\frac{s^2}{2D} \,. \label{gy10}$$ Now, to obtain ${\cal Z}(y)$ one should integrate the exponent in (\[gaa16\]) (with ${\cal L}_{\rm r}$) over $\rho_a$, $m_a$ and $s$. To examine (\[gy10\]) it is convenient to pass to new variables $\phi_a=O_{ab}\rho_b$ and $\tilde m_a=O_{ab}m_b$ where $\hat O$ is an orthogonal matrix. We make the following transformation $$\begin{aligned} && \phi_1=\sqrt{\frac{3}{d(d^2-1)}}\, \left[(d-1)\rho_1+(d-3)\rho_2+\dots +(1-d)\rho_d\right] \,, \label{phi} \\ && \phi_2= \dots \,, \quad \dots \,, \quad \phi_d=\frac1{\sqrt{d}}\, \left[\rho_1+\rho_2+\dots +\rho_d\right] \,. \nonumber \end{aligned}$$ Then the expression (\[gy10\]) will be rewritten as $${\rm i}{\cal L}_{\rm r} = {\rm i} \sum_{a=1}^{d}\tilde m_a\partial_t\phi_a +{\rm i}\sqrt{d}\tilde m_d s-\frac{s^2}{2D} -\frac{Dd}{2}\sum\limits_a \tilde m_a^2 +{\rm i}\frac{Dd}{2}\sqrt{\frac{d(d^2-1)}{3}}\tilde m_1 \,. \label{gy13}$$ The Lagrangian (\[gy13\]) is a sum over different degrees of freedom. The dynamics of $\phi_1$ is ballistic whereas the dynamics of $\phi_a$ for $a>1$ is purely diffusive. We will see that times determining the main contribution to the generating functional are large enough so that for the relevant region $\phi_1\gg\phi_a$. Therefore the potential $U$ (\[ga18\]) depends practically only on $\phi_1$ and it is possible to integrate explicitly over $s$, $\phi_a$ and $\tilde m_a$ for $a>1$. After that we stay only with one degree of freedom which is described by the Lagrangian $${\rm i}{\cal L}_{1}= {\rm i}\tilde m_1\left(\partial_t\phi_1 +\frac{Dd}{2}\sqrt{\frac{d(d^2-1)}{3}}\right) -\frac{Dd}{2}\tilde m_1^2 \,. \label{gy14}$$ Neglecting all $\phi_a$ for $a>1$ and performing the transformation inverse to (\[phi\]) we obtain $$\begin{aligned} && \rho_1\approx\sqrt{\frac{3(d-1)}{d(d+1)}}\,\phi_1 \,, \qquad \rho_a\approx\frac{d-2a+1}{d-1}\rho_1 \,. \label{conn} \end{aligned}$$ We will soon see that the characteristic value $\phi_1\gg1$. Therefore the characteristic value of $e^{\rho_1}$ is much larger than other $e^{\rho_a}$ and we conclude that the potential $U$ depends really only on $\rho_1$. For the case of the single-point statistics the characteristic value of the difference ${\bbox r}_1-{\bbox r}_2$ in (\[ga18\]) is $\Lambda^{-1}$. Then it follows from (\[gy1\],\[gy5\]) that the potential $U$ falls down from $P_2$ to zero near the point $\rho_1=\ln(L\Lambda)$ that is near the point $\phi_1=\phi_\Lambda$ where $$\phi_\Lambda= \sqrt {\frac{d(d+1)}{3(d-1)}}{\rm ln\,}(L\Lambda) \,. \label{jump}$$ For the difference the potential increases from zero to $2P_2$ at $\phi_1=\phi_R$ where $$\phi_R= \sqrt {\frac{d(d+1)}{3(d-1)}}{\rm ln\,}(L/r_0) \,, \label{jump1}$$ and then falls down from $P_2$ to zero near the point $\phi_1=\phi_\Lambda$. The expressions (\[jump\],\[jump1\]) determine the characteristic values of $\phi_1$ which are really large since $L\Lambda\gg1$ or $L/r_0\gg1$ what justifies our conclusions. Now we can employ the saddle-point approximation: $$\left. \ln{\cal Z}(y)\approx \int_{-\infty}^0 \!\! {\rm d}t\, \ \left( {\rm i}{\cal L}_1-\frac{y^2}{2}U\right)\right\|_{\rm inst} \,, \label{go}$$ where we should substitute solutions of the extrema conditions which we will call instantonic equations. The instantonic equations which can be found as extrema conditions for ${\rm i}{\cal L}_1-y^2 U/2$ are $$\begin{aligned} && \partial_t\phi_1+\frac{Dd}{2}\sqrt{\frac{d(d^2-1)}{3}} =-{\rm i}Dd\tilde m_1 \,, \label{go1} \\ && \partial_t\tilde m_1 ={\rm i}\frac{y^2}{2}\frac{\partial U}{\partial\phi_1} \,. \label{go2} \end{aligned}$$ The equations conserve the ‘energy’ $$\begin{aligned} && -{\rm i}\frac{Dd}{2}\tilde m_1\sqrt{\frac{d(d^2-1)}{3}} +\frac{Dd}{2}\tilde m_1^2+\frac{y^2}{2}U \,. \label{go3}\end{aligned}$$ The conservation law is satisfied since ${\rm i}{\cal L}_1-y^2U/2$ does not explicitly depend on $t$. The ‘energy’ (\[go3\]) is equal to zero since at $t\to-\infty$ the value of $\tilde m_1$ should tend to zero. The property can be treated as the extremum condition appearing at variation of ${\rm i}{\cal L}_{\rm r}-y^2 U/2$ over the initial value of $\phi_1$. Equating (\[go3\]) to zero we can express $\tilde m_1$ via $\phi_1$. Next, since (\[go3\]) is zero then the saddle-point value of ${\cal Z}(y)$ (\[go\]) can be written as ${\rm i}\int{\rm d}\phi_1\,\tilde m_1$, where the integral over $\phi_1$ goes from zero to infinity. Substituting into ${\rm i}\int{\rm d}\phi_1\,\tilde m_1$ the expression of $\tilde m_1$ in terms of $\phi_1$ we get for the single-point statistics $$\ln{\cal Z}(y)\simeq \frac{d(d+1)}{6} \left[1-\sqrt{1+\frac{12y^2 P_2}{Dd^2(d^2-1)}}\,\right] {\ln}(L\Lambda) \,. \label{gy15}$$ Note that the expression (\[gy15\]) has (as a function of $y$) two branch points $y=\pm{\rm i}y_{\rm sing}$ where $$y_{\rm sing}^2 = \frac{Dd^2(d^2-1)}{12 P_2} \,. \label{sing}$$ The same procedure can be done for the passive scalar difference, or, more precisely, for the object (\[diff\]). Taking into account the presence of the jumps (\[jump1\]) and (\[jump\]) in the potential $U$ we get an answer slightly different from (\[gy15\]) $$\begin{aligned} && \ln{\cal Z}(y)\simeq \frac{d(d+1)}6 \left[1-\sqrt{1+\frac{24y^2 P_2}{Dd^2(d^2-1)}}\,\right] {\ln}(r_0\Lambda) \,, \label{go4} \\ && y_{\rm sing}^2 = \frac{Dd^2(d^2-1)}{24 P_2} \,. \label{goo4} \end{aligned}$$ Note that (\[go4\]) does not depend on the pumping scale $L$ but still depend on the cutoff $\Lambda$. The characteristic value of $\phi_1$ is determined by the quantity (\[jump\]) which is much larger than unity. Then it follows from (\[conn\]) that $\exp(2\rho_j-2\rho_i)\ll1$, $i>j$, (excluding for a short initial stage of the evolution) and we see from (\[gy8\]) that fluctuations of the fields $\eta$ are suppressed in comparison, say, with $\rho_a$. This justifies neglecting the fields $\eta$ and $\mu$ leading to the reduced Lagrangian (\[gy10\]). Next, dynamics of $\phi_a$ for $a>1$ is diffusive and it follows from (\[gy13\]) that the characteristic value of $\phi_a$ can be estimated as $\sqrt{Dd\|t\|}$. As follows from (\[gy13\]) $\partial_t\phi_1\sim Dd^{5/2}$ and we find from (\[jump\]) the instantonic life time $$t_{\rm lt}=D^{-1}d^{-2}\ln(L\Lambda) \,, \label{life}$$ which determines times producing nonzero contribution to the effective action. At $\|t\|\sim t_{\rm lt}$ the characteristic values of $\phi_a$ for $a>1$ are of order $\sqrt{\ln(L\Lambda)/d}$ and we conclude that $$\begin{aligned} && \frac{\phi_a}{\phi_1}\sim \frac{1}{d\sqrt{\ln(L\Lambda)}}\ll1 \,, \label{goj} \end{aligned}$$ at times $\|t\|\sim t_{\rm lt}$. The inequality (\[goj\]) justifies passing to the Lagrangian (\[gy14\]). The same arguments can be applied to the generating functional for the passive scalar difference, the only modification is in the substitution $\ln(L\Lambda)\to{\ln}(r_0\Lambda)$. There are also additional applicability conditions for the answers (\[gy15\],\[go4\]). To establish the conditions one should go beyond the main order of the saddle-point approximation. It will be more convenient for us to develop an alternative scheme which enables to find the conditions simpler. That is the subject of the next subsection. Schrödinger equation -------------------- Here we present another way to get the answers (\[gy15\],\[go4\]). As previously we start from the path integral representation (\[gaa16\]) for the generation functional ${\cal Z}(y)$. Unfortunately it is impossible to get a closed equation for ${\cal Z}(y)$. To avoid the difficulty we introduce an auxiliary quantity $$\Psi(t,y,\rho_0,\eta_0)=\left.\int {\cal D}\rho{\cal D}\eta{\cal D}m{\cal D}\mu\, \exp\left[\int_{-t}^0 \!\! {\rm d}t' \, \left( {\rm i}{\cal L}-\frac{y^2}{2}U\right)\right] \right\|_{\rho(-t)=\rho_0,\eta(-t)=\eta_0} \,. \label{ga22}$$ It follows from the definition (\[ga22\]) that $${\cal Z}(y)=\lim_{t\to\infty}\int\prod {\rm d}\rho_a \, {\rm d}\eta_{ij}\, \Psi(t,y,\rho,\eta) \,. \label{ga23}$$ The equation for the function $\Psi$ can be obtained from the expression (\[gy8\]) and the definition (\[ga22\]): $$\begin{aligned} && \partial_t \Psi = \frac{Dd}2 \left[ \sum\limits_{i=1}^{d}\frac{\partial^2}{\partial\rho_i^2} -\frac{1}d\left(\sum\limits_{i=1}^{d} \frac{\partial}{\partial\rho_i}\right)^2 -\sum\limits_{i=1}^{d}(d-2i+1)\frac{\partial}{\partial\rho_i} \right. \nonumber \\ && + 2\sum\limits_{i<j}\exp(2\rho_j-2\rho_i) \frac{\partial^2}{\partial\eta_{ij}^2} +4\sum\limits_{i<k<j} \exp(2\rho_k-2\rho_i) \frac{\partial}{\partial\eta_{ij}} \frac{\partial}{\partial\eta_{ik}}\eta_{kj} \nonumber \\ && \left. + 2\sum\limits_{i<k<m,n} \exp(2\rho_k-2\rho_i) \frac{\partial}{\partial\eta_{im}} \frac{\partial}{\partial\eta_{in}}\eta_{km}\eta_{kn}\right] \Psi -\frac{y^2 U}2 \Psi \,. \label{gyy9}\end{aligned}$$ We see that the equation (\[gyy9\]) for $\Psi$ resembles the Schrödinger equation. The initial condition to the equation can be found directly from the definition (\[ga22\]): $$\Psi(t=0,y,\rho,\eta)=\prod\delta(\rho_a)\delta(\eta_{ij}) \,. \label{gaa22}$$ The value of ${\cal Z}$ in accordance with (\[ga23\]) is determined by the integral of $\Psi$ over $\eta$ and $\rho$. This integral is equal to unity at $t=0$ and then varies with increasing time $t$ due to $U\neq0$ since only the term with $U$ in (\[gaa22\]) brakes conservation of the integral. Thus to find ${\cal Z}$ we should establish an evolution of the function $\Psi$ from $t=0$ to large $t$. Below we concentrate on the single-point statistics. The scheme can be obviously generalized for the passive scalar difference. Let us first describe the evolution qualitatively. The initial condition (\[gaa22\]) shows that at $t=0$ the function $\Psi$ is concentrated at origin. Then it undergoes spreading in all directions except for $\rho_1+\dots+\rho_d$ since the operator in the right-hand side of (\[gyy9\]) commutes with $\rho_1+\dots+\rho_d$. This is a consequence of the condition ${\rm Det}\,\hat T=1$ (to be satisfied) which implies that during the evolution $\rho_1+\dots+\rho_d=0$. That means that we should treat a solution of (\[gyy9\]) which is $\Psi\propto\delta(\rho_1+\dots+\rho_d)$. The function $\Psi$ is smeared diffusively with time and also moves as a whole in some direction, which is determined by the term with the first derivative in (\[gyy9\]). The rate of the ballistic motion is $$\langle \partial_t \rho_i \rangle = D\,\frac{d(d-2i+1)}2 \,. \label{psi1}$$ Therefore $\Psi$ describes a cloud, centre of which moves according to the law $$\rho_i = D\,\frac{d(d-2i+1)}2 t\,. \label{ps1}$$ Effective diffusion coefficients for $\eta$’s fall down with increasing $t$ since in accordance with (\[ps1\]) the differences $\rho_k-\rho_i$ \[figuring in (\[gyy9\])\] are negative and grow by their absolute value. Therefore the diffusion over $\eta$ stops when the characteristic values of $\rho_i-\rho_k$ becomes greater than unity. Note that the ‘frozen’ values of $\eta$ do not depend on $y$ since $U$ can be considered as uniform during the initial stage of the evolution. After that $\eta$ are frozen, the diffusion continues only over $\rho$’s. If the cloud is inside the region where $U \simeq P_2$ then the evolution of the cloud is not influenced by $U$. After the period of time $t_{\rm lt}$ (\[life\]) the cloud reaches a barrier where the potential $U$ falls down from $P_2$ to $0$. The subsequent history depends on the value of $y$. For moderate $y$ the cloud passes this barrier and continues to move with the same rate. After this the integral of $\Psi$ will not change in time, and its value will determine the generating functional ${\cal Z}(y)$. Naive estimations would give the answer $\ln{\cal Z}(y)=-y^2 t_{\rm lt}/2$, which reproduces the value of the pair correlation function of $\theta$. A special consideration is needed if $\|y\|\gg y_{\rm sing}$ or if $y$ is close to $\pm {\rm i}y_{\rm sing}$ where $y_{\rm sing}$ is introduced by (\[sing\]). Just the last region determined the PDF’s and is consequently of a special interest. Note that $y=\pm {\rm i}y_{\rm sing}$ corresponds to appearance of a bound state near the pumping boundary (where $U$ falls down from $P_2$ to zero). If $y\gg y_{\rm sing}$ then the front side of the cloud reaches the jump of the potential much earlier than $t_\Lambda$. The residue of the cloud (living inside the potential well) is damped due to the term with $y$ and don’t give a contribution to ${\cal Z}(y)$. If $\|y\|\gg y_{\rm sing}$ then ${\cal Z}(y)\gg \exp(-y^2 t_\Lambda/2)$, really the asymptotics of ${\cal Z}(y)$ is exponential in the case. If $\|y \pm {\rm i}y_{\rm sing}\| \ll y_{\rm sing}$ then the cloud stays near the pumping boundary for a long time, that is the shape of $\Psi$ inside the region $U\simeq P_2$ varies in time comparatively slow. Besides, a part of $\Psi$ percolates out to the region where $U\simeq 0$ and the integral of $\Psi$ grows with increasing $\|t\|$. As $y$ come to ${\rm i}y_{\rm sing}$ closer this stage lasts longer. One can say that the back side of the cloud $\Psi$ gives the right answer for ${\cal Z}(y)$. The important point is that if $y$ is not very close to ${\rm i}y_{\rm sing}$ then during the time of exiting $\Psi$ from the potential the width of $\Psi$ in terms of diffusive degrees of freedom is much less than ${\rm ln\,}L\Lambda$. This means that the function $\Psi$ is really narrow, that justifies our consideration. For an quantitative analysis it is convenient to pass to the variables $\phi_i$ (\[phi\]). Since an $\eta$-dependence of $\Psi$ is frozen after an initial part of evolution, then it is possible to obtain an equation for the integral of $\Psi$ over $\eta$: $$\tilde\Psi(\phi_1, \dots, \phi_{d-1}) = \int{\rm d}\phi_d \prod {\rm d}\eta_{ij}\,\Psi \,, \label{int}$$ where we included also an integration over $\phi_d$ to remove the factor $\delta(\rho_1+\dots+\rho_d)$. The equation for the function (\[int\]) is $$\partial_t \tilde\Psi = \frac{Dd}2 \left[ \sum_{i=1}^{d-1} \frac{\partial^2}{\partial\phi_i^2} - \sqrt{\frac{d(d^2-1)}3} \frac{\partial}{\partial \phi_1} \right] \tilde\Psi - \frac{y^2 \tilde U}2 \tilde\Psi \,, \label{dpsi}$$ where $\tilde U$ is function of $\phi_a$ only which can be found by substituting into $U$ the ‘frozen’ values of $\eta$’s. Qualitatively $\tilde U$ has the same structure as $U$ itself. One can conclude from (\[dpsi\]) that the cloud described by $\tilde\Psi$ moves ballistically into the $\phi_1$ direction and spreads along other directions. We are going to treat the situation when the cloud remains narrow during the relevant part of the evolution. Then one can integrate $\tilde\Psi$ over all $\phi_i$, $i>1$ in a similar way as in the case with $\eta$’s and get an $1d$ equation for $$\begin{aligned} && \bar\Psi(\phi_1)= \int \prod\limits_{2}^{d-1} {\rm d}\phi_i\, \tilde\Psi \,. \nonumber \end{aligned}$$ The function $\bar\Psi$ satisfies the equation $$\partial_t \bar\Psi = \frac{Dd}2 \left[ \frac{\partial}{\partial\phi_1} - \sqrt{\frac{d(d^2-1)}3} \, \right] \frac{\partial}{\partial \phi_1} \bar\Psi - \frac{y^2 \bar U}2 \bar\Psi \,. \label{psi2}$$ The initial condition to Eq. (\[psi2\]) is $\bar\Psi(t=0)=\delta(\phi_1)$. The potential $\bar U$ is obtained from $\tilde U$ by substitution $\phi_a\to0$ for $a>0$. In fact, on the direction (\[ps1\]) we have that is the potential $\bar U$ depends only on $\rho_1$. The barrier is achieved when $\rho_1 \simeq {\rm ln\,}L\Lambda$. Passing to the variables $\phi_i$ we conclude that the potential $\bar U$ diminishes from $P_2$ at $\phi_1<\phi_\Lambda$ to zero at $\phi_1>\phi_\Lambda$ where $\phi_\Lambda$ is defined by (\[jump\]). The character of a solution of the equation (\[psi2\]) can be analyzed semiqualitatively in terms of the width $l$ of $\bar\Psi$ over $\phi_1$ and its amplitude $h$. When $\bar\Psi$ reaches the pumping boundary then it stops there for a period of time. Then the width $l$ and the amplitude $h$ governed by the equations $$\frac{{\rm d}l}{{\rm d}t} = - Dd\lambda + \frac{Dd}{l} \,, \quad \frac{{\rm d}h}{{\rm d}t} = - \frac{Dd h}{l^2} - \frac{y^2 P_2 h}2 \,,$$ where $\lambda = \sqrt{d(d^2-1)/12}$, $Dd\lambda$ is the rate of the cloud motion along $\phi_1$ direction (when $U={\rm const}$), and $Dd$ is the diffusion coefficient for $\phi_1$ direction. One can estimate from the 1st equation the width $l$ as $l \sim 1/\lambda$. Then from the 2nd equation the height $h$ falls down or grows in time depending on $y$. The characteristic $y$ where the regime changes is of the order $\|y_{\rm sing}\|^2 \sim Dd\lambda^2/P_2$. We’ll show this by consistent calculations. The equation (\[psi2\]) can be solved analytically, e.g., by the Laplace transform over time $t$. Making the Laplace transform one gets $$p \bar\Psi(p) - \delta(\phi_1) = \frac{Dd}2 \left[ \frac{\partial}{\partial\phi_1} - \sqrt{\frac{d(d^2-1)}3} \, \right] \frac{\partial}{\partial \phi_1} \bar\Psi(p) - \frac{y^2}{2}\bar U(\phi_1) \bar\Psi(p) \,. \label{psi2p}$$ We are interesting in the bound state describing by the equation. Solutions for $\bar\Psi(p)$ in the intervals $(-\infty, 0)$, $(0,\phi_\Lambda)$, $(\phi_\Lambda, \infty)$ are exponential, one should match them. The function $\bar\Psi(p)$ as a function of $p$ has two branch points at $$\begin{aligned} && p_1 = -\frac{Dd^2(d^2-1)}{24} - y^2 P_2/2 \,, \qquad p_2= -\frac{Dd^2(d^2-1)}{24} \,, \label{bpp}\end{aligned}$$ coming from the regions $\phi_1 < \phi_\Lambda$ and $\phi_1>\phi_\Lambda$, respectively. When one of this branch points passes $p = 0$ then $\bar\Psi$ starts to grow exponentially in time. This happens when $y$ passes $\pm{\rm i}y_{\rm sing}$ from (\[sing\]) moving along the imaginary axis. The value of the generating functional is determined in accordance with (\[ga23\]) by large time behavior of $\Psi(t)$. This means that we should be interested in the behavior of $\Psi(p)$ at small $p$. Really the function $\int{\rm d}\phi_1\,\hat\Psi(p)$ entering (\[ga23\]) has a pole at $p=0$ related to the asymptotic behavior $$\begin{aligned} && \bar\Psi(p)\propto\exp\left(-\frac{2p}{Dd} \sqrt{\frac{3}{d(d^2-1)}}\phi_1\right) \,, \nonumber \end{aligned}$$ at $\phi_1>\phi_\Lambda$ and small $p$, the behavior can be found from (\[psi2p\]). Just the residue of $\int{\rm d}\phi_1\,\bar\Psi(p)$ at the pole determines ${\cal Z}(y)$. To find the value of the residue we should analyze the behavior of $\bar\Psi(p)$ at $0<\phi_1<\phi_\Lambda$. At small $p$ there are two contributions into $\bar\Psi$ which behave as $$\begin{aligned} && \propto\exp \left\{ \left( \sqrt{\frac{d(d^2-1)}{12}} \pm\sqrt{\frac{d(d^2-1)}{12}+\frac{y^2P_2}{Dd}}\right) \phi_1\right\} \,, \label{inter} \end{aligned}$$ as follows from (\[psi2p\]) at $p=0$. Therefore the residue which is determined by the integral $\int{\rm d}\phi_1\,\bar\Psi(p)$ over the region $\phi_1>\phi_\Lambda$ is proportional to $$\begin{aligned} && \exp\left\{\left(\sqrt{\frac{d(d^2-1)}{12}}+ \sqrt{\frac{d(d^2-1)}{12} +\frac{y^2P_2}{Dd}}\right)\phi_\Lambda\right\} \,, \label{go8} \end{aligned}$$ Substituting here (\[jump\]) we reproduce (\[gy15\]). Let us now establish the applicability condition of the above procedure. The expression (\[go8\]) implies that the exponent with the sign minus in (\[inter\]) gives a negligible contribution to $\Psi(p)$ at $\phi_1=\phi_\Lambda$. The condition is satisfied if $$\begin{aligned} && \|y^2+y_{\rm sing}^2\|\phi_\Lambda^2\gg \frac{Dd}{P_2} \,. \nonumber \end{aligned}$$ Substituting here (\[jump\],\[sing\]) we obtain $$\left\| \frac{y \pm {\rm i}y_{\rm sing}}{y_{\rm sing}}\right\| \gg \left( d^4\,{\ln}^2 L\Lambda \right)^{-1} \,. \label{go9}$$ For $y$ closer to $\pm {\rm i}y_{\rm sing}$ one should be careful since then a fine analytical structure of ${\cal Z}(y)$ will be relevant. As an analysis for $d=2$ shows [@98Ste] ${\cal Z}(y)$ has a system of poles along the imaginary semiaxis starting from $\pm {\rm i}y_{\rm sing}$ and the parameter $\left( d^4{\ln}^2 L\Lambda \right)^{-1}$ just determines the separation between the poles. The assertion about the cut made in the previous subsection is related to the restrictions of the saddle-point approximation which cannot feel this fine pole structure, it gives a picture averaged over the interpole distances, it is precisely the cut. This approach is correct just at the condition (\[go9\]). Note that the same criterion (\[go9\]) justifies our assumption that the cloud described by $\Psi$ is narrow during the relevant part of the evolution. Namely, the duration of the part is determined by the time $t_{\rm exit}=p_1^{-1}$ (see (\[bpp\])). This is the time which the cloud stays near the barrier. For $y$ close to $\pm iy_{\rm sing}$ the time can be estimated as $t_{\rm exit}^{-1}\sim P_2\|y_{\rm sing}\| \|y\mp iy_{\rm sing}\|$. Then the diffusive width $\sqrt{Dd t_{\rm exit}}$ of $\Psi$ in the directions $\phi_a$ for $a>1$ is much less than $\phi_\Lambda$ just if (\[go9\]) is satisfied. Principally the diffusive dynamics at $d>2$ could modify the noted fine pole structure of ${\cal Z}$, the problem needs a separate investigation. The same procedure can be done for the passive scalar differences. The cloud $\Psi$ should pass the region $\rho_1<\ln(L/r_0)$ before it reaches the potential. Then it enters the region $\bar U =2P_2$ with some finite diffusive width. One can note, however, that it is irrelevant. The only characteristics of potential which are needed are its value (here $2P_2$ instead of $P_2$) and the length of the path inside it (which is $\Delta\rho_1 = \ln(r_0\Lambda)$ instead of $\ln(L\Lambda)$).The evolution of $\bar\Psi$ goes in the same way as in the case of single-point statistics. Again, we get the answer (\[go4\]) and the criterion analogous to (\[go9\]). In the subsection we presented the analysis based on the dynamical equation (\[gyy9\]) for the auxiliary object $\Psi$. The results obtained can be reproduced also on an alternative language: For this we should introduce another auxiliary object the equation for which is stationary. The corresponding scheme which could be interesting from the methodical point of view is sketched in Appendix. Calculation of PDF {#sec:pdf} ================== In this section we calculate the PDF’s ${\cal P}$ for the objects (\[theta\]) and (\[diff\]). The most convenient way to do it is in using the relation $${\cal P}(\vartheta)=\int\frac{{\rm d}y}{2\pi} \exp(-{\rm i}y\vartheta) {\cal Z}(y) \,, \label{gy16}$$ where $\vartheta$ is $$\vartheta=\int {\rm d}{\bbox r}\, \beta({\bbox r}) \theta(0,{\bbox r}) \,. \label{ga4}$$ Let us remind that knowing ${\cal P}(\vartheta)$ one can restore also moments of $\vartheta$: $$\begin{aligned} && \langle\|\vartheta\|^n\rangle =\int{\rm d}\vartheta\,\|\vartheta\|^n{\cal P}(\vartheta) \,. \label{mom} \end{aligned}$$ The generating functional in (\[gy16\]) is determined by (\[gy15\]) or (\[go4\]). Being interested in the main exponential dependence of the PDF’s for the objects (\[theta\]) and (\[diff\]) we can forget about preexponents. Then $$\begin{aligned} && {\cal P}(\vartheta)= \int \frac{{\rm d}y}{2\pi} {\rm exp} \left( - {\rm i}y\vartheta + q \left[ 1 - \sqrt{1+ y^2/y_{\rm sing}^2 } \right] \right) \,, \label{ygy16} \end{aligned}$$ where $$\begin{aligned} && y_{\rm sing}^2 = \frac{Dd^2(d^2-1)}{12 P_2}\ \ ({\rm for\ scalar}), \quad y_{\rm sing}^2 = \frac{Dd^2(d^2-1)}{24 P_2}\ \ ({\rm for\ difference}), \\ && q=\frac{d(d+1)}6 \, \ln(L\Lambda)\ \ ({\rm for\ scalar}), \quad q=\frac{d(d+1)}6 \, \ln(r_0\Lambda)\ \ ({\rm for\ difference}). \label{qu}\end{aligned}$$ Since both $q$ defined by (\[qu\]) are regarded to be much larger than unity the integral (\[ygy16\]) can be calculated in the saddle-point approximation. The saddle-point value is $$\begin{aligned} && y_{\rm sp} = {\rm i}\frac{y_{\rm sing}} {1+q^2/y_{\rm sing}^2 \vartheta^2} \,. \label{sadd} \end{aligned}$$ Then $${\rm ln\,} {\cal P}(\vartheta) \simeq q \left( 1 - \sqrt{1+\frac{y_{\rm sing}^2 \vartheta^2}{q^2}} \right) \,. \label{lnpdf}$$ The expression leads to the exponential tail $$\begin{aligned} && {\rm ln\,} {\cal P}(\vartheta) \simeq -y_{\rm sing}\|\vartheta\| \,, \label{tail} \end{aligned}$$ realized at $\|\vartheta\|\gg q/y_{\rm sing}$. The coefficient $y_{\rm sing}$ in (\[tail\]) determined by (\[sing\]) is in accordance with the result obtained in [@98BGK]. The expression (\[lnpdf\]) enables one to find the following averages in accordance with (\[mom\]) $$\begin{aligned} && \langle\theta_\Lambda^2\rangle= \frac{2P_2}{d(d-1)D}\ln(L\Lambda) \,, \quad \langle(\Delta\theta_\Lambda)^2\rangle= \frac{4P_2}{d(d-1)D}\ln(r_0\Lambda) \,. \label{pair} \end{aligned}$$ The expressions (\[pair\]) can be obtained also by direct expansion of ${\cal Z}(y)$ from (\[gy15\]) or (\[go4\]). The universal tail (\[tail\]) is realized if $$\begin{aligned} && \theta_\Lambda\gg \sqrt{\langle\theta_\Lambda^2\rangle}\, d\ln(L\Lambda) \,, \quad \Delta\theta_\Lambda\gg \sqrt{\langle(\Delta\theta_\Lambda)^2\rangle}\, d\ln(r_0\Lambda) \,. \label{crit} \end{aligned}$$ Since both logarithms are supposed to be large we conclude that there exists a relatively wide region where the statistics of $\vartheta$ is approximately Gaussian, the region is determined by the inequalities inverse to (\[crit\]). Let us discuss the applicability conditions of the expression (\[lnpdf\]). First, if one calculates the passive scalar PDF by the saddle point method then the position of the saddle point is determined by (\[go9\]) if $$\vartheta \ll d^2 \sqrt{\frac{P_2}D} \, {\ln}^2 (L\Lambda) \,. \label{remo}$$ The applicability domain of the saddle-point method overlaps the region of validity of expression (\[gy15\]) for the generation function ${\cal Z}(y)$. The above inequalities are correct for $\theta_\Lambda$, for $\Delta\theta_\Lambda$ one should substitute $\ln(L\Lambda)$ by $\ln(r_0\Lambda)$. Second, fluctuations of $y$ have to be small compared to the distance between $y_{\rm sp}$ and $y_{\rm sing}$. This gives the same criterion (\[remo\]). Let us stress that though formally our procedure is incorrect at $\vartheta \gtrsim d^2\sqrt{P_2/D} \,{\ln}^2 (L\Lambda)$ the answer will be the same: the PDF will be determined by the exponential tail (\[tail\]). The point is that the character of the integral (\[gy16\]) at such extremely large $\vartheta$ will be determined by the position of the singular point of ${\cal Z}(y)$ nearest to the real axis. This is just ${\rm i}y_{\rm sing}$ leading to (\[tail\]). To conclude, only the character of the preexponent in ${\cal P}(\vartheta)$ is changed at $\vartheta \sim d^2 \sqrt{P_2/D} \,{\ln}^2 (L\Lambda)$ whereas the principal exponential behavior of ${\cal P}(\vartheta)$ remains unchanged there. Conclusion {#conclusion .unnumbered} ========== The single-point statistics of the passive scalar $\theta$ and the statistics of its difference $\Delta\theta$ are traditional objects which carry an essential information about correlation functions of the passive scalar in the convective interval. We examined the passive scalar in the large-scale turbulent flow where the correlation functions logarithmically depend on scale. Since really the logarithms are not very large it is useful to have the whole PDF’s of $\theta$ and $\Delta\theta$. That was the main purpose of our investigation which was performed in the frame of the Kraichnan model. The single-point PDF for the passive scalar and the PDF for the passive scalar differences can be obtained from (\[lnpdf\]) if to substitute $\Lambda\to r_{\rm dif}^{-1}$ where $r_{\rm dif}$ is the diffusive length. Though both the advecting velocity and the pumping force in the Kraichnan model are considered as $\delta$-correlated in time we hope that our answer are universal that is are true in the limit when the size of the convective interval tends to infinity for arbitrary temporal behavior of the velocity and of the pumping. The reason is that the spectral transfer time grows with increasing the convective interval and in the limit is much larger than the correlation times of the velocity and of the pumping. We believe also that the analytical scheme proposed in our work could be extended for other problems related to the passive scalar statistics. Note as an example the work [@98GK] where a modification of the scheme enabled to find the statistics of the passive scalar dissipation. It is also useful for investigating the large-scale statistics (on scales larger that the pumping length) of the passive scalar [@98BFL]. We hope also that it is possible to go beyond the case of the large-scale velocity field using the perturbation technique of the type proposed in [@95SS; @96PSS; @97BFL]. We are grateful to E. Balkovsky, M. Chertkov, G. Falkovich, K. Gawedzki and M. Olshanetsky for useful discussions. This work was supported in part by Einstein and Minerva Centers at the Weizmann Institute, by the grants of Minerva Foundation, Germany and Israel Science Foundation, by Russian Foundation for Basic Research (I.K., M.S., gr. 98-02-17814), by Soros Foundation (M.S., gr. a98-674) and by INTAS (M.S., gr. 96-0457) within the program of ICFPM. Here we present an alternative way to obtain the answers (\[gy15\],\[go4\]). We will use an auxiliary quantity $$\Xi(y,\rho_0,\eta_0)=\left.\int {\cal D}\rho{\cal D}\eta{\cal D}m{\cal D}\mu\, \exp\left[\int_{-\infty}^0 \!\! {\rm d}t \, \left( {\rm i}{\cal L}-\frac{y^2}{2}U\right)\right] \right\|_{\rho(0)=\rho_0,\eta(0)=\eta_0} \,, \label{gu1}$$ then $${\cal Z}(y)=\Xi(y,0,0) \,. \label{gu2}$$ Function $\Xi$ can be also defined as $$\Xi(y,\rho_0,\eta_0) = \lim_{t\to\infty}\int\prod {\rm d}\rho_a \, {\rm d}\eta_{ij}\, \Psi(t,y,\rho,\eta) \,, \label{gu2p5}$$ where $\Psi$ is governed by equation (\[gyy9\]) with initial condition $\Psi(t=0,y,\rho,\eta)=\delta(\rho - \rho_0)\delta(\eta - \eta_0).$ The equation for the function $\Xi$ can be found from (\[gy8\],\[gu1\]): $$\begin{aligned} && \left[ \sum\limits_{i=1}^{d}\frac{\partial^2}{\partial\rho_i^2} -\frac{1}d\left(\sum\limits_{i=1}^{d} \frac{\partial}{\partial\rho_i}\right)^2 +\sum\limits_{i=1}^{d}(d-2i+1)\frac{\partial}{\partial\rho_i} \right. \nonumber \\ && + 2\sum\limits_{i<j}\exp(2\rho_j-2\rho_i) \frac{\partial^2}{\partial\eta_{ij}^2} +4\sum\limits_{i<k<j} \exp(2\rho_k-2\rho_i)\eta_{kj} \frac{\partial}{\partial\eta_{ij}} \frac{\partial}{\partial\eta_{ik}} \nonumber \\ && \left. + 2\sum\limits_{i<k<m,n} \exp(2\rho_k-2\rho_i)\eta_{km}\eta_{kn} \frac{\partial}{\partial\eta_{im}} \frac{\partial}{\partial\eta_{in}}\right] \Xi -\frac{y^2 U}{Dd} \Xi =0 \,. \label{gu3}\end{aligned}$$ The boundary condition for the equation (\[gu3\]) follows from the definition (\[gu1\]): For large enough $\rho_i,\,\eta_i$ the potential $U=0$ for $t=0$ and remains zero also at finite times $t$. Therefore the integral (\[gu1\]) should be equal to unity in the case. Thus $\Xi(y,\rho,\eta)$ should tend to unity where $\rho,\,\eta\to\infty$. Let us rewrite the equation (\[gu3\]) in terms of the variables (\[phi\]): $$\begin{aligned} && \big( \hat\Gamma_1 + \hat\gamma \big) (\Xi_1 + \xi) = 0 \,, \quad \Xi = \Xi_1 + \xi \,, \label{gu4} \\ && \hat\Gamma_1 = \frac{\partial^2}{\partial\phi_1^2} + \sqrt{\frac{d(d^2-1)}3} \frac{\partial}{\partial\phi_1} - \frac{y^2 U}{Dd} \,, \\ && \hat\gamma=\sum_{i=2}^{d-1} \frac{\partial^2}{\partial\phi_i^2} + 2\sum\limits_{i<k}\exp(2\rho_k-2\rho_i) \frac{\partial^2}{\partial\eta_{ik}^2} + 4\sum\limits_{i<k<n} \exp(2\rho_k-2\rho_i)\eta_{kn} \frac{\partial}{\partial\eta_{in}} \frac{\partial}{\partial\eta_{ik}} \nonumber \\ && + 2\sum\limits_{i<k<m,n} \exp(2\rho_k-2\rho_i)\eta_{km}\eta_{kn} \frac{\partial}{\partial\eta_{im}} \frac{\partial}{\partial\eta_{in}}\,. \label{gu5}\end{aligned}$$ Here $U$ as a function of $\phi_1$ is equal to $P_2$ inside a region restricted by $\phi_\Lambda^-$ and $\phi_\Lambda^+$ (where $\phi_\Lambda^\pm$ are functions of variables $\phi_2, \dots, \phi_d, \eta$) and tends to zero outside the region. We will solve the equation (\[gu4\]) using the perturbation theory over $\hat\gamma$, $\xi$. Then the zero order equation is $$\begin{aligned} && \hat\Gamma_1 \Xi_1 =0 \,. \label{zero} \end{aligned}$$ The equation (\[zero\]) can be easily solved at $\phi_\Lambda^- < \phi_1 < \phi_\Lambda^+$, the answer is $$\Xi_1 \simeq \frac{2\lambda}{\sqrt{\lambda^2 + \frac{y^2 P_2}{Dd}} + \lambda} {\rm exp} \left( - \left( \sqrt{\lambda^2 + \frac{y^2 P_2}{Dd}} - \lambda \right) (\phi_\Lambda^+ - \phi_1) \right) \,, \label{res}$$ where $\lambda = \sqrt{d(d^2-1)/12}$, $Dd\lambda$ is the rate of the cloud motion along $\phi_1$ direction. The result (\[res\]) can be obtained using the inequality $\sqrt{\lambda^2 + y^2 P_2/Dd} \, {\rm ln\,}L\Lambda \gg 1$. The derivative $\partial \Xi_1 / \partial \phi_1 = 0$ at $\phi_1<\phi_\Lambda^-$. However, $\Xi_1 \ne 1$ in this region. This is due to the following fact: this region corresponds to the evolution of $\Psi$ when its initial position is to the left of potential $U$ (see (\[gu2p5\])). During the evolution cloud $\Psi$ passes the region of $U$ and its integral over $\rho$, $\eta$ changes. Then $\Xi$ is not equal to $1$. Only when the distance between initial position and potential is of order ${\rm ln}^2 L\Lambda$ the diffusion of the cloud leads to smallness of the part of $\Psi$ which passes the potential $U$, and $\Xi$ becomes closer to unity. Thus, function $\Xi$ has long tail from potential with direction to negative $\phi_1$ where it is not equal to $1$. The procedure of finding $\Xi$ from the equation (\[zero\]) corresponds to the geometrical optics approximation (taking into account only derivatives in propagating direction; this allows one to get the fact of propagation). This tail of $\Xi$ in this approximation is nothing else than the shadow of potential $U$. Higher orders of perturbation theory over the transverse derivatives correspond to diffraction corrections. Now let us consider the correction $\xi$. The equation for it looks like $( \hat\Gamma_1 + \hat\gamma) \xi = -\hat\gamma \Xi_1$. Again let us neglect $\hat\gamma$ in l.h.s. and solve the equation. $\Xi_1$ is some exponential function with scale of the order 1. Then $\hat\gamma \Xi_1 \sim \Xi_1$. Not that $\hat\gamma \Xi_1$ is almost equal to zero at $\phi_1 > \phi_\Lambda^+$. To estimate $\xi$ one should construct Green function $G(\phi_1\|\phi_0)$ for operator $\hat\Gamma_1$: $$G(0\|\phi_0) \simeq \frac1{2\lambda} \, {\rm exp} \left( - \left( \sqrt{\lambda^2 + \frac{y^2 P_2}{Dd}} - \lambda \right) \phi_0 \right) \left( 1 - C {\rm exp} \left( -2 \sqrt{\lambda^2 + \frac{y^2 P_2}{Dd}} (\phi_\Lambda^+ - \phi_0) \right) \right) \,, \label{ccc}$$ where $$C =\left(\sqrt{\lambda^2 + y^2 P_2/Dd}-\lambda\right)/ \left(\sqrt{\lambda^2 + y^2 P_2/Dd}+\lambda\right) \,.$$ The unity in brackets in (\[ccc\]) gives the correction for $\Xi$ which has the same exponential factor as $\Xi_1$. Thus $\xi$ does not change the answer with the logarithmic accuracy. The second term in the brackets gains while $\phi_0$ is close to $\phi_\Lambda^+$. It is due to nonzero width of the cloud $\Xi$ and to dependence of $t_{\rm lt}$ on other variables. Again, it does not change the exponent. The case of the passive scalar differences can be considered in a similar way. G. K. Batchelor, J. 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--- abstract: 'A short review of the problems which arise in the generalization of the Lifshitz theory of van der Waals force in the case of forces inside dielectric media is presented, together with some historical remarks. General properties of the stress tensor of equilibrium electromagnetic field in media are discussed, and the importance of the conditions of mechanical equilibrium is stressed. The physical meaning of the repulsive van der Waals interaction between bodies immersed in a liquid is discussed.' author: - 'Lev P. Pitaevskii' title: On the problem of van der Waals forces in dielectric media --- Introduction ============ The quantum theory of the long range van der Waals interaction between neutral objects has a long and instructive history [@footnote1]. The existence of attractive forces between atoms was established by van der Waals on the basis of analysis of experimental data on equations of state of real gases. It was F. London who understood the electric nature of these forces and built in 1930 a seminal quantum-mechanical theory of the forces at large distances, deriving the famous $1/R^6$ law for the energy of interaction [@London30]. The next important step was performed by Casimir and Polder. Using quantum electrodynamics, they derived a more general expression for the energy of the atom-atom interaction and showed that, due to retardation effects, London’s law at large distances is changed to a $1/R^7$ law[@CP48]. In the same paper the authors considered for the first time the problem of including a macroscopic body. They calculated a force between an atom and perfect metal plate. The interaction between perfect metal plates was calculated by Casimir [@Casimir48]. The most general theory of the van der Waals interaction between any condensed bodies was developed by E. Lifshitz in 1954-1955 [@Lifshitz54; @Lifshitz56]. His theory is applicable to a body with arbitrary dielectric properties at any temperature. It also automatically takes into account relativistic retardation effects. To calculate forces one must know the dielectric and magnetic permeabilities of the bodies and solve the Maxwell equations for their given configuration. It was assumed in the Lifshitz theory (as well as in previous theories) that the space between the bodies was a vacuum. A generalization of the theory where the gap between bodies is filled with some medium was a natural next step. However, to make this step one must overcome some essential difficulties. To clarify the nature of these difficulties, let us recall the main points of the Lifshitz approach. This approach is based on averaging of the Maxwell stress tensor in vacuum, $$\sigma_{ik}^{(vac)}= \frac{ %\left\langle E_{i}E_{k}+B_{i}B_{k}\; , %\right\rangle }{4\pi }- \frac{ %\left\langle E^{2}+B^{2} %\right\rangle }{8\pi }\delta _{ik}. \label{vac}$$ with respect to electromagnetic fluctuations in thermodynamic equilibrium. Because the theory of equilibrium electromagnetic fluctuations in arbitrary media was already developed by Rytov [@Rytov] and Landau and Lifshitz [@LL8], the tensor could be averaged for equilibrium conditions and the forces calculated. The Rytov theory has a semi-phenomenological character. Rytov considered the fluctuations as created by the Langevin-like sources, namely fluctuating electric and magnetic polarizations. It was assumed that these polarizations at two points $\mathbf{r_1}$ and $\mathbf{r_2}$ of a medium are correlated only when the two points coincide, i. e. their correlation functions are proportional to $\delta(\mathbf{r_1}-\mathbf{r_2})$. The coefficients of proportionality were chosen to obtain the correct density of black-body radiation from the bodies. During preparation of the book [@LL8], Landau and Lifshitz derived equations of the Rytov theory using the exact fluctuation-dissipation theorem, established by Callen and Welton in 1951. It was natural to think that the generalization for the case of bodies separated by a medium could be obtained if one could find a general expression for the electromagnetic stress tensor for arbitrary time-dependent fields in a medium. Because the Rytov theory describes electromagnetic fluctuations also inside a medium, it would then be possible to perform the average of the tensor. Thus the first step was to calculate the tensor. Note that this problem was formulated in the book [@LL8], and Landau believed that it could be solved. One can read in the end of §61: “Considerable interest attaches to the determination of the (time) average stress tensor giving the forces on matter in a variable electromagnetic field. The problem is meaningful for both absorbing and non absorbing media, whereas that concerning the internal energy can be proposed only if absorption is neglected. The corresponding formulae, however, have not yet been derived.” I had an opportunity to read proofs of the book when I joined the Landau department in the Institute for Physical Problems in Moscow as a Ph.D. student in 1955. After reading this paragraph I decided to derive “the corresponding formulae” for the tensor. It was, of course, a quite ambitious goal. After approximately three months of work I met Igor Dzyaloshinskii. Because the authors asked both him and me to help in the proof-reading of Ref. [@LL8], we naturally discussed the topics of the book and I discovered that Dzyaloshinskii had also been working on the tensor problem even longer than I. We decided to join our efforts. Our attempts were based on the use of the second-order quantum mechanical perturbation theory to calculate the quadratic contribution of the fields into the tensor. One can obtain formal equations; however, for a medium with dissipation we could not express them in terms of dielectric and magnetic permeabilities. Oddly enough, one could easily obtain an equation for the total force acting on a body in vacuum. However, trying to obtain from this equation the force density inside the body, one necessary violated the condition of the symmetry of the stress tensor, or other conditions implied for the tensor. I also tried to develop a thermodynamic approach. However, the entropy increase due to the dissipation makes this approach meaningless. We worked long enough and finally decided that the problem was hopeless. (In any case it was my opinion.) However, after a period of disappointment, I suddenly recognized that the problem of the van der Waals forces for the case of a liquid film can be solved without involving the general tensor problem. Because, as we will see below, these considerations actually were not employed and never were published, I would like to present them here. Let us consider a liquid film of thickness $d$ in vacuum. Its free energy per unit of area can be presented in the form $${\cal F}(T,d)=\varphi_0(T)d+{\cal F}^{(elm)}(T,d) \label{F}$$ where $\varphi_0(T)$ is the density of free energy for a bulk body and ${\cal F}^{(elm)}(T,d)$ is the contribution of the van der Waals forces, which can be normalized in a such way that ${\cal F}^{(elm)} \to 0$ at $d \to \infty$. The goal of the theory in this case is to calculate ${\cal F}^{(elm)}(T,d)$. However, this quantity can be calculated on the basis of the Lifshitz theory. Let us consider our film at the distance $l$ from a half-space of the same liquid (see Fig. 1). ![ Scheme for calculation of the chemical potential of a film in terms of the stress tensor in vacuum. []{data-label="fig:film"}](Fig1-film.pdf){width="7"} The theory permits the calculation of the force $F(d,l)$ between the film and the half-space and the change of the free energy when the film moves from the surface of the bulk liquid to infinity: $$\Delta {\cal F}(T,d)=\int_0^{\infty}Fdl \label{DF}\; .$$ This quantity is just ${\cal F}^{(elm)}(T,d)$. Indeed, if the film is near the surface of the half space, the configuration corresponds to a bulk body. Actually, the integral (\[DF\]) is divergent at small $l$. However, this infinite contribution does not depend on $d$ and can be omitted altogether with a $d$-independent constant. We discussed the idea with Landau and he agreed with the argumentation. We immediately began calculations of the force $F$ for the three-boundary configuration of Fig. 1. E. M. Lifshitz, who was interested in the film problem very much, joined us. However, Landau cooled our ardor. He said that it is meaningless to solve a three boundary problem to find an answer for a two-boundary one and that the integration (\[DF\]) must be performed in a general form, “in some symbolic way” as he said. As a result, Dzyaloshinskii and I began to look for a different approach, being sure this time that the answer exists. It was a lucky coincidence that just in this period Dzyaloshinskii, in collaboration with Abrikosov and Gorkov, worked on developing the Matsubara diagram technique for the solution of equilibrium problems in the quantum many-body theory. We decided to try this approach. The first attempt was very successful. We immediately recognized which diagrams are important. As an intermediate result we obtained an equation where the van der Waals contribution to the free energy was expressed in terms of an integration with respect to the charge of the electron. This equation was correct, but not very useful because the dielectric permeability can, of course, be measured only for the actual value of the charge. Finally, however, we discovered that in the Matsubara technique the variation of the free energy with respect to the dielectric permeability can be expressed in terms of the Matsubara Green’s function of imaginary frequency. This permitted us to calculate the tensor explicitly [@DP]. The reason why our previous attempts were in vain became clear now. One can obtain the tensor of the equilibrium electromagnetic field in a medium, that is, the tensor of van der Waals forces, but not the tensor of arbitrary electromagnetic fields. The possibility to derive the tensor of the equilibrium fluctuation electromagnetic field in an absorbing medium does not mean, of course, the possibility to determine the tensor for an arbitrary variable field. Even if this quantity has physical meaning, according to the Landau conjecture quoted above, it does not mean that it can be expressed in terms of the electric and magnetic permeabilities $\varepsilon(\omega)$ and $\mu(\omega)$. I believe that this is impossible. In any case, the tensor, obtained by direct calculation for a plasma, where the problem can be explicitly solved using the Boltzmann kinetic equation, cannot be expressed in such a way [@Perel68]. The difficulty, of course, is the energy dissipation. In a transparent medium, where dissipation is absent, the tensor can be obtained for arbitrary non-equilibrium fields[@Pitaevskii61]. Before discussing concrete results, let us discuss general properties of the stress tensor in thermal equilibrium. It can be presented in the form $$\sigma_{ik}=-P_{0}(T,\rho)\delta _{ik}+ \sigma_{ik}^{(elm)} \; , \label{gen}$$ where $P_{0}(T,\rho)$ can be defined as the pressure of a uniform infinite liquid at given density $\rho$ and temperature $T$ and $\sigma_{ik}^{(elm)}$ is the contribution from the electromagnetic fluctuations, i.e., the van der Waals interaction. This contribution must satisfy several important conditions: i\) The tensor must be symmetric: $\sigma_{ik}^{(elm)}=\sigma_{ki}^{(elm)}$. ii\) The van der Waals part of the force, acting on the liquid, must be derivable from a potential: $$F^{(elm)}_{i}=\partial_{k} \sigma_{ik}^{(elm)} =-\rho \partial_i \zeta^{(elm)} \; . \label{Fgen}$$ Actually $\zeta^{(elm)} $ is just the contribution of the van der Waals interaction to the chemical potential of the fluid. The first condition is a direct consequence of the symmetry of the microscopic energy-momentum tensor. The tensor $(-\sigma_{ik})$ is its averaged spatial part. Condition ii) ensures the possibility of mechanical equilibrium of the fluid in the presence of the van der Waals interaction. Indeed, the condition for such an equilibrium is $F_{i}=\partial_{k} \sigma_{ik}=0$. Taking into account that $dP_0=\rho d\zeta_0(\rho,t)$, we can rewrite this equation as $$\partial_{i} \zeta_0- F^{(elm)}_{i}/\rho=0 \;. \label{eq0}$$ which implies (\[gen\]) for arbitrary configurations of interactive bodies. Thus violation of the condition ii) would result in permanent flow of the liquid in equilibrium and actually would permit us to build the notorious Perpetual Motion machine. For an analogous reason, on the boundary between a fluid and a solid the tangential components of the tensor must be continuous. This condition is satisfied automatically by virtue of the boundary conditions for the fluctuating fields. If the normal to the surface is directed along $z$, it must be the case that $\sigma_{\alpha z}^{elm(1)}=\sigma_{\alpha z}^{elm(2)}, \alpha=x,y.$ Violation of this condition would result in the existence of permanent flow of the liquid near a solid boundary. Equation (\[eq0\]) can be written as a condition of constancy of the total chemical potential $\zeta$: $$\zeta(\rho,T)=\zeta_0(\rho,t)+ \zeta^{(elm)}(\rho,T)=const \;. \label{equzeta}$$ Notice that, neglecting the change in density of the liquid under the influence of the van der Waals forces, one can express this condition also as $$P_{0}(\rho,T)/\rho+ \zeta^{(elm)}=const \;. \label{equ1}$$ For calculations of the forces in the state of mechanical equilibrium one can omit this part of the tensor and exclude the pressure, i. e. use instead of (\[gen\]) the tensor $${\sigma}\prime_{ik}= \sigma_{ik}^{(elm)}+ \rho \zeta^{(elm)}(T,\rho)\delta_{ik} \; . \label{equ}$$ Notice that the tensor ${\sigma}\prime_{ik}$ by definition satisfies the equation $$\partial_{k}{\sigma}\prime_{ik}=0 \; . \label{ds0}$$ This obvious property results sometimes in misunderstandings (Ref. [@Comment]). Free energy of the equilibrium electromagnetic field in an absorbing medium =========================================================================== I present now a simplified version of our deviation of the force tensor in a liquid obtained for the first time in [@DP]. It is sufficient to take into account only the electromagnetic interaction in the system. Nuclear forces are obviously irrelevant to our problem. Then the correction due to interaction to the free energy can be presented in the Matsubara technique as a set of “ring” diagrams (see Fig. 2), where the dashed lines represent the Matsubara Green’s functions of the electromagnetic field ${\cal D}_0$ without interaction[@AGD]. Every “bead” is the polarization operator $\Pi$, which includes all diagrams which cannot be separated into parts, connected by one dashed line. It is important that the diagrams of Fig. 2 cannot be summed up into the exact Green’s function, because of the extra factor $1/n$ in each term, where $n$ is the number of the ”beads”. ![ Diagrammatic representation of corrections to the free energy. []{data-label="fig:diag"}](Fig2-film.pdf){width="10"} Let us calculate now the variation of the free energy with respect to a small change $\delta \Pi$ of the density of the liquid. The crucial point is that the factor $1/n$ will be canceled as a result of the variation and the parts of the diagrams with the non-variated beads will be summed up to the exact Green’s function. The result is $$\begin{aligned} \delta {\cal F} = \delta {\cal F}_0 - {\sum_{s=0}^{\infty}}~^\prime \int{\cal D}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s})\delta \Pi_{ik}(\mathbf{r},\mathbf{r}';\xi_{s})d\mathbf{r}d\mathbf{r}'\; , \label{deltaF}\end{aligned}$$ where $\xi_s=2s\pi T$ and the term with $s=0$ is taken with a factor (1/2)[@footnote]. This is an exact equation of quantum electrodynamics. It is valid also in a vacuum, where $\Pi$ describes the radiation correction to ${\cal D}$. However, it is practically worthless, because explicit expressions for ${\cal D}$ and $\Pi$ cannot be obtained. It also contains ultraviolet divergences. However, these divergences are due to contributions from the short wave-length fluctuations, whereas we are interested in effects due to the inhomogeneity of the medium. i. e., the presence of boundaries of bodies etc. This permits us to produce a renormalization of this equation. To do this, let us write the ${\cal D}$-function as $$\begin{aligned} {\cal D}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s}) = \left [ {\cal D}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s}) - \bar{{\cal D}}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s}) \right ]+\bar{{\cal D}}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s}) \; , \label{ren}\end{aligned}$$ where $\bar{{\cal D}}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s})$ is the Green’s function of an auxiliary homogeneous infinite medium whose permeabilities are the same as that of the actual medium at the point $\mathbf{r}'$. After substitution into (\[deltaF\]) the third term can be absorbed in the term $\delta {\cal F}_{0}$. This term acquires the meaning of the variation of the free energy of this uniform medium. Then one gets instead (\[deltaF\]): $$\begin{aligned} \delta {\cal F} = \delta {\cal F}_0 - {\sum_{s=0}^{\infty}}~^{\prime} \int\left [ {\cal D}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s}) - \bar{{\cal D}}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s}) \right ]\delta \Pi_{ik}(\mathbf{r},\mathbf{r}';\xi_{s})d\mathbf{r}d\mathbf{r}'\; . \label{dF1}\end{aligned}$$ In equation (\[dF1\]) the important fluctuations are those whose wavelengths are of the same order of magnitude as the inhomogeneities of the system (e.g., the thickness of films and separations of bodies). These lengths are assumed to be large compared to interatomic dimensions. However, these long-wavelength fluctuations can be described by macroscopic electrodynamics. According to the general theory, the Matsubara Green’s function ${\cal D}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s})=D_{ik}^{R}(\mathbf{r},\mathbf{r}';i\xi_{s})$, where $D^{R}$ is the “usual” retarded Green’s function for the vector-potential of the electromagnetic field. Accordingly, ${\cal D}$ satisfies in the macroscopic limit the explicit equation $$\begin{aligned} \left [\partial_i\partial_l -\delta_{il}\Delta+ (\xi_{s}^2/c_2)\varepsilon(i\|\xi_s\|,\bf{r})\delta_{il}\right ]{\cal D}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s})=-4\pi\hbar\delta_{ik}\delta(\bf{r}-\bf{r}')\; . \label{D}\end{aligned}$$ The equation for $\bar{\cal D}$ can be obtained by changing in (\[D\]) the permeability $\varepsilon(i\|\xi_{s}\|,\bf{r})$ to $\varepsilon(i\|\xi_{s}\|,\bf{r}')$. However, in the majority of practical problems the excluding of the divergences can be achieved simply by omitting terms which do not depend of the spatial parameters, e.g., on the distances between bodies. Taking this into account, and to avoid complications of the equations, we will denote in the future the difference (\[ren\]) as ${\cal D}$. It is worth noting that the left-hand side of (\[D\]) coincides with the Maxwell equations for the vector potential $\bf{A}$ of the electromagnetic field with the frequency $i\xi_{s}$ in the gauge where the scalar potential $\phi=0$. The final results of the theory do not depend, of course, on the gauge. The Green’s function ${\cal D}_0$ satisfies the same equation (\[D\]) with $\varepsilon =1$. Let us write ([\[D\]]{}) symbolically as $\hat {\cal D}^{-1}{\cal D} =\delta_{il}\delta(\bf{r}-\bf{r}')$ and the equation for ${\cal D}_0$ as $\hat {\cal D}_0^{-1}{\cal D}_0 =\delta_{il}\delta(\bf{r}-\bf{r}')$. Then from the definition $\Pi_{ik}=\left [ {\cal D}^{-1}_{0ik} -{\cal D}^{-1}_{ik} \right ]$ we find an equation for $\Pi$: $$\begin{aligned} \label{Pi} \Pi_{kl}(\xi_{s};\bf{r}_1,\bf{r}_2)=\frac{\xi_{s}^2}{4\pi}\delta_{kl}\delta(\bf{r}-\bf{r}')\left [\varepsilon(i\xi_{s};\bf{r}_1)-1 \right ] \;.\end{aligned}$$ This permits us to express the variation of the free energy in terms of $\varepsilon$: $$\begin{gathered} \delta {\cal F}= \delta {\cal F}_0 - \frac{T}{4\pi} {\sum_{s=0}^{\infty}}~^{\prime} \int\left [\xi^2_s {\cal D}_{ll}(\mathbf{r},\mathbf{r};\xi_{s}) %- \bar{{\cal D}}_{ll}(\mathbf{r},\mathbf{r};\xi_{s}) \right ]\delta \varepsilon(i\xi_{s};\mathbf{r})d\mathbf{r}\; . \label{dFfin}\end{gathered}$$ It is convenient for further equations to introduce a new function $$\begin{gathered} {\cal D}^E_{ik}(\mathbf{r},\mathbf{r}';\xi_{s}) =-\xi^2_s {\cal D}_{ik}(\mathbf{r},\mathbf{r}';\xi_{s}) \; . \label{DE}\end{gathered}$$ Function ${\cal D}$ describes the quadratic fluctuations of the vector potential in the Matsubara technique. Accordingly, the function ${\cal D}^E$ describes fluctuations of the electric field. We will also use the function $$\begin{gathered} {\cal D}^H_{ik}(\mathbf{r},\mathbf{r}';\xi_{s}) ={\rm curl}_{il}{\rm curl}'_{km} {\cal D}_{lm}(\mathbf{r},\mathbf{r}';\xi_{s}) \label{DE}\end{gathered}$$ describing fluctuations of the magnetic field. Now we can rewrite (\[DE\]) as $$\begin{gathered} \delta {\cal F} = \delta {\cal F}_0 + \frac{T}{4\pi} {\sum_{s=0}^{\infty}}~^{\prime} \int{\cal D}^{E}_{ll}(\mathbf{r},\mathbf{r};\xi_{s}) \delta \varepsilon(i\xi_{s};\mathbf{r})d\mathbf{r}\; . \label{dFfin1}\end{gathered}$$ Stress tensor of the van der Waals interaction inside an absorbing medium. ========================================================================== We can use equation (\[dFfin1\]) to calculate the tensor of van der Waals forces in a fluid. It is instructive, however, as a first step to compare the equation for a free energy variation for given sources of field in a dielectric in the absence of dispersion (Ref. [@LL8], equation (15.19)): $$\begin{gathered} \delta {\cal F} = \delta {\cal F}_0 - \int\frac{E^2}{8\pi}\delta\varepsilon d\mathbf{r} \label{dFstat}\end{gathered}$$ This equation permits the calculation of the force $\mathbf{f}$ and finally to find the stress tensor (see Ref. [@LL8], (15.9) and (35.2), in the presence of both electric and magnetic fields we must take the sum of these equations): $$\begin{gathered} \sigma_{ik}^{A}=-P_{0}\delta _{ik}+ \frac{\varepsilon E_{i}E_{k}+H_{i}H_{k}}{4\pi }% \nonumber \\ -\frac{E^{2}}{8\pi }\left[ \varepsilon -\rho \left( \frac{\partial \varepsilon }{\partial \rho }\right) _{T}\right] \delta _{ik}-\frac{H^{2}}{% 8\pi }\delta _{ik}. \label{Abr}\end{gathered}$$ This equation was derived by M. Abraham around 1909 and is one of the most important results of the electrodynamics of continuous media. Now we can write the tensor of the van der Waals forces by direct analogy with (\[Abr\]). Indeed, the functions ${\cal D}^E_{ik}(\mathbf{r},\mathbf{r}';\xi_{s})$ and ${\cal D}^H_{ik}(\mathbf{r},\mathbf{r}';\xi_{s})$ satisfy equations which are similar to the products $E_{i}(\mathbf{r})E_k(\mathbf{r}')$ and $H_{i}(\mathbf{r})H_k(\mathbf{r}')$. The presence of the $\delta$-function term on the right-hand side of (\[D\]) is not important because this term in any case will be eliminated in the course of the renormalization. Thus, the general expression for the stress tensor for a fluid with $\mu =1$ is [@DP]: $$\begin{gathered} \sigma_{ik}=-P_{0}\delta _{ik}-\frac{\hbar T}{2\pi }\Big \{ \sum_{s=0}^{\infty}~^{\prime}\Big( \varepsilon D_{ik}^{E}+D_{ik}^{H}\nonumber \\ -\frac{1}{2}D_{ii}^{E}\Big [ \varepsilon -\rho \Big ( \frac{\partial \varepsilon }{\partial \rho }\Big ) _{T}\Big ] \delta _{ik}-\frac{1}{2}D_{ii}^{H}\delta _{ik}\Big ) \Big \} , \label{DP}\end{gathered}$$ where $D_{ik}^{E}$ and $D_{ik}^{H}$ were defined above, $\varepsilon =\varepsilon \left( \rho ,T,i\zeta _{s}\right) ,\zeta _{s}=2sT/\hbar $, and $P_{0}\left( \rho ,T\right) $ is the pressure as a function of density and temperature in the absence of an electric field. Equation $\left( \ref{DP}\right) $ assumes the system to be in thermal, but still not in [mechanical ]{} equilibrium. As was said before, the last condition can be formulated as a condition of the constancy of the chemical potential $\zeta$, which can be defined by the equation $\delta F=\int \zeta \delta \rho d\mathbf{r}$. The variation must be taken at fixed boundaries of the bodies. One has from (\[dFfin1\]) $$\begin{gathered} \zeta(\rho, T) = \zeta_0(\rho,T) + \frac{\hbar T}{4\pi} {\sum_{s=0}^{\infty}}~^{\prime} {\cal D}^{E}_{ll}(\mathbf{r},\mathbf{r};\xi_{s}) \frac{\partial \varepsilon(i\xi_{s};\mathbf{r})}{\partial \rho}\; . \label{zeta}\end{gathered}$$ The condition of mechanical equilibrium means that $\zeta(\rho, T)=const$. Let the fluid have uniform density in the absence of the van der Waals forces. Taking into account that $dP_0=\rho d\zeta_0$ and neglecting in the second term any change of $\rho$ due the van der Waals interaction, we can rewrite the condition of equilibrium as: $$P_{0}+\frac{\hbar T}{4\pi }\sum\limits_{s}~^{\prime}D_{ii}^{E}\rho \left( \frac{% \partial \varepsilon }{\partial \rho }\right) _{T}=const \; . \label{mech}$$ This equation can be used to calculate the perturbation $\delta \rho$ of the density of the liquid due to the van der Waals forces. Expanding the first term with respect to $\delta \rho$, we easily find $$\delta \rho=-\frac{\hbar T}{4\pi }\sum \limits_{s}~^{\prime}D_{ii}^{E}\rho \left( \frac{% \partial \varepsilon }{\partial P }\right) _{T} \; . \label{drho}$$ Equation (\[mech\]) implies that a part of the stress tensor $\left( \ref{DP}% \right) $ is constant through the fluid, being a uniform compressing or expanding pressure. This part can be omitted in many problems, for example in the calculation of the full force acting on a body embedded in the fluid. Subtracting the constant tensor $\left[ -P_{0}-\frac{\hbar T}{4\pi }% \sum\limits_{n}D_{ii}^{E}\rho \left( \frac{\partial \varepsilon }{\partial \rho }\right) _{T}\right] \delta _{ik}$ from $\left( \ref{DP}\right) $, one arrives to the “contracted” tensor, which was obtained for the first time in [@DLP] (see also Ref. [@DLP2]): $$\sigma_{ik}^{\prime}=-\frac{\hbar T}{2\pi }\left\{ \sum_{s}~^{\prime}\left( \varepsilon {\cal D}_{ik}^{E}+{\cal D}_{ik}^{H}-\frac{1}{2}\varepsilon {\cal D}_{ii}^{E}\delta _{ik}-\frac{1}{% 2}{\cal D}_{ii}^{E}\delta _{ik}\right) \right\} \; . \label{elmM}$$ I would like to stress that the “$P_0$” term in the tensor (\[DP\]) plays an important role. Ignoring this term would lead to wrong results. In this connection it is appropriate to quote Landau and Lifshitz’s remark (see Ref. [@LL8], §15): “The problem of calculating the forces (called pondermotive forces) which act on a dielectric in an arbitrary non-uniform electric field is fairly complicated...” Notice that $\partial \sigma_{ik}^{\prime}/\partial x_{k}=0$ and hence $\oint \sigma_{ik}'dS_{k}=0$ for integration over any closed surface, surrounding a volume of a uniform fluid, just due to the fact that in mechanical equilibrium electromagnetic forces are compensated by a pressure gradient. Analogous integration over any surface surrounding a solid body gives the total force acting on the body. Equivalent theories of the force between bodies separated by a liquid were developed by Barash and Ginzburg [@BG] and Schwinger, DeRead and Milton [@SDM]. The method of [@BG] is based on a very interesting and new physical idea. I cannot discuss it here. Notice only, that the method permits us to calculate forces on the basis of the solution of the imaginary frequencies “dispersion relation” ${\cal D}^{-1}(i\xi_s)=0$, without an actual calculation of ${\cal D}$. This results in further simplification of calculations. The authors of [@SDM] performed the free energy variation assuming actually the condition of the mechanical equilibrium from the very beginning and obtained directly equation (\[elmM\]). Van der Waals forces between bodies separated by a liquid ========================================================= Now we can calculate the force acting on bodies separated by a dielectric liquid. It is worth noticing, however, that even for bodies in vacuum the method, based on using the imaginary-frequencies Green’s functions, involves simpler calculations than the original Lifshitz method, because the solution of the equation for the Green’s functions is simpler than the procedure of averaging of the stress tensor. It was shown in [@DLP] that if the problem has been solved for bodies in a vacuum, the answer for bodies in liquid can be found by a simple scaling transformation. Let us denote the dielectric permeability of the liquid as $\varepsilon$. If we perform a coordinate transformation $\mathbf{r}=\tilde{\mathbf{r}}/{\varepsilon}^{1/2}$ and introduce the new functions ${\cal D}_{ik}=\tilde{{\cal D}}_{ik}/\varepsilon^{1/2}$ and ${\cal D}^E_{ik}=\tilde{{\cal D}}^E_{ik}\varepsilon^{1/2}$, ${\cal D}^H_{ik}=\tilde{{\cal D}}^H_{ik}\varepsilon^{3/2}$, then $$\sigma_{ik}^{\prime}=-\frac{\hbar T}{2\pi }\left\{ \sum_{s=0}^{\infty}~^{\prime}{\varepsilon}^{3/2}\left( \tilde{{\cal D}}_{ik}^{E}+\tilde{\cal D}_{ik}^{H}-\frac{1}{2} \tilde{\cal D}_{ii}^{E}\delta _{ik}-\frac{1}{% 2}\tilde{\cal D}_{ii}^{E}\delta _{ik}\right) \right\} \; . \label{stilde}$$ One can see easily that the new functions $\tilde{{\cal D}}_{ik}$ satisfy in the new coordinates $\tilde{\mathbf{r}}$ equations of the same form (\[D\]) for bodies in vacuum, while the permeabilities $\varepsilon_{\alpha}$ of the bodies were changed to $\varepsilon_{\alpha}/\varepsilon$. One can usually neglect the influence of the temperature on the forces between bodies in a liquid. Then one can change $T{\sum}_{s=0}^{\infty}~^{\prime}... \to \frac{\hbar}{2\pi}\int_{0}^{\infty}....d\xi_s$. We will consider below only this case. We also will consider only the small “London” distances where the characteristic distance between bodies $l \ll \lambda$, where $\lambda $ is the characteristic wavelength of the absorption spectra of the media. In this case one can neglect the magnetic Green’s function ${\cal D}_{ik}^{H}$ and the electric function can be presented as ${\cal D}_{ik}^{E}=\hbar\partial_i\partial_k\prime\phi$, where the “electrostatic” Green’s function $\phi$ satisfies equation [@Volokitin; @Pitaevskii08] $$\partial_i[\varepsilon(i\xi;\mathbf{r})\partial_i\phi(\xi;\mathbf{r},\mathbf{r}')] =-4\pi\delta(\mathbf{r}-\mathbf{r}') \; . \label{phi}$$ Thus $\phi$ is just the potential of a unit charge placed at point $\mathbf{r}'$. We will present here results for two important problems. Interaction of a small sphere with a plane body ----------------------------------------------- As a first example we consider a dielectric sphere in the vicinity of a plane surface of a bulk body. Let the radius $R$ of the sphere be small in comparison with the distance $l$ between the sphere and the surface. We consider first the problem in vacuum. Then the energy of interaction can be obtained directly from (\[dFfin1\]), taking into account that the change of the dielectric permeability due to the presence of the sphere at point $\mathbf{r}_0$ is $\delta \varepsilon (\omega)= 4\pi\alpha (\omega) \delta(\mathbf{r}-\mathbf{r}_0)$, where $\alpha (\omega)$ is the polarizability of the sphere. In the zero-temperature London regime we get $$V(l)=\frac{\hbar}{2\pi}\int_{0}^{\infty}\alpha(i\xi)\left [{\cal D}^E_{ll}(\xi;(\mathbf{r},\mathbf{r}')\right ]_{\mathbf{r} \to \mathbf{r} \to \mathbf{r}_0} \label{Vl}d\xi \; .$$ The “potential” $\phi$ can be taken from [@LL8], §7, Problem 1. A simple calculation then gives $${\cal D}^E_{ll}(\xi;(\mathbf{r}_0,\mathbf{r}_0) =-\frac{\hbar}{2l^3}\frac{\epsilon_1(i\xi)-1}{\epsilon_1(i\xi)+1}\; .$$ Taking into account that $$\alpha(i\xi) =R^3\frac{\varepsilon_2(i\xi)-1}{\varepsilon_2(i\xi)+2} \; ,\label{alpha}$$ we find $$V(l)=-\frac{\hbar R^3}{4\pi l^3} \int_{0}^{\infty}\frac{(\varepsilon_2(i\xi)-1)(\epsilon_1(i\xi)-1)}{(\varepsilon_2(i\xi)+2)(\epsilon_1(i\xi)+1)}d\xi \; . \label{Vvac}$$ The force acting on the sphere is $$F(l)=-\frac{dV}{dl}=-\frac{3\hbar R^3}{4\pi l^4}\int_{0}^{\infty}\frac{(\varepsilon_2(i\xi)-1)(\epsilon_1(i\xi)-1)}{(\varepsilon_2(i\xi)+2)(\epsilon_1(i\xi)+1)}d\xi \; . \label{Fvac}$$ One must be careful when rewriting this equation for a case of bodies separated by liquid. The transformation was formulated for the tensor $\sigma'$. Taking into account that $F=\int \sigma'_{zz}dxdy$, we conclude that it is enough to change $\epsilon_1 \to \epsilon_1/\epsilon$ and $\epsilon_2 \to \epsilon_2/\epsilon$: $$F(l)=\frac{3\hbar R^3}{4\pi l^4}\int_{0}^{\infty}\frac{(\varepsilon_2(i\xi)-\varepsilon(i\xi))(\epsilon_1(i\xi)-\varepsilon(i\xi))}{(\varepsilon_2(i\xi)+\varepsilon(i\xi))(\epsilon_1(i\xi)+\varepsilon(i\xi))}d\xi \; . \label{Fliq}$$ Interaction between two parallel plates --------------------------------------- Let us consider now the force between solid bodies 1 and 2 separated by very small distances. It should be noted that, for a rigorous statement of the problem, it is necessary to consider at least one of the bodies as being of finite size and surrounded by the liquid. Then $F_i=\oint \sigma_{ik}'dS_{k}$ is the total force acting on the body. However, since the van der Waals forces decrease very quickly with distance, the integrand is actually different from zero only inside the gap and the force can be calculated as $F=F_z=\int \sigma_{zz}'dxdy$. Notice that, due to equation (\[ds0\]), the quantity $\sigma'_{zz}$ does not depend on $z$. Finally the force per unit area can be expressed as[@DLP] $$F=\frac{\hbar}{16\pi l^3}\int_0^{\infty}\int_0^{\infty}x^2\left [\frac{(\varepsilon_1+\varepsilon)(\varepsilon_2+\varepsilon)}{(\varepsilon_1-\varepsilon)(\varepsilon_1-\varepsilon)}e^x-1 \right ] dxd\xi \label{plates}$$ where the dielectric permeabilities must be taken as functions of the imaginary frequency $i\xi$. Remarks about repulsive interactions ==================================== It is well known that forces between bodies in vacuum are attractive. In the cases considered in the previous section it is obvious, because for any body $\varepsilon(i\xi)>1$ for $\xi >0$. It also follows from Eqs. (\[Fliq\]) and (\[plates\]) that forces are attractive for bodies of the same media ($\varepsilon_1=\varepsilon_2$). If, however, the bodies are different, the force can be either attractive or repulsive. It is clear from (\[Fliq\]) and (\[plates\]) that if the differences $\varepsilon_1-\varepsilon$ and $\varepsilon_2-\varepsilon$ have different signs in the essential region of values $\xi$, we have $F<0$, that is, the bodies repel one another. To understand better the physical meaning of this repulsion, let us consider the problem of the body-sphere interaction and assume that the materials of both the sphere and the liquid (but not the body) are optically rarefied, i. e., that $\varepsilon_2-1 \ll 1$ and $\varepsilon -1 \ll 1$. Then (\[Fliq\]) can be simplified as $$F(l)\approx \frac{3\hbar R^3}{8\pi l^4} \int_{0}^{\infty}\frac{(\varepsilon_2(i\xi)-1)}{(\varepsilon_2(i\xi)+1)}(\epsilon_1(i\xi)-\varepsilon(i\xi))d\xi \; . \label{Frar}$$ The force is now expressed as a difference of two terms, with clear physical meaning. The first term is the force that acts on the sphere in vacuum. The second term is the force that would act in vacuum on an identical sphere, but with optical properties of the liquid. This second term is an exact analogy of the Archimedes’ buoyant force, which acts on a body embedded in a liquid in a gravitational field. This remark again stresses the importance of the condition of the mechanical equilibrium in the liquid. Of course, such a simple interpretation is possible only in the limit of rarefied media. The existence of the repulsive van der Waals forces, predicted in [@DLP] was confirmed in several experiments (see[@Capasso; @Capasso2] and references therein). See also the chapter by Capasso [et al.]{} in this volume. Corresponding experiments are, however, quite difficult. Forces at large distances are quite small, while at small distances the atomic structure of the media becomes essential. Liquid films ============ The van der Waals forces play an important part in the physics of surface phenomena, and in the properties of thin films in particular. A fundamental problem here is the dependence of the chemical potential $\zeta$ on the thickness $d$ of a film. For example, the thickness of a film on a solid surface in equilibrium with the vapour at pressure $P$ is given by the equation $$\zeta(P,d)=\zeta_0(P)+\frac{T}{m}\ln\frac{P}{P_{sat}} \; . \label{sat}$$ If the thickness $d$ of the film is large compared to interatomic distances, this dependence is defined mainly by the van der Waals forces. Actually the contribution of these forces to the chemical potential is given by the general equation (\[zeta\]). However, this equation cannot be used directly, because it gives the chemical potential $\zeta$ in terms of the density $\rho$, while (\[sat\]) requires $\zeta$ as a function of the pressure $P$. According to the conditions of mechanical equilibrium, the normal component $\sigma_{zz}$ of the stress tensor must be continuous at the surface of the film $-P_0(\rho,T)+\sigma^{(elm)}_{zz}=-P$. Then $\rho(P_0)=\rho(P+\sigma^{(elm)}_{zz}) \approx \rho(P)+(\partial \rho)/\partial P) \sigma^{(elm)}_{zz}$ and $\zeta_0(\rho) \approx \zeta_0(P)+ (\partial \zeta/\partial \rho)(\partial \rho/\partial P) \sigma^{(elm)}_{zz}= \zeta_0(P)+ \sigma^{(elm)}_{zz}/\rho$, where we took into account that $(\partial \zeta/\partial \rho)=1/\rho$. Thus we have $$\begin{aligned} \zeta(P,d)&=&\zeta_0(P)+\sigma^{(elm)}_{zz}/\rho+\frac{\hbar T}{4\pi} {\sum_{s=0}^{\infty}}~^{\prime} {\cal D}^{E}_{ll}(\mathbf{r},\mathbf{r};\xi_{s}) \frac{\partial \varepsilon(i\xi_{s};\mathbf{r})}{\partial \rho} \nonumber\\ &=&\zeta_0(P)+\sigma_{zz}'/\rho = \zeta_0(P)+F(d)/\rho\; , \label{zetaP}\end{aligned}$$ where $F(d)$ is the force, which in the London regime is given by (\[plates\]) with $\varepsilon_2=1, l \rightarrow d$. (As far as electromagnetic properties of the vapour are concerned, we can treat it as a vacuum.) One can now rewrite (\[sat\]) in the form $$F(d)=-\frac{T}{m}\ln\frac{P}{P_{sat}} \; . \label{sat1}$$ If the film is placed on a solid wall situated vertically in the gravitational field, the dependence of the film thickness on the altitude is given by the equation $$F(d)=\rho g x \; , \label{g}$$ where $x$ is the height. In conclusion, let us consider a “free” film in vacuum. Then the chemical potential can be written as $$\zeta(P,T,d)=\zeta_0(P,T)+F(d)/\rho \; \label{free}$$ where $F$ can be obtained from (\[plates\]) with $\varepsilon_1=\varepsilon_2=1, l \to d$: $$F=\frac{\hbar}{16\pi d^3}\int_0^{\infty}\int_0^{\infty}x^2\left [\frac{(1+\varepsilon)^2}{(1-\varepsilon)^2}e^x-1 \right ] dxd\xi\; . \label{freeplates}$$ Note that this is just the quantity which can be calculated by integration of the force in the three-boundary geometry of Fig. 1. However, these calculations have never been performed, and the correctness of the corresponding considerations has not been proved. I thank R. Scott for critical reading of this paper and useful suggestions. [99]{} I use the generic term “van der Waals forces” for long-range forces between neutral objects in any conditions. Thus I do not distinguish between the London, Casimir, Casimir-Polder and Lifshitz forces. F. London: Theory and system of molecular forces. Z. Phys. 63, 245 (1930). H. B. Casimir and D. Polder: The influence of retardation of the London-van der Waals forces. Phys. Rev. 73, 360 (1948). H. B. Casimir: On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet. 51, 793 (1948). E. M. Lifshitz: Theory of molecular attraction forces between condensed bodies. Doclady Akademii Nauk SSSR 97, part 4, 643 (1954); Influence of temperature on molecular attraction forces between condensed bodies. 100, Part 5, 879 (1955). E. M. Lifshitz: The theory of molecular attractive forces between solids. Sov. Phys. JETP 2, 73 (1956). S. M. Rytov, Theory of the Electric Fluctuations and Thermal Radiation \[in Russian\], Publication of Acad. of Sciencies of USSR, Moscow, 1953. L. D. Landau and E. M. Lifshitz, [Electrodynamics of Continuous Media]{}, Pergamon Press, Oxford, 1960. Russian edition was published in 1957. I. E. Dzyaloshinskii and L. P. Pitaevskii: Van der Waals forces in an inhomogeneous dielectric. Sov. Phys. JETP [9]{}, 1282 (1959). V. I. Perel and Ya. M. Pinskii: Stress tensor for a plasma in a high frequency electromagnetic field with account of collisions. Sov. Phys. JETP [27]{}, 1014 (1968). L. P. Pitaevskii: Electric forces in a transparent dispersive medium. Sov. Phys. JETP [12]{}, 1008 (1961). L. P. Pitaevskii: Comment on “Casimir force acting on magnetodielectric bodies embedded in media”, Phys. Rev. A [73]{}, 047801 (2006). A. A.Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, 1963. Here and below I put $k_B=1$. Likewise I put $\hbar=1$ in intermediate equations. I use the CGSE system of electromagnetic units and for simplicity neglect the magnetic properties of media, i. e. put $\mu=1$. I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii: Van der Waals forces in liquid films. Sov. Phys. JETP [10]{}, 161 (1960). I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii: The general theory of van der Waals forces. Adv. Phys. [10]{}, 165 (1961). Yu. S. Barash and V. L. Ginzburg: Electromagnetic fluctuations in matter and molecular (van der Waals) forces between them. Sov. Phys. Uspekhi 18, 305 (1975). J. Schwinger, L. L. DeRaad, and K. A. Milton: Casimir effect in dielectrics. Ann. Phys. (N.Y.) 115, 1 (1978). A. I. Volokitin and B. N. P. Persson: Radiative heat transfer between nanostructures. Phys. Rev. B 63, 205404 (2001). L. P. Pitaevskii: Thermal Lifshitz force between an atom and a conductor with small density of carriers. Phys. Rev. Lett. 101, 163202 (2008). J. N. Munday, Federico Capasso, and V. Adrian Parsegian: Measured long-range repulsive Casimir-Lifshitz forces. Nature 457, 170 (2009) J. N. Munday, Federico Capasso: Repulsive Casimir and van der Waals forces: from measurements to future technologies. In: Quantum Field Theory under the Influence of External Conditions, p. 127. Ed. K. A. Milton and M. Bordag, Word Scientific, New Jersey, 2010.
--- abstract: 'Hamiltonian Monte Carlo (HMC) improves the computational efficiency of the Metropolis algorithm by reducing its random walk behavior. Riemannian Manifold HMC (RMHMC) further improves HMC’s performance by exploiting the geometric properties of the parameter space. However, the geometric integrator used for RMHMC involves implicit equations that require costly numerical analysis (e.g., fixed-point iteration). In some cases, the computational overhead for solving implicit equations undermines RMHMC’s benefits. To avoid this problem, we propose an explicit geometric integrator that replaces the momentum variable in RMHMC by velocity. We show that the resulting transformation is equivalent to transforming Riemannian Hamilton dynamics to Lagrangian dynamics. Experimental results show that our method improves RMHMC’s overall computational efficiency. All computer programs and data sets are available online (<http://www.ics.uci.edu/~babaks/Site/Codes.html>) in order to allow replications of the results reported in this paper.' author: - 'Shiwei Lan[^1], Vassilios Stathopoulos[^2], Babak Shahbaba$^{}$, Mark Girolami$^{\dagger}$' title: Lagrangian Dynamical Monte Carlo --- Introduction ============ Hamiltonian Monte Carlo (HMC) [@duane87] reduces the random walk behavior of Metropolis by proposing samples that are distant from the current state, but nevertheless have a high probability of acceptance. These distant proposals are found by numerically simulating Hamiltonian dynamics for some specified amount of fictitious time [@neal10]. Hamiltonian dynamics can be represented by a function, known as the Hamiltonian* function, of model parameters $\boldsymbol\theta$ and fictitious momentum parameters ${\bf{p}}\sim N({\bf 0, M})$ (with the same dimension as $\boldsymbol\theta$) as follows: $$\begin{aligned} \label{hamiltonian} H({\boldsymbol\theta}, {\bf p}) & = & -\log p({\boldsymbol\theta}) + \frac12 {\bf p}^{\textsf T}{\bf M}^{-1}{\bf p} \end{aligned}$$ where $\bf M$ is a symmetric, positive-definite mass matrix. Hamilton’s equations, which involve differential equations of $H$, determine how ${\boldsymbol\theta}$ and ${\bf p}$ change over time. In practice, however, solving these equations exactly is too hard, so we need to approximate them by discretizing time, using some small step size ${\varepsilon}$. For this purpose, the leapfrog method is commonly used. As the dimension grows, the system becomes increasingly restricted by its smallest eigen-direction, requiring smaller step sizes to maintain the stability of numerical discretization. [@girolami11] proposed a new method, called Riemannian Manifold HMC (RMHMC), that exploits the geometric properties of the parameter space to improve the efficiency of standard HMC. Simulating from the resulting dynamic, however, is computationally intensive since it involves solving two implicit equations, which require additional iterative numerical analysis (e.g., fixed-point iteration). To increase RMHMC’s speed, we propose a new integrator that is completely explicit: we propose to replace momentum with velocity in Riemannian Manifold Hamilton dynamics. As we will see, this is equivalent to using Lagrangian dynamics as opposed to Hamiltonian dynamics. By doing so, we eliminate one of the implicit steps in RMHMC. Next, we construct a time symmetric integrator to remove the remaining implicit step in RHHMC. This leads to a sampling scheme, called e-RMHMC, that involves explicit equations only. In what follows, we start with a brief review of RMHMC and its geometric integrator. Section 3 introduces our proposed semi-explicit integrator based on defining Hamiltonian dynamics in terms of velocity as opposed to momentum. Next, in Section 4, we eliminate the remaining implicit equation and propose a fully explicit integrator. In Section 5, we use simulated and real data to evaluate our methods’ performance. Finally, in Section 6, we discuss some possible future research directions. Riemannian Manifold Hamiltonian Monte Carlo =========================================== ![The first 10 iterations in sampling from a banana shaped distribution with random walk Metropolis (RWM), Hamiltonian Monte Carlo (HMC), and Riemannian Manifold HMC (RMHMC). For all three methods, the trajectory length (i.e., step size times number of integration steps) is set to 1. Solid red lines are the sampling path, and black circles are the accepted proposals.[]{data-label="fig:illust"}](illust.pdf){width="6.5in"} As discussed above, although HMC explores the parameter space more efficiently than random walk Metropolis does, it does not fully exploits the geometric properties of parameter space defined by the density $p({\boldsymbol\theta})$. Indeed, [@girolami11] argue that dynamics over Euclidean space may not be appropriate to guide the exploration of parameter space. To address this issue, they propose a new method, called Riemannian Manifold HMC (RMHMC), that exploits the Riemannian geometry of the parameter space [@amari00] to improve standard HMC’s efficiency by automatically adapting to the local structure. They do this by using a position-specific mass matrix $\bf{M} = {\bf G}({\boldsymbol\theta})$. More specifically, they set ${\bf G}({\boldsymbol\theta})$ to the Fisher information matrix. As a result, ${\bf p} = {\bf G}({\boldsymbol\theta})\dot {\boldsymbol\theta} \sim \mathcal N(0, {\bf G}({\boldsymbol\theta}))$, and Hamiltonian is defined as follows: $$\label{rmhamiltonp} H({\boldsymbol\theta}, {\bf p}) = -\log p({\boldsymbol\theta}) +\frac12 \log\det {\bf G}({\boldsymbol\theta}) + \frac12 {\bf p}^{\textsf T}{\bf G}({\boldsymbol\theta})^{-1}{\bf p} = \phi({\boldsymbol\theta}) + \frac12 {\bf p}^{\textsf T} {\bf G}({\boldsymbol\theta})^{-1}{\bf p}$$ where $\phi({\boldsymbol\theta}):= -\log p({\boldsymbol\theta}) +\frac12 \log\det {\bf G}({\boldsymbol\theta})$. Based on this dynamic, [@girolami11] propose the following HMC on Riemmanian manifold: $$\begin{aligned} \begin{array}{lcrcr } \displaystyle \dot {\boldsymbol\theta} & = & \nabla_{\bf p} H({\boldsymbol\theta}, {\bf p}) & = & {\bf G}({\boldsymbol\theta})^{-1}{\bf p} \\ [12pt] \displaystyle \dot {\bf p} & = & -\nabla_{\boldsymbol\theta} H({\boldsymbol\theta}, {\bf p})& = & -\nabla_{\boldsymbol\theta} \phi({\boldsymbol\theta}) + \frac12 {\boldsymbol\nu}({\boldsymbol\theta},{\bf p}) \end{array}\label{rmhd}\end{aligned}$$ Using the shorthand notation ${\partial}_i = {{\partial}}/{{\partial}{\boldsymbol\theta}_i}$ for partial derivative, the $i$th element of the vector ${\boldsymbol\nu}({\boldsymbol\theta},{\bf p})$ is $$\begin{aligned} ({\boldsymbol\nu}({\boldsymbol\theta},{\bf p}))_i = -{\bf p}^{\textsf T} {\partial}_i ( {\bf G}({\boldsymbol\theta})^{-1}){\bf p} = ({{\bf G}({\boldsymbol\theta})^{-1}\bf p})^{\textsf T} {\partial}_i {\bf G}({\boldsymbol\theta}) {{\bf G}({\boldsymbol\theta})^{-1}\bf p}\end{aligned}$$ The above dynamic is non-separable (it contains products of $\boldsymbol{\theta}$ and $p$), and the resulting map $({\boldsymbol\theta},{\bf p})\to ({\boldsymbol\theta}^,{\bf p}^)$ based on the standard leapfrog method is neither time-reversible nor symplectic. Therefore, the standard leapfrog algorithm cannot be used for the above dynamic [@girolami11]. Instead, we can use the Stömer-Verlet [@verlet67] method as follows: $$\begin{aligned} \label{gleapfrog} {\bf p}^{(n+1/2)} & = & {\bf p}^{(n)} - \frac{{\varepsilon}}{2} \left[\nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})-\frac12 {\boldsymbol\nu}({\boldsymbol\theta}^{(n)},{\bf p}^{(n+1/2)})\right] \label{gleapfrog:imp1}\\ {\boldsymbol\theta}^{(n+1)} & = & {\boldsymbol\theta}^{(n)} + \frac{{\varepsilon}}{2} \left[{\bf G}^{-1}({\boldsymbol\theta}^{(n)}) + {\bf G}^{-1}({\boldsymbol\theta}^{(n+1)})\right]{\bf p}^{(n+1/2)}\label{gleapfrog:imp2}\\ {\bf p}^{(n+1)} & = & {\bf p}^{(n+1/2)} - \frac{{\varepsilon}}{2}\left[\nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n+1)})-\frac12 {\boldsymbol\nu}({\boldsymbol\theta}^{(n+1)},{\bf p}^{(n+1/2)})\right]\end{aligned}$$ This is also known as generalized leapfrog [@leimkuhler04]. The above series of transformations are (i) deterministic (ii) reversible and (iii) volume-preserving. Therefore, the effective proposal distribution is a delta function $\delta(({\boldsymbol\theta}^{(1)}, {\bf p}^{(1)}), ({\boldsymbol\theta}^{(L)}, {\bf p}^{(L)})$ and the acceptance probability is as follows: $$\frac{\exp(-H({\boldsymbol\theta}^{(L)}, {\bf p}^{(L)}))}{\exp(-H({\boldsymbol\theta}^{(1)}, {\bf p}^{(1)}))} \times \frac {\delta(({\boldsymbol\theta}^{(L)}, {\bf p}^{(L)}), ({\boldsymbol\theta}^{(1)}, {\bf p}^{(1)}))} {\delta(({\boldsymbol\theta}^{(1)}, {\bf p}^{(1)}), ({\boldsymbol\theta}^{(L)}, {\bf p}^{(L)}))} = \exp(H({\boldsymbol\theta}^{(1)}, {\bf p}^{(1)})-H({\boldsymbol\theta}^{(L)}, {\bf p}^{(L)}))$$ Here, $({\boldsymbol\theta}^{(1)}, {\bf p}^{(1)})$ is the current state, and $({\boldsymbol\theta}^{(L)}, {\bf p}^{(L)})$ is the proposal after $L$ leapfrog steps. As an illustrative example, Figure \[fig:illust\] shows the sampling paths of random walk Metropolis (RWM), HMC, and RMHMC for an artificially created banana-shaped distribution [See @girolami11 discussion by Luke Bornn and Julien Cornebise]. For this example, we fixed the trajectory and chose the step sizes such that the acceptance probability for all three methods remains around 0.7. RWMH moves slowly and spends most of iterations at the distribution’s low-density tail, and HMC explores the parameter space in a tortuous way, while RMHMC moves directly to the high-density region and explores the distribution more efficiently. One major drawback of this geometric integrator, which is both time-reversible and volume-preserving, is that it involves two implicit functions: Equations and . These functions require extra numerical analysis (e.g. fixed-point iteration), which results in higher computational cost and simulation error. To address this problem, we propose an alternative approach that uses velocity instead of momentum. Moving from Momentum to Velocity {#useV} ================================ In the Hamiltonian dynamic , the product of ${\bf G}({\boldsymbol\theta})^{-1}$ and ${\bf p}$ is in fact velocity, ${\bf v}={\bf G}({\boldsymbol\theta})^{-1}{\bf p}$. This motivates us to define the dynamic in terms of ${\bf v}$ instead of ${\bf p}$. The transformation ${\bf p}\mapsto {\bf v}$ changes the Hamiltonian dynamics to the following form (derivation in Appendix \[derRMLD\]): $$\begin{aligned} \begin{array}{lcl } \displaystyle \dot {\boldsymbol\theta} & = & {\bf v} \\[12pt] \displaystyle \dot {\bf v} & = & - {\boldsymbol\eta}({\boldsymbol\theta},{\bf v}) - {\bf G}({\boldsymbol\theta})^{-1} \nabla_{\boldsymbol\theta} \phi({\boldsymbol\theta}) \end{array}\label{rmld}\end{aligned}$$ where ${\boldsymbol\eta}({\boldsymbol\theta},{\bf v})$ is a vector whose $k$th element is $\sum\nolimits_{i,j}\Gamma^k_{ij}({\boldsymbol\theta}){\bf v}^i {\bf v}^j$. Here, $\Gamma^k_{ij}({\boldsymbol\theta}) := \frac12 \sum_l {\bf g}^{kl} ({\partial}_i {\bf g}_{lj} + {\partial}_j {\bf g}_{il} - {\partial}_l {\bf g}_{ij}) $ is Christoffel symbol whose $(i,j)$th element is ${\bf G}({\boldsymbol\theta})= ({\bf g}_{ij})$. Further, ${\bf G}({\boldsymbol\theta})^{-1}=({\bf g}^{ij})$. This transformation moves the Hamiltonian dynamic’s complexity from its first equation for ${\boldsymbol\theta}$ to its second equation. In this way, we resolve one implicit function in the generalized leapfrog method and develop a semi-explicit integrator as follows: $$\begin{aligned} \label{GLv0} {\bf v}^{(n+1/2)} & = & {\bf v}^{(n)} - \frac{{\varepsilon}}{2}[({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n)}) {\bf v}^{(n+1/2)} + {\bf G}({\boldsymbol\theta}^{(n)})^{-1} \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})] \label{GLv:1}\\ {\boldsymbol\theta}^{(n+1)} & = & {\boldsymbol\theta}^{(n)} + {\varepsilon}{\bf v}^{(n+1/2)} \label{GLv:2}\\ {\bf v}^{(n+1)} & = & {\bf v}^{(n+1/2)} - \frac{{\varepsilon}}{2}[({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n+1)}) {\bf v}^{(n+1/2)} + {\bf G}({\boldsymbol\theta}^{(n+1)})^{-1} \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n+1)})] \label{GLv:3}\end{aligned}$$ Note that updating ${\bf v}$ remains implicit (more details are available in Appendix \[semi-RMLMC\]). In general, the new dynamic cannot be recognized as a Hamiltonian dynamic of $({\boldsymbol\theta}, {\bf v})$. Nevertheless, it remains a valid proposal-generating mechanism, which preserves the original Hamiltonian $H({\boldsymbol\theta}, {\bf p}={\bf G}({\boldsymbol\theta}){\bf v})$ (proof in Appendix \[derRMLD\]); thus, the acceptance probability is only determined by a discretization error from the numerical integration, as before. Because ${\bf p}\sim \mathcal N({\bf 0},{\bf G}({\boldsymbol\theta}))$, the distribution of ${\bf v}={\bf G}({\boldsymbol\theta})^{-1}{\bf p}$ is $ \mathcal N({\bf 0},{\bf G}({\boldsymbol\theta})^{-1})$. Therefore, we have $$\begin{aligned} p({\bf v}) = \frac{1}{(\sqrt{2\pi})^{D}\sqrt{\det({\bf G}({\boldsymbol\theta})^{-1})}} \exp\left\{-\frac12 {\bf v}^{\textsf T} {\bf G}({\boldsymbol\theta}) {\bf v} \right\} \propto (\det {\bf G}({\boldsymbol\theta}))^{1/2} \exp\left\{-\frac12 {\bf v}^{\textsf T} {\bf G}({\boldsymbol\theta}) {\bf v} \right\}\end{aligned}$$ We define the energy function ${\bf E}({\boldsymbol\theta}, {\bf v})$ as the sum of the potential energy, $U({\boldsymbol\theta})$ and $K({\boldsymbol\theta}, {\bf v})$, where $K({\boldsymbol\theta}, {\bf v}) = -\log(p({\bf v}))$. Analogous to RMHMC, thebacceptance probability is calculated based on ${\bf E}({\boldsymbol\theta}, {\bf v})$, which is the negative log of the joint density of parameter ${\boldsymbol\theta}$ and the new auxiliary variable ${\bf v}$. Therefore, we can apply the generalized leapfrog scheme to this new dynamic and derive a semi-implicit method. (See more details in Appendix \[semi-RMLMC\].) Although the resulting integrator is not symplectic, we nonetheless have detailed balance with volume correction. Algorithm \[Alg:RMLMC\] shows the corresponding steps for implementing this method. Initialize ${\boldsymbol\theta}^{(1)} = \textrm{current}\; {\boldsymbol\theta}$ Sample new velocity ${\bf v}^{(1)}\sim \mathcal N(0,{\bf G}^{-1}({\boldsymbol\theta}^{(1)}))$ Calculate current ${\bf E}({\boldsymbol\theta}^{(1)}, {\bf v}^{(1)})$ according to equation % Update the velocity with fixed point iterations ${\bf\hat v}^{(0)} = {\bf v}^{(n)}$ ${\bf\hat v}^{(i)} = {\bf v}^{(n)} - \frac{{\varepsilon}}{2}{\bf G}({\boldsymbol\theta}^{(n)})^{-1}[({\bf\hat v}^{(i-1)})^{\textsf T}\tilde\Gamma({\boldsymbol\theta}^{(n)}) {\bf\hat v}^{(i-1)} + \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})]$ ${\bf v}^{(n+1/2)} = {\bf\hat v}^{(last\; i)}$ % [Update the position only with simple one step]{} $\Delta\log\det_n = \log \det ({\bf I}-{\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n+1)})) - \log \det ({\bf I}+{\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n)}))$ Update the velocity exactly ${\bf v}^{(n+1)} = {\bf v}^{(n+1/2)} - \frac{{\varepsilon}}{2}{\bf G}({\boldsymbol\theta}^{(n+1)})^{-1}[({\bf v}^{(n+1/2)})^{\textsf T}\tilde\Gamma({\boldsymbol\theta}^{(n+1)}) {\bf v}^{(n+1/2)} + \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n+1)})]$ Calculate proposed ${\bf E}({\boldsymbol\theta}^{(L+1)}, {\bf v}^{(L+1)})$ according to equation logRatio = $-\textrm{ProposedH}+\textrm{CurrentH} + \sum_{n=1}^{N}\Delta\log\det_n$ Accept or reject according to Metropolis ratio In Appendix \[connection2L\], we show that the new dynamic is essentially a Lagrangian dynamic. Therefore, we refer to the derived algorithm (Algorithm \[Alg:RMLMC\]) as Riemannian Manifold Lagrangian Monte Carlo (RMLMC), which explores the parameter space along the path on a Riemannian manifold that minimizes the total Lagrangian. We can use this new proposal-generating mechanism, RMLMC, which is based on an Euler-Lagrange system of $({\boldsymbol\theta}, {\bf v})$ instead of the original Hamiltonian system defined in terms of $({\boldsymbol\theta}, {\bf p})$. The two methods use equivalent dynamics but differ numerically in the following way: RMHMC augments parameter space with momentum, while RMLMC augments parameter space with velocity. Later, we will show that switching to velocity leads to substantial improvement in computational efficiency. Explicit Riemannian Manifold Lagrangian Monte Carlo =================================================== We now propose a fully explicit integrator for Lagrangian dynamics as follows: $$\begin{aligned} \label{TSvsolved} {\bf v}^{(n+1/2)} & = & [{\bf I} + \frac{{\varepsilon}}{2} {\bf\Omega}({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})]^{-1} [{\bf v}^{(n)} - \frac{{\varepsilon}}{2} {\bf G}({\boldsymbol\theta}^{(n)})^{-1} \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})] \label{TSvsolved:1}\\ {\boldsymbol\theta}^{(n+1)} &=& {\boldsymbol\theta}^{(n)} + {\varepsilon}{\bf v}^{(n+1/2)} \\ {\bf v}^{(n+1)} & = & [{\bf I} + \frac{{\varepsilon}}{2} {\bf\Omega}({\boldsymbol\theta}^{(n + 1)},{\bf v}^{(n + \frac 1 2)})]^{-1}[{\bf v}^{(n+1/2)} - \frac{{\varepsilon}}{2} {\bf G}({\boldsymbol\theta}^{(n+1)})^{-1}\nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n+1)})] \label{TSvsolved:2}\end{aligned}$$ where ${\bf\Omega}({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})$ is a matrix whose $(i,j)$th element is $\sum_k {\bf v}^{(n)}_k\Gamma_{kj}^i({\boldsymbol\theta}^{(n)})$. This integrator is (i) reversible and (ii) energy-preserving up to order ${\cal O}({\varepsilon})$, where ${\varepsilon}$ is the stepsize. The resulting map, however, is not volume-preserving and as such the effective proposal distribution will be the product of a delta function and the determinant of the transformation, $$\frac{\exp(-{\bf E}({\boldsymbol\theta}^{(L)}, {\bf v}^{(L)}))}{\exp(-{\bf E}({\boldsymbol\theta}^{(1}, {\bf v}^{(1)}))} \times \frac {\delta(({\boldsymbol\theta}^{(L)}, {\bf v}^{(L)}), ({\boldsymbol\theta}^{(1)}, {\bf v}^{(1)}))} {\delta(({\boldsymbol\theta}^{(1)}, {\bf v}^{(1)}), ({\boldsymbol\theta}^{(L)}, {\bf v}^{(L)}))} \times \det {\bf J}$$ which simplifies to $$\label{acptsv} \exp({\bf E}({\boldsymbol\theta}^{(1)}, {\bf v}^{(1)})-{\bf E}({\boldsymbol\theta}^{(L)}, {\bf v}^{(L)})) \times \det {\bf J}$$ with ${\bf J}$ the Jacobian matrix of $({\boldsymbol\theta}^{(1)}, {\bf v}^{(1)})\to ({\boldsymbol\theta}^{(L)}, {\bf v}^{(L)})$. Detailed derivations and proofs are given in Appendix \[xplct-RMLMC\]. Initialize ${\boldsymbol\theta}^{(1)} = \textrm{current}\; {\boldsymbol\theta}$ Sample new velocity ${\bf v}^{(1)}\sim \mathcal N(0,{\bf G}({\boldsymbol\theta}^{(1)})^{-1})$ Calculate current ${\bf E}({\boldsymbol\theta}^{(1)},{\bf v}^{(1)})$ according to equation $\Delta \log\det = 0$ $\Delta \log\det = \Delta \log\det - \det({\bf G}({\boldsymbol\theta}^{(n)})+{\varepsilon}/2 {\bf\tilde\Omega}({\boldsymbol\theta}^{(n)},{\bf v}^{(n)}))$ Update the velocity [explicitly]{} with a half step: ${\bf v}^{(n+1/2)}\! =\! [{\bf G}({\boldsymbol\theta}^{(n)})\!+\!\frac{{\varepsilon}}{2} {\bf\tilde\Omega}({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})]^{-1}[{\bf G}({\boldsymbol\theta}^{(n)}){\bf v}^{(n)}\!-\!\frac{{\varepsilon}}{2} \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})]$ $\Delta \log\det = \Delta \log\det + \det({\bf G}({\boldsymbol\theta}^{(n)})-{\varepsilon}/2 {\bf\tilde\Omega}({\boldsymbol\theta}^{(n)},{\bf v}^{(n+1/2)}))$ Update the position with a full step: $\Delta \log\det = \Delta \log\det - \det({\bf G}({\boldsymbol\theta}^{(n+1)})+{\varepsilon}/2 {\bf\tilde\Omega}({\boldsymbol\theta}^{(n+1)},{\bf v}^{(n+1/2)}))$ Update the velocity [explicitly]{} with a half step: ${\bf v}^{(n+1)}\! =\! [{\bf G}({\boldsymbol\theta}^{(n+1)})\!+\!\frac{{\varepsilon}}{2} {\bf\tilde\Omega}({\boldsymbol\theta}^{(n+1)},{\bf v}^{(n+1/2)})]^{-1}[{\bf G}({\boldsymbol\theta}^{(n+1)}){\bf v}^{(n+1/2)}\!-\!\frac{{\varepsilon}}{2} \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n+1)})]$ $\Delta \log\det = \Delta \log\det + \det({\bf G}({\boldsymbol\theta}^{(n+1)})-{\varepsilon}/2 {\bf\tilde\Omega}({\boldsymbol\theta}^{(n+1)},{\bf v}^{(n+1)}))$ Calculate proposed ${\bf E}({\boldsymbol\theta}^{(L+1)},{\bf v}^{(L+1)})$ according to equation logRatio = $-\textrm{Proposed{\bf E}}+\textrm{Current{\bf E}} + \Delta \log\det$ Accept or reject the proposed state according to We refer to this approach as explicit Riemannian Manifold Lagrangian Monte Carlo (e-RMLMC). Algorithm \[Alg:e-RMLMC\] shows the corresponding steps for this method. In this algorithm, we use ${\bf\tilde\Omega}({\boldsymbol\theta},{\bf v})$ to denote ${\bf G}({\boldsymbol\theta}){\bf\Omega}({\boldsymbol\theta},{\bf v})$ whose $(k,j)$th element is equal to $\sum_{i} {\bf v}^{i}\tilde\Gamma_{ij}^{k}({\boldsymbol\theta})$. Our proposed e-RMLMC does not involve implicit functions for updating $({\boldsymbol\theta},{\bf v})$ in RMHMC. Because we remove multiple fixed-point iteration steps, we reduce the computation time by $\mathcal O(D^{2})$ where $D$ is the dimension of the parameters. Additionally, using this explicit updating, we resolve the convergence issue faced by fixed-point iterations. The connection terms $\tilde \Gamma({\boldsymbol\theta})$ in ${\bf\tilde\Omega}$ do not add substantial computational cost since they are obtained from permuting three dimensions of the array ${\partial}{\bf G}({\boldsymbol\theta})$, which is computed in RMHMC. However, besides ${\bf G}({\boldsymbol\theta})^{-1}$, which is required for $\nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta})$, e-RMLMC has two extra matrix inversions to update ${\bf v}$, whose complexity in general is $\mathcal O(D^{3})$. Therefore, as dimension grows, the efficiency gained by removing multiple fixed-point iterations may be overwhelmed by this additional overhead. This is evident from our experimental results presented in Section \[experiments\]. Faster matrix inversion algorithms could be used to alleviate this issue. Experimental Results {#experiments} ==================== In this section, we use simulated and real data to evaluate our methods, RMLMC and e-RMLMC, compared to RMHMC. Following [@girolami11], we use a time-normalized effective sample size (ESS) to compare these methods. For $B$ posterior samples we calculate ESS = $B[1 + 2{\varSigma}_{k}\gamma(k)]^{-1}$ for each parameter and choose the minimum as the measure of sampling efficiency, where ${\varSigma}_{k}\gamma(k)$ is the sum of the $K$ monotone sample autocorrelations estimated by the initial monotone sequence estimator [@geyer92]. All computer programs and data sets discussed in this paper are available online at <http://www.ics.uci.edu/~babaks/Site/Codes.html>. Data method AP s ESS min(ESS)/s ------------- --------- ------ ---------- --------------------- ------------ RMHMC 0.74 2.40E-02 (8561,9595,10262) 23.77 Australian RMLMC 0.75 1.90E-02 (8038,10488,11468) 28.24 D=14,N=690 e-RMLMC 0.75 1.55E-02 (9636,10443,11268) 41.34 RMHMC 0.77 5.63E-02 (15000,15000,15000) 17.76 German RMLMC 0.73 4.32E-02 (15000,15000,15000) 23.17 D=24,N=1000 e-RMLMC 0.70 3.59E-02 (13762,15000,15000) 25.57 RMHMC 0.77 1.65E-02 (7050,8369,8905) 28.57 Heart RMLMC 0.78 1.09E-02 (10847,11704,12405) 66.25 D=13,N=270 e-RMLMC 0.76 1.01E-02 (10347,10724,11773) 68.18 RMHMC 0.81 1.25E-02 (4325,4622,4980) 23.10 Pima RMLMC 0.82 7.27E-03 (4713,5448,5576) 43.20 D=7,N=532 e-RMLMC 0.82 7.04E-03 (4839,5193,5539) 45.85 RMHMC 0.78 8.39E-03 (15000,15000,15000) 119.20 Ripley RMLMC 0.76 5.07E-03 (13498,15000,15000) 177.43 D=2,N=250 e-RMLMC 0.79 4.77E-03 (12611,15000,15000) 176.37 : Comparing alternative methods using five binary classification problems discussed in [@girolami11]. For each dataset, the number of predictors, $D$, and the number of observations, $N$, are specified. For each method, we provide the acceptance probability (AP), the CPU time (s) for each iteration, and the time-normalized ESS.[]{data-label="realLR"} Logistic Regression Models -------------------------- We start by evaluating our methods based on five binary classification problems used in [@girolami11]. These are Australian Credit data, German Credit data, Heart data, Pima Indian data, and Ripley data. For each problem, we use a logistic regression model and run 20000 MCMC iterations. Results (after discarding the initial 5000 iterations) are summarized in Table \[realLR\], and show that in general our methods improve the sampling efficiency measured in terms of ESS per second compared to RMHMC. Simulated Logistic Regression ----------------------------- Next, we construct some synthetic datasets with variable numbers of observations and increasing dimensionality for logistic regression. The simulated results are summarized in Table \[simLR\]. In general, our methods RMLMC and e-RMLMC improve RMHMC in minimal ESS per second, but as expected, such an advantage gradually diminishes as the dimension increases. [l\|l\|cccc]{} Data & Method & AP & s & ESS & min(ESS)/s\ & RMHMC & 0.84 & 1.85e+02 & (4837, 4902, 4968) & 26.20\ & RMLMC & 0.86 & 4.44e+01 & (5000, 5000, 5000) & 112.60\ & e-RMLMC & 0.83 & 7.42e+01 & (3792, 4310, 4671) & 51.11\ & RMHMC & 0.82 & 9.56e+02 & (4727, 4893, 5000) & 4.95\ & RMLMC & 0.80 & 1.28e+02 & (4680, 4819, 4857) & 36.44\ & e-RMLMC & 0.81 & 2.59e+02 & (2964, 3543, 3968) & 11.46\ & RMHMC & 0.82 & 3.15e+03 & (4691, 4983, 5000) & 1.49\ & RMLMC & 0.82 & 6.88e+02 & (4749, 4836, 4960) & 6.91\ & e-RMLMC & 0.81 & 1.09e+03 & (2902, 3636, 4127) & 2.65\ & RMHMC & 0.81 & 9.87e+03 & (3712, 4515, 4950) & 0.38\ & e-RMLMC & 0.83 & 4.64e+03 & (4002, 4672, 4919) & 0.86\ & e-RMLMC & 0.80 & 1.19e+04 & (2565, 3415, 4081) & 0.22\ & RMLMC & 0.79 & 1.63e+05 & (3160, 3959, 4464) & 0.02\ & RMLMC & 0.83 & 1.44e+05 & (3458, 4221, 4676) & 0.02\ & e-RMLMC & 0.80 & 1.20e+05 & (2708, 3548, 4156) & 0.02\ Simulating a banana-shaped distribution --------------------------------------- The banana-shaped distribution, which we used above for illustration, can be constructed as the posterior distribution of $\theta=(\theta_{1},\theta_{2})\|y$ based on the following model: $$\begin{aligned} y\|\theta & \sim & N(\theta_{1}+\theta_{2}^{2},\sigma^{2}_{y})\\ \theta & \sim & N(0,\sigma^{2}_{\theta})\end{aligned}$$ The data $\{y_{i}\}_{i=1}^{100}$ are generated with $\theta_{1}+\theta_{2}^{2}=1,\sigma_{y}=2$. We set $\sigma_{\theta}=1$. We want to investigate how the three algorithms, RMHMC, RMLMC, e-RMLMC, explore the parameter space. Fig.\[fig:comp3\] shows the first 10 iterations for each algorithm using fixed trajectory length of 1.45. ![The first 10 iterations in sampling from the banana-shaped distribution with Riemannian Manifold HMC (RMHMC), Riemannian Manifold Lagrange Monte Carlo (RMLMC) and explicit RMLMC (e-RMLMC). For all three methods, the trajectory length (i.e., step size times number of integration steps) is set to 1.45. Solid red lines show the sampling path, and each point represents an accepted proposal.[]{data-label="fig:comp3"}](comp3.pdf){width="6.5in"} We can see that RMLMC and e-RMLMC explore the parameter space according to its curvature and mix quickly, similar to RMHMC. Table \[toy\] compares the performances of these algorithms based on 5000 MCMC iterations (after burning the initial 1000 iteration). For this example, RMLMC has the highest ESS/s. As discussed above, the additional overhead of matrix inversion for e-RMLMC occasionally overwhelms its gain in computational efficiency. Method AP s ESS min(ESS)/s --------- ------ ---------- ----------------- ------------ RMHMC 0.73 8.21e-03 (729,1117,1506) 17.76 RMLMC 0.79 5.36e-03 (857,1317,1777) 31.99 e-RMLMC 0.78 5.61e-03 (585,1085,1585) 20.85 : Comparing alternative methods using a banana-shaped distribution. For each method, we provide the acceptance probability (AP), the CPU time (s) for each iteration, and the time-normalized ESS.[]{data-label="toy"} Finite Mixture of Gaussians --------------------------- Finally we consider finite mixtures of univariate Gaussian components of the form $$p(x_i\|\boldsymbol{\theta}) = \sum_{k=1}^K\pi_k\mathcal{N}(x_i\|\mu_k,\sigma^2_k) \label{eq:likelihood}$$ where $\boldsymbol{\theta}$ is the vector of size $D=3K$ of all the parameters $\pi_k$, $\mu_k$ and $\sigma^2_k$ and $\mathcal{N}(\cdot\|\mu,\sigma^2)$ is a Gaussian density with mean $\mu$ and variance $\sigma^2$. A common choice of prior takes the form $$p(\boldsymbol{\theta}) = \mathcal{D}(\pi_1,\dots,\pi_K \| \lambda)\prod_{k=1}^K \mathcal{N}(\mu_k\|m,\beta^{-1}\sigma_k^2)\mathcal{IG}(\sigma^2_k\| b,c) \label{eq:prior}$$ where $ \mathcal{D}(\cdot\|\lambda)$ is the symmetric Dirichlet distribution with parameter $\lambda$ and $\mathcal{IG}(\cdot\|b,c)$ is the inverse Gamma distribution with shape parameter $b$ and scale parameter $c$. Although the posterior distribution associated with this model is formally explicit, it is computationally intractable, since it can be expressed as a sum of $K^N$ terms corresponding to all possible allocations of observations $x_i$ to mixture components [@marin05 chap. 9]. We want to use this model to test the efficiency of posterior sampling $\boldsymbol{\theta}$ using the three methods. A more extensive comparison of Riemannian Manifold MCMC and HMC, Gibbs sampling and standard Metropolis-Hastings for finite Gaussian mixture models can be found at [@Stathopoulos11]. Due to the non-analytic nature of the expected Fisher Information, $\boldsymbol{I}(\boldsymbol{\theta})$, we use the empirical Fisher information as metric tensor, defined in [@mclachlan00 chap. 2]: $$\boldsymbol{G}(\boldsymbol{\theta}) = \boldsymbol{S}^T\boldsymbol{S} - \frac{1}{N}{\boldsymbol{s}}{\boldsymbol{s}}^T \label{eq:eFI}$$ where $N\times D$ score matrix $\boldsymbol{S}$ has elements $S_{i,d}=\frac{\partial \log p(x_i\|\boldsymbol{\theta})}{\partial \theta_d}$ and ${\boldsymbol{s}}=\sum_{i=1}^N\boldsymbol{S}^T_{i,\cdot}$. Depending on allocations of $\pi$, we show several classical mixtures showing in the following Table \[tab:dens\] and Figure \[fig:dens\]. Their sampling efficiency is compared in Table \[MOG\]. As before, our two algorithms outperform RMHMC. ---------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------ Dataset Density function Num. of name parameters Kurtotic $ \frac{2}{3}\mathcal{N}(x \| 0,1)+\frac{1}{3}\mathcal{N}\left(x\|0,\left(\frac{1}{10}\right)^2 \right)$ 6 Bimodal $\frac{1}{2}\mathcal{N}\left(x \| -1,\left(\frac{2}{3}\right)^2\right)+\frac{1}{2}\mathcal{N}\left(x\|1,\left(\frac{2}{3}\right)^2 \right) $ 6 Skewed $ \frac{3}{4}\mathcal{N}\left(x \| 0,1 \right)+\frac{1}{4}\mathcal{N}\left(x\|\frac{3}{2},\left(\frac{1}{3}\right)^2 \right)$ 6 Trimodal $\frac{9}{20}\mathcal{N}\left(x \| -\frac{6}{5},\left(\frac{3}{5}\right)^2 \right)+\frac{9}{20}\mathcal{N}\left(x\|\frac{6}{5},\left(\frac{3}{5}\right)^2 \right)+\frac{1}{10}\mathcal{N}\left(x\|0,\left(\frac{1}{4}\right)^2 \right) $ 9 Claw $\frac{1}{2}\mathcal{N}\left(x \|0,1\right)+\sum_{i=0}^4\frac{1}{10}\mathcal{N}\left(x\|\frac{i}{2}-1,\left(\frac{1}{10}\right)^2 \right) $ 18 ---------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------ : Densities used for the generation of synthetic Mixture of Gaussian data sets. \[tab:dens\] ![Densities used to generate synthetic datasets. From left to right the densities are in the same order as in Table \[tab:dens\]. The densities are taken from [@mclachlan00][]{data-label="fig:dens"}](kurtotic "fig:"){width="19.00000%"}![Densities used to generate synthetic datasets. From left to right the densities are in the same order as in Table \[tab:dens\]. The densities are taken from [@mclachlan00][]{data-label="fig:dens"}](bimodal "fig:"){width="19.00000%"} ![Densities used to generate synthetic datasets. From left to right the densities are in the same order as in Table \[tab:dens\]. The densities are taken from [@mclachlan00][]{data-label="fig:dens"}](skewed "fig:"){width="19.00000%"}![Densities used to generate synthetic datasets. From left to right the densities are in the same order as in Table \[tab:dens\]. The densities are taken from [@mclachlan00][]{data-label="fig:dens"}](trimodal "fig:"){width="19.00000%"} ![Densities used to generate synthetic datasets. From left to right the densities are in the same order as in Table \[tab:dens\]. The densities are taken from [@mclachlan00][]{data-label="fig:dens"}](claw "fig:"){width="19.00000%"} [l\|l\|cccc]{} Data & Method & AP & s & ESS & min(ESS)/s\ & RMHMC & 0.80 & 2.54e+03 & (1524, 3474, 4586) & 0.60\ & RMLMC & 0.86 & 1.88e+03 & (2531, 4332, 5000) & 1.35\ & e-RMLMC & 0.82 & 1.46e+03 & (2436, 3455, 4608) & 1.67\ & RMHMC & 0.79 & 4.97e+02 & (4701, 4928, 5000) & 9.46\ & RMLMC & 0.82 & 2.01e+02 & (4978, 5000, 5000) & 24.77\ & e-RMLMC & 0.80 & 2.42e+02 & (4899, 4982, 5000) & 20.21\ & RMHMC & 0.85 & 2.55e+02 & (5000, 5000, 5000) & 19.63\ & RMLMC & 0.82 & 1.13e+02 & (4698, 4940, 5000) & 41.68\ & e-RMLMC & 0.84 & 1.26e+02 & (4935, 5000, 5000) & 39.09\ & RMHMC & 0.82 & 2.36e+02 & (5000, 5000, 5000) & 21.20\ & RMLMC & 0.85 & 1.27e+02 & (5000, 5000, 5000) & 39.34\ & e-RMLMC & 0.81 & 1.35e+02 & (5000, 5000, 5000) & 36.90\ & RMHMC & 0.86 & 2.69e+02 & (5000, 5000, 5000) & 18.56\ & RMLMC & 0.81 & 1.03e+02 & (4935, 4996, 5000) & 48.00\ & e-RMLMC & 0.85 & 1.08e+02 & (5000, 5000, 5000) & 46.43\ Conclusions and Discussion ========================== Following the method of [@girolami11] for more efficient exploration of parameter space, we have proposed new sampling schemes to reduce the computational cost associated with using a position-specific mass matrix. To this end, we have developed a semi-explicit (RMLMC) integrator and a fully explicit (e-RMLMC) integrator for RMHMC and demonstrated their advantage in improving computational efficiency over the generalized leapfrog (RMHMC) method used by [@girolami11]. It is easy to show that for ${\bf G}({\boldsymbol\theta})= {\bf I}$, our method degenerates to standard HMC. Future directions could involve splitting Hamiltonian [@dullweber87; @sexton92; @neal10; @shahbabaSplitHMC] to develop explicit geometric integrators. For example, one could split a non-separable Hamiltonian dynamics into several smaller dynamics some of which can be analytically solved. A similar idea has been explored by [@chin09], where the Hamiltonian, instead of the dynamic, is split. Another possible research direction could be to approximate the mass matrix (Christofell Symbols). For many large-dimensional problems, the mass matrix could be appropriately approximated by a highly sparse matrix. This could further improve our method’s computational efficiency. Amari, S. and H. Nagaoka (2000). . Oxford University Press. Bishop, Richard L. and Samuel I. Goldberg (2000). . Dover Publications, Inc. Chin, S. A. (2009). . [80]{}, 037701. Christensen, O.F., G.O. Roberts, J.S. Rosenthal (2005, April). . [67]{}(2), 253–268. Duane, S., A. D. Kennedy, B. J. Pendleton, and D. Roweth (1987). Hybrid monte carlo. [195]{}(2), 216 – 222. Dullweber, A., B. Leimkuhler, R. McLachlan (1997). Split-Hamiltonian Methods for Rigid Body Molecular Dynamics. [107]{}(15), 5840 – 5852. Geyer, C. J. (1992). Practical markov chain monte carlo. [7]{}(4), 473–483. Girolami, M. and Calderhead, B. (2011), , (with discussion) 73, 123–214. Hoffman, M. D. and A. Gelman (2011). The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. arXiv:1111.4246v1. Leimkuhler, B. and S. Reich (2004). . Cambridge University Press. Liu, J. S. (2001). . In Liu, Jun S., [Monte Carlo Strategies in Scientific Computing]{}. Springer-Verlag. Marin, J.M., K.L. Mengersen and C. Robert (2005). . In D. Dey and C.R. Rao, [Handbook of Statistics]{} [Volume 25]{}. Elsevier. McLachlan, G.J. and D. Peel (2000). . John Wiley $\&$ Sons, Inc., New York. Neal, R. M. (2010). . In S. Brooks, A. Gelman, G. Jones, and X. L. Meng (Eds.), [Handbook of Markov Chain Monte Carlo]{}. Chapman and Hall/CRC. Sexton, J. C. and D.H. Weingarten (1992). . [380]{}(3), 665–677. Shahbaba, B., S. Lan, W. Johnson, and R. Neal (2011). Split hamiltonian monte carlo. arXiv:1106.5941. Stathopoulos, V. and Girolami, M. (2011) . Verlet, Loup (1967). . [159]{}, 98–103. [Appendix: Derivations and Proofs]{} In what follows, we show the detailed derivations of our methods. We adopt Einstein notation (summation convention), so whenever the index appears twice in a mathematical expression, we sum over it: e.g., $a_{i}b^{i}:=\sum_{i}a_{i}b^{i}$, $\Gamma_{ij}^{k}v^{i}v^{j}:=\sum_{i,j}\Gamma_{ij}^{k}v^{i}v^{j}$. A lower index is used for the covariant tensor, whose components vary by the same transformation as the change of basis�e.g., gradient�whereas the upper index is reserved for the contravariant tensor, whose components vary in the opposite way as the change of basis in order to compensate: e.g. velocity vector. Interested readers should refer to [@bishop80]. Transformation of Hamiltonian Dynamics {#derRMLD} ====================================== To derive the dynamic from the Hamiltonian dynamic , the first equation in is directly obtained from the assumed transformation: $\dot {\boldsymbol\theta}^k = {\bf g}^{kl}{\bf p}_l={\bf v}^k$. For the second equation in , we have $$\dot {\bf p}_{l} = \frac{d ({\bf g}_{lj}({\boldsymbol\theta}){\bf v}^{j})}{dt} =\frac{{\partial}{\bf g}_{lj}}{{\partial}{\boldsymbol\theta}_{i}}\dot {\boldsymbol\theta}^{i}{\bf v}^{j} + {\bf g}_{lj}\dot {\bf v}^{j} = {\partial}_{i} {\bf g}_{lj}{\bf v}^{i}{\bf v}^{j} + {\bf g}_{lj}\dot {\bf v}^{j}$$ Further, from Equation we have $$\begin{split} \dot {\bf p}_{l} & = -{\partial}_{l} \phi({\boldsymbol\theta}) + \frac{1}{2}{\bf v}^{\textsf T}{\partial}_{l} {\bf G}({\boldsymbol\theta}) {\bf v} = -{\partial}_{l} \phi + \frac{1}{2}{\bf g}_{ij,l}{\bf v}^{i}{\bf v}^{j}\\ & = {\partial}_{i} {\bf g}_{lj}{\bf v}^{i}{\bf v}^{j} + {\bf g}_{lj}\dot {\bf v}^{j} \end{split}$$ which means $${\bf g}_{lj}\dot {\bf v}^{j} = -({\partial}_{i} {\bf g}_{lj} - \frac{1}{2}{\partial}_{l} {\bf g}_{ij}){\bf v}^{i}{\bf v}^{j} -{\partial}_{l} \phi$$ By multiplying ${\bf G}^{-1}= ({\bf g}^{kl})$ on both sides, we have $$\label{dvij} \dot {\bf v}^{k} = \delta^{k}_{j}\dot {\bf v}^{j} = -{\bf g}^{kl}({\partial}_{i} {\bf g}_{lj} - \frac{1}{2}{\partial}_{l} {\bf g}_{ij}){\bf v}^{i}{\bf v}^{j} -{\bf g}^{kl}{\partial}_{l} \phi$$ Since $i, j$ are symmetric in the first summand, switching them gives the following equations: $$\label{dvji} \dot {\bf v}^{k} = -{\bf g}^{kl}({\partial}_{j} {\bf g}_{li} - \frac{1}{2}{\partial}_{l} {\bf g}_{ji}){\bf v}^{i}{\bf v}^{j} -{\bf g}^{kl}{\partial}_{l} \phi$$ which in turn gives the final form of Equation after adding equations and and dividing the results by two: $$\dot {\bf v}^k = -\Gamma^k_{ij}({\boldsymbol\theta}){\bf v}^i {\bf v}^j - {\bf g}^{kl}({\boldsymbol\theta}){\partial}_{l} \phi({\boldsymbol\theta})$$ Here, $\Gamma^k_{ij}({\boldsymbol\theta}) := \frac12 {\bf g}^{kl}({\partial}_{i} {\bf g}_{lj} + {\partial}_{j} {\bf g}_{il} - {\partial}_{l} {\bf g}_{ij})$ is Christoffel Symbol of second kind. Note that the new dynamic still preserves the original Hamiltonian $H({\boldsymbol\theta}, {\bf p}={\bf G}({\boldsymbol\theta}){\bf v})$. This is of course intuitive, but it also can be proven as follows: $$\begin{split} \frac{d}{dt} H({\boldsymbol\theta}, {\bf G}({\boldsymbol\theta}){\bf v}) & = \dot {\boldsymbol\theta}^{\textsf T} \frac{{\partial}}{{\partial}{\boldsymbol\theta}} H({\boldsymbol\theta}, {\bf G}({\boldsymbol\theta}){\bf v}) + \dot {\bf v}^{\textsf T} \frac{{\partial}}{{\partial}{\bf v}} H({\boldsymbol\theta}, {\bf G}({\boldsymbol\theta}){\bf v})\\ & = {\bf v}^{\textsf T}\left[ \nabla_{\boldsymbol\theta} \phi({\boldsymbol\theta}) + \frac12 {\bf v}^{\textsf T}{\partial}{\bf G}({\boldsymbol\theta}) {\bf v}\right] + \left[-{\bf v}^{\textsf T}\Gamma({\boldsymbol\theta}) {\bf v} - {\bf G}({\boldsymbol\theta})^{-1}\nabla_{\boldsymbol\theta} \phi({\boldsymbol\theta})\right]^{\textsf T} {\bf G}({\boldsymbol\theta}){\bf v}\\ & = {\bf v}^{\textsf T}\nabla_{\boldsymbol\theta} \phi({\boldsymbol\theta}) - \left(\nabla_{\boldsymbol\theta} \phi({\boldsymbol\theta})\right)^{\textsf T}{\bf v} + \frac12{\bf v}^{\textsf T}\left( {\bf v}^{\textsf T}{\partial}{\bf G}({\boldsymbol\theta}) {\bf v}\right)- ({\bf v}^{\textsf T}\tilde\Gamma({\boldsymbol\theta}) {\bf v})^{\textsf T} {\bf v}\\ & = 0+ 0 =0 \end{split}$$ where ${\bf v}^{\textsf T}\Gamma({\boldsymbol\theta}) {\bf v}$ is a vector whose $k$th element is $\Gamma^k_{ij}({\boldsymbol\theta}){\bf v}^i {\bf v}^j$. The second 0 is due to the triple form $({\bf v}^{\textsf T}\tilde\Gamma({\boldsymbol\theta}) {\bf v})^{\textsf T} {\bf v}= \tilde\Gamma_{ijk}{\bf v}^{i}{\bf v}^{j}{\bf v}^{k}=\frac12 {\partial}_{k} {\bf g}_{ij}{\bf v}^{i}{\bf v}^{j}{\bf v}^{k}$, where $\tilde \Gamma$ is Christoffel Symbol of first kind with elements $\tilde\Gamma_{ijk}({\boldsymbol\theta}) := {\bf g}_{kl}\Gamma_{ij}^{l}({\boldsymbol\theta}) = \frac12 ({\partial}_{i} {\bf g}_{kj} + {\partial}_{j} {\bf g}_{ik} - {\partial}_{k} {\bf g}_{ij})$. Derivation of semi-explicit Riemannian Manifold Lagrangian Monte Carlo (RMLMC) {#semi-RMLMC} ============================================================================== Consider the following generalized leapfrog integration scheme: $$\begin{aligned} {\bf p}^{(n+1/2)} & = & {\bf p}^{(n)} - \frac{{\varepsilon}}{2}\frac{{\partial}H}{{\partial}{\boldsymbol\theta}}({\boldsymbol\theta}^{(n)},{\bf p}^{(n+1/2)})\\ {\boldsymbol\theta}^{(n+1)} & = & {\boldsymbol\theta}^{(n)} + \frac{{\varepsilon}}{2}\left[ \frac{{\partial}H}{{\partial}{\bf p}}({\boldsymbol\theta}^{(n)},{\bf p}^{(n+1/2)}) + \frac{{\partial}H}{{\partial}{\bf p}}({\boldsymbol\theta}^{(n+1)},{\bf p}^{(n+1/2)}) \right] \\ {\bf p}^{(n+1)} & = & {\bf p}^{(n+1/2)} - \frac{{\varepsilon}}{2}\frac{{\partial}H}{{\partial}{\boldsymbol\theta}}({\boldsymbol\theta}^{(n)},{\bf p}^{(n+1/2)})\end{aligned}$$ We composite implicit steps for velocity and explicit step for position within a leapfrog step to integrate dynamic and derive the following semi-explicit integrator: $$\begin{aligned} \label{GLv} {\bf v}^{(n+1/2)} & = & {\bf v}^{(n)} - \frac{{\varepsilon}}{2}[({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n)}) {\bf v}^{(n+1/2)} + {\bf G}({\boldsymbol\theta}^{(n)})^{-1} \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})] \label{GLv:1}\\ {\boldsymbol\theta}^{(n+1)} & = & {\boldsymbol\theta}^{(n)} + {\varepsilon}{\bf v}^{(n+1/2)} \label{GLv:2}\\ {\bf v}^{(n+1)} & = & {\bf v}^{(n+1/2)} - \frac{{\varepsilon}}{2}[({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n+1)}) {\bf v}^{(n+1/2)} + {\bf G}({\boldsymbol\theta}^{(n+1)})^{-1} \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n+1)})] \label{GLv:3}\end{aligned}$$ The time-reversibility of this integrator can be shown by switching $({\boldsymbol\theta},{\bf v})^{(n+1)}$ and $({\boldsymbol\theta},{\bf v})^{(n)}$ and negating velocity. The resulting integrator, however, is no longer volume-preserving (see subsection \[detadj1\]). Nevertheless, based on proposition \[dbwda\], we can still have detailed balance after determinant adjustment. [See @liu01 for more details.] \[dbwda\] Denote ${\bf z}=({\boldsymbol\theta},{\bf p})$, ${\bf z}'=\hat T_{L}({\bf z})$ for some time reversible integrator $\hat T_{L}$ to the Lagrangian dynamic. If the acceptance probability is adjusted in the following way: $$\tilde\alpha({\bf z},{\bf z'}) =\min\left\{1,\frac{\exp(-H({\bf z'}))}{\exp(-H({\bf z}))}\|\det \hat T_{L}\|\right\}$$ then the detailed balance condition still holds $$\tilde\alpha({\bf z},{\bf z'}) \mathbb P(d{\bf z}) = \tilde\alpha({\bf z'},{\bf z}) \mathbb P(d{\bf z'})$$ $$\begin{split} \tilde\alpha({\bf z},{\bf z'}) \mathbb P(d{\bf z}) & =\min\left\{1,\frac{\exp(-H({\bf z'}))}{\exp(-H({\bf z}))}\left\|\frac{d{\bf z'}}{d{\bf z}}\right\|\right\} \exp(-H({\bf z}))d{\bf z}\\ & \overset{{\bf z}= \hat T_{L}^{-1}({\bf z'})}= \min\left\{\exp(-H({\bf z})),\exp(-H({\bf z'}))\left\|\frac{d{\bf z'}}{d{\bf z}}\right\|\right\} \left\|\frac{d{\bf z}}{d{\bf z'}}\right\|d{\bf z'}\\ & = \min\left\{1,\frac{\exp(-H({\bf z}))}{\exp(-H({\bf z'}))}\left\|\frac{d{\bf z}}{d{\bf z'}}\right\|\right\} \exp(-H({\bf z'})) {\bf z'} = \tilde\alpha({\bf z'},{\bf z}) \mathbb P(d{\bf z'}) \end{split}$$ Therefore, the acceptance probability could be calculated based on $H({\boldsymbol\theta},{\bf G}({\boldsymbol\theta}){\bf v})$. However, it also could be also calculated as follows based on the energy function ${\bf E}({\boldsymbol\theta},{\bf v})$ defined in section \[useV\], $$\label{energyv} {\bf E}({\boldsymbol\theta}, {\bf v}) = U({\boldsymbol\theta}) + K({\boldsymbol\theta}, {\bf v}) = -\log p({\boldsymbol\theta}) -\frac12 \log\det {\bf G}({\boldsymbol\theta}) + \frac12 {\bf v}^{\textsf T}{\bf G}({\boldsymbol\theta}){\bf v}$$ To show their equivalence, we note that $\displaystyle \left\|\frac{{\partial}({\boldsymbol\theta}',{\bf p}')}{{\partial}({\boldsymbol\theta},{\bf p})}\right\| = \frac{\det({\bf G}({\boldsymbol\theta}'))}{\det({\bf G}({\boldsymbol\theta}))} \left\|\frac{{\partial}({\boldsymbol\theta}',{\bf v}')}{{\partial}({\boldsymbol\theta},{\bf v})}\right\|$ and proved our claim as follows: $$\begin{split} \tilde\alpha & = \min\left\{1,\frac{\exp(-H({\boldsymbol\theta}',{\bf p}'))}{\exp(-H({\boldsymbol\theta},{\bf p}))} \left\|\frac{{\partial}({\boldsymbol\theta}',{\bf p}')}{{\partial}({\boldsymbol\theta},{\bf p})}\right\| \right\} = \min\left\{1,\frac{\exp(-H({\boldsymbol\theta}',{\bf G}({\boldsymbol\theta}'){\bf v}'))}{\exp(-H({\boldsymbol\theta},{\bf G}({\boldsymbol\theta}){\bf v}))} \left\|\frac{{\partial}({\boldsymbol\theta}',{\bf p}')}{{\partial}({\boldsymbol\theta},{\bf p})}\right\| \right\} \\ & = \min\left\{1,\frac{\exp\{-(\log p({\boldsymbol\theta}') +\frac12 \log\det {\bf G}({\boldsymbol\theta}') + \frac12 {\bf v'}^{\textsf T}{\bf G}({\boldsymbol\theta}'){\bf v'})\}}{\exp\{-(\log p({\boldsymbol\theta}) +\frac12 \log\det {\bf G}({\boldsymbol\theta}) + \frac12 {\bf v}^{\textsf T}{\bf G}({\boldsymbol\theta}){\bf v})\}} \frac{\det({\bf G}({\boldsymbol\theta}'))}{\det({\bf G}({\boldsymbol\theta}))} \left\|\frac{{\partial}({\boldsymbol\theta}',{\bf v}')}{{\partial}({\boldsymbol\theta},{\bf v})}\right\| \right\}\\ & = \min\left\{1,\frac{\exp\{-(\log p({\boldsymbol\theta}') -\frac12 \log\det {\bf G}({\boldsymbol\theta}') + \frac12 {\bf v'}^{\textsf T}{\bf G}({\boldsymbol\theta}'){\bf v'})\}}{\exp\{-(\log p({\boldsymbol\theta}) -\frac12 \log\det {\bf G}({\boldsymbol\theta}) + \frac12 {\bf v}^{\textsf T}{\bf G}({\boldsymbol\theta}){\bf v})\}} \left\|\frac{{\partial}({\boldsymbol\theta}',{\bf v}')}{{\partial}({\boldsymbol\theta},{\bf v})}\right\| \right\}\\ & = \min\left\{1,\frac{\exp(-{\bf E}({\boldsymbol\theta}',{\bf v}'))}{\exp(-\bf{E}({\boldsymbol\theta},{\bf v}))} \left\|\frac{{\partial}({\boldsymbol\theta}',{\bf v}')}{{\partial}({\boldsymbol\theta},{\bf v})}\right\| \right\} \end{split}$$ Volume Correction {#detadj1} ----------------- To adjust volume, we must derive the Jacobian determinant, $\det {\bf J}:=\left\|\frac{{\partial}({\boldsymbol\theta}^{(L+1)},{\bf v}^{(L+1)})}{{\partial}({\boldsymbol\theta}^{(1)},{\bf v}^{(1)})}\right\|$ using wedge products. Differential one-form $\alpha: TM^{D}\to \mathbb R$ on a differential manifold $M^{D}$ is a smooth mapping from tangent space $TM^{D}$ to $\mathbb R$, which can be expressed as linear combination of differentials of local coordinates: $\alpha = f_{i}dx^{i}=: f\cdot dx$. For example, if $f: \mathbb R^{D}\to \mathbb R$ is a smooth function, then its directional derivative along a vector $v\in \mathbb R^{D}$, denoted by $df(v)$ is given by $$df(v) = \frac{{\partial}f}{{\partial}z_{i}} v^{i}$$ then $df(\cdot)$ is a linear functional of $v$, called the differential of $f$ at $z$ and is an example of a differential one-form. In particular, $dz^{i}(v)=v^{i}$, thus $$df(v) = \frac{{\partial}f}{{\partial}z_{i}} dz^{i}(v), \quad then \; df = \frac{{\partial}f}{{\partial}z_{i}} dz^{i}$$ Wedge Product of two one-form $\alpha, \beta$ is a 2-form $\alpha\wedge\beta$ anti-symmetric bilinear function on tangent space which has the following properties ($\alpha,\beta,\gamma$ one-forms, $A$ be a square matrix of same dimension $D$): - $\alpha\wedge\alpha=0$ - $\alpha\wedge(\beta+\gamma) = \alpha\wedge\beta+\alpha\wedge\gamma$ (thus $\alpha\wedge\beta=-\beta\wedge\alpha$) - $\alpha\wedge A\beta=A^{\textsf T}\alpha\wedge\beta$ The following proposition enables us to calculate the Jacobian determinant denoted as $\det {\bf J}$. \[wedge\] Let $T_{L}: ({\boldsymbol\theta}^{(1)},{\bf v}^{(1)})\to ({\boldsymbol\theta}^{(L+1)},{\bf v}^{(L+1)})$ be evolution of a smooth flow, then $$d{\boldsymbol\theta}^{(L+1)}\wedge d{\bf v}^{(L+1)} = \frac{{\partial}({\boldsymbol\theta}^{(L+1)},{\bf v}^{(L+1)})}{{\partial}({\boldsymbol\theta}^{(1)},{\bf v}^{(1)})} d{\boldsymbol\theta}^{(1)}\wedge d{\bf v}^{(1)}$$ Note that the Jacobian determinant $\det {\bf J}$ can also be regarded as Radon-Nikodym derivative of two probability measures: $\det {\bf J} = \displaystyle \frac{\mathbb P(d{\boldsymbol\theta}^{(L+1)},d{\bf v}^{(L+1)})}{\mathbb P(d{\boldsymbol\theta}^{(1)}, d{\bf v}^{(1)})}$, where $\mathbb P(d{\boldsymbol\theta},d{\bf v})=p({\boldsymbol\theta},{\bf v})d{\boldsymbol\theta}d{\bf v}$. We have $$\begin{aligned} d{\bf v}^{(n+1/2)} & = & d{\bf v}^{(n)} - {\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n)}) d{\bf v}^{(n+1/2)}+ ()d{\boldsymbol\theta}^{(n)}\\ d{\boldsymbol\theta}^{(n+1)} & = & d{\boldsymbol\theta}^{(n)} + {\varepsilon}d{\bf v}^{(n+1/2)}\\ d{\bf v}^{(n+1)} & = & d{\bf v}^{(n+1/2)} - {\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n+1)}) d{\bf v}^{(n+1/2)} +()d{\boldsymbol\theta}^{(n+1)}\end{aligned}$$ where ${\bf v}^{\textsf T} \Gamma({\boldsymbol\theta})$ is a matrix whose $(k,j)$th element is ${\bf v}^{i}\Gamma_{ij}^{k}({\boldsymbol\theta})$. Therefore, $$\begin{split} d{\boldsymbol\theta}^{(n+1)} \wedge d{\bf v}^{(n+1)} & = [{\bf I}-{\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n+1)})]^{\textsf T} d{\boldsymbol\theta}^{(n+1)} \wedge d{\bf v}^{(n+1/2)}\\ & = [{\bf I}-{\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n+1)})]^{\textsf T} d{\boldsymbol\theta}^{(n)} \wedge d{\bf v}^{(n+1/2)}\\ & = [{\bf I}-{\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n+1)})]^{\textsf T} [{\bf I}+{\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n)})]^{-T} d{\boldsymbol\theta}^{(n)} \wedge d{\bf v}^{(n)} \end{split}$$ For volume adjustment, we must use the following Jacobian determinant accumulated along leap frog steps: $$\label{detglv} \det {\bf J}_{LMC} := \left\|\frac{{\partial}({\boldsymbol\theta}^{(L+1)},{\bf v}^{(L+1)})}{{\partial}({\boldsymbol\theta}^{(1)},{\bf v}^{(1)})}\right\| = \prod_{n=1}^{L} \frac{\det(I-{\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n+1)}))}{\det(I+{\varepsilon}({\bf v}^{(n+1/2)})^{\textsf T} \Gamma({\boldsymbol\theta}^{(n)}))}\\[12pt]$$ As a result, the acceptance probability becomes $$\alpha_{LMC} = \min\{1,\exp(-{\bf E}({\boldsymbol\theta}^{(L+1)},{\bf v}^{(L+1)})+{\bf E}({\boldsymbol\theta}^{(1)},{\bf v}^{(1)}))\|\det {\bf J}_{LMC}\|\}$$ Using this acceptance probability, we are able to derive a semi-explicit integrator for RMLMC as shown in Algorithm \[Alg:RMLMC\]. In this approach, the updates for ${\boldsymbol\theta}$ are explicit, while updating ${\bf v}$ remains implicit. Connection to Lagrangian Dynamics {#connection2L} ================================= We now show that the above dynamic is indeed Lagrangian dynamic. We define [Lagrangian]{} as follows: $${\bf L} = \frac12 {\bf v}^{\textsf T}{\bf G}({\boldsymbol\theta}){\bf v} - \phi({\boldsymbol\theta}) \;$$ Using variation calculus to minimize the Lagrangian, we obtain a Euler-Lagrange equation of the second kind, $$\frac{{\partial}{\bf L}}{{\partial}{\boldsymbol\theta}}=\frac{d}{dt} \frac{{\partial}{\bf L}}{{\partial}\dot{\boldsymbol\theta}} \;$$ which is $$\label{variation} \ddot {\boldsymbol\theta} = - \dot{\boldsymbol\theta}^{\textsf T} \Gamma({\boldsymbol\theta}) {\bf v} - {\bf G}({\boldsymbol\theta})^{-1} \nabla_{\boldsymbol\theta} \phi({\boldsymbol\theta})$$ This is equivalent to the new dynamic by taking the time derivative on the first equation and equating it with the time derivative on the second. Derivation of explicit Riemannian Manifold Lagrangian Monte Carlo (e-RMLMC) {#xplct-RMLMC} =========================================================================== We now propose an additional modification of Algorithm \[Alg:RMLMC\] to resolve the remaining implicit equation , while keeping time-reversibility to ensure the ergodicity of the induced Markov chain. We do this by modifying the symmetric quadratic form in equations , $$\begin{aligned} \label{TSv} {\bf v}^{(n+1/2)} & = & {\bf v}^{(n)} - \frac{{\varepsilon}}{2} [({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)}) {\bf v}^{(n+1/2)} + {\bf G}({\boldsymbol\theta}^{(n)})^{-1}\nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})] \label{TSv:1}\\ {\boldsymbol\theta}^{(n+1)} & = & {\boldsymbol\theta}^{(n)} + {\varepsilon}{\bf v}^{(n+1/2)} \label{TSv:2}\\ {\bf v}^{(n+1)} & = & {\bf v}^{(n+1/2)} - \frac{{\varepsilon}}{2} [({\bf v}^{(n+1/2)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n+1)}) {\bf v}^{(n+1)} + {\bf G}({\boldsymbol\theta}^{(n+1)})^{-1}\nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n+1)})] \label{TSv:3}\end{aligned}$$ The resulting integrator is completely explicit since both updates of velocity and can be solved as follows: $$\begin{aligned} \label{TSvsolved} {\bf v}^{(n+1/2)} & = & [{\bf I} + \frac{{\varepsilon}}{2} ({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)})]^{-1} [{\bf v}^{(n)} - \frac{{\varepsilon}}{2} {\bf G}({\boldsymbol\theta}^{(n)})^{-1} \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})] \label{TSvsolved:1}\\ {\bf v}^{(n+1)} & = & [{\bf I} + \frac{{\varepsilon}}{2} ({\bf v}^{(n+1/2)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n+1)})]^{-1}[{\bf v}^{(n+1/2)} - \frac{{\varepsilon}}{2} {\bf G}({\boldsymbol\theta}^{(n+1)})^{-1}\nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n+1)})] \label{TSvsolved:2}\end{aligned}$$ Convergence of Numerical Solution {#convg} --------------------------------- We now show that the discretization error $e_n=\Vert {\bf z}(t_n) - {\bf z}^{(n)}\Vert =\Vert ({\boldsymbol\theta}(t_n),{\bf v}(t_n)) - ({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})\Vert$ (i.e. the difference between the true solution and the numerical solution) accumulated over final time interval $[0,T]$, is bounded and goes to zeros as the stepsize ${\varepsilon}$ goes to zero. (See [@leimkuhler04] for a similar proof for the generalized leapfrog method.) Here, we assume that $F({\boldsymbol\theta},{\bf v}):= {\bf v}^{\textsf T}\Gamma({\boldsymbol\theta}) {\bf v} + {\bf G}({\boldsymbol\theta})^{-1} \nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta})$ is smooth; hence, $F$ and its derivatives are uniformly bounded as $({\boldsymbol\theta},{\bf v})$ evolves within finite time duration $T$. We expand the true solution ${\bf z}(t_{n+1})$ at $t_n$: $$\begin{split} {\bf z}(t_{n+1}) & = {\bf z}(t_n)+\dot {\bf z}(t_n){\varepsilon}+\frac{1}{2}\ddot {\bf z}(t_n){\varepsilon}^2 +o({\varepsilon})\\ & = \begin{bmatrix}{\boldsymbol\theta}(t_n)\\{\bf v}(t_n)\end{bmatrix} + \begin{bmatrix}{\bf v}(t_n)\\-F({\boldsymbol\theta}(t_n),{\bf v}(t_n))\end{bmatrix}{\varepsilon}+ \frac12 \begin{bmatrix}-F({\boldsymbol\theta}(t_n),{\bf v}(t_n))\\-\frac{{\partial}F}{{\partial}{\boldsymbol\theta}}{\bf v}(t_n)+\frac{{\partial}F}{{\partial}{\bf v}}F({\boldsymbol\theta}(t_n),{\bf v}(t_n))\end{bmatrix}{\varepsilon}^2 +o({\varepsilon})\\ & = \begin{bmatrix}{\boldsymbol\theta}(t_n)\\{\bf v}(t_n)\end{bmatrix} + \begin{bmatrix}{\bf v}(t_n)\\-F({\boldsymbol\theta}(t_n),{\bf v}(t_n))\end{bmatrix}{\varepsilon}+ O({\varepsilon}^2) \end{split}$$ Next, we simplify the expression of the numerical solutions ${\bf z}^{(n+1)} = \begin{bmatrix}{\boldsymbol\theta}^{(n+1)}\\{\bf v}^{(n+1)}\end{bmatrix}$ for the fully explicit integrator and compare it to the above true solutions. To this end, we rewrite equation as follows: $$\begin{split} {\bf v}^{(n+1/2)} & = [{\bf I} + \frac{{\varepsilon}}{2} ({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)})]^{-1}[{\bf v}^{(n)} - \frac{{\varepsilon}}{2} {\bf G}({\boldsymbol\theta}^{(n)})^{-1}\nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})]\\ & = {\bf v}^{(n)} - [{\bf I} + \frac{{\varepsilon}}{2} ({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)})]^{-1}[({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)}) {\bf v}^{(n)} + \frac{{\varepsilon}}{2} {\bf G}({\boldsymbol\theta}^{(n)})^{-1}\nabla_{\boldsymbol\theta}\phi({\boldsymbol\theta}^{(n)})]\\ & = {\bf v}^{(n)} - [{\bf I} + \frac{{\varepsilon}}{2} ({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)})]^{-1} \frac{{\varepsilon}}{2} F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})\\ & = {\bf v}^{(n)} - \frac{{\varepsilon}}{2} F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)}) + \frac{{\varepsilon}^2}{4}[{\bf I} + \frac{{\varepsilon}}{2} ({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)})]^{-1} [({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)})] F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})\\ & = {\bf v}^{(n)} - \frac{{\varepsilon}}{2} F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)}) + O({\varepsilon}^2) \end{split}$$ Similarly, from equation we have $${\bf v}^{(n+1)} = {\bf v}^{(n+1/2)} - \frac{{\varepsilon}}{2} F({\boldsymbol\theta}^{(n+1)},{\bf v}^{(n+1/2)}) + O({\varepsilon}^2)$$ Substituting ${\bf v}^{(n+1/2)}$ in the above equation, we obtain ${\bf v}^{(n+1)}$ as follows: $$\begin{split} {\bf v}^{(n+1)} & = {\bf v}^{(n)} - \frac{{\varepsilon}}{2} F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)}) - \frac{{\varepsilon}}{2} F({\boldsymbol\theta}^{(n+1)},{\bf v}^{(n)}) + O({\varepsilon}^2)\\ & = {\bf v}^{(n)} - F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)}){\varepsilon}+ \frac{{\varepsilon}}{2}[ F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})- F({\boldsymbol\theta}^{(n)}+O({\varepsilon}),{\bf v}^{(n)} ] + O({\varepsilon}^2)\\ & = {\bf v}^{(n)} - F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)}){\varepsilon}+ O({\varepsilon}^2) \end{split}$$ From , , and , we have the following numerical solution: $${\bf z}^{(n+1)} = \begin{bmatrix}{\boldsymbol\theta}^{(n+1)}\\{\bf v}^{(n+1)}\end{bmatrix} = \begin{bmatrix}{\boldsymbol\theta}^{(n)}\\{\bf v}^{(n)}\end{bmatrix} +\begin{bmatrix}{\bf v}^{(n)}\\- F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})\end{bmatrix}{\varepsilon}+ O({\varepsilon}^2)$$ Therefore, the local error is $$\begin{split} e_{n+1} & =\Vert {\bf z}(t_{n+1}) - {\bf z}^{(n+1)}\Vert = \left\Vert \begin{bmatrix}{\boldsymbol\theta}(t_n)-{\boldsymbol\theta}^{(n)}\\{\bf v}(t_n)-{\bf v}^{(n)}\end{bmatrix} + \begin{bmatrix}{\bf v}(t_n)-{\bf v}^{(n)}\\-[F({\boldsymbol\theta}(t_n),{\bf v}(t_n))- F({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})]\end{bmatrix}{\varepsilon}+ O({\varepsilon}^2) \right\Vert\\ & \leq (1+M{\varepsilon}) e_{n} + O({\varepsilon}^2) \end{split}$$ where $M = c\sup_{t\in [0,T]}\Vert \nabla F({\boldsymbol\theta}(t),{\bf v}(t))\Vert$ for some constant $c>0$. Accumulating the local errors by iterating the above inequality for $L=T/{\varepsilon}$ steps provides the following global error: $$\begin{split} e_{n+1} & \leq (1+M{\varepsilon}) e_{n} + O({\varepsilon}^2) \leq (1+M{\varepsilon})^2 e_{n-1} + 2 O({\varepsilon}^2)\leq \cdots \leq (1+M{\varepsilon})^n e_1 + n O({\varepsilon}^2)\\ & \leq (1+M{\varepsilon})^L {\varepsilon}+ L O({\varepsilon}^2) \leq (e^{MT}+T){\varepsilon}\to 0,\quad as\; {\varepsilon}\to 0 \end{split}$$ Volume Correction {#tsvdetadj} ----------------- As before, using wedge product calculation on the system , , and , the Jacobian matrix is $$\begin{split} \frac{{\partial}({\boldsymbol\theta}^{(n+1)},{\bf v}^{(n+1)})}{{\partial}({\boldsymbol\theta}^{(n)},{\bf v}^{(n)})} = & [{\bf I}+\frac{{\varepsilon}}{2} ({\bf v}^{(n+1/2)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n+1)})]^{-T} [{\bf I}-\frac{{\varepsilon}}{2} ({\bf v}^{(n+1)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n+1)})]^{\textsf T} \cdot\\ & [{\bf I}+\frac{{\varepsilon}}{2} ({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)})]^{-T} [{\bf I}-\frac{{\varepsilon}}{2} ({\bf v}^{(n+1/2)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)})]^{\textsf T} \end{split}$$ As these new equations show, our derived integrator is not symplectic so the acceptance probability needs to be adjusted by the following Jacobian determinant, $\det {\bf J}$, in order to preserve the detailed balance condition: $$\begin{aligned} \begin{array}{ll} & \det {\bf J}_{e-LMC} := \displaystyle \left\|\frac{{\partial}({\boldsymbol\theta}^{(L+1)},{\bf v}^{(L+1)})}{{\partial}({\boldsymbol\theta}^{(1)},{\bf v}^{(1)})}\right\|\\ & = \displaystyle \prod_{n=1}^{L} \frac{\det(I-{\varepsilon}/2 ({\bf v}^{(n+1)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n+1)})) \det(I-{\varepsilon}/2 ({\bf v}^{(n+1/2)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)}))}{\det(I+{\varepsilon}/2 ({\bf v}^{(n+1/2)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n+1)})) \det(I+{\varepsilon}/2 ({\bf v}^{(n)})^{\textsf T}\Gamma({\boldsymbol\theta}^{(n)}))}\\[12pt] & = \displaystyle \prod_{n=1}^{L} \frac{\det({\bf G}({\boldsymbol\theta}^{(n+1)})-{\varepsilon}/2 ({\bf v}^{(n+1)})^{\textsf T}\tilde\Gamma({\boldsymbol\theta}^{(n+1)})) \det({\bf G}({\boldsymbol\theta}^{(n)})-{\varepsilon}/2 ({\bf v}^{(n+1/2)})^{\textsf T}\tilde\Gamma({\boldsymbol\theta}^{(n)}))}{\det({\bf G}({\boldsymbol\theta}^{(n+1)})+{\varepsilon}/2 ({\bf v}^{(n+1/2)})^{\textsf T}\tilde\Gamma({\boldsymbol\theta}^{(n+1)})) \det({\bf G}({\boldsymbol\theta}^{(n)})+{\varepsilon}/2 ({\bf v}^{(n)})^{\textsf T}\tilde\Gamma({\boldsymbol\theta}^{(n)}))} \end{array}\label{dettsv}\end{aligned}$$ As a result, the acceptance probability is $$\alpha_{e-LMC} = \min\{1,\exp(-{\bf E}({\boldsymbol\theta}^{(L+1)},{\bf v}^{(L+1)})+{\bf E}({\boldsymbol\theta}^{(1)},{\bf v}^{(1)}))\|\det {\bf J}_{e-LMC}\|\}$$ We can now derive a completely explicit integrator for RMLMC defined in terms of $({\boldsymbol\theta},{\bf v})$. We refer to this integrator as e-RMLMC for which the corresponding steps are presented in Algorithm \[Alg:e-RMLMC\]. In both algorithms \[Alg:RMLMC\] and \[Alg:e-RMLMC\], the position update is relatively simple while the computational time is dominated by choosing the “right” direction (velocity) using the geometry of parameter space. Finally, it is easy to show that for ${\bf G}({\boldsymbol\theta})= {\bf I}$, our method degenerates to standard HMC. [^1]: Department of Statistics, University of California, Irvine, USA. [^2]: Department of Statistical Science, University College London, UK.
--- abstract: 'We consider a trapped repulsive one-dimensional (1D) Bose gas at very low temperature. In order to study the collective modes of this strongly interacting system, we use a hydrodynamic approach, where the gas is locally described by the Lieb-Liniger model of bosons interacting via a repulsive delta potential. Solving the corresponding linearized hydrodynamic equations, we obtain the collective modes and concentrate more specifically on the lowest compressional mode. This is done by finding models, approaching very closely the exact equation of state of the gas, for which the linearized hydrodynamic equations are exactly solvable. Results are in excellent agreement with those of the sum rule approach of Menotti and Stringari.' address: - \| Laboratoire Kastler Brossel, Ecole Normale Supérieure,\ 24 rue Lhomond, 75231 Paris Cedex 05, France\ email: fuchs@lkb.ens.fr - \| Laboratoire de Physique Statistique, Ecole Normale Supérieure,\ 24 rue Lhomond, 75231 Paris Cedex 05, France author: - 'J. N. Fuchs' - 'X. Leyronas and R. Combescot' date: Received title: \| Collective modes of a trapped Lieb-Liniger gas:\ a hydrodynamic approach --- INTRODUCTION ============ Cold atomic gases of reduced dimensionality are currently being investigated experimentally [@Schreck; @Gorlitz; @Zurich]. A typical situation is that of a dilute 3D Bose-Einstein condensate (BEC) confined in a very anisotropic trap. If the confinement is strong enough in two perpendicular directions, the gas behaves microscopically as a 1D Bose gas in the remaining perpendicular direction. A very interesting model of such an interacting system is that of Lieb and Liniger [@Lieb]: it consists of 1D bosons interacting via repulsive zero-range interactions of variable strength. In the homogeneous case, this model can be solved exactly, whatever the strength of the interactions, by means of Bethe ansatz [@Lieb]. In the present paper, we propose to study the lowest lying collective modes of the trapped Lieb-Liniger gas by means of a hydrodynamic approach [@CL; @MS]. The linearized hydrodynamic equations are solved using the recently developed method of Combescot and Leyronas [@CL]. The results, already presented in [@FLC], are compared with those of Menotti and Stringari [@MS], which were obtained from a sum rule approach. TRAPPED LIEB-LINIGER GAS ======================== The physical system we consider is a 3D dilute BEC at very low temperature ($T\simeq 0$) confined in an elongated cigar-shaped trap. The trap is taken to be harmonic with radial frequency $\omega_{\perp}=\omega_{x}=\omega_{y}$ much greater than the axial frequency $\omega_{z}$. At very low temperature, repulsive interactions between atoms are fully characterized by the 3D s-wave scattering length $% a>0 $. The temperature $T$ and the chemical potential $\mu$ are much smaller than the radial quantum of oscillation $\hbar \omega_{\perp}$, so that the radial motion is frozen and the gas behaves microscopically as a 1D system along the axial direction of the trap. Olshanii [@Olshanii] showed that in that case the corresponding effective 1D scattering length $a_{1}$ is given by $a_{1}=a_{\perp}^2/a$ [@sign] when $a_{\perp}\equiv\sqrt{% \hbar/m\omega_{\perp}} \gg a$, which we assume for simplicity. In order to study such a trapped 1D Bose gas, we assume that it is locally described by the Lieb-Liniger model [@Lieb] of hard-core bosons, of mass $m$, interacting trough a repulsive contact potential $g_{1}\delta(z)$, where $g_{1}=2\hbar^2/ma_{1}$ [@Olshanii]. In the Lieb-Liniger model, the relevant parameter measuring the strength of the interactions is the dimensionless coupling constant $\gamma\equiv mg_{1}/\hbar^2 n_{1}=2/n_{1}a_{1}$, where $n_{1}$ is the 1D density. It is the only dimensionless intensive parameter proportional to the coupling constant $g_{1} $ [@Lieb]. It shows that the gas is weakly interacting $\gamma \ll 1$ in the dense limit $n_{1}a_{1} \gg 1$ and strongly interacting $\gamma \gg 1$ in the dilute limit $n_{1}a_{1}\ll 1$. In the dense limit, the homogeneous gas is a quasi-condensate, i.e. a condensate with a fluctuating phase. Nevertheless, in a trapped gas, when the temperature is well below the degeneracy temperature [@Petrov], the phase is almost constant over the whole sample and the gas behaves as a true condensate: it is, for example, well described by a 1D Gross-Pitaevskii equation [@GP]. In this limit, we say that the gas is in the “1D mean-field” regime and the chemical potential is given by $\mu=g_{1}n_{1}=2\hbar \omega_{\perp} a n_{1}$ (with the convention that $\mu=0$ when $n_{1}=0$). The low density limit corresponds to the “Tonks-Girardeau” regime, i.e. to a gas of impenetrable bosons. Girardeau [@Girardeau] showed that in this limit there is an exact mapping between the gas of impenetrable bosons and the 1D free Fermi gas. In the Tonks-Girardeau regime, the chemical potential is given by $\mu=(\hbar \pi n_{1})^2/2m$. In the whole range between the 1D mean-field and the Tonks-Girardeau regime, the equation of state $\mu(n_{1})$ can be obtained from the exact solution of the homogeneous Lieb-Liniger model [@Lieb]. Menotti and Stringari numerically evaluated this equation of state and made their result available [@MS; @Chiara]. As we will see, the equation of state is the essential ingredient needed in the hydrodynamic approach in order to compute the collective modes. In the following, we use reactive hydrodynamics, where dissipation is negligible and thermal effects can be omitted. This is valid at low enough temperature. The hydrodynamic equations are: $$\begin{aligned} \frac{\partial n_{1}}{\partial t} &+& \partial_{z} (n_{1}v)=0 \label{continuity} \\ m\frac{d v}{d t} &=& -\partial_{z} \big[ \mu(n_{1})+m \omega_{z}^2 z^2/2 % \big] \label{Euler}\end{aligned}$$ where $n_{1}(z,t)$ is the 1D density at position $z$ and time $t$ and $% v(z,t) $ is the velocity along the $z$ axis. The first equation is the continuity equation, the second is the Euler equation at zero temperature. The equation of state for the homogeneous gas $\mu(n_{1})$ appears as a needed input in the hydrodynamic equations: it is provided by the Lieb-Liniger microscopic theory. Here we briefly discuss the validity of the hydrodynamic approach. First, in the 1D mean-field regime, a trapped gas with a finite number of particles behaves as a true condensate at low enough temperature [@Petrov]. It is therefore well described by a 1D Gross-Pitaevskii equation. As in the 3D case [@dgps], neglecting the quantum pressure (which is valid when the number of particle is large), the Gross-Pitaevskii equation can be rewritten in the form of reactive hydrodynamics with the equation of state given by the mean-field result $\mu(n_{1})=g_{1}n_{1}$. Second, for the whole range of densities (i.e. of coupling constant $\gamma$), the low lying excitations of the homogeneous Lieb-Liniger gas are known to be phonon-like with a sound velocity depending on the coupling constant [@Lieb]. This property allows the description of the system in term of an effective Luttinger liquid theory [@Haldane]. In the case of the trapped gas, when studying collective modes of long wavelength and low energy, the chemical potential is locally given by its value in the homogeneous gas at the same density $\mu[n_{1}(z)]$. This allows to extend the hydrodynamic approach initially developed for condensates in the mean-field regime [@dgps] to the strongly interacting regime [@CL; @MS]. COLLECTIVE MODES : SOLVABLE MODELS FOR HYDRODYNAMICS ==================================================== In this section we use the method recently developed by Combescot and Leyronas [@CL] in order to solve the linearized hydrodynamic equations. The basic idea is to replace the equation of state $\mu(n_{1})$ by a model $% \mu_{mod}(n_{1})$ for which the linearized hydrodynamic equations can be solved analytically or quasi-analytically. The model has several parameters that allows to fit the actual equation of state as closely as possible. Details of the fitting procedure are given in [@FLC]. We start by briefly reviewing the method of Combescot and Leyronas in the 1D case. At equilibrium the particle density $n_{1}^0(z) $ satisfies: $$\mu [n_{1}^0(z)] + m\omega_{z}^2z^2/2 = \tilde{\mu} \label{equilibrium}$$ where $\tilde{\mu }$ is the constant value over the system of the global chemical potential. For small density fluctuations $\delta n_{1}(z,t)=n_{1}(z,t)-n_{1}^0(z)$ we introduce the departure of the chemical potential from its equilibrium value $w(z,t) = \tilde{\mu }(z,t)-\tilde{\mu}% = (\partial \mu/\partial n_{1}^0) \delta n_{1}(z,t)$. For a fluctuation occurring at frequency $\omega $, the preceding equation gives $i\omega \delta n_{1}=\partial_{z}(n_{1}^0 v) =n_{1}^0\partial_{z}v +v\partial_{z}(n_{1}^0)$ which, together with Euler equation (\[Euler\]) leads to: $$n_{1}^0\partial_{z}^{2} w+ \partial_{z}n_{1}^0\partial_{z}w +m\omega^{2}(\partial n_{1}^0 / \partial \mu) w=0 \label{eq1}$$ The equilibrium relation (\[equilibrium\]) implies $(\partial \mu/\partial n_{1}^0)(\partial n_{1}^0 / \partial z)=-m\omega_{z}^2z$ which gives: $$zw^{\prime\prime}+ zL^{\prime}(z)w^{\prime}-\nu^2 L^{\prime}(z) w = 0 \label{eq2}$$ where $\nu^2\equiv \omega^2/\omega_{z}^2$ is the squared reduced mode frequency and we defined $L(z) \equiv \ln [n_{1}^0(z)]$ with $L^{\prime}(z) \equiv dL/dz $. Let us introduce the function $v(z)$ by $w(z)=z^{l}v(z)$, with $l=0$ for an even mode with respect to $z$ and $l=1$ for an odd mode, and restrict to $z>0$. With this change of function, equation (\[eq2\]) becomes: $$zv^{\prime\prime}+ [ 2l+ zL^{\prime}(z)] v^{\prime}-(\nu^2-l) L^{\prime}(z) v = 0 \label{eq4}$$ This equation is invariant under the replacement $z \rightarrow z/R $. We therefore take the cloud radius $R$ as unity in the following. The cloud radius is given by $\mu[n_{1}^0(0)]=m\omega_{z}^2 R^2/2$. Next we make the change of variable $y=z^{\alpha}$ to obtain: $$y \frac{d ^{2}v}{ dy ^{2}} + (\Delta + y \frac{dL}{ dy}) \frac{dv}{ dy} - \frac{\nu ^{2} - l }{\alpha } \frac{dL}{ dy} v = 0 \label{eq6}$$ where $\Delta = 1+ (2l-1)/\alpha$. Finally, we define the reduced density $% \bar{n}_{1}(z)\equiv n_{1}(z)/n_{1}(0)$ and the normalized local chemical potential $\bar{\mu }(z) \equiv \mu [n_{1}(z)]/\mu [n_{1}(0)]=1-z^{2}$. We note that in the mode equation (\[eq6\]), the equation of state only enters trough the quantity $dL/dy$. The simplest model introduced by Combescot and Leyronas is the two parameter $\alpha-p$ model [@CL]. It consists in replacing the quantity $dL/dy$ by the model $-p/(1-y)$ in equation (\[eq6\]), which then reduces to the hypergeometric differential equation: $$y (1-y)\frac{d^{2}v}{dy^{2}} + [\Delta-y(p+\Delta)] \frac{dv}{ dy}+ p\frac{% \nu^{2}-l}{\alpha}v=0 \label{eqhyper}$$ The solution to this equation (with proper boundary conditions) is a polynomial [@CL]. The corresponding mode frequencies are: $$\nu^2 =l+\frac{\alpha}{p} \: n \: (n+p+\frac{2l-1}{\alpha}) \label{eqfreq}$$ where $n$ is a positive integer and $l=0$ or $1$. The $\alpha-p$ model corresponds explicitely to the equilibrium density $\bar{n}_{1}(z)=(1-z^\alpha)^p$ and to the equation of state $\bar{\mu}=1-(1-\bar{n}_{1}^{1/p})^{2/\alpha}$. We can check on equation (\[eqfreq\]) that the “dipole mode” ($n=0$ and $l=1$) occurs at frequency $\omega=\omega_{z}$ in agreement with Kohn’s theorem. The particular case $\alpha=2$ corresponds to the power law equation of state $\bar{\mu }=\bar{n}_{1}^{1/p}$. This is precisely the functional dependence of the equation of state in the two limiting regimes that we discussed above: the Tonks-Girardeau regime corresponds to $p=1/2$ and the 1D mean-field regime to $p=1$ . In the Tonks-Girardeau case, the mode frequencies are given by $\omega=k\omega_{z}$, where $k\equiv2n+l$ is any positive integer (it is convenient to notice that $l^2=l$). This result was already obtained in [@Minguzzi; @MS]. In the 1D mean field regime, the mode frequencies are given by $\nu^2=k(1+k)/2$, where $k\equiv2n+l$ is any positive integer, in agreement with [@MS]. Combescot and Leyronas introduced other models which lead to a generalized hypergeometric differential equation with quasi-polynomial solutions [@CL]. Here, we only describe the “3 parameters quasi-polynomial” model. The quantity $dL/dy$ is modeled by $-(p_{0}+p_{1}y)/(1-y)$ depending on the three parameters $\alpha$, $p_{0}$ and $p_{1}$. It is solution of: $$y(1-y)\frac{d ^{2}v}{dy^{2}}+(-p_{1}y^{2}-(\Delta+p _{0})y + \Delta) \frac{dv% }{ dy} + \frac{\nu^2-l}{\alpha}(p_{1}y +p_{0}) v = 0 \label{equasipol}$$ This differential equation admits solutions that are very rapidly converging series which can safely be truncated above some order and that we call quasi-polynomials [@CL]. The mode frequencies are easily obtained numerically. Details are given in [@FLC]. From now on, we will concentrate on the lowest compressional mode ($n=1$ and $l=0$). In order to obtain the mode frequency, we will use the following procedure: (i) fit the actual equation of state by a model (either $\alpha-p$ or 3 parameters quasi-polynomial) to obtain the best set of parameters; (ii) calculate the mode frequency using this set of parameters (with equation (\[eqfreq\]) in the case of the $\alpha-p$ model, for example). In the next section, we show how to improve on this zero-order result for the mode frequency by using perturbation theory. PERTURBATION THEORY =================== In order to account for the difference between the actual equation of state and the model equation of state we can rewrite the mode equation (\[eq6\]) in the form of a Schrödinger-like equation and use standard perturbation theory. This allows to correct the results obtained for the mode frequency calculated using the model equation of state and to come closer to the mode frequency corresponding to the actual equation of state. The following change of function $\psi(y)=v(y)\big[y^\Delta n_{1}^0(y)\big]$ is made in equation (\[eq6\]) leading to: $$\begin{aligned} -\frac{d ^{2} \psi }{ dy ^{2}} + \Big[\frac{1}{2} L^{\prime\prime}+ \frac{1}{% 4} L^{^{\prime}2} + (\frac{\nu ^{2} - l }{\alpha }+ \frac{ \Delta }{2} ) \frac{ L^{\prime}}{y}+ \frac{ \Delta ( \Delta - 2)}{4 y ^{2}}\Big] \psi = 0 \label{eq7t}\end{aligned}$$ which is a Schrödinger equation with an effective potential (term between brackets $[$ $]$) and corresponding to zero energy (and $\hbar ^{2}/2m = 1$). The effective potential can be decomposed into a zero-order term corresponding to the model equation of state and a perturbation term corresponding to the difference of effective potential between the actual and the model equation of state. The perturbation is treated using standard first order perturbation theory. This leads to a first order change of the energy in the Schrödinger equation. In order to keep the energy equal to zero, we have to give a compensating variation $\delta \nu ^{2}$ for the frequency: $$\delta \nu ^{2}=\alpha \frac{\int_{0}^{1} \! dy \: \delta L^{\prime}\, n _{1}^0 \, y ^{\Delta -1} v [\frac{\nu ^{2} - l }{\alpha } v - y v^{\prime}]% } {\int_{0}^{1} \! dy \: (-n_{1}^{0\prime}) \, y ^{\Delta -1}v^{2}} \label{eqcorr}$$ where $\delta L^{\prime}\equiv L^{\prime}-L_{mod}^{\prime}$. This result is proven in [@FLC]. LOWEST COMPRESSIONAL MODE ========================= We apply now the previously developed method to study the lowest compressional mode ($n=1$ and $l=0$), which physically corresponds to a breathing motion of the gas along the axis. The needed input is the equation of state $\mu(n_{1})$ of the Lieb-Liniger model. Using the numerical evaluation of this equation of state by Menotti and Stringari, we compute the mode frequency as follows: for each value of the dimensionless density at the center of the trap $n_1(0)a_{1}$, (i) we fit the equation of state (for $n_1(z)$ varying between $0$ and $n_1(0)$) with either the $\alpha-p$ or the 3 parameters quasi-polynomial models; (ii) the zero order mode frequency is then obtained by inserting the value of the best set of parameters in the formula giving the mode frequency for the model; (iii) we then compute the first order perturbation correction to the mode frequency with equation (\[eqcorr\]). Figure \[nu2pro\] shows the results of four different calculations of the lowest compressional mode using respectively the $\alpha-p$ model and the 3 parameters quasi-polynomial model without perturbative correction, and the same models corrected by first order perturbation theory. Actually, at this scale, it is hard to see any difference between these calculations, except for the $\alpha-p$ model which gives a slightly lower mode frequency. to 70mm[ ]{} In order to compare the various calculations, we plot the same data in a different manner, see figure \[diffpro\]. As discussed in [@FLC] we expect the corrected 3 parameters model to be the most precise of our results. We therefore take this corrected model as a reference and plot the differences between the mode frequencies calculated within the three other approaches ($\alpha-p$ , corrected $\alpha-p$ and 3 parameters quasi-polynomial models) and this reference, as shown in figure \[diffpro\]. We can see on this figure that, once corrected, the results of the $\alpha-p$ and the 3 parameters quasi-polynomial models agree remarkably well, at the absolute $10^{-3}$ level for $\nu^2$. As already mentioned, the $\alpha-p$ model is exact in the 1D mean-field and in the Tonks-Girardeau limits where it predicts $\nu^2=3$ and $\nu^2=4$, respectively. In between these limits, the uncorrected $\alpha-p$ model has some difficulties in predicting precisely the correct value of the mode frequency. For a discussion of this issue see [@FLC]. to 70mm[ ]{} Next we want to compare our most precise result (the corrected 3 parameters quasi-polynomial model) with the results of Menotti and Stringari [@MS], who calculated the lowest compressional mode frequency using a sum rule technique [@dgps], which is basically a variational approach. This method is known to provide an exact upper bound to the lowest mode frequency. Both results are shown in figure \[nu2mspro\]. It is seen that they agree very well over the whole range of $n_1 (0)a_{1}$. to 70mm[ ]{} DISCUSSION ========== In this paper, we used a recently developed method to solve the linearized hydrodynamic equations of the trapped Lieb-Liniger gas at $T \simeq 0$. We obtained the collective modes with high precision and especially studied the lowest compressional mode which is relevant to current experiments. Moritz et al. [@Zurich] recently measured this mode in a trapped 1D Bose gas in the mean-field regime. They found $\nu^2\simeq 3$ when the gas is degenerate, in agreement with our calculations. As first emphasized in [@MS], measuring the lowest compressional mode should allow to reveal the transition from the 1D mean-field to the Tonks-Girardeau regime, where fermionization is expected. More generally, we showed, in the particular case of the trapped Lieb-Liniger gas, how the hydrodynamic approach allows to obtain collective modes even in a strongly interacting system. We also checked that the sum rule approach used by Menotti and Stringari [@MS] indeed gives very good results for the trapped 1D Bose gas. For a discussion of a system where this is not necessarily true see [@FLC]. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== We are grateful to Sandro Stringari, Chiara Menotti and Dima Gangardt for many stimulating discussions. Laboratoire Kastler Brossel and Laboratoire de Physique Statistique are Unités de Recherche de l’École Normale Supérieure (ENS) et des Universités Paris 6 et Paris 7, associées au Centre National de la Recherche Scientifique (CNRS). Schreck, F., et al., 2001, Phys. Rev. Lett., [87]{}, 080403. Görlitz, A., et al., 2001, Phys. Rev. Lett., [87]{} , 130402. Moritz, H., Stöferle, T., Köhl, M., and Esslinger, T., 2003, arXiv:cond-mat/0307607. Lieb, E.H., and Liniger, W., 1963, Phys. Rev., [130]{}, 1605. Combescot, R., and Leyronas, X., 2002, Phys. Rev. Lett., [89]{}, 190405. Menotti, C., and Stringari, S., 2002, Phys. Rev. A, [66]{}, 043610. Fuchs, J.N., Leyronas, X., and Combescot, R., 2003, arXiv:cond-mat/0305647, to appear in Phys. Rev. A. Olshanii, M., 1998, Phys. Rev. Lett., [81]{}, 938. We use a definition of the effective 1D scattering length similar to that of Menotti and Stringari [@MS]. There is a difference in sign with the definition of Olshanii [@Olshanii]. Petrov, D.S., Shlyapnikov, G.V., Walraven, J.T.M., 2000, Phys. Rev. Lett., [85]{}, 3745. Gross, E.P., 1961, Nuovo Cimento, [20]{}, 454; Pitaevskii, L.P., 1961, J. Exp. Theor. Phys., [40]{}, 646. Girardeau, M., 1960, J. Math. Phys., [1]{}, 516. We thank Chiara Menotti for providing us with her data. Dalfovo, F., Giorgini, S., Pitaevskii, L.P., and Stringari, S., 1999, Rev. Mod. Phys., [71]{}, 463. Haldane, F.D.M., 1981, Phys. Rev. Lett., [47]{}, 1840. Minguzzi, A., Vignolo, P., Chiofalo, M.L., Tosi, M.P., 2001, Phys. Rev. A, [64]{}, 033605.
--- abstract: 'This paper presents a millisecond-level look-ahead control algorithm for energy storage. The algorithm connects the optimal control with the Lagrangian multiplier associated with the state-of-charge constraint. It is compared to solving look-ahead control using a state-of-the-art convex optimization solver. The paper include discussions on sufficient conditions for including the non-convex simultaneous charging and discharging constraint, and provide upper and lower bounds for the primal and dual results under such conditions. Simulation results show that both methods obtain the same control result, while the proposed algorithm runs up to 100,000 times faster and solves most problems within one millisecond. The theoretical results from developing this algorithm also provide key insights into designing optimal energy storage control schemes at the centralized system level as well as under distributed settings.' author: - 'Bolun Xu, Magnus Korpås, Audun Botterud, Francis O’Sullivan [^1]' bibliography: - 'IEEEabrv.bib' - 'literature.bib' title: A Lagrangian Policy for Optimal Energy Storage Control --- Energy systems, Numerical algorithm, Predictive control for nonlinear systems Introduction ============ Energy storage devices such as batteries are key resources in future energy systems due to their flexibility and fast response speed, and their convenient installations as either large-scale bulk units or as distributed resources. Most real-time energy storage operations are optimized using predictive control, with applications such as economic dispatch [@korpas2006operation], frequency control [@li2016connecting], voltage control [@zhang2015optimal], renewable integration [@khalid2010model], energy arbitrage [@krishnamurthy2018energy], peak shaving [@shi2018using], electric vehicle charging [@xu2016hierarchical], or a combination of several aforementioned applications [@megel2014scheduling]. These predictive control strategies solve a multi-period optimization problem over a look-ahead horizon at each control step, obtaining the control and state profile over the entire horizon but only applies the first control result, the problem is then updated with a new horizon and state information for the next control step. The challenge of using look-ahead control in practice is trading off optimality with computational tractability, as a longer look-ahead horizon incorporates more future information and thus improves solution optimality, but increases the computational challenge significantly. For example, real-time economic dispatches in power systems are typically solved over a single period or with a look-ahead horizon less than one hour [@epri_2016]. However, power system operations have strong daily patterns due to load and weather variations, such as charging storage from solar power during the day and discharge during the night. Thus, being able to incorporate a look-ahead horizon over one day or even longer is crucial for the future power system, but solving such problems over the scale of a realistic power system is extremely computationally challenging [@zhao2018multi], especially binary variables must be introduced in certain application to prevent simultaneous charging and discharging, making the problem non-convex [@castillo2013profit]. In addition, future uncertainties in power systems are often modeled with scenarios [@wang2011wind], and modeling uncertainties from multiple sources can easily lead to hundreds of scenarios that makes it almost impossible to solve look-ahead economic dispatch with conventional optimization solvers. While methods such as stochastic dual dynamic programming [@papavasiliou2018application] reduce the solution complexity by introducing inter-temporal and scenario decomposition, the computation is still difficult and requires significant memory usage. On the other hand, the optimal control problem must be solved within a reasonable timescale to fully utilize the fast response speed of energy storage devices. For example, a battery ramps from zero to full discharge power within milliseconds [@gao2002dynamic] thus, a scheme that takes seconds or even minutes to update the control decision is not appropriate for controlling batteries. Solving storage control from the dual problem is more effective than dealing with the primal problem directly since the storage has only a single state variable with upper and lower bounds. Cruise et a l. [@cruise2014optimal] concluded the storage control problem can be solved using a search algorithm based on the binding conditions on the state-of-charge, and Hashimi et al [@hashmi2017optimal] has developed an algorithm for energy storage price arbitrage with quadratic time complexity, based on solving the dual problem. Comparably, the technical contributions of this paper and the main advantages of the proposed algorithm is summarized as follows: 1. We show that energy storage control with a generalized time-varying objective functions can be solved in worst-case linear time complexity and constant space complexity, with respect to the look-ahead horizon. 2. We conclude the optimal control condition for energy storage without having to go through the full look-ahead horizon, i.e., the current control is optimal with respect to any future realizations that may not be included in the current look-ahead window. 3. We derive a sufficient condition for the occurrence of simultaneous charging and discharging, and provide upper and lower bounds for the prime and dual results under such non-convex conditions. The rest of this paper is organized as follows: Section II formulates the problem; Section III presents main analytical results and the algorithm; Section IV demonstrates numerical results; and Section V concludes the paper. Formulation and Preliminaries ============================= Problem Formulation ------------------- We consider a time period $t\in \mathcal{T}=\{1,\dotsc, T\}$ where $t=1$ is the current control step and $t=2$ to $T$ is the look-ahead horizon. The optimal control profile $p^_t$ is a minimizer to the following multi-period optimization problem \[mso\] $$\begin{gathered} p^_t \in \arg \min_{p_t}\; \sum_{t=1}^T O_t(p_t) + C_T(e_T) \label{mso:obj}\\ \text{s.t.}\nonumber\\ p_t = p^+_t - p^-_t \label{mso:c1}\\ p^+_t = 0 \text{ or } p^-_t = 0 \;\forall t\in \mathcal{T} \label{mso:c0}\\ e_t - e_{t-1} = -p_t^+/\eta + p_t^-\eta: \theta_{t-1} \label{mso:c2} \end{gathered}$$ $$\label{mso:c3} \begin{aligned} p^+_t &\geq 0 : \underline{\mu}^+_t\,, \\ p^-_t &\geq 0 : \underline{\mu}^-_t\,, \\ e_t &\geq 0 : \underline{\nu}_t\,, \end{aligned} \quad \begin{aligned} p^+_t &\leq P : \overline{\mu}^+_t \\ p^-_t &\leq P : \overline{\mu}^-_t \\ e_t &\leq E : \overline{\nu}_t \end{aligned}$$ where 1. $O_t(\cdot)\|\mathbb{R}\to\mathbb{R}$ is a scalar time-varying convex objective function. Its derivative is denoted as $o_t = \dot{O}_t$. 2. $C_T(\cdot)\|\mathbb{R}\to\mathbb{R}$ is the terminal cost function of the end state of charge $e_T$. $C_T$ is also convex and its derivative is denoted as $c_T = \dot{C}_T$. Note that $C_T$ can also be used to model the operation beyond $T$ via dynamic programming [@bertsekas2005dynamic]. 3. $p_t$ is the control decision variable and it is the energy dispatched from the storage during the time period $t$. 4. $p^+_t$ is the positive (discharge) component of $p_t$. 5. $p^-_t$ is the negative (charge) component of $p_t$. 6. $e_t$ is the state of charge (SoC) at the end of time period $t$, subjects to an initial value of $e_0$. 7. $\eta \in (0,1]$ is the storage charge and discharge efficiency. 8. $P \in \mathbb{R}^+$ is the maximum energy that can be charged or discharged into the storage during a single period. 9. $E \in \mathbb{R}^+$ is the maximum energy that can be stored in the storage. 10. $p^_t$ is a set of minimizers to the optimization problem. 11. $\theta_{t}$ is the Lagrangian multiplier associated with the SoC dynamic, its physical meaning is the marginal value of SoC at the end of time $t$ over the future operation $[t+1, T]$. 12. $\underline{\mu}^+_t$, $\overline{\mu}^+_t$, $\underline{\mu}^-_t$, $\overline{\mu}^-_t$, $\underline{\nu}_t$, $\overline{\nu}_t$ are positive dual variables associated with inequality constraints. The objective function minimizes the total operating cost over the period $[1, T]$. Constraint divides the control $p_t$ into a positive component $p^+_t$ and a negative component $p^-_t$ in order to model the efficiency difference during charge and discharge in the SoC evolution constraint. is the non-simultaneous charging and discharging constraint that enforces the storage to only charge or discharge at any given time point. models the SoC evolution subjects to efficiencies. Power and energy ratings are modeled in . \ \ Karush-Kuhn-Tucker conditions ----------------------------- The results in this paper are primarily based on the use of the Karush-Kuhn-Tucker (KKT) conditions [@boyd2004convex], which are listed below for ( the non-simultaneous charging and discharging constraint is non-convex and is excluded from the KKT condition below, treatment of this constraint will be discussed later): \[mfv\] $$\begin{aligned} o_t(p^_t) + \overline{\mu}^+_t - \underline{\mu}^+_t + \theta_{t-1}/\eta &= 0 : p^+_t \label{mfv_eq2}\\ -o_t(p^_t) + \overline{\mu}^-_t - \underline{\mu}^-_t - \theta_{t-1}\eta &= 0 : p^-_t\label{mfv_eq3}\\ \theta_{t-1} + \overline{\nu}_t - \underline{\nu}_t - \theta_{t} &= 0 : e_t \label{mfv_eq1}\\ \theta_{T} + c_T(e^_T) &= 0 : e_T \label{mfv_eq4}\end{aligned}$$ and the complimentary slackness conditions associated with the inequality dual variables: $$\label{csc} \begin{aligned} \underline{\mu}^+_t [p^_t]^+ &= 0 \,, \\ \underline{\mu}^-_t [-p^_t]^+ &= 0 \,, \\ \underline{\nu}_t e^_t &= 0\,, \end{aligned} \quad \begin{aligned} \overline{\mu}^+_t (P-[p^_t]^+) &= 0 \,, \\ \overline{\mu}^-_t (P-[-p^_t]^+) &= 0 \,, \\ \overline{\nu}_t (E-e^_t) &= 0\,, \end{aligned}$$ and $\underline{\mu}^+_t$, $\overline{\mu}^+_t$, $\underline{\mu}^-_t$, $\overline{\mu}^-_t$, $\underline{\nu}_t$, $\overline{\nu}_t \geq 0$. Note that we replaced the use of $p^+_t$ and $p^-_t$ with $p_t$ since $\partial p^+_t = \partial p_t$, $\partial p^-_t = -\partial p_t$, $p^+_t = [p_t]^+$ and $p^-_t = [-p_t]^+$, where $[x]^+ = \max\{0,x\}$ is the positive value function. Main Results ============ We start by relaxing constraint so that the rest of the problem is convex. Thus we establish a closed-form connection between the primal and dual problem in Proposition \[pro:pi\]. We then present Theorem \[th1\] on identifying the equality relationship between $\theta_0$ and any real number $x\in\mathbb{R}$ using numerical simulation, and develop a binary search algorithm that finds the dual result $\theta_0$ and , thus, the primal result $p^_1$. Then we discuss how we can bound the result when it is necessary to incorporate the non-convex non-simultaneous charging and discharging constraint using the proposed algorithm. Optimal Control Policy ---------------------- We define the policy $\pi$ that calculates a storage control decision $p{^\mathrm{\pi}}_t(x)$ for time $t$ from an input $x\in\mathbb{R}$ as $$\begin{gathered} p^+_t(x) = \Big[\varphi_t(-x/\eta)\Big]^P_0,\; p^-_t(x) = \Big[-\varphi_t(-x\eta)\Big]^P_0 \\ p^{\pi}_t(x) = p^+_t(x) - p^-_t(x)\label{es:policy}\end{gathered}$$ where $[x]^y_z = \max\{\min\{x,y\},z\}$ saturates $x$ between $y$ and $z$ ($z\leq y$), and $\varphi_t(x) : \mathbb{R}\to \mathbb{R}$ is the inverse function of $o_t$ (derivative of $O_t$) as $$\begin{aligned} \varphi_t(x) &= \sup\{y\in\mathbb{R} \| o_t(y) \leq x\}\,.\end{aligned}$$ Note that $\varphi_t$ is an alternative definition of the inverse function to $o_t$ while compatible with a piecewise linear $O_t$. The following proposition states that we can obtain the optimal control $p^_t$ by using the Lagrangian multiplier $\theta_{t-1}$ as input to policy $\pi$: \[pro:pi\] Policy $\pi$ is a minimizer to problem when using the Lagrangian $\theta_{t-1}$ as the input, i.e., $p^_t = p^{\pi}_t(\theta_{t-1})$. We start by rewriting the KKT condition associated with $p^+_t$ as $$\begin{aligned} o_t(p^_t)+\theta_{t-1}/\eta &< 0 \text{ only if } p^_t = P \\ o_t(p^_t)+\theta_{t-1}/\eta &> 0 \text{ only if } p^_t = 0 \\ o_t(p^_t)+\theta_{t-1}/\eta &= 0 \text{ if } 0\leq \varphi_t(-\theta_{t-1}/\eta)\leq P\end{aligned}$$ where we substitute the complementary slackness condition into that replaces $\overline{\mu}^+_t$ and $\underline{\mu}^+_t$. It is now trivial to see that we can calculate $p^_t$ as $\varphi_t(-\theta_{t-1}/\eta)$ and limiting the result between 0 and $P$, hence $$\begin{aligned} p^+_t = \max\{0, \min\{P, \varphi_t(-\theta_{t-1}/\eta)\}\}\,.\end{aligned}$$ We repeat the similar process for $p^-_t$ with , and use $p_t = p^+_t-p^-_t$ which gives us the result in Proposition \[pro:pi\]. The following corollary supplements that with $\theta_{0}$ we can obtain $p^_1$ as well as a series of consecutive optimal control decisions by recording the accumulated sum of the control results $\sigma_t$ defined as $$\begin{aligned} \label{eq:th_soc} \sigma_t(x) =& \sigma_{t-1}(x) - [p^{\pi}_t(x)]^+/\eta - [p^{\pi}_t(x)]^-\eta\,,\end{aligned}$$ with the initial value $\sigma_0(x) = e_0$, where $[x]^+ = \max\{0,x\}$ is the positive value function, and $[x]^- = \min\{0,x\}$ is the negative value function. $\sigma_t(x)$ emulates the SoC evolution but using the control result $p{^\mathrm{\pi}}_t(x)$ which may not be optimal. Another difference is that $\sigma_t(x)$ is not limited between $[0,E]$, instead, whether any $\sigma_t(x)$ falls above $E$ or below $0$ is an indicator on the optimality of $p{^\mathrm{\pi}}_t(x)$, as defined by the following corollary: \[cor:pi\] $p^_t=p{^\mathrm{\pi}}_t(\theta_0)$ if $0 \leq \sigma_{\tau}(\theta_0) \leq E$ $\forall$ $\tau\in[1,t]$. Corollary \[cor:pi\] means that we can maintain optimal control by using $\theta_0$ as the input to for control steps beyond $t=1$ if all previous $\sigma_t(\theta_0)$ are within the SoC constraint. Corollary \[cor:pi\] is based on Proposition \[pro:pi\] and the KKT condition associated with $e_t$ in that the $\theta_t$ value will not change if both $\overline{\nu}_t$ and $\underline{\nu}_t$ are zeros, indicating $0\leq e^_t \leq E$ and $\sigma_t(x) = e^_t$. This corollary is thus proved. Main Theorem on Finding Lagrangian Dual --------------------------------------- \[th1\]Given $x\in\mathbb{R}$, its equality relationship with respect to $\theta_0$ can be determined as 1. If* a) $\sigma_t(x)$ reaches upper bound first, i.e., $\exists \tau\in\mathcal{T} \text{ s.t. } \sigma_{\tau}(x) > E \text{ and } 0\leq\sigma_{\gamma}(x) \leq E$ $\forall \gamma \in [1,\tau)$; or b) $\sigma_t(x)$ reached neither bound and $x > -c_T(\sigma_T(x))$; then $x\geq \theta_0$; 2. If a) $\sigma_t(x)$ reaches lower bound first, i.e., $\exists \tau\in\mathcal{T} \text{ s.t. } \sigma_{\tau}(x) < 0 \text{ and } 0\leq\sigma_{\gamma}(x) \leq E$ $\forall \gamma \in [1,\tau)$; or b) $\sigma_t(x)$ reached neither bound and $x < -c_T(\sigma_T(x))$; then $x\leq \theta_0$; 3. If $\sigma_t(x)$ reached neither bound and $x = -c_T(\sigma_T(x))$, then $x = \theta_0$. Proof of this theorem is deferred to Appendix. The intuition is that the Lagrangian dual is the price of the stored energy, its value does not change despite the change with the SoC evolution, except reaching either the upper or lower SoC bound. The SoC series driven by the optimal dual value should never exceed the SoC bounds as the dual value itself reflects the constrained storage capacity. If the SoC exceed the upper SoC bound, it means SoC value is over estimated as the storage does not have enough capacity to store the excessive energy, hence we picked an $x$ that is higher than the optimal Lagrangian dual value. Vice versa, if the SoC exceed the lower bound, meaning we under estimated the dual value. Solution Algorithm {#app:alg} ------------------ We design a binary search algorithm that finds $\theta_0$ according to Theorem \[th1\], thus we find $p^_1=p{^\mathrm{\pi}}_1(\theta_0)$ (Proposition \[pro:pi\]) as well as some consecutive optimal control actions (Corollary \[cor:pi\]) without needing to explicitly solve Problem . The algorithm requires a preset search accuracy $\epsilon$ and is described as follows: 1. Initialize a search range $L$ and $R$ with which we are confident that $L \leq \theta_0 \leq R$; 2. Set $x$ to $(L+R)/2$. If $R-L < \epsilon$, return $x$ as the optimal Lagrangian dual value and $p{^\mathrm{\pi}}_t(x)$ as the optimal storage control up to time step $t$; 3. Run the following iterative simulation 1. Set $1\to t$ and $e_0\to \sigma_0(x)$; 2. Calculate $p{^\mathrm{\pi}}_t(x)$ using Eq. ; 3. Calculate $\sigma_t(x)$ using Eq. ; 4. If $\sigma_t(x) \geq E$, set $x \to R$, go to Step 2); 5. If $\sigma_t(x) \leq 0$, set $x \to L$, go to Step 2); 6. If $t<T$, set $t+1 \to t$ and go to Step b); 7. If $x \geq -c_T(\sigma_T(x))$, set $x \to R$; 8. If $x < -c_T(\sigma_T(x))$, set $x \to L$; 4. Go to Step 2). An example of a confident search range is that we can assume stored energy always has a positive value and choose $L = 0$ and $R = \max\{o_t(p)/\eta\|t\in\mathcal{T}, p\in [-P, P]\}$. This algorithm achieves the following complexity results: 1. Constant space complexity:* The algorithm achieves $O(1)$ space complexity with respect to the search range and the look-ahead duration $T$, because the equality relationship between $x$ and $\theta_0$ can be identified using only the current simulation result $\sigma_t(x)$ so that previous simulation results are not required to be stored. 2. Worst-case linear run-time complexity: The algorithm achieves a worst-case $O(n)$ complexity with respect to the look-ahead horizon $T$ since the worst-case scenario is to simulate all operations steps from $t=1$ to $T$ during each search, but may terminate before reaching $T$ as stated in step 3-d and 3-e. It also achieves $O(\log n)$ time complexity with respect to the search range for using a binary search algorithm. Non-simultaneous Charging and Discharging ----------------------------------------- By far we have concluded the optimal storage control when relaxing the non-simultaneous charging and discharging constraint . A sufficient condition for relaxing this constraint without sacrificing result optimality is illustrated in the following proposition: A sufficient condition for simultaneous charging and discharging to happen is the Lagrangian dual being negative, i.e., if $\theta_{t-1} \geq 0$ then $p^+_tp^-_t = 0$. From , it is trivial to see that for simultaneous charging and discharge to happen, both terms in must take non-zero values, hence $$\begin{gathered} \varphi_t(-x\eta) < 0 < \varphi_t(-x/\eta)\end{gathered}$$ and since $\varphi_t$ is a monotonic increasing function and $0 < \eta \leq 1$, it follows $$\begin{aligned} -x\eta < -x/\eta\end{aligned}$$ hence $x$ must be less than zero. Recall that $\theta_t$ is the value of the stored energy, hence $\theta_t$ being negative indicates the stored energy has a negative value, i.e., we have an intention to store as less energy as possible. Thus, when , the storage can charging and discharging at the same time and use round-trip efficiency loss to consume excessive energy even when the storage is full and has no more storage space. This intuition is useful when deciding whether simultaneous charging and discharging should be considered when formulating the problem, for example, this constraint should be considered in price arbitrage for markets with frequent negative prices. A common method for enforcing non-simultaneous charging and discharging is to add auxiliary binary variables $\bm{v}=\{v_t\}$ such that the storage can only charge or discharge at one time, as $$\begin{aligned} \label{mso:c0b} p^+_t <= Pv_t,\; p^-_t <= P(1-v_t),\; v_t \in \{0,1\}\end{aligned}$$ making a mixed-integer programming problem, the Lagrangian dual can thus be calculated given a set of fixed $\bm{v}$. We assume set $\mathcal{V}$ as the set of all reasonable charging status results, in which $v_t$ may be either 0 or 1 during all periods when simultaneous charging and discharge occur, i.e., when must be enforced. It is worth noting that the optimal result for $v\bm{v}$ must be in $\mathcal{V}$. We denote $\theta_0(\bm{v})$ as the resulting Lagrangian dual associated with the charging status $\bm{v}$, then the following proposition stands: \[pro:dual\_b\] For all $\bm{v}\in\mathcal{V}$, $\underline{\theta}_0 \leq \theta_0(\bm{v})\leq \overline{\theta}_0$, where $\underline{\theta}_0$ is calculated using Algorithm 1 by replacing with $$\begin{aligned} p^{\pi-}_t(x) = p^+_t(x)\mathbf{1}_{[p^-_t(x) == 0]} - p^-_t(x)\label{es:policy2} \end{aligned}$$ and $\underline{\theta}_0$ is calculated using Algorithm 1 by replacing with $$\begin{aligned} p^{\pi+}_t(x) = p^+_t(x) - p^-_t(x)\mathbf{1}_{[p^+_t(x) == 0]}\label{es:policy3} \end{aligned}$$ where $\mathbf{1}_{[x]} = \{\text{ 0 if $x$ is true and 1 otherwise} \}$ is the indicator function. First note that compared to , enforces the storage to charge whenever charging and discharging components are both non-zero. Thus when using to simulate the battery operation in Algorithm 1, the resulting SoC must always be lower than using any $\bm{v}\in\mathcal{V}$. Thus according to Theorem \[th1\], the resulting dual $\theta_0$ must be no smaller than any dual $ \theta_0(\bm{v})$ using charging status $\bm{v}\in\mathcal{V}$. Vice versa, when using , the battery prefer discharging over charging, resulting in a lower bound for $\theta_0$. Note that in and , $p^{\pi}_t(x)$ is still a monotonic decreasing function to $x$ since $p^+_t(x) \geq 0$ and $p^-_t(x) \geq 0$ for all $x$, $p^+_t(x)$ is an decreasing function and $p^-_t(x)$ is an increasing function. Hence the SoC series is still monotonic increasing with respect to $x$, and the convergence optimality of Algorithm 1 will not be effected. And the result on the dual binding can be extended to bind the primal control results: \[pro:prim\_b\] Let $\underline{p}_t = p^{\pi-}_1(\underline{\theta}_{0})$ and $\overline{p}_t = p^{\pi+}_t(\overline{\theta}_{0})$, and $p^_t$ is the optimal control, then $\underline{p}_t \leq p^_t \leq \overline{p}_t$ for all $t\in[1,\;\tau]$. First given the dual $\theta_0$, the optimal control $p^_1$ is either $p^+_t(\theta_0)$ or $-p^-_t(\theta_0)$ as in , since the battery either charges or discharges. Then according to Proposition \[pro:dual\_b\], we know the dual must be within $[\underline{\theta}, \overline{\theta}]$. Then we have $$\begin{aligned} \underline{p}_t \leq -p^-_t(\theta_0) \leq p^+_t(\theta_0) \leq \overline{p}_t\end{aligned}$$ which proved this Proposition. Hence Proposition \[pro:dual\_b\] and Proposition \[pro:prim\_b\] provide bounds for the primal and dual result of the non-convex battery control problem using the proposed algorithm that has worst-case linear time complexity. Numerical Simulation ==================== We use randomly generated data sets to compare the proposed algorithm to solving Problem \[mso\] with different objectives using Gurobi [@gurobi] (model generated using CVX [@cvx]). All simulations are performed in Matlab [@matlab] on a 2.3 GHz machine with 16GB memory.The storage parameter is set as $P = 1$ p.u., $E = 4$ p.u., $e_0 = 2$ p.u., $\eta = 0.92$, and the terminal cost function is set to $C_T(e_T)= (E-e_T)^2/2$. The accuracy of the search algorithm is set to $\epsilon = 10^{-3}$. Piece-wise linear objectives ---------------------------- [l r r r r r r]{} ------------------------------------------------------------------------ & &\ Trials & $\theta_0$ & $p^_1$ & cpu \[ms\] & $\theta_0$ & $p^_1$ & cpu \[ms\]\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ &\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ 1 & 17.53 & 0.6 & 408.8 & 17.53 & 0.6 & 0.1\ 2 & 16.82 & 0.0 & 278.1 & 16.82 & 0.0 & 0.1\ 3 & 14.59 & 0.3 & 274.5 & 14.59 & 0.3 & 0.1\ 4 & 14.21 & 0.1 & 281.7 & 14.21 & 0.1 & 0.1\ 5 & 7.27 & -0.5 & 283.7 & 7.27 & -0.5 & 0.1\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ &\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ 6 & 18.67 & 0.0 & 1851.5 & 18.66 & 0.0 & 0.1\ 7 & 20.28 & 0.8 & 1791.5 & 20.28 & 0.8 & 0.2\ 8 & 19.50 & 0.0 & 1788.2 & 19.50 & 0.0 & 0.1\ 9 & 17.44 & -0.7 & 1868.4 & 17.44 & -0.7 & 0.2\ 10 & 19.00 & 0.0 & 1868.7 & 19.00 & 0.0 & 0.2\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ &\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ 11 & 19.09 & 0.6 & 18497.7 & 19.09 & 0.6 & 0.1\ 12 & 19.95 & 0.8 & 19108.2 & 19.95 & 0.8 & 0.2\ 13 & 19.44 & 0.0 & 18786.1 & 19.44 & 0.0 & 0.1\ 14 & 18.91 & -1.0 & 19263.2 & 18.91 & -1.0 & 0.2\ 15 & 19.55 & 0.0 & 19080.4 & 19.55 & 0.0 & 0.2\ \[tab:sim4\] In this section the proposed algorithm is compared with Gurobi using piecewise linear objective function inspired by the supply curves in power system economic dispatches [@kirschen2018fundamentals]. The derivative of the objective function $o_t$ is written as $$\begin{aligned} o_t(p) = c_j \text{ if } q_{j-1} \leq p < q_j\end{aligned}$$ where $j\in [1,J]$ is the piecewise segment index, $J$ is the number of segments, $c_j$ is the marginal cost (derivative) of the system when $p$ is between quantities $q_{j-1}$ and $q_j$, and the objective is convex if $c_i \leq c_j$ and $q_i \leq q_j$ for all $i<j$, $i,j\in [1,J]$. Some examples of the generated cost curve are plotted in Fig. \[fig:pwl\_example\]. \ Similar to the quadratic results, we test the proposed algorithm and Gurobi using different settings and the results are demonstrated in Table \[tab:sim4\], where trials 1–5 have 10 time steps and 100 cost segments $T=10$, $J=100$, trials 6–10 have 10 time steps and 1,000 cost segments $T=10$, $J=1000$, trials 11–15 have 100 time steps and 1,000 cost segments $T=100$, $J=1000$. The result shows the proposed algorithm obtains the same results in all trials compared to Gurobi, while being hundreds or even thousands of times faster. In particular, in trials 11-15 Gurobi needs around 18 seconds to complete the computation, while the proposed algorithm finishes below 1ms. In Fig. \[fig:comp\], we further test the computation speed of both methods in solving look-ahead economic dispatches over the size of realistic power systems with 5000 cost segments per five minute dispatch interval. The result shows that the computation time of Gurobi increases significantly with respect to the look-ahead horizon, and in particular at the 6 hour look-ahead, the problem takes more than 5 minutes to solve which is not feasible since the economic dispatch must be calculated within 5 minutes. In contrast, with our proposed algorithm, the average solution speed is below 0.5 milliseconds for look-ahead horizons up to 8 hours, providing a speed-up up to 100,000 times. Negative Lagrangian dual example -------------------------------- [l r r r r r r]{} ------------------------------------------------------------------------ & &\ Trials & $\underline{\theta}_0$ & ${\theta}^_0$ & $\overline{\theta}_0$ & $\underline{p}_1$ & $p^_1$ & $\overline{p}_1$\ ------------------------------------------------------------------------ ------------------------------------------------------------------------ 1 & -10.7 & -10.6 & -10.2 & -0.14 & 0.10 & 0.11\ 2 & -12.9 & -12.5 & -11.8 & 0.49 & 0.53 & 0.56\ 2 & -8.5 & -8.2 & -8.2 & -0.17 & -0.17 & -0.16\ 4 & -4.4 & -4.4 & -4.0 & -1.00 & -1.00 & -1.00\ 5 & -15.0 & -14.6 & -14.5 & 0.89 & 0.91 & 0.97\ \[tab:sim1\] We consider the following quadratic objective function $$\begin{aligned} O_t = \frac{\alpha_t}{2} (\beta_t - p)^2\end{aligned}$$ where $\alpha_t$ are randomly generated between $[0, 10]$, and $\beta_t$ between $[-10,0]$. Recall that negative sign is for charging the battery, hence this is a generation tracking problem where the storage wishes to absorb as much energy as possible. This will result in negative dual prices for the storage and simultaneous charging and discharging will occur if not constrained. Table \[tab:sim1\] shows the simulation result for five trails including upper and lower bounds on the dual and control, where ${\theta}^_0$ and $p^_1$ is calculated using Gurobi with mixed-integer quadratic programming under default settings, solving using the integer constraint for enforcing constraint . In all test trails, the primal and dual result fall within the calculated range. In terms of computation speed, the proposed method all solves less than 1 millisecond, while the benchmark method using Gurobi may need up to several minutes to solve the problem depending on the problem size (thousands of steps), due to solving a mixed-integer quadratic programming problem. Conclusion ========== This paper proposed a novel algorithm for solving look-ahead control for energy storage. The numerical results illustrate that the algorithm provides computation speed in milliseconds for controlling a single energy storage device over an extended planning period. In future research, we plan on expanding this method to controlling multiple energy storage devices subject to network constraints. Moreover, using the generalized terminal state function we plan on incorporating this algorithm into scenario-based stochastic programming or dynamic programming. In addition, our results connects the optimal control with the Lagrangian multiplier associated with the state-of-charge constraint, which we will further explore to provide key insights into designing future electricity pricing and distributed control schemes. Proof of Theorem 1 ================== We start by showing that when moving to the next control step, the value of the Lagrangian $\theta_t$ only changes after $e^_t$ is reaching the upper or lower SoC bound, or more specifically: \[eq:theta\] $$\begin{aligned} \theta_{t-1} = \theta_t\quad &\text{if $\overline{\nu}_{\tau} = \underline{\nu}_{\tau} = 0$}\\ \theta_{t-1} < \theta_t\quad &\text{if $\overline{\nu}_{\tau} > 0$}\\ \theta_{t-1} > \theta_t\quad &\text{if $\underline{\nu}_{\tau} > 0$} \end{aligned}$$ Hence, it is trivial to see that if $0 \leq \sigma_{\tau}(\theta_0) \leq E$ $\forall$ $\tau\in[1,t]$ then $\theta_{t-1} = \theta_0$, leading $p{^\mathrm{\pi}}_t(\theta_{t-1}) = p{^\mathrm{\pi}}_t(\theta_{0}) = p^_t$ according to Proposition \[pro:pi\]. We will do the proof separately for three possible cases of $e^_t$: 1) $e^_t$ never reaches upper or lower bound with all $\underline{\nu}_t$ and $\overline{\nu}_t$ equal to zero; 2) $e^_t$ reached upper bound first; 3) $e^_t$ reached lower bound first. These three cases are illustrated in Fig. \[fig:th1\]. ### $0<e^_t < E$ $\forall$ $t\in [1,T]$ This cover the cases when $e^_t$ never reached the upper or lower bound, which from we know $\theta_0 = \theta_1 = \dotsc = \theta_T$, hence $p^_t = p{^\mathrm{\pi}}_t(\theta_0)$ and $e^_t = \sigma_t(\theta_0)$ for all $t\in[1,T]$, and in particular for $t=T$ we have $$\begin{aligned} \label{eq:th1_T} \theta_0 = \theta_T = -c_T(e^_T) = -c_T(\sigma_T(\theta_0))\end{aligned}$$ according to and the aforementioned result. Since $O_t$ and $c_T$ are convex, $o_t$, $c_T$, and $\varphi_t$ (inverse of $o_t$) are monotonic increasing functions, it follows \[eq:th1a\] $$\begin{aligned} x &\geq \theta_0 \\ p{^\mathrm{\pi}}_t(x) &\leq p{^\mathrm{\pi}}_t(\theta_0) : \text{convexity, see \eqref{pro:pi}} \\ \sigma_t(x) &\geq \sigma_t(\theta_0) : \text{see \eqref{eq:th_soc}} \\ c_T(\sigma_t(x)) &\geq c_T(\sigma_t(\theta_0)) : \text{$c_T(\cdot)$ both sides } \\ c_T(\sigma_T(x)) &\geq -\theta_0 : \eqref{eq:th1_T} \\ c_T(\sigma_T(x)) &\geq -x : -\theta_0 \geq -x\\ x &\geq -c_T(\sigma_T(x))\end{aligned}$$ meaning if $x \geq -c_T(\sigma_T(x))$ then $x \geq \theta_0 $, thus we proved condition 1-b in the Theorem. Similarly we can prove condition 2-b starting with $x\leq \theta_0$. Also it is trivial to see that if any $\sigma_t(x)$ goes above $E$ then in this case we know $\sigma_t(x) > \sigma_t(\theta_0)$ hence $x > \theta_0$, and vice versa for $\sigma_t(x)$ goes below 0, hence we proved condition 1-a and 2-a. It is also trivial to see that if $x = c_T(\sigma_T(x))$, then the KKT condition is satisfied and $x=\theta_0$, which proves condition 3. Thus all conditions in this theorem are proved for this this case. ### $\exists \tau\in[1,T]$ s.t. $\overline{\nu}_{\tau} > 0$ and $0<e^_t < E$ $\forall$ $t\in [1,\tau)$ This covers the cases when $e^_t$ reached the upper bound first. An example of this case in shown in Fig. \[fig:th1\_example\]. Now from we can conclude if $\exists \tau\in[1,T]$ s.t. $\overline{\nu}_{\tau} > 0$ and $0<e^_t < E$ $\forall$ $t\in [1,\tau)$, then the same condition must be satisfied for all $x\geq \theta_0$, hence condition 1-a is proved. And from we know after reaching the upper bound, all the following $\theta$ values will be greater than $\theta_0$ until $e^_t$ reaches the lower bound or till the end of the operation $T$ (i.e., $e^_t$ never reaches the lower bound). Without loss of generality, let $\gamma$ be the time that $e^_t$ first reaches the lower bound or the end of the operation period, i.e., $e^_t > 0$ $\forall$ $t\in [1,\gamma)$, it follows $$\begin{aligned} x &\leq \theta_0 \\ x &\leq \theta_t\; \forall\; t\in [1,\gamma)\\ p{^\mathrm{\pi}}_t(x) &\geq p{^\mathrm{\pi}}_t(\theta_{t-1}) : \text{see \eqref{pro:pi}} \\ p{^\mathrm{\pi}}_t(x) &\geq p^_t : \text{Proposition~\ref{pro:pi}} \\ \sigma_t(x) &\leq e^_t\end{aligned}$$ which means $\sigma_t(x)$ either will go below $0$ (condition 2-a) or $\sigma_T(x)\leq e^_T$ which leads to $x \leq c_T(\sigma_T(x))$ (condition 2-b) according to , hence we proved this theorem for this case. ### $\exists \tau\in[1,T]$ s.t. $\underline{\nu}_{\tau} > 0$ and $0<e^_t < E$ $\forall$ $t\in [1,\tau)$ This covers the case when $e^_t$ reaches the lower bound first. This is a mirror proof to the previous case while inverting the upper and lower bound logic, hence this proof is omitted. [^1]: B. Xu, A. Botterud, and F. O’Sullivan are with Massachusetts Institute of Technology, MA, USA. M. Korpås is with Norwegian University of Science and Technology, Trondheim, Norway. Contact: {xubolun, audunb,frankie}@mit.edu, magnus.korpas@ntnu.no.
--- abstract: 'The Higgs particle can have dominantly invisible decay in a large class of Majoron models as well as some SUSY models. The LEP signal and mass limit for a Higgs particle undergoing invisible decay are explored. They are found to be very similar to those for the standard model decay. The best signatures and discovery limit for an invisibly decaying Higgs particle at the LHC are also discussed.' --- =-1.85cm =-2.8cm CERN-TH.7221/94 ------------------------------------------------------------------------ width 0pt \ \ [D.P. ROY]{}\ Theory Division, CERN, CH–1211 Geneva 23, Switzerland\ and Theoretical Physics Group, Tata Institute of Fundamental Research, Bombay 400 005, India[^1] CERN-TH.7221/94\ April 1994 (Invited Talk at the XXIXth Rencontres de Moriond on QCD and High Energy Hadronic Interactions, Méribel, France, 19–26 March 1994.) The signatures for Higgs particle detection at LEP and the future hadron colliders (LHC/SSC) have been extensively studied in the framework of the standard model (SM) and its supersymmetric (SUSY) extensions$^{1,2)}$. There exist some extensions of the SM, however, with a qualitatively different signature for the Higgs particle. These extensions are generically called the Majoron models (MM)$^{3-7)}$ and have been quite popular, e.g. in the context of generating neutrino mass. They are characterized by the existence of a Goldstone boson (the Majoron). Since the coupling of this Goldstone boson to the Higgs particle is not required to be small on any theoretical or phenomenological grounds, the Higgs particle could decay into an invisible channel containing a Majoron pair$^{7-9)}$. Indeed the importance of extending the Higgs search to this invisible decay channel has been repeatedly emphasized over the past decade$^{8,9)}$. However, quantitative investigations along this line have only started very recently$^{10-12)}$. The key features shared by essentially all Majoron models is a spontaneously broken global $U(1)$ symmetry and a complex $SU(2) \times U(1)$ singlet scalar field $\eta$ transforming non-trivially under the global $U(1)$. The spontaneous breaking of the global $U(1)$ generates a massless Goldstone boson, the Majoron $J \equiv$ Im $ \eta/\sqrt 2$, and a massive scalar $\eta_R \equiv \mathrm{Re}~ \eta/\sqrt 2$. The latter mixes with the massive neutral component $\phi_R$ of the standard Higgs doublet through a quartic term $\phi^\dagger \phi \eta^\dagger \eta$ in the scalar potential. Thus one has two massive physical scalars $$H = \cos \theta \phi_R + \sin \theta \eta_R \quad \mathrm{and} \quad S = \cos \theta \eta_R - \sin \theta \phi_R ~, \eqno (1)$$ where the mixing angle can always be chosen to lie in the range 0–45$^\circ$, so that the $H$ and $S$ have dominant doublet and singlet components respectively. The above quartic term also generates the following couplings of $H$ and $S$ to the massless Goldstone boson $J$ : $${\cal L} = {(\sqrt 2 G_F)^{1/2} \over 2} \tan \beta \left[M_S^2 \cos \theta S J^2 - M_H^2 \sin \theta H J^2 \right]~, \eqno (2)$$ where $\tan \beta = \langle\phi\rangle/\langle\eta\rangle$ is the ratio of the two vacuum expectation values$^{8)}$. The resulting decay widths of $H, S$ into the invisible channel $(JJ)$ relative to the dominant SM channel $(b \bar b)$ are $$\Gamma_{H \rightarrow JJ}/\Gamma_{H \rightarrow b \bar b} \simeq {1 \over 12} \left({M_H \over m_b}\right)^2 \tan^2 \theta \tan^2 \beta \left(1 - {4m^2_b \over M_H^2} \right)^{-3/2}~, \eqno (3)$$ $$\Gamma_{S \rightarrow JJ}/\Gamma_{S \rightarrow b \bar b} \simeq {1 \over 12} \left({M_S \over m_b}\right)^2 \cot^2 \theta \tan^2 \beta \left(1 - {4m^2_b \over M_S^2} \right)^{-3/2}~. \eqno (4)$$ The large mass ratio $(M_{H,S}/m_b)^2$ on the r.h.s. implies that the invisible decay channel could dominate for $S$ as well as $H$ over a large range of the parameters $\tan \theta$ and $\tan \beta$.[^2] Although eqs. (1)–(4) above were derived for the simplest model$^{3)}$ having 1 singlet and 1 doublet scalar fields, similar considerations hold for those having a larger Higgs content$^{5-8)}$ or a larger global symmetry group than $U(1)$$^{9)}$. It may be added here that the Higgs particles can also decay invisibly in the SUSY models via a pair of lightest superparticles (LSP). For the minimal supersymmetric standard model (MSSM) this invisible decay mode has been shown to dominate only over a tiny range of parameters for the scalar Higgs particles but over a larger range for the pseudoscalar$^{13)}$. Thus it is important to extend the Higgs search strategies at LEP and the LHC to cover the possibility of a dominantly invisible decay. This is simple at LEP, since the dominant channel for the Higgs search is the same for the SM and the invisible decays – i.e. the missing energy channel with one or two jets$^{10-12)}$. It corresponds to the Bjorken production process $$e^+ e^- \buildrel Z \over \rightarrow Z^\ast H \eqno (5)$$ followed by $Z^\ast \rightarrow \nu \bar \nu ~, ~H \rightarrow b \bar b$ for the SM decay and $Z^\ast \rightarrow q \bar q ~,~ H \rightarrow JJ$ for the invisible decay. Indeed the larger branching fraction of $Z^\ast$ into quarks implies a larger event rate for the latter case. Figure 1 shows the expected number of signal events for the two cases$^{11)}$ for the ALEPH data sample of ref. \[2\]. The cuts reduce the signals by only a factor of $\sim 2/3$ in either case while completely eliminating the background. Thus the signal size for the invisible Higgs decay is a factor of 2–3 higher than the SM decay; and the corresponding 95% CL mass limit is higher by $\sim 6$ GeV. This would imply an $H$ mass bound somewhere between these two limits, depending on the relative size of the two decay models (eq. (3)). On the other hand the production cross-section (5) would be suppressed by a factor of $\cos^2 \theta~(\geq 0.5)$ since the $Z$ couples only to the doublet component of the Higgs field. Combining the two effects leads to a $M_H$ bound in the Majoron models, which is within $\pm 6$ GeV of the SM value, irrespective of the model parameters, i.e. $48 \pm 6$ GeV $^{2,11)}$ going up to $60 \pm 6$ GeV with the new ALEPH data$^{12)}$. A similar correlation between the Higgs signatures and discovery limits for the two models is expected to hold for LEP II as well. It may be noted that the dominantly singlet Higgs $S$ of the MM can be arbitrarily light for sufficiently small mixing angle $\theta$, since the production cross-section is suppressed by a factor of $\sin^2\theta$. At the LHC, the missing energy is not measurable in view of the large energy loss along the beam pipe. So one has to convert it into a missing-$p_T~(p\!\!/_T)$ signature by looking at one of the following associated production processes at large $p_T$: $$(i)~H + {\rm jet}, \;\;\; (ii)~H + Z, \;\;\; (iii)~H+W \;\; \mathrm{and} \; (iv)~H + t\bar t~, \eqno (6)$$ followed by the invisible decay of $H$. While the first process has too large a background from $Z (\longrightarrow \nu\bar \nu) +~\mathrm{jet}$, the second and third processes are expected to give viable signatures at the LHC$^{14)}$. Figure 2 shows the Higgs signal from $(ii)$ along with the dominant background in the $\ell^+\ell^-p\!\!/_T$ channel$^{14)}$. For $p\!\!/_T > 200$ GeV, one gets a viable signal size $(\sim 2.5~$fb) and signal/background ratio ($\sim 1/2)$. Figure 3 shows the signal from $(iii)$ along with the irreducible background in the $\ell p\!\!/_T$ channel$^{14)}$. Here the signal size is $\sim 5~$fb and the signal/background ratio is $\sim 1$ for $p\!\!/_T > 200$ GeV. Thus one should be able to probe the intermediate mass range of Higgs (100–200 GeV) even after making allowance for the suppression factor of $\cos^2 \theta$ at the production vertex. For $M_H > 200$ GeV, the $H \rightarrow WW, ZZ$ decay modes are expected to dominate over the invisible $(JJ)$ mode. The fourth process has also been shown to give a signal/background ratio of $\sim 1$ at the LHC$^{16)}$. But the signal size is relatively small $(\sim 0.5~$fb). Besides, this signal is far more demanding on the detector performance, as it requires good $b$ identification as well as reconstruction of $W$ and $t$ masses from hadronic jets. Finally, it may be noted that the signals of Figs. 1–3 are equally applicable to the invisible decay of Higgs scalar into a pair of LSP in the SUSY models. In fact they should be exact in this case since there is no singlet Higgs scalar $S$, i.e. $\cos \theta = 1$. However they do not apply to the pseudoscalar Higgs boson $A$ of SUSY models, since it does not couple to gauge bosons. Only the last process mentioned above is applicable to the invisible decay of $A$ $^{16)}$. 1. See e.g. J.F. Gunion, H. Haber, G.L. Kane and S. Dawson, The Higgs Hunter’s Guide (Addison-Wesley, Reading, MA, 1990). 2. ALEPH Collaboration : D. Decamp et al., Phys. Rep. 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Lett. [B312]{} (1993) 240. 11. B. Brahmachari, A.S. Joshipura, S.D. Rindani, D.P. Roy and K. Sridhar, Phys. Rev. [D48]{} (1993) 4224. 12. ALEPH Collaboration : D. Buskulic et al., Phys. Lett. [B313]{} (1993) 312. 13. A. Djouadi, J. Kalinowski and P.M. Zerwas, Z. Phys. [C57]{} (1993) 569. 14. Debajyoti Choudhury and D.P. Roy, Phys. Lett. [B322]{} (1994) 368. 15. M. Diemoz, F. Ferroni, E. Longo and G. Martinelli, Z. Phys. [C39]{} (1988) 21. 16. J.F. Gunion, Phys. Rev. Lett. [72]{} (1994) 199. - The expected Higgs signals for the SM and MM (invisible) decay modes corresponding to the published ALEPH data$^{2)}$. - The $HZ$ signal (dotted and dashed lines) and the $ZZ$ background (solid line) cross-sections for the dilepton + missing-$p_T$ channel at the LHC, calculated using the DFLM structure functions$^{15)}$. - The $HW$ signal (dotted and dashed lines) and the $WZ$ background (solid line) cross-sections for the lepton + missing-$p_T$ channel at the LHC, calculated using the DFLM structure functions$^{15)}$. [^1]: Permanent address [^2]: Both parameters depend on the scale of the global $U(1)$ breaking relative to the $SU(2) \times U(1)$ breaking scale, on which there are no severe phenomenological constraints.
--- abstract: 'By describing strong interactions between hadrons via a relativistic supermultiplet scheme and regarding weak interactions as a perturbation thereof, we derive expressions for nonleptonic weak decay amplitudes in terms of constituent quark masses and CKM angles, with no other parameters. Application of this method leads to $\Delta I=1/2$ dominance in some pseudoscalar meson decays if one scales down the couplings of heavy particles by $\sqrt{M}$ mass factors, in keeping with heavy quark theory expectations. However, certain B and D decay processes to kaons are badly predicted and point to substantial soft gluon renormalization effects in W-quark interactions.' address: \| School of Mathematics and Physics, University of Tasmania,\ Hobart, Australia 7001 author: - 'R. Delbourgo$^$ and Dongsheng Liu$^{}$' title: On nonleptonic decays of Supermultiplets --- =10000 Introduction ============ With so much data available for the decays of heavy quark composites, a real industry has grown, devoted to calculating nonleptonic weak amplitudes. A standard picture has evolved (Neubert 1994, 1997; Neubert and Stech 1997; Stech 1997) from the work of Bauer, Stech and Wirbel (1987), which parametrises flavour-changing decays in terms of a number of effective four-quark operators, such as $${\cal H} = \frac{G_F}{2\sqrt{2}}V_{UD}V^_{ud}[c_1(\mu)(\bar{d}\gamma u)_L. (\bar{U}\gamma D)_L + c_2(\mu)(\bar{U}\gamma u)_L.(\bar{d}\gamma D)_L],$$ and associated parameters $c_i(\mu)$; the idea is then to connect the running $c(\mu)$ values via QCD and the renormalization group with heavy quark theory (Isgur and Wise 1989, 1990) schemes. This description (Ali et al 1998, 1999; Lu 1999) introduces many parameters and requires the input of meson decay constants $f$ (which are extracted from experiment). Even after introducing so many phenomenological constants, a certain amount of confusion remains in this subject, mainly connected with the role of non-factorizable contributions (Beneke 2000, Neubert 2000); indeed some of the elaborations are so Byzantine in their intricacies that one yearns for a simpler and more direct approach to this problem. The only striking and mysterious fact remains the dominance of the $\Delta I=1/2$ rule in kaon and hyperon decays. In this paper we will attempt to shed some light on this topic by trying a different approach, one which has long been applied to electromagnetic interactions; namely we will regard the Lagrangian of electroweak theory as a perturbation of the strong interactions. The difficulty with such an approach lies in knowing the values of strong interaction amplitudes, since this is to be the starting point for applying perturbation theory. Strong amplitudes are of course given reliably by QCD at high energy, but at low energy we are obliged to resort to models which respect the symmetries of QCD, rather than QCD itself, because of unknown effects produced by colour confinement and soft multigluon exchanges that enhance the quark interactions at low mass scales. One of the simplest such models is the linear (or nonlinear) sigma-type model (Scadron 1997) because it embodies full chiral symmetry for massless bare quarks. Another favourite scheme is the spin-flavour supermultiplet model, which not only applies to heavy quarks but to light ones as well (Salam et al 1965; Sakita and Wali 1965), provided their masses are dressed to their constituent values. We explore this route and take it that strong amplitudes are quite well described—often to better than 10%—by supermultiplet tree interactions (Delbourgo and Liu 1996); then we shall add electroweak interactions as a small perturbation[^1]. Semileptonic decays are well-understood via weak-boson exchange, provided the hadronic matrix element $\langle f\|J^{weak}\|i\rangle$ of the weak current $J^{weak}$ is extracted from experiment or estimated reliably from theory, using heavy quark symmetry, dispersion relations, sum rules, $1/N_c$ expansions or whatever other tool one can make use of. We will have nothing to say about semileptonic processes; rather, our focus is on nonleptonic processes where weak and strong interactions are linked. Our attitude towards flavour changing weak amplitudes is that, to first order in the Fermi coupling $G_F$, the process can be construed as a set of W-exchange loop diagrams between the hadronic participants, with the [hadronic]{} amplitude satisfactorily determined by higher spin-flavour symmetry interactions. The computational rules are thus fixed and our answers can only depend on the masses of the constituent quarks and the CKM mixing angles. This assumes the Feynman integrals are convergent—which they are, thanks to the unitarity of the CKM matrix[^2]. Hence there are no adjustable parameters in our scheme. Note that we do not consider gluon corrections and resultant ‘penguin diagrams’ in the first place, since gluons are already assumed to be incorporated into dressing the bare quarks and producing the constituent fields which then interact via simple higher symmetry rules. The layout of the paper is as follows. We summarise the working rules for strong interactions and their weak perturbations in the next section. Then we characterise the types of diagram that should be calculated in the following section. It turns out that there are three types and they are evaluated one by one in Sections IV, V and VI. Next we see to what extent the results are altered by introducing form factors into the weak interactions (Section VII). Finally we apply the ideas to typical pseudoscalar mesons decays, to check whether we are on the right track; we find that the top quark contributions especially are overestimated unless the couplings are scaled down by $1/\sqrt{M}$ factors where $M$ is the mass of the heavy quark, as indicated by heavy quark symmetry. With such damping included, many of the amplitudes fall in the right ballpark—within a factor of two or three. The ability of the scheme to reproduce the ‘$\Delta I =1/2$ rule’ in most cases is an encouraging sign. However having said that, there are a number of cases in which the supermultiplet predictions are dreadful, indicating that we have overlooked some important feature or that our approach is fundamentally awry: these are processes involving the decays of D-mesons or B-mesons to K-mesons. It is entirely possible that for such heavy/strange objects our hope of absorbing all soft gluon effects into the dressing of the quarks is misplaced, and that they play a very significant role in modifying the weak vertices themselves. If this puzzle can be unravelled, the prospects for calculating other nonleptonic amplitudes, such as initial heavy vector mesons or baryons, are good. Strong and Electroweak Interactions =================================== The starting point of our model is that hadrons are reasonably well described by multiquark, constituent composites with supermultiplet wavefunctions $A=\alpha a$, where $\alpha$ is a Dirac spinor label and $a$ stands for flavour: $$\Phi(p)_A^B = [(1+\gamma.v)(\gamma_5P +\gamma^\mu V_\mu)]_A^B;\quad m=m_a+m_b,\,p=mv$$ $$\begin{aligned} \Psi_{(ABC)}(p)&=&[(1+\gamma.v)\gamma^\mu C]_{\alpha\beta}u_{\mu(abc)}(p)+ \nonumber \\ & &\{[(1+\gamma.v)\gamma_5C]_{\alpha\beta}\epsilon_{abd}u^d_{c\gamma}(p) +{\rm perms} \};\quad m=m_a+m_b+m_c,\,p=mv\end{aligned}$$ of ground state mesons ($0^-, 1^-$) and baryons ($1/2^+, 3/2^+$), with tree-level interactions given by momentum-conserving effective Lagrangians like, $$g\bar{\Psi}^{(ABC)}\Psi_{(ABD)}\Phi_C^D,\quad f\Phi_A^B\Phi_B^C\Phi_C^A.$$ Such algebraic contractions of indices can be represented pictorially as a joining of the flavour quantum numbers via duality diagrams, with the remaining contractions over spinorial indices serving to provide the Lorentz structure of the amplitude, in keeping with higher symmetry requirements. In (3), with the normalizations used in (1) and (2), the coupling constant $g$ is dimensionless while $f$ has dimensions of mass; we shall return to this point presently. Now consider the addition of electroweak perturbations on the above sorts of effective Lagrangians, by invoking the standard electroweak Lagrangian, $${\cal L}=\sum_{\rm flavour}\bar{\psi}[\gamma.(i\partial+{\cal W})-g_HH]\psi - \vec{W}^{\mu\nu}\cdot\vec{W}_{\mu\nu}/4 + {\cal L}_H + \ldots,$$ where the supermultiplet boson field is $${\cal W} = -eQA + g_W VW(1-i\gamma_5)/\sqrt{8} + \ldots; \quad V = {\rm CKM~matrix},\,\,Q = {\rm charge}.$$ By using the Particle Data Group parametrization ($c_{ij}\equiv\cos\theta_{ij}, s_{ij}\equiv \sin\theta_{ij}$), $$\left( \begin{array}{ccc} V_{ud} & V_{us} & V_{ub}\\ V_{cd} & V_{cs} & V_{cb}\\ V_{td} & V_{ts} & V_{tb} \end{array} \right)\! =\! \left( \begin{array}{ccc} c_{12}c_{13} & s_{12}c_{13} & s_{13}\exp(-i\delta)\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}\exp(i\delta) & c_{12}c_{23}- s_{12}s_{23}s_{13}\exp(i\delta) & s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}\exp(i\delta) & -c_{12}s_{23}- s_{12}c_{23}s_{13}\exp(i\delta) & c_{23}c_{13} \end{array} \right),$$ one guarantees that unitarity is respected and is not subject to errors by picking specific values for certain matrix elements $V_{ij}$, without regard to other elements. In practice we shall use the ‘average’ values, $\theta_{12}= 0.223,\theta_{13}=0.0031,\theta_{23}=-0.039$ and disregard CP-violation effects which lie outside the scope of this article, by setting $\delta = 0$. In (4) we have not bothered to incorporate the Z-interactions because our main concern is flavour-changing processes. The most significant aspect of (4) is that it is normally applied to current quark fields, not the more massive constituent quark fields which are dynamically induced via strong interactions. It becomes questionable then whether (5) and (6) are applicable to the effective quark fields present in (3). Indeed previous experience shows that we must anticipate nontrivial corrections of around 30% due to renormalization effects; for instance the chiral interaction should be changed from the left-projection $P_L=(1 - i\gamma_5)/2$ to about $(1-\frac{3}{4}\gamma_5)/2$ so that the axial vector component for the nucleon is reduced from 5/3 (arising the higher symmetry D/F ratio of 3/2) to the experimental value of $g_A/g_V\simeq 5/4$. The principal goal of this paper is to see if one can get sensible estimates of [all]{} nonleptonic amplitudes without invoking extra parameters, so we will ignore relatively small renormalization effects on the axial current but not the more substantial dependence on the mass of the participating constituent quarks in the hadrons. Refinements can come later. Flavour-changing decay amplitudes are governed by virtual W-exchange between the quarks. We can recognise three types of contribution, where - a charge-conserving transition occurs on one of the quark lines, either at the start or at the finish (wave-function-like renormalizations) or via a vertex-like correction; see Figures 1a,b,c. - the charge exchange takes place between the participating quarks in the hadron, again either as a self-energy or a pair of vertex corrections; see Figures 2a,b,c. (If these participating quarks comprise a meson, it must be uncharged.) - the W acts as a quark annihilation intermediary (see Figure 3) and is just what one encounters in semileptonic processes. It requires a charged meson to be one of the participating particles, of course. For some processes, there might be a missing type; for example there is no annihilation diagram in K$^0\rightarrow \pi^0\pi^0$ decay. Since there is a well-defined prescription for treating the strong vertex—where the quark tramlines join up—it only remains to estimate the Feynman integrals corresponding to the W-loop exchange. This we shall presently do, using the Feynman-’tHooft gauge field propagator, $$\Delta_{\mu\nu}(k) = -\eta_{\mu\nu}/(k^2-m_W^2).$$ The left matrix $\gamma_{L\mu}=\gamma_\mu(1-i\gamma_5)/2$ multiplying the propagator has a number of properties which come in handy during the calculations; aside from the obvious utility of the left-handed projection, there is the bonus that Fierz identities can be used to shuffle $\gamma_L\otimes\gamma_L$ from one pair of fermion lines into another pair, thereby relating disparate amplitudes. This paper will concentrate on the pseudoscalar mesons decays, to verify if the ideas have any sort of validity. We do not seek complete accuracy to any number of decimal places but will content ourselves if we can capture most if not all the amplitudes to within about 30% or better. (If the scheme is successful, the generalisation to vector meson and baryon non-leptonic decays presents itself as the next obvious step.) Throughout we will track flavour indices by making great use of quark line diagrams; sewing the Dirac spinorial indices for each diagram then provides the Lorentz structure. Now, disregarding weak interactions, the three-meson vertex is fixed by the coupling factor $f$ and, because $0^-$ is forbidden to decay into two other pseudoscalars via strong interactions, one must look to the strong decay $\rho\rightarrow 2\pi$, say, in order to get the value of $f$. Two quark line diagrams determine $g_{\rho\pi\pi}$ and each of them contains a flavour normalization factor $1/\sqrt{2}$, but with opposite sign. Referring to Figure 4, with all momenta $p_i$ taken as incoming, one therefore arrives at the effective interaction Lagrangian, $$\begin{aligned} {\cal L}_{\rho\pi\pi}&=&f\,{\rm Tr}[\Phi_P(p_3)\Phi_P(p_2)\Phi_V(p_1) /\sqrt{2} - (1 \leftrightarrow 2)]/4 \nonumber \\ &=&f\,{\rm Tr}[(1+\gamma.v_3)\gamma_5(1+\gamma.v_2)\gamma_5(1+\gamma.v_1) \gamma.\epsilon_1-(1\leftrightarrow 2)]/4\sqrt{2} \nonumber \\ &=&(p_3-p_2).\epsilon_1 f(m_1+m_2+m_3)/\sqrt{2}m_2m_3 \equiv g_{\rho\pi\pi}(p_3-p_2).\epsilon_1\end{aligned}$$ Experimentally, the $\rho$-decay width tells us that the dimensionless coupling $g_{\rho\pi\pi} \simeq 6.03$. Thus $f(6\hat{m})/4\hat{m}^2 = \sqrt{2}g_{\rho\pi\pi}$, or $f=\sqrt{8}\hat{m}g_{\rho\pi\pi}/3 \sim 2$ GeV. Now $f(m_1+m_2+m_3)$ is a ubiquitous factor in relativistic supermultiplet theory, so we shall adopt this quantity as the ‘universal’ value which equals about 4.1 GeV$^2$, based on the assumption that the quark masses are not far from $m_u\simeq m_d \equiv \hat{m} \sim 0.34$ GeV. Type I - Quark Line Transitions =============================== First we shall deal with changes of flavour (but not of charge) on a single quark, which can take place as self-energy-like diagrams at each of the hadronic legs (Figures 1a, 1c) or as a vertex correction across the legs (Figure 1b). In both cases, one must sum over the internal quark flavours; a generic case (Delbourgo and Scadron 1985; Fuchs and Scadron 1986) is the $s-d$ transition, where we have to sum over $u,c,t$. Quite generally, between $i$ and $k$ quarks, the self-energy part is $\Sigma_{ik}(p) \equiv \sum_jV_i^jV_k^{j}\Sigma_j(p)$ where $$\Sigma_j(p) = i\frac{g_W^2}{2} \int \frac{\bar{d}^4k}{k^2-m_W^2}\gamma_{L\mu} \frac{\eta^{\mu\nu}}{\gamma.(p+k)-m_j} \gamma_{L\nu} \equiv p.\gamma_L {\cal F}_j(p^2,m_W,m_j).$$ Now ${\cal F}$ is potentially troublesome because, [neglecting form factors]{}, it contains a logarithmic divergence: $${\cal F}_j = -i\frac{g_W^2}{p^2}\int \frac{\bar{d}^4k\,\,p.(p+k)} {[k^2-m_W^2][(k+p)^2-m_j^2]} =\int_0^1\!\!dx\int \frac{(1-x)\,\bar{d}^4k}{[k^2+p^2x(1-x)-m_j^2x -m_W^2(1-x)]^2}.$$ However, through unitarity of the CKM matrix and the fact that we are studying flavour changing transitions ($i\neq k$), we can subtract off the dangerous divergent part, contained in the limit $m_j\rightarrow 0$, and happily use the convergent difference instead of ${\cal F}$: $${\cal F}_j(p^2,m_W,m_j)-{\cal F}_j(p^2,m_W,0)=-\frac{g_W^2}{16\pi^2} \int_0^1dx\,(1-x)\ln\left[1 + \frac{m_j^2 x}{(1-x)(m_W^2-p^2x)}\right].$$ This can then be suitably approximated; thus for $p^2 \ll m_W^2$ which applies to all [external]{} hadrons not containing a top quark, $${\cal F}_j(m_j)-{\cal F}_j(0)\!\simeq\! -\frac{g_W^2}{16\pi^2}\int_0^1\!\! dx\,\,x\ln\left[1+\frac{m_j^2(1-x)} {xm_W^2}\right]\!=\! -\frac{g_W^2}{32\pi^2}\frac{J}{1-J} \left[1 + \frac{J\ln J}{1-J}\right];\,J\equiv \frac{m_j^2}{m_W^2}.$$ In particular, for $m_j\ll m_W$ or $J\ll 1$, we get a very good estimate of (9) with $${\cal F}_j(m_j)-{\cal F}_j(0) \simeq -\frac{g_W^2J}{32\pi^2}=-\frac{1}{2}\left(\frac{g_Wm_j}{4\pi m_W}\right)^2.$$ But for the top quark one must be more careful in approximating (9). Taking the experimental value of $m_t$ and $m_W$ as inputs we estimate $${\cal F}_t(m_t)\!-\!{\cal F}_t(0)\simeq -\frac{1.23g_W^2}{32\pi^2}\simeq -\frac{0.13g_W^2m_t^2}{16\pi^2m_W^2},$$ which is almost exactly 1/4 of the expression (10). Remembering that $G_F/\sqrt{2} = g_W^2/8m_W^2$ = 8.25$\times 10^{-6}$ GeV$^{-2}$, we may finally write the transition element, $$\Sigma_{ik}(p) \equiv p.\gamma_L{\cal F}_{ik} = -\frac{G_Fp.\gamma_L}{4\sqrt{2}\pi^2}\sum_j V_i^jV_k^{j}m_j^2\rho_j,$$ where the weight factor $\rho_q$=1 for all but the top quark, when $\rho_t \simeq 1/4$. Evaluation of the transition factors, ${\cal F}_{ik}=-\sum_j V_i^jV_k^{j} \,G_Fm_j^2\rho_j/4\sqrt{2}\pi^2$, is reasonably straightforward, using fairly well-known values of mixing angles and the (GeV) values $m_u\simeq m_d \simeq 0.34$,$m_s\simeq 0.48$, $m_c\simeq 1.5$, $m_b\simeq 4.7$, $m_t\simeq 175$. The values are listed in the left-hand columns of Table I, and it should be noted that transition elements for down-type quarks depend significantly on the contribution from the intermediate top quark; the effect is smallest for the $s-d$ transition, but even so the $t$ quark competes well with the $u,c$ contributions; mostly it dominates the other contributions, in spite of the fact that off-diagonal $V_t^q$ terms are quite small. Next we turn to Figure 1b, which produces the (matrix) vertex integral, $$\begin{aligned} \Gamma_{ik}&=&-i\sum_jV_i^jV_k^{j}\frac{g_W^2}{2}\int \frac{\bar{d}^4k}{k^2-m_W^2}\gamma_{\mu L} \frac{1}{\gamma.(p_k+k)-m_j}\frac{1}{\gamma.(p_i+k)-m_j} \gamma^\mu_L \nonumber \\ &=&ig_W^2\sum_jV_i^jV_k^{j}\int\frac{m_j\,(p_k+p_i+2k).\gamma_L\,\bar{d}^4k} {[(p_k+k)^2-m_j^2][k^2-m_W^2][(p_i+k)^2-m_j^2]}.\end{aligned}$$ where $p_i$ is the incoming momentum of the $i$-quark and $p_k$ is the outgoing momentum of the $k$-quark. Introducing Feynman parameters and assuming that $p_{i,k}\ll m_W$ —as is true for all top-free external hadrons—we may get the one-dimensional parametric representation, $$\begin{aligned} \Gamma_{ik}&=&\sum_jV_i^jV_k^{j}m_j(p_i+p_k).\gamma_L\frac{g_W^2}{16\pi^2} \int_0^1 \frac{x(1-x)\,\,dx}{m_W^2x+m_j^2(1-x)} \nonumber \\ &=&\sum_jV_i^jV_k^{j}m_j(p_i+p_k).\gamma_L\frac{g_W^2}{16\pi^2m_W^2} \left[\frac{J\ln{J}}{(1-J)^3}+\frac{1+J}{2(1-J)^2}\right];\quad J\equiv\frac{m_j^2}{m_W^2}.\end{aligned}$$ When the intermediate $j$-quark is considerably lighter than the W, we can make the reasonable approximation, $J\ll 1$ and find $$\Gamma_{ik} \simeq \frac{1}{2}\left(\frac{g_W}{4\pi m_W}\right)^2 (p_i+p_k).\gamma_L\,\sum_j V_i^jV_k^{j}m_j,$$ but for the top quark, the numerical value of $J \sim 4.7$, means that the contribution is about 1/8 of what the light quark approximation above gives. We shall therefore write the vertex correction in the form $$\Gamma_{ik}\equiv \frac{1}{2}(p_i+p_k).\gamma_L{\cal G}_{ik} = \frac{G_F(p_i+p_k).\gamma_L}{4\sqrt{2}\pi^2}\sum_jV_i^jV_k^{j}m_j\sigma_j,$$ where the weight factor $\sigma_q =1$ for all quarks but the top, when $\sigma_t\simeq 1/8$. The magnitudes of the vertex transition elements ${\cal G}_{ik}$ are listed in the second columns of Table I; the effect from the top is not so significant as for the self-energy elements, but it still dominates the $b\leftrightarrow s,d$ transitions. We should point out that the above results were derived on the assumption that the W-coupling to the constituent quarks has no form factors; but this assumption is obviously not correct. A more refined calculation ought to include form factors of the pole type $F(k^2)\sim M^2/(M^2-k^2)$ or something fancier. We shall return to this point later. It remains to sum up the terms arising from Figures 1a, 1b and 1c. Since hadronic supermultiplets consist of constituent quarks on their mass shells sharing the total momentum according to their mass (they all have equal velocity) with negligible binding, we shall evaluate the result between free spinors $\bar{u}(p_k)..u(p_i)$. In doing so we must be careful to [halve]{} the self-energy contributions on external quark lines, since they are eventually associated with $Z^{1/2}$ renormalization factors. Hence, between spinors, the sum equals $$\begin{aligned} {\cal T}^I_{ik}&=&-\frac{1}{2(\gamma.p_i-m_k)}\Sigma_{ik}(p_i)+ \Gamma_{ik}(p_i,p_k)-\Sigma_{ik}(p_k)\frac{1}{2(\gamma.p_k-m_i)}\nonumber\\ &=&-\frac{{\cal F}_{ik}}{2(\gamma.p_i-m_k)}p_i.\gamma_L+ \frac{1}{2}(p_i+p_k).\gamma_L{\cal G}_{ik} -p_k.\gamma_L \frac{{\cal F}_{ik}}{2(\gamma.p_k-m_i)} \nonumber \\ &=&-\frac{{\cal F}_{ik}}{4}\left[1 + i\gamma_5\frac{m_k-m_i}{m_k+m_i}\right] + \frac{{\cal G}_{ik}}{4}\left[(m_k+m_i) + i\gamma_5(m_i-m_k)\right].\end{aligned}$$ The last step is to contract ${\cal T}^I$ over the hadronic wavefunctions. For three 0$^-$ mesons, the only relevant part of ${\cal T}^I$ is the one containing $\gamma_5$ so as to get a non-vanishing trace. Therefore, with the flavour labels of Figure 1, we get the generic amplitude: $$\begin{aligned} M^I_{sdcu}&=&f\,{\rm Tr}[\Phi_P(p_2){\cal T}^I\Phi_P(p_1)\Phi_P(p_3)] /m_1m_2m_3 \nonumber \\ &=&i\frac{(m_d-m_s)f}{4m_1m_2m_3}\left[{\cal G}_{sd}+ \frac{{\cal F}_{sd}}{m_s+m_d}\right] {\rm Tr}[(\gamma.p_2+m_2)(\gamma.p_1+m_1)(\gamma.p_3-m_3)]\nonumber\\ &=&i\frac{f(m_1+m_2+m_3)(m_s-m_d)[m_3^2-(m_1-m_2)^2]}{2m_1m_2m_3} \left[{\cal G}_{sd} + \frac{{\cal F}_{sd}}{m_s+m_d}\right].\end{aligned}$$ Recall that the $m_i$ in (14) are the sums of the constituent quark masses comprising the hadron. When analysing any other flavour changing amplitude of type I, it is a simple matter of substituting the appropriate flavour labels in (14) above. We shall frequently be doing so in section VI. Type II - W-Exchange across quarks ================================== We now turn to the graphs contained in Figure 2. The first of these actually corresponds to a $u\bar{c}$ transition into two mesons, dominated by an intermediate $d\bar{s}$ state. Neglecting the small momentum carried by the W meson relative to its mass, this particular contribution is given by the generic contraction, $$\begin{aligned} M^{IIA}_{udbsc}&=&-\frac{g_W^2\,f}{32m_W^2}V_u^dV_c^{s}{\rm Tr} [(\Phi_P(p_2)(\gamma.p_d\!+\!m_d) \gamma_{L\mu}\Phi_P(p_1)\gamma_L^\mu(\gamma.p_s\!+\!m_s)\Phi_P(p_3)]\nonumber \\ &=&\frac{g_W^2m_dm_sfV_d^uV_s^{c}}{32m_W^2m_1m_2m_3}{\rm Tr} \left[(\gamma.p_2\!-\!m_2)\left(1\!+\!\frac{\gamma.p_1}{m_s\!+\!m_d}\right) i\gamma_L.p_1\left(1\!-\!\frac{\gamma.p_1}{m_s\!+\!m_d}\right) (\gamma.p_3\!+\!m_3) \right] \nonumber \\ &=&\frac{iG_Ff(m_1\!+\!m_2\!+\!m_3)m_dm_sV_u^dV_c^{s}} {2\sqrt{2}m_1m_2m_3}\left[1\!-\!\left(\frac{m_u\!\!+\!m_c}{m_s\!\!+\!m_d} \right)^2\right](m_d\!\!-\!m_s)(m_1\!-\!m_2\!-\!m_3),\end{aligned}$$ because in the intermediate state, $p_d=p_1m_d/M_1, p_s=-p_1m_s/M_1; M_1=m_s+m_d$. Competing with this answer are the vertex corrections of Figures 2b,2c. The latter involve Feynman integrals which are more difficult to calculate analytically. An interesting technical aspect of the evaluation is that they contain cancelling logarithmic divergences, irrespective of CKM matrix unitarity. Figure 2b and 2c yield, respectively $$M^{IIB}_{udbsc}\!=\!-i\frac{g_W^2f}{8}\int\frac{V_u^dV_c^{s}\,\bar{d}^4k} {k^2-m_W^2}{\rm Tr}\left[\Phi_P(p_2)\frac{1} {\gamma.(p_u\!+\!k)\!-\!m_d}\gamma_{L\mu}\Phi_P(p_1) \frac{1}{\gamma.(p_s\!-\!k)\!-\!m_c}\gamma_L^\mu\Phi_P(p_3)\right],$$ $$M^{IIC}_{udbsc}\!=\!-i\frac{g_W^2f}{8}\int\frac{V_u^dV_c^{s}\,\bar{d}^4k} {k^2-m_W^2}{\rm Tr}\left[\Phi_P(p_2)\gamma_{L\mu} \frac{1}{\gamma.(p_d\!-\!k)\!-\!m_u}\Phi_P(p_1)\gamma_L^\mu \frac{1}{\gamma.(p_c\!+\!k)\!-\!m_s}\Phi_P(p_3)\right],$$ where now $p_s=m_sp_3/m_3, p_u=m_up_1/m_1, p_d=-m_dp_2/m_2, p_c=-m_cp_1/m_1$. Simplifying the sum, $$\begin{aligned} M^{IIBC}_{udbsc}&=&\frac{g_W^2f}{8}\int\frac{V_u^dV_c^{s}\,\bar{d}^4k} {k^2-m_W^2}\,\,{\rm Tr}\nonumber \\ & &\left[\Phi_P(p_2)\!\!\left(\frac{1}{\gamma.(p_u\!+\!k)\!-\!m_d} \frac{\gamma.(k\!-\!p_c)\!-\!m_s}{(k\!-\!p_s)^2\!-\!m_c^2} - \frac{\gamma.(k\!-\!p_u)\!-\!m_d}{(k\!-\!p_d)^2\!-\!m_u^2} \frac{1}{\gamma.(k\!+\!p_c)\!-\!m_s}\right)\!\!\Phi_P(p_3) \right].\end{aligned}$$ This calculation is messy—suggesting numerical methods as a last resort—if all external momenta are religiously kept within the Feynman integral. However, we can achieve an reasonable estimate of the result by going to the soft limit, i.e. neglecting the $p$-dependence within the propagators, relative to the large momentum $k$ carried by the W line. Using the supplementary integral, $$\begin{aligned} \int\frac{i\,\bar{d}^4k}{k^2\!-\!m_W^2}\left[ \frac{1}{(k\!-\!p_u)^2\!-\!m_d^2}\!\right.&-& \left.\!\frac{1}{(k\!-\!p_c)^2\!-\!m_s^2}\right] =\int_0^1\frac{dx}{16\pi^2}\ln\left[ \frac{m_d^2x\!+\!m_W^2(1\!-\!x)\!-\!m_u^2x(1\!-\!x)} {m_s^2x\!+\!m_W^2(1\!-\!x)\!-\!m_c^2x(1\!-\!x)}\right] \nonumber \\ &\simeq&\frac{1}{32\pi^2m_W^2}\left[m_c^2+2m_s^2\ln\left(\frac{m_s^2}{m_W^2} \right)-m_u^2-2m_d^2\ln\left(\frac{m_d^2}{m_W^2}\right)\right],\end{aligned}$$ valid when $m_i^2\ll m_W^2$, and keeping leading logarithms, one may estimate $$\begin{aligned} M^{IIBC}_{udbsc}\simeq\frac{g_W^2f}{8}\!\int\!\bar{d}^4k& &V_u^dV_c^{s} \frac{k^2-m_dm_s}{k^2-m_W^2}\\ &&{\rm Tr}\left[\Phi_P(p_2)\left(\frac{1}{(k^2-m_c^2)(k^2-m_d^2)}- \frac{1}{(k^2-m_u^2)(k^2-m_s^2)}\right)\Phi_P(p_3)\right]\end{aligned}$$ $$\begin{aligned} =-\frac{iG_FV_u^dV_c^{s}\,f}{16\sqrt{2}\pi^2m_2m_3} &&\left[m_1^2-(m_2+m_3)^2\right] \nonumber\\ &&\left[(m_u^2-m_c^2)\ln\left(\frac{m_c^2m_u^2}{m_W^4}\right) +\frac{1}{2}(m_s^2-m_d^2)\ln\left( \frac{m_s^2m_d^2m_c^2m_u^2}{m_W^8} \right) \right].\end{aligned}$$ The significant point about this last result is that it is of the same order of magnitude as $M^{IIA}$; even though there is a suppression factor of $1/4\pi^2$ from the integration in eq (18), it is compensated by a number of logarithms which are typically in the range $\ln(m_W^2/m_s^2) \sim 10$. In fact, from our viewpoint, cancellations between these sorts of terms are responsible for the small size of the $\Delta I = 3/2$ amplitude in K decays. TYPE III - W-Annihilation ========================= Evidently, this process only applies to charged mesons, which may be incoming or outgoing. A typical example is drawn in Figure 3. It is easy to calculate, from what has gone before. Basically, we take advantage of the fact that through Fierz reshuffling, $(\bar{u}_1\gamma_L^\mu u_2).(\bar{u}_3\gamma_{L\mu} u_4)= -(\bar{u}_3\gamma_L^\mu u_2).(\bar{u}_1\gamma_{L\mu} u_4)$, aside from colour factors; and because we have an extra fermion loop in Figure 3 relative to Figure 2a, the extra (-) sign is effectively swallowed up. Thus, on the assumption that constituent quarks interact left-handedly to a first approximation, Figure 3 gives the same answer as Figure 2a so far as the Lorentz contraction is concerned. Transcribing the flavour labels, and incorporating a colour factor of 3, we get the generic amplitude, $$M^{III}_{sdbcu}=\frac{3iG_Ff(m_1\!+\!m_2\!+\!m_3)m_dm_cV_u^sV_c^{d}} {2\sqrt{2}m_1m_2m_3}\left[1\!-\!\left(\frac{m_s\!\!+\!m_u}{m_c\!\!+\!m_d} \right)^2\right](m_d\!\!-\!m_c)(m_1\!-\!m_2\!-\!m_3).$$ Form factor corrections ======================= The integrals in sections III-V were derived on the assumption that the coupling of the W-boson to the quarks was pointlike. In fact the weak current must be affected by intermediate vector and pseudoscalar bosons that can latch on to the quark fields and this will produce a natural damping. (This phenomenon is very familiar in QED and is responsible for the finiteness of photonic corrections to the proton neutron mass difference, aside from the contribution due to the intrinsic $u-d$ mass difference. It is typically governed by a strong interaction scale of about 1.1 GeV.) We may estimate the effects of mediating mesons by incorporating the form factor $M^2/(M^2-k^2)$ at each W-leg, where $M$ is some kind of geometric mean of the intermediate mesons on each side of the W-line. The consequence is that the integral (8) say is finite, regardless of CKM unitarity, because it gets modified to $$\begin{aligned} {\cal F}_j &=& -i\frac{g_W^2}{p^2}\int \frac{\bar{d}^4k\,\,p.(p+k)} {[k^2-m_W^2][(k+p)^2-m_j^2]}\left(\frac{M^2}{M^2-k^2}\right)^2 \nonumber \\ &=& \frac{g_W^2L^2}{32\pi^2}\left[\frac{1}{1-L}\left( \frac{1}{L}+\frac{\log L}{1-L}+\frac{J^2\log J}{(L-1)(J-1)^2}\right)\right. +\nonumber \\ & & \qquad\quad\left.\frac{J}{(L-J)^2}\left(\frac{1}{J-1}+\frac{J}{L(L-1)}+ \frac{J\log(J/L)}{(1-L)^2}+\frac{2J\log J}{(L-1)(L-Q)}\right)\right],\end{aligned}$$ where $L\equiv M^2/m_W^2, J\equiv m_J^2/m_W^2$. The main effect is to enhance the contribution from the top quark; this can be quite substantial and having included it, we must rescale the universal constant $f$ down appropriately so that the plain results (no form factors), which should be governed by $G_F$, do not go badly out of line. Once this is done there is no more room for maneouvre. The same sort of modification arises in the vertex integral ${\cal G}$ but we shall not bother to exhibit the dependence on $L$ and $J$ because the consequences are very minor; the point is that each ${\cal G}$ contribution is given by a well-behaved finite integral. The form factor effects here, in contrast to those on ${\cal F}_t$, are very tame and amount to corrections of just a few per cent. Typical Results and Difficulties ================================ Let us now describe some of the consequences of the supermultiplet scheme and the ensuing problems. To keep the discussion clean, we shall ignore channels which involve uncharged mesons that can mix, like $\eta$ and $\eta'$, so we will focus on decays where the outgoing particles are pions, kaons and heavy mesons like D and D$_s$. Below we define $M$ to be the magnitude of the decay amplitude (it has dimensions of mass), which is derived from the partial decay width via $$\Gamma_{m\rightarrow m_1m_2} = \|M\|^2\Delta/16\pi m^3;\qquad \Delta \equiv \sqrt{[m^2-(m_1-m_2)^2][m^2-(m_1+m_2)^2]}.$$ Throughout we have disregarded the $u-d$ mass difference and used Mathematica to compute the integrals numerically, as required. First let us consider the time-honoured example of kaon decay. Two typical cases are K$^+ \rightarrow \pi^+\pi^0$, corresponding to $\Delta I =3/2$, and K$_s \rightarrow \pi^+\pi^-$ which amounts to $\Delta I =1/2$. It is of course known that $M^{expt}_{K_s\pi^+\pi^-} = 3.91\times 10^{-7}$ is about twenty times larger than $M^{expt}_{K^+\pi^+\pi^-} = 1.83\times 10^{-8}$ and this is the main feature to be ‘explained’. When we tackle these cases via the supermultiplet scheme, we find that with the longer-lived K$^+$, the type I contributions cancel, as they must, but for the short-lived K$^0$ there is a significant type I contribution (because the $s-d$ transition is $\Delta I=1/2$); thus $$M_{K_s^0\rightarrow \pi^+\pi^0}=\sqrt{2}[M^I_{sdud}+M^{II}_{sudud}+ M^{III}_{uudsd}]$$ $$M_{K^+\rightarrow\pi^+\pi^0}=[M^{II}_{usudu}+M^{III}_{usuud}]/\sqrt{2}$$ have a ratio of about 4, which is still a factor of 5 too small; but this is easily remedied by including form factors[^3] using a weak cutoff of about 1.25$m_t$. In this way we can obtain values which are rather close to experiment: $$\|M_{K_s^0\rightarrow \pi^+\pi^0}\|\simeq 3.9\times 10^{-7}{\rm~GeV};\quad \|M_{K^+\rightarrow\pi^+\pi^0}\|\simeq 1.9\times 10^{-8}{\rm~GeV}.$$ If one turns to D-meson decays, there are some good results, but there are also some dreadful ones. For example, we get the nice answer $$\|M_{D^0\rightarrow \bar{K}^0\pi^0}\|=\|M^{II}_{duucs}-M^{II}_{csddu}\|/\sqrt{2} \simeq 1.79\times 10^{-6} {\rm~GeV};\quad {\rm cf~} \|M_{D^0\rightarrow\bar{K}^0\pi^0}^{\rm expt}\|\simeq 1.85\times 10^{-6} {\rm~GeV},$$ as well as the ridiculous value (in GeV) $$\|M_{D^+\rightarrow \pi^0\pi^+}\|=\|M^{I}_{cuud}-M^{II}_{dudcd}-M^{III}_{dsdcu}\| /\sqrt{2} \simeq 6.08\times 10^{-5}; \,{\rm cf~} \|M_{D^+\rightarrow \bar{K}^0\pi^+}^{\rm expt}\|\simeq 1.35\times 10^{-6}.$$ In fact most of the results involving heavy mesons are predicted to be too large in the raw supermultiplet scheme; it is not hard to trace the reason for this effect and thereby cure it. The point is that the supermultiplet interactions which we wrote down previously did not take account of the $1\sqrt{M}$ diminution of matrix elements which are expected from heavy quark theory, in order to give symmetry results at equal velocity, [not equal momentum]{}. These factors have little effect on the light quark composites but play a substantial role for the heavy mesons. When the appropriate factors $\sqrt{2m_u/(m_i+m_k)}$ for a meson composed of quarks $i$ and $k$ are incorporated, they depress the nonleptonic decay amplitudes of heavy mesons; thus $M_{D^+\rightarrow\pi^0\pi^+}$ goes down to the acceptable value 1.14$\times 10^{-6}$ GeV and likewise for many other matrix elements. Nevertheless there remain some processes whose reduction is excessive, such as $$\|M_{D_s^+\rightarrow K^0\bar{K}^0}\|=\|M^{II}_{duscs}+M^{III}_{cusds}\|\simeq 4.3\times 10^{-9}{\rm~GeV};\,{\rm cf~} \|M_{D_s^+\rightarrow K^0\bar{K}^0}^{\rm expt}\|=2.41\times 10^{-6}{\rm~GeV},$$ and others which stubbornly resist reduction, such as $$\|M_{B^+\rightarrow K^0\pi^+}\|=\|M^{I}_{sbdu}+M^{III}_{sbduu}\|\simeq 2.3\times 10^{-4}{\rm ~GeV};\,{\rm cf~} \|M_{B^+\rightarrow K^0\pi^+}^{\rm expt}\| \simeq 0.5\times 10^{-7}{\rm~GeV}.$$ The cure is therefore only partial and this is most disappointing. In spite of a number of silly predictions, we believe that our approach makes sound philosophical sense even if the way we have applied the idea has not been wholly successful: weak interactions should be regarded as a perturbation of the strong interactions and not the other way round. Maybe others will find errors with our numerical work and/or will be able to tackle the problem better than we have. We have perhaps been over-ambitious in thinking that we could get away without introducing any parameters and it is possible that with the introduction of many more couplings one can get suitable agreement with all the experimental results: the weak vertices do receive substantial renormalization corrections from the strong interactions. To conclude on a more optimistic note, if one could resolve the difficulties, it would be a simple step to generalize our work to the vector mesons and the baryonic supermultiplet. [Acknowledgments]{} This work was supported by the Australian Research Council under grant number A69800907. [References]{} $^$ Email: Bob.Delbourgo@utas.edu.au $^{}$ Email: D.Liu@utas.edu.au Ali, A., Kramer, G., and Lu, C-D. (1998).[Phys. Rev.]{} [D58]{}, 094009. Ali, A., Kramer, G., and Lu, C-D. (1999).[Phys. Rev.]{} [D59]{}, 014005. Bauer, M., Stech, B., and Wirbel, M. (1987). [Z. Phys.]{} [C34]{}, 103. Beneke, M. (2000) “Conceptual aspects of QCD factorization in hadronic B decays”, hep-ph/0009328. Delbourgo, R., and Liu, D. (1996). [Phys. Rev.]{} [D53]{}, 6576. Delbourgo, R. and Liu, D. (1998). “ Nonleptonic decays: amplitude analysis and supermultiplet schemes”, pp 249-260 in ‘Nonperturbative Methods in Quantum Field Theory’, (ed. by A.W. Schreiber, A.G. Williams and A.W. Thomas), (World Scientific, Singapore). Delbourgo, R., and Scadron, M.D. (1985). [Nuovo Cim. Lett.]{} [44]{}, 193. Fuchs, N.H., and Scadron, M.D. (1986). [Nuovo Cim.]{} [A93]{}, 205. Isgur, N., and Wise, M.B. (1989). [Phys. Lett.]{} [B232]{}, 113. Isgur, N., and Wise, M.B. (1990). [Phys. Lett.]{} [B237]{}, 527. Lu, C-D. (1999). [Nucl. Phys. Proc. Suppl.]{} [74]{}, 227. Neubert, M. (1994). [Phys. Rept.]{} [245]{}, 259. Neubert, M. (1997). “B-decays and heavy quark expansion” in ‘Heavy Flavours II’, pp 239-293 (World Scientific, Singapore). Neubert, M. (2000). “Application of QCD Factorization in Hadronic B-Decays”, Cornell preprint. Neubert, M., and Stech, B. (1997). “Nonleptonic weak decays of B-mesons”, in ‘Heavy Flavours II’, pp 294-344 (World Scientific, Singapore). Sakita, B., and Wali, K.C. (1965). [Phys. Rev.]{} [139]{}, B1355. Salam, A., Delbourgo, R., and Strathdee, J. (1965). [Proc. R. Soc. London]{}, [A284]{}, 146. Scadron, M.D. (1997). [Nuovo Cim.]{} [110A]{}, 865. Stech, B. (1997). “Twenty Beautiful Years of Bottom Physics”, (Chicago Univ. Press), hep-ph/9709280. ----------------- ----------------------- ----------------- ----------------------- ----------------- ----------------------- ${\cal F}_{sd}$ $7.91\times 10^{-7}$ ${\cal G}_{sd}$ $-1.08\times 10^{-7}$ ${\cal H}_{sd}$ $2.11\times 10^{-7}$ ${\cal F}_{sb}$ $-5.96\times 10^{-5}$ ${\cal G}_{sb}$ $3.17\times 10^{-7}$ ${\cal H}_{sb}$ $-2.80\times 10^{-6}$ ${\cal F}_{db}$ $1.86\times 10^{-5}$ ${\cal G}_{db}$ $-1.00\times 10^{-7}$ ${\cal H}_{db}$ $9.96\times 10^{-7}$ ${\cal F}_{uc}$ $-4.30\times 10^{-9}$ ${\cal G}_{uc}$ $1.15\times 10^{-8}$ ${\cal H}_{uc}$ $2.29\times 10^{-9}$ ${\cal F}_{ut}$ $-1.44\times 10^{-8}$ ${\cal G}_{ut}$ $6.08\times 10^{-9}$ ${\cal H}_{ut}$ $1.50\times 10^{-9}$ ${\cal F}_{ct}$ $1.78\times 10^{-7}$ ${\cal G}_{ct}$ $-6.88\times 10^{-8}$ ${\cal H}_{ct}$ $-1.70\times 10^{-8}$ ----------------- ----------------------- ----------------- ----------------------- ----------------- ----------------------- : Flavour changing self-energy transition elements estimated on the basis of eq (10), all in GeV units. The combination ${\cal H}_{ij} \equiv{\cal G}_{ij}+{\cal F}_{ij}/(m_i+m_j)$ is needed in eq (15). \[table1\] (460,100)(10,60) (12,146)(38,146) (18,156)\[\][$s$]{} (14,143)\[\] (38,146)[2.0]{} (38,146)(68,146) (92,138)(12,138)(18,130)\[\][$u$]{} (53,146)(15,0,180)[2]{}[10]{}(53,170)\[\][$W$]{} (68,146)(130,146) (125,156)\[\][$d$]{} (128,143)\[\] (68,146)[2.0]{} (92,90)(92,138)(84,100)\[\][$u$]{} (130,138)(100,138)(125,130)\[\][$c$]{} (100,138)(100,90)(110,100)\[\][$c$]{} (96,94)\[\] (74,65)\[\][(a)]{} (143,142)\[\][$+$]{} (153,146)(199,146) (163,156)\[\][$s$]{} (155,143)\[\] (199,146)[2.0]{} (199,146)(229,146) (229,146)[2.0]{} (229,146)(265,146) (257,156)\[\][$d$]{} (263,143)\[\] (214,146)(15,10,170)[2]{}[10]{}(214,170)\[\][$W$]{} (210,138)(153,138)(163,130)\[\][$u$]{} (265,138)(218,138)(257,130)\[\][$c$]{} (210,90)(210,138)(202,100)\[\][$u$]{} (218,138)(218,90)(226,100)\[\][$c$]{} (215,94)\[\] (214,65)\[\][(b)]{} (279,142)\[\][$+$]{} (288,146)(360,146) (298,156)\[\][$s$]{} (290,143)\[\] (360,146)[2.0]{} (360,146)(390,146) (390,146)[2.0]{} (390,146)(420,146) (416,156)\[\][$d$]{} (418,143)\[\] (375,146)(15,10,170)[2]{}[10]{}(375,170)\[\][$W$]{} (328,138)(288,138)(298,130)\[\][$u$]{} (328,90)(328,138)(320,100)\[\][$u$]{} (336,138)(336,90)(346,100)\[\][$c$]{} (420,138)(336,138)(416,130)\[\][$c$]{} (333,94)\[\] (364,65)\[\][(c)]{} (600,265)(10,-100) (12,146)(38,146) (18,156)\[\][$u$]{} (14,138)\[\] (38,146)[2.0]{} (38,146)(68,146) (92,130)(38,130) (38,130)(12,130)(18,122)\[\][$c$]{} (38,146)(38,130)[2]{}[3]{}(54,138)\[\][$W$]{} (68,146)(130,146) (125,156)\[\][$d$]{} (128,138)\[\] (38,130)[2.0]{} (92,80)(92,130)(84,90)\[\][$s$]{} (130,130)(100,130)(125,120)\[\][$b$]{} (100,130)(100,80)(110,90)\[\][$b$]{} (97,84)\[\] (74,65)\[\][(a)]{} (143,142)\[\][$+$]{} (153,146)(199,146) (163,156)\[\][$u$]{} (155,138)\[\] (178,146)[2.0]{} (199,146)(229,146) (210,105)[2.0]{} (219,146)(265,146) (257,156)\[\][$d$]{} (263,138)\[\] (178,146)(210,105)[2]{}[6]{}(198,138)\[\][$W$]{} (210,130)(153,130)(163,120)\[\][$c$]{} (265,130)(218,130)(257,120)\[\][$b$]{} (210,80)(210,130)(202,90)\[\][$s$]{} (218,130)(218,80)(226,90)\[\][$b$]{} (215,84)\[\] (214,65)\[\][(b)]{} (279,142)\[\][$+$]{} (288,146)(316,146) (298,156)\[\][$u$]{} (290,138)\[\] (316,146)(360,146) (316,130)[2.0]{} (360,146)(390,146) (394,146)[2.0]{} (390,146)(420,146) (416,156)\[\][$d$]{} (412,138)\[\] (360,110)(50,45,155)[2]{}[10]{}(346,172)\[\][$W$]{} (328,130)(288,130)(298,120)\[\][$c$]{} (328,80)(328,130)(320,90)\[\][$s$]{} (336,130)(336,80)(346,90)\[\][$b$]{} (420,130)(336,130)(416,120)\[\][$b$]{} (333,84)\[\] (354,65)\[\][(c)*]{} (600,100)(-150,40) (8,146)(38,146) (18,156)\[\][$s$]{} (10,138)\[\] (38,138)(8,-90,90) (74,138)(8,90,270) (46,138)[2.0]{} (38,130)(8,130)(18,122)\[\][$u$]{} (46,138)(66,138)[2]{}[4]{}(56,148)\[\][$W$]{} (74,146)(140,146) (135,156)\[\][$d$]{} (138,138)\[\] (66,138)[2.0]{} (92,130)(74,130) (92,80)(92,130)(84,90)\[\][$c$]{} (140,130)(100,130)(135,120)\[\][$b$]{} (100,130)(100,80) (97,84)\[\] (110,90)\[\][$b$]{} [^1]: A preliminary version of this method was presented in Delbourgo and Liu (1998), but publication requirements meant that the articles were so compressed as to be difficult to follow; the details are fleshed out here and the scope greatly expanded. [^2]: But even if they were not, one should be aware that form factors, which are inevitably present, will serve to damp out bare integrals; see section VI. [^3]: and renormalizing the strong coupling because of the enhancement factors.
--- abstract: \| Error correction of sequenced reads remains a difficult task, especially in single-cell sequencing projects with extremely non-uniform coverage. While existing error correction tools designed for standard (multi-cell) sequencing data usually come up short in single-cell sequencing projects, algorithms actually used for single-cell error correction have been so far very simplistic. We introduce several novel algorithms based on Hamming graphs and Bayesian subclustering in our new error correction tool . While was designed for single-cell sequencing, we demonstrate that it also improves on existing error correction tools for multi-cell sequencing data while working much faster on real-life datasets. We benchmark on both $k$-mer counts and actual assembly results with the genome assembler. address: \| (1)Algorithmic Biology Laboratory, Academic University, St. Petersburg, Russia\ (2)St. Petersburg State University, Russia\ (3)Department of Computer Science and Engineering, University of South Carolina, Columbia, SC, USA author: - 'Sergey I. Nikolenko$^1$ , Anton I. Korobeynikov$^{1,2}$, and Max A. Alekseyev$^{1,3}$' title: 'BayesHammer: Bayesian clustering for error correction in single-cell sequencing' --- =1 \[1995/12/01\] Background {#background .unnumbered} ========== Single-cell sequencing [@Grindberg2011; @ChitsazEtAl11] based on the Multiple Displacement Amplification (MDA) technology [@Grindberg2011; @Ishoey2008] allows one to sequence genomes of important uncultivated bacteria that until recently had been viewed as unamenable to genome sequencing. Existing metagenomic approaches (aimed at genes rather than genomes) are clearly limited for studies of such bacteria despite the fact that they represent the majority of species in such important studies as the Human Microbiome Project [@Gill2006; @Hamady2009] or discovery of new antibiotics-producing bacteria [@LV09]. Single-cell sequencing datasets have extremely non-uniform coverage that may vary from ones to thousands along a single genome (Fig. \[fig-covplot\]). For many existing error correction tools, most notably [@KSS10], uniform coverage is a prerequisite: in the case of non-uniform coverage they either do not work or produce poor results. Error correction tools usually attempt to correct the set of $k$-character substrings of reads called $k$-mers and then propagate corrections to whole reads which are important to have for many assemblers. Error correction tools often employ a simple idea of discarding rare $k$-mers, which obviously does not work in the case of non-uniform coverage. Medvedev et al. [@MSP11] recently presented a new approach to error correction for datasets with non-uniform coverage. Their algorithm makes use of the Hamming graph (hence the name) on $k$-mers (vertices of the graph correspond to $k$-mers and edges connect pairs of $k$-mers with Hamming distance not exceeding a certain threshold). employs a simple and fast clustering technique based on selecting a central $k$-mer in each connected component of the Hamming graph. Such central $k$-mers are assumed to be error-free (i.e., they are assumed to actually appear in the genome), while the other $k$-mers from connected components are assumed to be erroneous instances of the corresponding central $k$-mers. However, may be overly simplistic: in connected components of large diameter or connected components with several $k$-mers of large multiplicities, it is more reasonable to assume that there are two or more central $k$-mers (rather than one as in ). Biologically, such connected components may correspond to either repeated regions with similar but not identical genomic sequences (repeats) which would be bundled together by existing error correction tools (including ); or artificially united $k$-mers from distinct parts of the genome that just happen to be connected by a path in the Hamming graph (characteristic to ). In this paper, we introduce the error correction tool that does not rely on uniform coverage. uses the clustering algorithm of as a first step and then refines the constructed clusters by further subclustering them with a procedure that takes into account reads quality values (e.g., provided by Illumina sequencing machines) and introduces Bayesian (BIC) penalties for extra subclustering parameters. subclustering aims to capture the complex structure of repeats (possibly of varying coverage) in the genome by separating even very similar $k$-mers that come from different instances of a repeat. also uses a new approach for propagating corrections in $k$-mers to corrections in the reads. All algorithms in are heavily parallelized whenever possible; as a result, gains a significant speedup with more processing cores available. These features make a perfect error correction tool for single-cell sequencing. We remark that produces only a set of central $k$-mers but does not correct reads, making it incompatible with most genome assemblers. does correct reads but has severe memory limitations for large $k$ and assumes uniform coverage. In contrast, <span style="font-variant:small-caps;">EULER-SR</span> [@CP08] and <span style="font-variant:small-caps;">Camel</span> [@ChitsazEtAl11] correct reads and do not make strong assumptions on coverage (both tools have been used for single-cell assembly projects [@ChitsazEtAl11]) which makes these tools suitable for comparison to . Our benchmarks show that outperforms these tools in both single-cell and standard (multi-cell) modes. We further couple with a recently developed genome assembler [@Bankevich2012] and demonstrate that assembly of -corrected reads significantly improves upon assembly with reads corrected by other tools for the same datasets, while the total running time also improves significantly. is freely available for download as part of the genome assembler at <http://bioinf.spbau.ru/spades/>. Methods {#methods .unnumbered} ======= Notation and outline {#notation-and-outline .unnumbered} -------------------- Let $\Sigma=\{A, C, G, T\}$ be the alphabet of nucleotides (discards $k$-mers with uncertain bases denoted $N$). A $k$-mer is an element of $\Sigma^k$, i.e., a string of $k$ nucleotides. We denote the $i^{\texttt{th}}$ letter (nucleotide) of a $k$-mer $x$ by $x[i]$, indexing them from zero: $0\leq i\leq k-1$. A subsequence of $x$ corresponding to a set of indices $I$ is denoted by $x[I]$. We use interval notation $[i, j]$ for intervals of integers $\{i, i+1, \ldots, j\}$ and further abbreviate $x[i, j]=x\left[\{i, i+1, \ldots, j\}\right]$; thus, $x=x[0, k-1]$. Input reads are represented as a set of strings $R\subset\Sigma^$ along with their [quality values]{} $(q_r[i])_{i=0}^{\|r\|-1}$ for each $r\in R$. We assume that $q_r[i]$ estimates the probability that there has been an error in position $i$ of read $r$. Notice that in practice, the `fastq` file format [@Cock2010] contains characters that encode probabilities on a logarithmic scale (in particular, products of probabilities used below correspond to sums of actual quality values). Below we give an overview of workflow (Fig. \[fig-workflow\]) and refer to subsequent sections for further details. On Step (1), $k$-mers in the reads are counted, producing a triple $statistics(x)=(count_x, quality_x, \textbf{error}_x)$ for each $k$-mer $x$. Here, $count_x$ is the number of times $x$ appears as a substring in the reads, $quality_x$ is its total quality expressed as a probability of sequencing error in $x$, and $\textbf{error}_x$ is a $k$-dimensional vector that contains products of error probabilities (sums of quality values) for individual nucleotides of $x$ across all its occurrences in the reads. On Step (2), we find connected components of the Hamming graph constructed from this set of $k$-mers. On Step (3), the connected components become subject to Bayesian subclustering; as a result, for each $k$-mer we know the center of its subcluster. On Step (4), we filter subcluster centers according to their total quality and form a set of solid* $k$-mers which is then iteratively expanded on Step (5) by mapping them back to the reads. Step (6) deals with reads correction by counting the majority vote of solid $k$-mers in each read. In the iterative version, if there has been a substantial amount of changes in the reads, we run the next iteration of error correction; otherwise, output the corrected reads. Below we describe specific algorithms employed in the pipeline. (init) at (10,11) [Set of reads]{}; (counts) at (10,10) [(1) Compute $k$-mer statistics from reads]{}; (init) – (counts); (hammer) at (10,9) [(2) Construct connected components of Hamming graph]{}; (counts) – (hammer); (bayes) at (10,8) [(3) Bayesian subclustering of the connected components]{}; (hammer) – (bayes); (filter) at (10,7) [(4) Select solid $k$-mers from subcluster centers]{}; (bayes) – (filter); (solid) at (10,5.7) [A set of solid $k$-mers]{}; (filter) – (solid); (iterative) at (10,5) [(5) Iteratively expand the set of solid $k$-mers]{}; (solid) – (iterative); (iterres) at (10,3.7) [Set of solid $k$-mers changed?]{}; (iterative) – (iterres); (iterres) – node\[near start,above\] [yes]{} ++(4.75cm,0) \|- (iterative); (reconstruct) at (10,2.8) [(6) Correct reads]{}; (iterres) edge node\[left\] [no]{} (reconstruct); (finalif) at (10,1.5) [Reads substantially changed?]{}; (results) at (10,0.5) [A set of corrected reads]{}; (reconstruct) – (finalif); (finalif) – node\[above, near start\] [yes]{} ++(4.95cm,0) \|- (counts); (finalif) edge node\[left\] [no]{} (results); Algorithms {#algorithms .unnumbered} ---------- ### Step (1): computing $k$-mer statistics. {#sub-count .unnumbered} To collect $k$-mer statistics, we use a straightforward hash map approach [@CLR09] that does not require storing instances of all $k$-mers in memory (as excessive amount of RAM might be needed otherwise). For a certain positive integer $N$ (the number of auxiliary files), we use a hash function $h:\Sigma^k\to\mathbb Z_N$ that maps $k$-mers over the alphabet $\Sigma$ to integers from $0$ to $N-1$. compute $h(x)$ and write $x$ to $\file_{h(x)}$. sort $\file_i$ with respect to the lexicographic order; reading $\file_i$ sequentially, compute $statistics(s)$ for each $k$-mer $s$ from $\file_i$. ### Step (2): Constructing connected components of Hamming graph. {#sub-hammer .unnumbered} Step (2) is the essence of the approach [@MSP11]. The Hamming distance between $k$-mers $x,y\in\Sigma^k$ is the number of nucleotides in which they differ: $$\ham(x, y) = \left\|\{ i\in[0,k-1]\,:\, x[i] \neq y[i] \}\right\|.$$ For a set of $k$-mers $X$, the Hamming graph $\HG_\tau(X)$ is an undirected graph with the set of vertices $X$ and edges corresponding to pairs of $k$-mers from $X$ with Hamming distance at most $\tau$, i.e., $x, y\in X$ are connected by an edge in $\HG_\tau(X)$ iff $\ham(x, y)\le\tau$ (Fig. \[fig-restricted\]). To construct $\HG_\tau(X)$ efficiently, we notice that if two $k$-mers are at Hamming distance at most $\tau$, and we partition the set of indices $[0, k-1]$ into $\tau+1$ parts, then at least one part corresponds to the same subsequence in both $k$-mers. Below we assume with little loss of generality that $\tau+1$ divides $k$, i.e., $k = \sigma\cdot (\tau+1)$ for some integer $\sigma$. (reads) at (1.5, 6) [ Reads ]{}; (read1) at (1.5, 5) [ ]{}; (read2) at (1.5, 3.5) [ ]{}; (read3) at (1.5, 2) [ ]{}; (kmers) at (3, 6) [ $k$-mers ]{}; (kmer11) at (3, 5) [ ]{}; (kmer12) at (3, 4.6) [ ]{}; (kmer13) at (3, 4.2) [ ]{}; (kmer21) at (3, 3.5) [ ]{}; (kmer22) at (3, 3.1) [ ]{}; (kmer23) at (3, 2.7) [ ]{}; (kmer31) at (3, 2) [ ]{}; (kmer32) at (3, 1.6) [ ]{}; (kmer33) at (3, 1.2) [ ]{}; (kmers) at (5.8, 6) [ $\HG_1(X)$ ]{}; (k11) at (4.4, 3.5) [ ]{}; (k12) at (6.3, 3.6) [ ]{}; (k13) at (4.4, 1.5) [ ]{}; (k21) at (4.2, 5) [ ]{}; (k22) at (7.6, 3.3) [ ]{}; (k23) at (5.4, 2.7) [ ]{}; (k31) at (5.5, 4.2) [ ]{}; (k32) at (7.3, 2.6) [ ]{}; (k33) at (6.5, 1.5) [ ]{}; (k11) – (k21); (k11) – (k31); (k21) – (k31); (k11) – (k23); (k13) – (k23); (k12) – (k22); (k32) – (k22); (k12) – (k32); (kmers) at (10.5, 6) [ $\HG_2(X)$ ]{}; (k11) at (9.1, 3.5) [ ]{}; (k12) at (10.9, 3.6) [ ]{}; (k13) at (9.0, 1.5) [ ]{}; (k21) at (8.8, 5) [ ]{}; (k22) at (12.2, 3.3) [ ]{}; (k23) at (10.0, 2.7) [ ]{}; (k31) at (10.2, 4.2) [ ]{}; (k32) at (11.9, 2.6) [ ]{}; (k33) at (11.2, 1.5) [ ]{}; (k11) – (k21); (k11) – (k31); (k21) – (k31); (k11) – (k23); (k13) – (k23); (k33) – (k23); (k13) – (k33); (k12) – (k22); (k32) – (k22); (k12) – (k32); (k31) – (k23); (k11) – (k13); (k11) edge (k33); (k33) – (k23); (k13) – (k33); For a subset of indices $I\subseteq [0, k-1]$, we define a partial lexicographic ordering $\prec_I$ as follows: $x\prec_I y$ iff $x[I]\prec y[I]$, where $\prec$ is the lexicographic ordering on $\Sigma^$. Similarly, we define a partial equality $=_I$ such that $x=_I y$ iff $x[I]=y[I]$. We partition the set of indices $[0, k-1]$ into $\tau+1$ parts of size $\sigma$ and for each part $I$, sort a separate copy of $X$ with respect to $\prec_I$. As noticed above, for every two $k$-mers $x, y\in X$ with $\ham(x, y)\le\tau$, there exists a part $I$ such that $x=_I y$. It therefore suffices to separately consider blocks of equivalent $k$-mers with respect to $=_I$ for each part $I$. If a block is small (i.e., of size smaller than a certain threshold), we go over the pairs of $k$-mers in this block to find those with Hamming distance at most $\tau$. If a block is large, we recursively apply to it the same procedure with a different partition of the indices. In practice, we use two different partitions of $[0, k-1]$: the first corresponds to contigious subsets of indices (recall that $\sigma=\tfrac{k}{\tau+1}$): $$I^{\textrm{cnt}}_s=\{s\sigma, s\sigma+1, \ldots, s\sigma+\sigma-1\}, \quad s=0, \ldots, \tau,$$ while the second corresponds to strided subsets of indices: $$I^{\textrm{str}}_s=\{s, s+\tau+1, s+2(\tau+1), \ldots, s+(\sigma-1)(\tau+1)\}, \quad s=0, \ldots, \tau.$$ uses a two-step procedure, first splitting with respect to $\{I^{\textrm{cnt}}_s\}_{s=0}^{\tau}$ (Fig. \[fig-subkmers\]) and then, if an equivalence block is large, with respect to $\{I^{\textrm{str}}_s\}_{s=0}^{\tau}$. On the block processing step, we use the disjoint set data structure [@CLR09] to maintain the set of connected components. Step (2) is summarized in Algorithm \[step2\]. Init components with singletons $\cal X = \{ \{x\}: x\in X\}$. $\mathrm{ProcessExhaustively}(Z, \cal X)$ $\mathrm{ProcessExhaustively}(Y, \cal X)$. sort a copy of $X$ with respect to $\prec_{I_s}$, getting $X_s$. output the set of equiv. blocks $\{Y\}$ w.r.t. $=_{I_s}$. join their sets in $\cal X$: $\cal X:=\cal X\cup\{ Z_x\cup Z_y \}\setminus \{Z_x, Z_y\}$. (1kmers) at (3, 6) [ $X$* ]{}; (1kmer1) at (3, 5) [ `1 ` ]{}; (1kmer2) at (3, 4.6) [ `2 ` ]{}; (1kmer3) at (3, 4.2) [ `3 ` ]{}; (1kmer4) at (3, 3.8) [ `4 ` ]{}; (1kmer5) at (3, 3.4) [ `5 ` ]{}; (1kmer6) at (3, 3) [ `6 ` ]{}; (2kmers) at (5, 6) [ $X[I^{\textrm{cnt}}_0]$ ]{}; (2kmer1) at (5, 5) [ `1 ` ]{}; (2kmer2) at (5, 4.6) [ `2 ` ]{}; (2kmer3) at (5, 4.2) [ `3 ` ]{}; (2kmer4) at (5, 3.8) [ `4 ` ]{}; (2kmer5) at (5, 3.4) [ `5 ` ]{}; (2kmer6) at (5, 3) [ `6 ` ]{}; (3kmers) at (6.5, 6) [ $X[I^{\textrm{cnt}}_1]$ ]{}; (3kmer1) at (6.5, 5) [ `1 ` ]{}; (3kmer2) at (6.5, 4.6) [ `2 ` ]{}; (3kmer3) at (6.5, 4.2) [ `3 ` ]{}; (3kmer4) at (6.5, 3.8) [ `4 ` ]{}; (3kmer5) at (6.5, 3.4) [ `5 ` ]{}; (3kmer6) at (6.5, 3) [ `6 ` ]{}; (3akmers) at (8, 6) [ $X[I^{\textrm{cnt}}_2]$ ]{}; (3akmer1) at (8, 5) [ `1 ` ]{}; (3akmer2) at (8, 4.6) [ `2 ` ]{}; (3akmer3) at (8, 4.2) [ `3 ` ]{}; (3akmer4) at (8, 3.8) [ `4 ` ]{}; (3akmer5) at (8, 3.4) [ `5 ` ]{}; (3akmer6) at (8, 3) [ `6 ` ]{}; (4kmers) at (9.5, 6) [ $X[I^{\textrm{cnt}}_0]$ ]{}; (4kmerss) at (9.5, 5.5) [ sorted ]{}; (4kmer1) at (9.5, 4.6) [ `1 ` ]{}; (4kmer2) at (9.5, 3.8) [ `2 ` ]{}; (4kmer3) at (9.5, 3) [ `3 ` ]{}; (4kmer4) at (9.5, 5) [ `4 ` ]{}; (4kmer5) at (9.5, 4.2) [ `5 ` ]{}; (4kmer6) at (9.5, 3.4) [ `6 ` ]{}; (5kmers) at (11, 6) [ $X[I^{\textrm{cnt}}_1]$ ]{}; (5kmerss) at (11, 5.5) [ sorted ]{}; (5kmer1) at (11, 4.2) [ `1 ` ]{}; (5kmer2) at (11, 5) [ `2 ` ]{}; (5kmer3) at (11, 3.8) [ `3 ` ]{}; (5kmer4) at (11, 3.4) [ `4 ` ]{}; (5kmer5) at (11, 4.6) [ `5 ` ]{}; (5kmer6) at (11, 3) [ `6 ` ]{}; (3bkmers) at (12.5, 6) [ $X[I^{\textrm{cnt}}_2]$ ]{}; (5bkmerss) at (12.5, 5.5) [ sorted ]{}; (3bkmer1) at (12.5, 4.2) [ `1 ` ]{}; (3bkmer2) at (12.5, 3.4) [ `2 ` ]{}; (3bkmer3) at (12.5, 5) [ `3 ` ]{}; (3bkmer4) at (12.5, 3.8) [ `4 ` ]{}; (3bkmer5) at (12.5, 3) [ `5 ` ]{}; (3bkmer6) at (12.5, 4.6) [ `6 ` ]{}; (10.3,5.3) – (10.3, 4.55) – (11.7, 4.55) – (11.7, 5.3) – (10.3,5.3); (10.3,4.5) – (10.3, 2.9) – (11.7, 2.9) – (11.7, 4.5) – (10.3,4.5); (11.8,4.5) – (11.8, 3.7) – (13.2, 3.7) – (13.2, 4.5) – (11.8,4.5); ### Step (3): Bayesian subclustering. {#sub-bayes .unnumbered} In ’s generative model [@MSP11], it is assumed that errors in each position of a $k$-mer are independent and occur with the same probability $\epsilon$, which is a fixed global parameter (used $\epsilon=0.01$). Thus, the likelihood that a $k$-mer $x$ was generated from a $k$-mer $y$ under ’s model equals $$L_{\Hammer}(x\mid y) = (1-\epsilon)^{k-\ham(x, y)}\epsilon^{\ham(x, y)}.$$ Under this model, the maximum likelihood center of a cluster is simply its consensus string [@MSP11]. In , we further elaborate upon ’s model. Instead of a fixed $\epsilon$, we use reads quality values that approximate probabilities $q_x[i]$ of a nucleotide at position $i$ in the $k$-mer $x$ being erroneous. We combine quality values from identical $k$-mers in the reads: for a multiset of $k$-mers $X$ that agree on the $j^{\textrm{th}}$ nucleotide, it is erroneous with probability $\prod_{x\in X}q_x[j]$. The likelihood that a $k$-mer $x$ has been generated from another $k$-mer $c$ (under the independent errors assumption) is given by $$L(x\mid c)=\prod_{j:\ x[j]\neq c[j]}q_{x}[j]\prod_{j:\ x[j]=c[j]}\left(1-q_{x}[j]\right),$$ and the likelihood of a specific subclustering $C=C_1\cup\ldots\cup C_m$ is $$L_m(C_1,\ldots,C_m)=\prod_{i=1}^m \prod_{x\in C_i} L(x\mid c_i)$$ where $c_i$ is the center (consensus string) of the subcluster $C_i$. In the subclustering procedure (see Algorithm \[step3\]), we sequentially subcluster each connected component of the Hamming graph into more and more clusters with the classical $k$-means clustering algorithm (denoted $m$-means since $k$ has different meaning). For the objective function, we use the likelihood as above penalized for overfitting with the Bayesian information criterion (BIC) [@Schwarz78]. In this case, there are $\|C\|$ observations in the dataset, and the total number of parameters is $3km+m-1$: - $m-1$ for probabilities of subclusters, - $km$ for cluster centers, and - $2km$ for error probabilities in each letter: there are $3$ possible errors for each letter, and the probabilities should sum up to one. Here error probabilities are conditioned on the fact that an error has occurred (alternatively, we could consider the entire distribution, including the correct letter, and get $3km$ parameters for probabilities but then there would be no need to specify cluster centers, so the total number is the same). Therefore, the resulting objective function is $$\ell_m := 2\cdot \log L_m(C_1,\ldots,C_m) - (3km + m-1)\cdot \log \|C\|$$ for subclustering into $m$ clusters; we stop as soon as $\ell_m$ ceases to increase. $m := 1$ $\ell_1 := 2\log L_1(C)$ (likelihood of the cluster generated by the consensus) $m := m+1$ do $m$-means clustering of $C=C_1\cup\ldots\cup C_m$ w.r.t. the Hamming distance; the initial approximation to the centers is given by $k$-mers that have the least error probability $\ell_m := 2\cdot \log L_m(C_1,\ldots,C_m) - (3km + m-1)\cdot \log \|C\|$ output the best found clustering $C=C_1\cup\ldots\cup C_{m-1}$ ### Steps (4) and (5): selecting solid $k$-mers and expanding the set of solid $k$-mers. {#sub-filter .unnumbered} We define the quality of a $k$-mer $x$ as the probability that it is error-free: $p_{x} = \prod_{j=0}^{k-1}\left(1-q_x[j]\right).$ The $k$-mer qualities are computed on Step (1) along with computing $k$-mer statistics. Next, we (generously) define the quality of a cluster $C$ as the probability that at least one $k$-mer in $C$ is correct: $$p_C = 1 - \prod_{x\in C}\left(1-p_x\right).$$ In contrast to , we do not distinguish whether the cluster is a singleton (i.e., $\|C\|=1$); there may be plenty of superfluous clusters with several $k$-mers obtained by chance (actually, it is more likely to obtain a cluster of several $k$-mers by chance than a singleton of the same total multiplicity). Initially we mark as solid the centers of the clusters whose total quality exceeds a predefined threshold (a global parameter for , set to be rather strict). Then we expand the set of solid $k$-mers iteratively: if a read is completely covered by solid $k$-mers we conclude that it actually comes from the genome and mark all other $k$-mers in this read as solid, too (Algorithm \[step5\]). mark all $k$-mers in $r$ as solid Return $\true$ if $X$ has increased and $\false$ otherwise. ### Step (6): reads correction. {#sub-reconstruct .unnumbered} After Steps (1)-(5), we have constructed the set of solid $k$-mers that are presumably error-free. To construct corrected reads from the set of solid $k$-mers, for each base of every read, we compute the consensus of all solid $k$-mers and solid centers of clusters of all non-solid $k$-mers covering this base (Fig. \[fig-reconstruct\]). This step is formally described as Algorithm \[step6\]. reads$R$,solid$k$-mers$X$,clusters$\mathcal C$. init consensus array $v:[0, \|r\|-1]\times \{A, C, G, T\}\to\mathbb N$ with zeros: $v(j, x[i]) := 0$ for all $i=0,\dots,\|r\|-1$ and $j=0,\dots,k-1$ $v(j, r[i]) := v(j, r[i]) + 1$ let $x$ be the center of $C$ $v(j, x[i]) := v(j, x[i]) + 1$ $r[i] := \amax_{a\in \Sigma} v(i, a)$. (read) at (3, 5.3) [ [ACGTGTGATGCATGATCG ]{} ]{}; (kmer1) at (3, 4.6) [ ]{}; (kmer2) at (3, 4.2) [ ]{}; (kmer3) at (3, 3.8) [ ]{}; (kmer4) at (3, 3.4) [ [TGTGATGCATG ]{} ]{}; (kmer5) at (3, 3.0) [ [GTGATGCATGA ]{} ]{}; (kmer6) at (3, 2.6) [ ]{}; (kmer7) at (3, 2.2) [ ]{}; (kmer8) at (3, 1.8) [ [ATGCATGATCG]{} ]{}; (read) at (7, 5.3) [ [ACGTGTGATGCATGATCG ]{} ]{}; (kmer21) at (7, 4.6) [ ]{}; (kmer22) at (7, 4.2) [ ]{}; (kmer23) at (7, 3.8) [ ]{}; (kmer24) at (7, 3.4) [ [TGTGATGCATG ]{} ]{}; (kmer25) at (7, 3.0) [ [GTGATGCATGA ]{} ]{}; (kmer26) at (7, 2.6) [ ]{}; (kmer27) at (7, 2.2) [ ]{}; (kmer28) at (7, 1.8) [ [ATGCATGATCG]{} ]{}; (read) at (7, 1.3) [ [ACGTGGATGCATGATCG ]{} ]{}; (4.3,4.3) – (5.8, 4.3); (4.5,3.9) – (6.0, 3.9); (5.1,2.7) – (6.6, 2.7); (,) – (,) – (,) – (,) – (,); Results and discussion {#results-and-discussion .unnumbered} ====================== Datasets {#datasets .unnumbered} -------- In our experiments, we used three datasets from [@ChitsazEtAl11]: a single-cell E. coli, a single-cell S. aureus, and a standard (multicell) E. coli dataset. Paired-end libraries were generated by an Illumina Genome Analyzer IIx from MDA-amplified single-cell DNA and from multicell genomic DNA prepared from cultured E. coli, respectively These datasets consist of $100$bp paired-end reads with insert size $220$; both E. coli datasets have average coverage $\approx 600\times$, although the coverage is highly non-uniform in the single-cell case. In all experiments, used $k=21$ (we observed no improvements for higher values of $k$). $k$-mer counts {#k-mer-counts .unnumbered} -------------- Correction tool Running time Reads ------------------- -------------- ------------------- ----------------- ------------------- ------------------------------------------ --------------------------------------- --------------------------- Total Genomic Non-genomic % of all genomic $k$-mers found in reads % genomic among all $k$-mers in reads % reads aligned to genome Uncorrected $187{,}580{,}875$ $4{,}543{,}684$ $183{,}037{,}191$ $99.99$ $2.4$ $99.05$ Quake $4{,}565{,}237$ $4{,}543{,}461$ $21{,}776$ $99.99$ $99.5$ $99.97$ HammerNoExpansion $30$m $58{,}305{,}738$ $4{,}543{,}674$ $53{,}762{,}064$ $99.99$ $8.4$ $95.59$ HammerExpanded $36$m $28{,}290{,}788$ $4{,}543{,}673$ $23{,}747{,}115$ $99.99$ $19.1$ $99.49$ BayesHammer $37$m $27{,}100{,}305$ $4{,}543{,}674$ $22{,}556{,}631$ $99.99$ $20.1$ $99.62$ Uncorrected $165{,}355{,}467$ $4{,}450{,}489$ $160{,}904{,}978$ $97.9$ $2.7$ $79.05$ Camel $2$h$29$m $147{,}297{,}070$ $4{,}450{,}311$ $142{,}846{,}759$ $97.9$ $3.0$ $81.25$ Euler-SR $2$h$15$m $138{,}677{,}818$ $4{,}450{,}431$ $134{,}227{,}387$ $97.9$ $3.2$ $81.95$ Coral $2$h$47$m $156{,}907{,}496$ $4{,}449{,}560$ $152{,}457{,}936$ $97.9$ $2.8$ $80.28$ HammerNoExpansion $37$m $53{,}001{,}778$ $4{,}443{,}538$ $48{,}558{,}240$ $97.8$ $8.3$ $81.36$ HammerExpanded $43$m $36{,}471{,}268$ $4{,}443{,}545$ $32{,}027{,}723$ $97.8$ $12.1$ $86.91$ BayesHammer $57$m $35{,}862{,}329$ $4{,}443{,}736$ $31{,}418{,}593$ $97.8$ $12.4$ $87.12$ Uncorrected $88{,}331{,}311$ $2{,}820{,}394$ $85{,}510{,}917$ $99.98$ $3.2$ $75.07$ Camel $5$h$13$m $69{,}365{,}311$ $2{,}820{,}350$ $66{,}544{,}961$ $99.97$ $4.1$ $75.27$ Euler-SR $2$h$33$m $58{,}886{,}372$ $2{,}820{,}349$ $56{,}066{,}023$ $99.97$ $4.8$ $75.24$ Coral $7$h$12$m $83{,}249{,}146$ $2{,}820{,}011$ $80{,}429{,}135$ $99.96$ $3.4$ $75.22$ HammerNoExpansion $58$m $37{,}465{,}296$ $2{,}820{,}341$ $34{,}644{,}955$ $99.97$ $7.5$ $71.63$ HammerExpanded $1$h$03$m $23{,}197{,}521$ $2{,}820{,}316$ $20{,}377{,}205$ $99.97$ $12.1$ $76.54$ BayesHammer $1$h$09$m $22{,}457{,}509$ $2{,}820{,}311$ $19{,}637{,}198$ $99.97$ $12.6$ $76.60$ Table \[tbl:kmers\] shows error correction statistics produced by different tools on all three datasets. For a comparison with , we have emulated with read correction by turning off Bayesian subclustering (HammerExpanded in the table) and both Bayesian subclustering and read expansion, another new idea of (HammerNoExpansion in the table). Note that despite its more complex processing, is significantly faster than other error correction tools (except, of course, for which is a strict subset of processing in our experiments and is run on code). also produces, in the single-cell case, a much smaller set of $k$-mers in the resulting reads which leads to smaller de Bruijn graphs and thus reduces the total assembly running time. Since trims only bad quality bases and does not, like , trim bases that it has not been able to correct (it has been proven detrimental for single-cell assembly in our experiments), it does produce a much larger set of $k$-mers than Quake on a multi-cell dataset. For a comparison of with other tools in terms of error rate reduction across an average read, see the logarithmic error rate graphs on Fig. \[fig:bherrors\]. Note that we are able to count errors only for the reads that actually aligned to the genome, so the graphs are biased in this way. Note how the first $21$ bases are corrected better than others in and both versions of since we have run it with $k=21$; still, other values of $k$ did not show a significant improvement in either $k$-mer statistics or, more importantly, assembly results. Assembly results {#assembly-results .unnumbered} ---------------- [\|l\[10]{}[\|p[26pt]{}]{}\|]{} Statistics & & & & & & & & & &\ &\ \# contigs ($\geq$ 1000 bp) & 191 & 158 & 276 & 224 & 231 & 150 & 195 & 282 & 242 & 173\ \# contigs & 521 & 462 & 675 & 592 & 578 & 375 & 529 & 655 & 592 & 477\ Largest contig & 269177 & 284968 & 179022 & 179022 & 267676 & 267676 & 268464 & 210850 & 210850 & 268464\ Total length & 4952297 & 4989404 & 5064570 & 5122860 & 4817757 & 4902434 & 4977294 & 5097148 & 5340871 & 5005022\ N50 & 110539 & 113056 & 45672 & 67849 & 74139 & 95704 & 97639 & 65415 & 84893 & 109826\ NG50 & 112065 & 118432 & 55073 & 87317 & 77762 & 108976 & 101871 & 68595 & 96600 & 112161\ NA50 & 110539 & 113056 & 45672 & 67765 & 74139 & 95704 & 97639 & 65415 & 84841 & 109826\ NGA50 & 112064 & 118432 & 55073 & 87317 & 77762 & 108976 & 101871 & 68594 & 96361 & 112161\ \# misassemblies & 4 & 6 & 9 & 12 & 6 & 8 & 4 & 4 & 7 & 7\ \# misassembled contigs & 4 & 6 & 9 & 10 & 6 & 8 & 4 & 4 & 7 & 7\ Misass. contigs length & 42496 & 94172 & 62114 & 150232 & 47372 & 149639 & 43304 & 26872 & 147140 & 130706\ Genome covered (%) & 96.320 & 96.315 & 96.623 & 96.646 & 95.337 & 95.231 & 96.287 & 96.247 & 96.228 & 96.281\ GC (%) & 49.70 & 49.69 & 49.61 & 49.56 & 49.90 & 49.74 & 49.68 & 49.64 & 49.60 & 49.68\ \# mismatches / 100kbp & 11.22 & 11.70 & 8.36 & 9.10 & 5.55 & 5.82 & 12.77 & 54.11 & 52.48 & 13.08\ \# indels / 100kbp & 1.07 & 8.26 & 9.17 & 12.76 & 0.52 & 47.80 & 0.91 & 1.17 & 7.96 & 8.69\ \# genes & 4065 + 124 part & 4079 + 110 part & 3998 + 180 part & 4040 + 143 part & 3992 + 140 part & 4020 + 107 part & 4068 + 123 part & 4034 + 152 part & 4048 + 136 part & 4078 + 111 part\ &\ \# contigs ($\geq$ 1000 bp) & 95 & 85 & 132 & 113 & 82 & 70 & 114 & 272 & 258 & 101\ Total length ($\geq$ 1000 bp) & 3019597 & 3309342 & 3055585 & 3066662 & 2972925 & 2993100 & 3033912 & 3389846 & 3405223 & 3509555\ \# contigs & 260 & 241 & 455 & 423 & 166 & 134 & 312 & 721 & 711 & 292\ Largest contig & 282558 & 328686 & 208166 & 208166 & 254085 & 535477 & 282558 & 148002 & 166053 & 328679\ Total length & 3081173 & 3368034 & 3160497 & 3166169 & 3008746 & 3020256 & 3111423 & 3575679 & 3594468 & 3584266\ N50 & 87684 & 145466 & 62429 & 90701 & 101836 & 145466 & 74715 & 30788 & 34943 & 131272\ NG50 & 112566 & 194902 & 87636 & 99341 & 108151 & 159555 & 88292 & 39768 & 45889 & 180022\ NA50 & 87684 & 145466 & 62429 & 89365 & 100509 & 145466 & 68711 & 30788 & 34552 & 112801\ NGA50 & 88246 & 148064 & 74452 & 90101 & 101836 & 145466 & 88289 & 35998 & 42642 & 148023\ \# misassemblies & 15 & 17 & 11 & 14 & 4 & 5 & 11 & 14 & 18 & 14\ \# misassembled contigs & 12 & 14 & 9 & 10 & 4 & 5 & 9 & 14 & 16 & 12\ Misass. contigs length & 340603 & 779785 & 478009 & 523596 & 377133 & 918380 & 402997 & 272677 & 324361 & 940356\ Genome covered (%) & 99.522 & 99.483 & 99.449 & 99.447 & 99.213 & 99.254 & 99.204 & 98.820 & 98.888 & 99.221\ GC (%) & 32.67 & 32.63 & 32.64 & 32.63 & 32.66 & 32.67 & 32.67 & 32.39 & 32.38 & 32.57\ \# mismatches per 100 kbp & 3.18 & 8.01 & 12.44 & 12.65 & 9.72 & 10.28 & 17.38 & 54.92 & 55.50 & 15.36\ \# indels per 100 kbp & 2.17 & 2.30 & 15.50 & 15.67 & 3.80 & 4.08 & 3.57 & 2.64 & 2.72 & 3.04\ \# genes & 2540 + 36 part & 2547 + 30 part & 2532 + 45 part & 2540 + 37 part & 2547 + 30 part & 2550 + 27 part & 2535 + 41 part & 2477 + 91 part & 2485 + 85 part & 2539 + 38 part\ [\|l\[10]{}[\|p[40pt]{}]{}\|]{} Statistics & & & & & & &\ &\ \# contigs ($\geq$ 500 bp) & 103 & 102 & 119 & 238 & 213 & 115 & 165\ \# contigs ($\geq$ 1000 bp) & 91 & 90 & 99 & 192 & 171 & 96 & 156\ Total length ($\geq$ 500 bp) & 4641845 & 4641790 & 4626515 & 4730338 & 4817457 & 4627067 & 4543682\ Total length ($\geq$ 1000 bp) & 4633361 & 4633306 & 4611745 & 4696966 & 4787210 & 4612838 & 4537565\ \# contigs & 122 & 121 & 146 & 325 & 303 & 141 & 204\ Largest contig & 285113 & 285113 & 218217 & 210240 & 210240 & 218217 & 165487\ Total length & 4647325 & 4647270 & 4635156 & 4756088 & 4844208 & 4635349 & 4555015\ N50 & 132645 & 132645 & 113608 & 59167 & 73113 & 113608 & 58777\ NG50 & 132645 & 132645 & 113608 & 59669 & 80085 & 113608 & 57174\ NA50 & 132645 & 132645 & 113608 & 59167 & 73113 & 113608 & 58777\ NGA50 & 132645 & 132645 & 113608 & 59669 & 80085 & 113608 & 57174\ \# misassemblies & 3 & 3 & 4 & 4 & 7 & 5 & 0\ \# misassembled contigs & 3 & 3 & 4 & 4 & 7 & 5 & 0\ Misassembled contigs length & 44466 & 44466 & 57908 & 15259 & 30901 & 60418 & 0\ Genome covered (%) & 99.440 & 99.440 & 99.383 & 98.891 & 98.925 & 99.385 & 98.747\ GC (%) & 50.78 & 50.77 & 50.77 & 50.73 & 50.71 & 50.77 & 50.75\ N’s (%) & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000\ \# mismatches per 100 kbp & 8.55 & 8.55 & 13.76 & 44.46 & 44.33 & 13.76 & 1.21\ \# indels per 100 kbp & 0.99 & 0.99 & 1.14 & 0.76 & 0.97 & 1.14 & 0.20\ \# genes & 4254+45 part & 4254+45 part & 4245+56 part & 4196+72 part & 4204+68 part & 4245+56 part & 4174+62 part\ Tables \[tbl:scres\] and \[tbl:mcres\] shows assembly results by the recently developed assembler [@Bankevich2012]; was designed specifically for single-cell assembly, but has by now demonstrated state-of-the-art results on multi-cell datasets as well. In the tables, N50 is such length that contigs of that length or longer comprise $\ge\tfrac12$ of the assembly; NG50 is a metric similar to N50 but only taking into account contigs comprising (and aligning to) the reference genome; NA50 is a metric similar to N50 after breaking up misassembled contigs by their misassemblies. NGx and NAx metrics have a more direct relevance to assembly quality than regular Nx metrics; our result tables have been produced by the recently developed tool QUAST [@Gurevich2012]. All assemblies have been done with . The results show that after correction, assembly results improve significantly, especially in the single-cell E. coli case; it is especially interesting to note that even in the multi-cell case, where loses to by $k$-mer statistics, assembly results actually improve over assemblies produced from -corrected reads (including genome coverage and the number of genes). Conclusions {#conclusions .unnumbered} =========== Single-cell sequencing presents novel challenges to error correction tools. In contrast to multi-cell datasets, for single-cell datasets, there is no pretty distribution of $k$-mer multiplicities; one therefore has to work with $k$-mers on a one-by-one basis, considering each cluster of $k$-mers separately. In this work, we further developed the ideas of from a Bayesian clustering perspective and presented a new tool that makes them practical and yields significant improvements over existing error correction tools. There is further work to be done to make our underlying models closer to real life; for instance, one could learn a non-uniform distribution of single nucleotide errors and plug it in our likelihood formulas. Another natural improvement would be to try and rid the results of contamination by either human or some other DNA material; we observed significant human DNA contamination in our single-cell dataset, so weeding it out might yield a significant improvement. Finally, a new general approach that we are going to try in our further work deals with the technique of minimizers introduced by Roberts et al. [@RHHMY04]. It may provide significant reduction in memory requirements and a possible approach to dealing with paired information. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Pavel Pevzner for many fruitful discussions on all stages of the project. We are also grateful to Andrei Prjibelski and Alexei Gurevich for help with the experiments and to the anonymous referees whose comments have benefited the paper greatly. This work was supported the Government of the Russian Federation, grant 11.G34.31.0018. Work of the first author was also supported by the Russian Fund for Basic Research grant 12-01-00450-a and the Russian Presidential Grant MK-6628.2012.1. Work of the second author was additionally supported by the Russian Fund for Basic Research grant 12-01-00747-a.
--- abstract: 'We present a novel deep neural network architecture for representing robot experiences in an episodic-like memory which facilitates encoding, recalling, and predicting action experiences. Our proposed unsupervised deep episodic memory model 1) encodes observed actions in a latent vector space and, based on this latent encoding, 2) infers most similar episodes previously experienced, 3) reconstructs original episodes, and 4) predicts future frames in an end-to-end fashion. Results show that conceptually similar actions are mapped into the same region of the latent vector space. Based on these results, we introduce an action matching and retrieval mechanism, benchmark its performance on two large-scale action datasets, 20BN-something-something and ActivityNet and evaluate its generalization capability in a real-world scenario on a humanoid robot.' author: - 'Jonas Rothfuss$^{\ast}{^\dagger}$, Fabio Ferreira$^{\ast}{^\dagger}$, Eren Erdal Aksoy $^{\ddagger}$, You Zhou$^{\dagger}$ and Tamim Asfour$^{\dagger}$[^1] [^2] [^3] [^4]' bibliography: - 'library.bib' title: \| Deep Episodic Memory: Encoding, Recalling, and Predicting\ Episodic Experiences for Robot Action Execution --- [^1]: [^2]: $^{\dagger}$The authors are with the Institute for Anthropomatics and Robotics, Karlsruhe Institute of Technology, Karlsruhe, Germany. [^3]: $^{\ddagger}$Eren E. Aksoy is with with the School of Information Technology, Halmstad University, Sweden [^4]: $^{\ast}$The first two authors contributed equally to this work.
--- abstract: 'We investigated the lock-in transition of charge density waves (CDWs) in quasi-one-dimensional conductors, based on McMillan’s free energy. The higher-order umklapp terms play an essential role in this study. McMillan’s theory was extended by Nakanishi and Shiba in order to treat multiple CDW vectors. Although their theories were aimed at understanding CDWs in quasi-two-dimensional conductors, we applied them to the quasi-one-dimensional conductors, including K$_{0.3}$MoO$_3$, NbSe$_3$, and $m$-TaS$_3$, and confirmed its validity for these cases. Then we discussed our previous experimental result of $o$-TaS$_3$, which revealed the coexistence of commensurate and incommensurate states. We found that the coexistence of multiple CDW vectors is essential for the lock-in transition to occur in $o$-TaS$_3$. The even- and odd-order terms in the free energy play roles for amplitude development and phase modulation, respectively. Moreover, consideration of the condition of being commensurate CDWs allowed us to relate it with that of the weak localization in random media.' author: - Katsuhiko Inagaki - Satoshi Tanda title: \| Lock-in transition of charge density waves in quasi-one-dimensional conductors:\ reinterpretation of McMillan’s theory --- I. Introduction =============== Lock-in transition between incommensurate and commensurate charge density waves (CDWs) has been studied since the mid-1970s [@Moncton1975; @McMillan1976; @Bak1976; @Nakanishi1977; @Suits1980; @Roucau1980; @Chen1981; @Roucau1983; @Tanda1984; @Fleming1984; @Tanda1985; @Fleming1985; @Inagaki2008; @Sakabe2017]. It is induced by the coupling of a lattice periodicity with a charge density wave. The transition is often accompanied with the formation of a discommensuration lattice between commensurate and incommensurate phases. Occurrence of discommensuration was predicted by theory [@McMillan1976] and found in quasi-two-dimensional conductors, e.g. in 2H-TaSe$_2$ [@Suits1980; @Chen1981] and 1T-TaS$_2$ [@Tanda1984; @Tanda1985; @Sakabe2017], both of which are typical quasi-two-dimensional conductors with Peierls transition. In contrast, the lock-in transition of CDWs in quasi-one-dimensional conductors remains unsubstantiated, although several experiments were reported [@Roucau1980; @Roucau1983; @Fleming1984; @Fleming1985]. We previously performed a synchrotron x-ray study in $o$-TaS$_3$ and suggested that the discommensuration lattice is formed when commensurate and incommensurate phases coexist [@Inagaki2008]. However, this preliminary report lacked theoretical interpretation. In this paper we review the theoretical treatments of the lock-in transition and apply them to the quasi-one-dimensional conductors, including K$_{0.3}$MoO$_3$, NbSe$_3$, and $m$-TaS$_3$. The validity of the theory is confirmed for these cases. We then go on to discuss the synchrotron data of $o$-TaS$_3$. The coexistence of the commensurate and incommensurate phases is found to be essential for the lock-in transition. The even- and odd-order terms in free energy play roles for amplitude development and phase modulation, respectively. Moreover, consideration of the condition of being commensurate CDWs allows us to relate it with that of the weak localization in random media [@Bergmann1983]. This explains why quantum interference phenomena have been observed in CDWs [@Tsubota2012; @Inagaki2016]. II. Previous experiments ======================== Let us take an overview of the lock-in transition in $o$-TaS$_3$. This quasi-one-dimensional conductor undergoes a Peierls transition at 220 K. At the transition, all the electrons on the Fermi surfaces contribute to form the Peierls gap and the system becomes an insulator, contrary to similar materials, such as NbSe$_3$ and $m$-TaS$_3$, in which remained electrons contribute to metallic conduction even after Peierls transitions occur. Hence, by absence of remained normal electrons, $o$-TaS$_3$ is one of the most appropriate materials for CDW studies. The first x-ray study was made by Tsusumi et al. who determined the CDW vector of $o$-TaS$_3$ [@Tsutsumi1978]. This work was followed by Roucau, who found that the CDW vector shifts from being $0.255c^$ (incommensurate) to $0.250c^$ (commensurate) at low temperatures [@Roucau1983]. The details of the lock-in transition were revealed by use of synchrotron diffraction [@Inagaki2008]. By lowering the temperature, the CDW vector shifts from being incommensurate closer to commensurate; however, it stops at $0.252c^$. The commensurate CDW independently appears at 130 K. In addition, coexistence of the commensurate and incommensurate CDWs was found in the temperature range down to 50 K, then the complete lock-in was observed at the lowest temperature, as shown in Fig. \[fig:phase\_diagram\] [@comment:phase_diagram]. The observed diffraction pattern of coexistence of two CDWs is clearly distinguished from those in the current-induced discommensuration lattice, which induces symmetric subpeaks at both sides of the main satellite [@LeBolloch2008; @Rojo-Bravo2016]. The observed CDW characteristics in $o$-TaS$_3$ differ from those in other quasi-one-dimensional conductors. Blue bronze K$_{0.30}$MoO$_3$ undergoes a Peierls transition at 180 K with $k_1=0.263b^$ [@Fleming1985]. By lowering the temperature to 100 K, its CDW wave vector shifts to be nearly commensurate with a slight residual incommensurability ($\sim\!0.250 b^$). Lock-in transition to the commensurate state does not occur in this conductor (incomplete lock-in). In NbSe$_3$ [@Fleming1984], as well as $m$-TaS$_3$ [@Roucau1980], there are three conducting chains, two of which contribute to form CDWs. Neither NbSe$_3$ nor $m$-TaS$_3$ exhibits lock-in transition. The first CDW wave vector in $m$-TaS$_3$ is independent of temperature with $q_1=0.245b^$ , whereas $q_1$ of that in NbSe$_3$ shifts from $0.245 b^$ at 150 K to $0.241 b^$ at low temperatures [@Moudden1990]. In these conductors, the second CDW ($q_2=0.247b^$ for $m$-TaS$_3$, $q_2=0.260b^$ for NbSe$_3$) appears at lower temperatures. As summarized in Table \[tb:transition\], it seems difficult to treat such various behaviors of CDWs in quasi-one-dimensional conductors with a simple theory, in particular, for such vector shift and complete/incomplete lock-in phenomena. Material CDW 1 CDW 2 Ground State ------------------ ---------------------------------- ------------- -------------- $o$-TaS$_3$ $0.255c^* \to 0.252c^$ $0.250c^$ C K$_{0.3}$MoO$_3$ $0.263b^* \to \ \sim\! 0.250b^$ $-$ NC $m$-TaS$_3$ $0.253b^$ $0.247b^$ IC NbSe$_3$ $0.245b^ \to 0.241b^$ $0.260 b^$ IC : Characteristics of quasi-one-dimensional CDW systems. The wave vectors of each material and its ground state are shown (commensurate, incommensurate, or nearly commensurate).[]{data-label="tb:transition"} III. Model and Results ====================== Theoretically, the lock-in transition in quasi-two-dimensional conductors has been discussed, initially by McMillan [@McMillan1976], and followed by Nakanishi and Shiba [@Nakanishi1977]. Their treatment is based on free energy with higher-order umklapp terms. McMillan’s free energy has the following form: $$F_1=F^0\int d^2 s[-\|\phi\|^2-\beta Y Re(\phi^3)+\frac{1}{2} \|\phi\|^4+\beta\|\vec\nabla\phi +i\phi\|^2+\gamma\|-\vec{q_1} \times\vec\nabla\phi \|^2 ] \label{eq:McMillan}$$ where $\phi$ is a phase of CDW defined as $\psi=\psi_0 e^{i{K}\cdot{r}/3}\phi({r})$, and $\psi$ is a complex order parameter. Here the coefficients $F^0$, $\beta$, $Y$, and $\gamma$ were the same as those defined in the original literature[@comment:coefficients]. Equation (\[eq:McMillan\]) was derived for commensurability index, namely, the ratio of CDW and lattice periodicities, $M=3$. The commensurability energy originates from the third-order umklapp term, proportional to the coefficient $\beta$. Though McMillan’s discussion aimed to understand the behavior of quasi-two-dimensional conductors, e.g., 2H-TaSe$_2$, it also includes quasi-one-dimensional cases. To apply their theories to our case, $M=4$, we should know what happens in McMillan’s free energy. By substituting $\psi=\psi_0 e^{i{K}\cdot{r}/4}\phi({r})$ for the order parameter, a simple calculation gives a result similar to Eq. (\[eq:McMillan\]); however, it lacks the $\beta Y Re(\phi^3)$ term, because the umklapp term becomes fourth-order in this case, namely, proportional to $\|\phi\|^4$. In contrary to the $M=3$ case, this calculation provides an unfamiliar result. The umklapp term gives no energy gain if a phase modulation $\phi=e^{-i\theta(x)}$ alone is considered as in McMillan. His calculation for the case $M=3$ provided the free energy as $F^0(-\frac{1}{2}+\beta(1-Y))$. A first-order lock-in transition takes place at the point $Y=1$. On the other hand, from our calculation for the case $M=4$, the free energy of the commensurate state is a constant value $F^0(-\frac{1}{2}+\beta)$, which is always larger than that of incommensurate state $F^0(-1/2)$. This explains the absence of the lock-in transition in the charge density wave of blue bronze, whose CDW vector becomes nearly-commensurate at low temperatures. On the contrary, the origin of the CDW vector shift from $0.263b^$ to $\sim 0.250b^$ remains unsolved. We will discuss this issue later. Nakanishi and Shiba’s extension of McMillan’s theory covers the systems with multiple CDW vectors [@Nakanishi1977]. They treated the lock-in transition of a two-dimensional conductor 1T-TaS$_2$, whose nesting vectors $k_i\ (i=1,2,3)$ satisfy a relation $3k_i-k_j=G_i$, where $G_i$ are reciprocal vectors, leading to the commensurability energy through the fourth term of umklapp processes. Also, after a simple calculation, this fourth-order term is found to give the energy gain only when coefficients of the nesting vectors in such a relation are odd numbers (1 or 3) for combining them to the reciprocal vector. This explains the absence of the lock-in transition in NbSe$_3$ and $m$-TaS$_3$, both of which have the nesting vectors satisfying $2k_1+2k_2=G$. As shown above, even-order processes in the free energy develop the amplitude, while odd-order processes induce phase-related phenomena. Figure \[fig:umklapp\] shows whether the fourth-order umklapp terms couple to the lattice periodicity or not. The (2,2) case, namely, $2k_1+2k_2=G$, which is satisfied in NbSe$_3$ and $m$-TaS$_3$, provides the same potential modulation as that in blue bronze. Therefore, the absence of lock-in transition in these conductors is found to be of the same origin. On the other hand, the (1,3) case provides sufficient contribution to the lock-in transition also in quasi-one-dimensional conductors. This case was discussed to explain the lock-in transition of an organic conductor, TTF-TCNQ [@Bak1976]. Now we will apply these theoretical considerations to our experimental results. CDWs in $o$-TaS$_3$ were not assumed as those in multiple chains, such as NbSe$_3$ and $m$-TaS$_3$. However, the coexistence of commensurate and incommensurate CDWs in $o$-TaS$_3$, as shown in Fig. \[fig:phase\_diagram\], suggests this possibility. By lowering the temperature, CDWs split into two kinds: commensurate and incommensurate ones. The commensurate CDW vector appears at $k_c=0.250c^$ from even-order terms in the free energy, whereas the incommensurate CDW vector remains at $k_{ic}=0.252c^$. The fourth-order umklapp term, which satisfies $k_c+3k_{ic}\simeq G$, couples to the lattice periodicity and obtains commensurability energy. At a temperature between 50 K and 30 K, a transition may occur, allowing the system to be complete lock-in. This scenario perfectly explains our synchrotron data [@Inagaki2008]. The incommensurate phase in the coexistence regime may have discommensurations, as discussed in the previous report, because a discommensuration state is energetically preferable to incommensurate CDW, according to McMillan [@McMillan1976]. In addition, the transition temperature coincides with that of occurrence of glasslike behavior [@Staresinic2002]. This behavior can be understood as a result of the lock-in transition, which freezes global motion of the CDWs. IV. Discussions =============== Our discussion does not rule out the possibility for the generation of an individual discommensuration, namely soliton, in commensurate CDWs. According to Bak and Emery [@Bak1976], a sinusoidal potential in CDW leads to the sine-Gordon equation, whose solution includes a phase soliton with the charge $e/M$. Moreover, such a sinusoidal modulation of potential can be derived only by commensurability [@Lee1974]. This agrees with previous experimental results, including the discrepancy between longitudinal and transverse conductivity at low temperatures [@Takoshima1980], the existence of unexpected carriers [@ZZ2006], and the nonlocal transportation [@Inagaki2010]. According to the microscopic theory [@Lee1974], the sinusoidal potential in commensurate CDWs is rooted in the condition $$\epsilon_{k+MQ}=\epsilon_k, \label{eq:Lee}$$ where $\epsilon_k$ is the energy of momentum $k$, and $Q=2k_F$ is a CDW vector. Equation (\[eq:Lee\]) means that the sum of each vector equals the reciprocal vector, i.e., $MQ=G$, and the energy of an electron-hole pair conserves after it is interacted $M$ times by the CDW momentum of $2k_F$. This leads to the phase dependence of the gap energy as $2\|\Delta\|^M(\cos M\phi-1)$ [@LRA]. If a system is purely one-dimensional, Eq. (\[eq:Lee\]) merely provides $M$’th order of umklapp process, whereas in two-dimensional systems, another interpretation becomes possible as follows: it is similar to that of Anderson localization, in particular, in the weak localization regime [@Bergmann1983]. Anderson localization results from self-interference of a wave function by multiple elastic scattering in random media. Bergmann’s condition for the localization to occur has a form $\sum_i g_i=0$, where $g_i$ denotes scattering vectors by impurities. It should be noted that a moment in the lattice can stay in any arbitrary Brillouin zone. Since all the scattering processes are elastic, the energy of the wave function conserves. Therefore, by considering Bergmann’s condition in substitution of $Q$ for $g_i$, one may obtain Eq. (\[eq:Lee\]), as shown in Fig. \[fig:wlr\]. This interpretation agrees with previous experimental results in $o$-TaS$_3$. At low temperatures, the system undergoes complete lock-in, as shown in Fig. \[fig:phase\_diagram\], where quantum interference phenomena were discovered in $o$-TaS$_3$ [@Tsubota2012; @Inagaki2016]. In particular, the localization phenomenon in the commensurate state suggests that CDWs have a two-dimensional correlation over the $b$-$c$ plane, and the closed path of CDW trajectory plays a crucial role [@Inagaki2016]. Finaly, here we will mention a limitation to our discussion. The lock-in energy has been found to relate with the odd-order terms in McMillan’s free energy. By applying this to quasi-one-dimensional conductors with $M=4$, most of characteristics summarized in Tabel \[tb:transition\] are explained within this framework, except for the vector shift observed in blue bronze. One plausible explanation is the excitation of soliton and antisoliton pairs [@Artemenko1981]. Each excitation of the soliton and antisoliton pair has been observed as a discrete step [@Zybtsev2010]. Since similar steps have also been observed in $o$-TaS$_3$ [@Zybtsev2016], further investigation must be necessary to clarify the lock-in transition of CDWs. V. Conclusion ============= In summary, we provide a unified view for the lock-in transition both in quasi-one- and two-dimensional conductors, based on the difference of roles between even- and odd-order terms in the free energy. The study of commensurate CDWs should be more focused, since it must contain far richer physics than previously thought. Acknowledgments {#acknowledgments .unnumbered} =============== The authors thank K. Ichimura, K. Yamaya, T. Honma, M. Hayashi, and K. Nakatsugawa for fruitful discussions, and M. Tsubota, T. Matsuura, S. Uji, N. Ikeda, and Y. Nogami for experimental support. [99]{} D.E. Moncton, J.D. Axe, and F.J. Disalvo, Phys. Rev. Lett. 34, 734 (1975). W. L. McMillan, Phys. Rev. B 14, 1496 (1976). P. Bak and V. J. Emery, Phys. Rev. Lett 36, 978 (1976). K. Nakanishi and H. Shiba, J. Phys. Soc. 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Śturm, and E, Tchernychova, J. Exper. Theor. Phys. 124, 665 (2017). ![ Phase diagram of $o$-TaS$_3$ deduced from this study. Solid circles represent the intensity of a commensurate CDW, $I_\mathrm{c}$, normalized by the total intensity of both the commensurate and incommensurate satellites, $I_\mathrm{ic}+I_\mathrm{c}$. The commensurate CDW begins to develop at around 180 K (solid line). The two $q$’s of the CDWs coexist until the system undergoes the lock-in transition at a temperature between 30 K and 50 K (broken line). The insets are diffraction profiles of satellite peaks at 30, 130, and 180 K.[]{data-label="fig:phase_diagram"}](./fig1.eps){width="80.00000%"} ![Visualization of the fourth-order umklapp terms: $\psi^4$, $\psi^2\psi^2$, and $\psi\psi^3$; charge density wave of commensurability index $M=4$, and lattice potential (from top to bottom).[]{data-label="fig:umklapp"}](./fig2.eps){width="80.00000%"} ![Umklapp process in $M=4$ CDW, responsible to commensurability (left). Schematic diagram of Bergmann’s condition (right) [@Bergmann1983].[]{data-label="fig:wlr"}](./fig3.eps){width="80.00000%"}
--- abstract: \| The kinematics of distant OB stars perpendicular to the Galactic plane, inferred from proper motions in the Hipparcos catalogue, are analysed and compared to the kinematic signature that would be induced by a long-lived Galactic warp. Previously we reported that the kinematics of the OB stars toward the anticenter were inconsistent with the assumption of a long-lived warp ([@SMA98]), showing negative systematic motions as opposed to the expected positive motions. Using a larger sample of OB stars, improved distances, and a more accurate model for the uncertainties, we confirm our previous result for a larger range of galactocentric radii. However, we note that the new model for errors in the photometric distances reveal an important bias that causes the observed systematic vertical motions to be smaller than their true values. Using synthetic catalogues we investigate the effect of this bias on the systematic vertical motions in conjunction with the possibility of a smaller warp amplitude and a warp precession rate, both of which can also lead to smaller systematic motions. Taken together these three effects can work to produce negative observed systematic vertical motions, similar to those detected in the data, though only with both excessively high precession rates ($-25 \kmskpc$) and very large photometric errors (1 magnitude). author: - 'R. Drimmel' - 'R.L. Smart' - 'M.G. Lattanzi' date: 'Received: /Accepted:' nocite: - '[@BLA93]' - '[@TOR97]' title: The Galactic warp in OB stars from Hipparcos --- 1[to 0pt[\1]{}]{} 1[1]{} Introduction ============ For some time now the existence of a warp in our Galaxy has been well known. First seen in neutral hydrogen emission, it’s presence in several other components have since been reported, i.e. IRAS point sources ([@DJO89]), CO gas ([@WOU90]), and dust ([@FRE94]). Studies of the Galactic warp have been mostly limited to it’s spatial structure (see Binney 1992 for review) because the kinematic signature of a warp primarily manifests itself in the component tangential to the line-of-sight (LOS) of an observer near the Galactic plane, the component that cannot be directly measured by radio observations. For stars, on the other hand, this component is in principle accessible via proper motions, but in practice this approach has been limited by the small magnitude of the expected systematic motion ($\sim1$–2 $\masyr$), which is smaller than the zonal systematic errors in traditional catalogues. The advent of astrometry from space, inaugurated with the Hipparcos satellite, signals a new era in the study of Galactic kinematics; the high precision ($\leq 1\masyr$) and accuracy ($<.1\masyr$) of the proper motions combined with and inertial frame means that it is now possible to detect the kinematic signature expected from a Galactic warp. Once the spatial structure of the warp is determined, its kinematic signature will yield direct information on the precession rate of the warp. Earlier we reported results from a subsample of the Hipparcos catalogue ([@SMA98]). In that study we looked exclusively at distant OB stars toward the anticenter (between 70 and 290 degrees in galactic longitude), expecting such stars to trace the motions of the gaseous component from which they were recently born. Unexpectedly it was found that their kinematics did not follow the predicted signature of a long-lived warp, either precessing or not. This led to the conclusion that either there are additional systematic vertical motions in the Galaxy’s gaseous component, or that the warp is not a long-lived structure. In this contribution we use a larger sample of OB stars from the Hipparcos catalogue to reinvestigate their systematic vertical motions. As before, our interpretation of the data is based upon a comparison between the observed kinematics of the selected sample of OB stars, and the “observe” kinematics of a sample created using a synthetic catalogue generator. The synthetic catalogues are generated based on an assumed statistical model of the (warped) stellar distribution, kinematics and observational errors. By making the identical selections and cuts on the simulated catalogues as are made on the actual data, any biases present or introduced in the sample of OB stars due to errors or selection criterion are likewise reproduced in the synthetic samples as well. In this way the modeled kinematics are translated into observed quantities which can be directly compared with those of the selected sample of stars. In essence, rather than attempting to remove or correct biases in the observed sample, we introduce the same biases into the model in order to determine what is actually detectable. In the following section the new sample of OB stars is described, while in Sect. 3 the model for the errors used for the synthetic catalogues is presented. Sect. 4 details an improved estimate of the distances and in Sect. 5, using a simple model for the Galactic warp, we find spatial parameters consistent the distribution of the Hipparcos OB stars. In Sect. 6 we discuss the estimation of the systematic vertical motions, derive the kinematic signature resulting from a long-lived precessing warp, and compare the observed and expected kinematic signatures. Sect. 7 describes the bias introduced in the estimation of the vertical motions by errors in the distance and how these, in conjunction with a smaller amplitude and precession, can produce smaller systematic motions than otherwise expected. We present our conclusions in Sect. 8. The Sample ========== The sample of OB stars used here to study the Galactic warp is taken from the 10544 OB stars in the Hipparcos catalogue; after excluding 3 stars whose spectral typing is no more specific than “O” or “B”, we find 4538 stars with a parallax $\pi \leq 2\mas$. Stars with measured parallaxes $\pi < 0$ are retained, as their exclusion would unfavorably bias the sample with respect to distance. In the Tycho catalogue there are 857 stars brighter than $V$ magnitude 7.5 which do not appear in the Hipparcos catalogue, which has a total of 24384 stars brighter than this magnitude. This suggests that the Hipparcos catalogue is approximately 97 percent complete to this magnitude limit. From our sample alone, based on the distribution of apparent magnitudes brighter than magnitude 7, we estimate that the subsample of 3840 OB stars with $m \leq 7.5$ is 98 percent complete ([@SMA99]). Of these brighter stars, 929 have a parallax $\pi \leq 2\mas$. Fig. \[lbdist\] shows the distribution of this subsample of distant OB stars on the sky. =6.0cm In our earlier work ([@SMA98]) purely photometric distances were used. In Sect. 4 we will improve upon these distance estimates, but nevertheless complete spectral classification is needed to calculate a photometric parallax. For the 7279 OB stars with complete spectral classification supplied by Hipparcos, the absolute magnitudes and colors from Schmidt-Kaler (1982) are used to calculate the photometric parallax. For 141 stars without luminosity classification H$\beta$ linewidths were used to determine luminosity class. The 3121 remaining stars without luminosity classification have photometric distances derived from their reddening and estimated absorptions. For further details of these procedures the reader should consult Smart et al. (1997). In order to understand the properties of the photometric error, the sample of all OB stars brighter than 7.5 will be considered in the following section. These are 3840 in number, with 789 not having an initial luminosity classification (nonLC stars). As we expect that our estimation of the luminosity classes for these stars will be more susceptible to error than those with full spectral classification, it is important to be aware of biases in their distribution. Indeed, because of the use of the Michigan Spectral Survey in the southern sky ([@MSS4] and references therein), 745 of the bright nonLC stars (94 %) are in the north ($\delta > -12.$ deg). This zonal bias results in a photometric error that is systematically larger in the northern part of the sky than the south, as the following section describes. =6.0cm =7.0cm To account for the higher photometric error introduced by the nonLC stars, it is necessary to describe their distribution not only on the sky, but also with respect to other relevant variables. Fig. \[nonlcsp\] shows the relative fraction of northern nonLC stars with respect to spectral type and Fig. \[nolclego\] the relative fraction of nonLC stars with respect to apparent magnitude for B9, B8, B5 and B3 spectral types. The fact that the nonLC stars are primarily restricted to these types, and not hardly seen in types B7, B6, and B4, indicates that the nonLC stars are in general typed at a lower resolution than those stars with luminosity class. For stars of spectral types earlier than B2 the relative fraction of nonLC stars does not vary strongly with apparent magnitude. While in Sect. 4 only a magnitude cut is applied, in Sects. 5 and 6 two additional cuts are employed; stars with a parallax of $\pi > 2\mas$ and a vertical velocity greater than 50 $\kms$ are removed. The parallax cut is imposed to avoid being biased by local structure, such as Gould’s Belt, which dominates the distribution of nearby OB stars. The velocity cut is astrophysically motivated by the presence of runaway stars, which through close encounters have reached escape velocities, and thus do not possess kinematics principally determined by the Galactic potential. The typical threshold detection value for the relative space velocity of runaway stars varies from 40 to 65 $\kms$ (Blaauw 1993, Torra et al. 1997), thus our value of 50 $\kms$ applied to one component is conservative, especially as it is approximately seven times greater than the velocity dispersion of this population. The velocity cut is applied to galactocentric radial bins, excluding those stars which deviate from each bin’s median velocity by more than $50\kms$, reducing the sample of bright, distant OB stars to 895 in number (less 34), 142 of which are nonLC stars. Meanwhile 255 are cut from all the stars with $\pi \leq 2$, leaving 4283 stars, 1291 of which are nonLC stars. The calculation of the individual and mean vertical velocity is discussed in Sect. 6. Simulated catalogues and improved error model ============================================= In order to interpret the observed kinematics of OB stars from the Hipparcos catalogue, a program was developed to generate synthetic catalogues based on a statistical model of the spatial and velocity distributions of the OB stars (Drimmel et al. 1997, 1999). The model distribution is non-axisymmetric, being described by spiral arm segments, most important of which is the local Orion arm, which has a pitch angle of 8 degrees and a spur projecting toward the outer Perseus spiral arm. The synthetic catalogue generator produces catalogues with distributions in galactic longitude and latitude ($l,b$), apparent magnitude, and proper motion, which closely resemble those of the distant ($\pi \leq 2\mas$) OB stars in the Hipparcos catalogue (see also [@SMA97A]). The synthetic catalogue generator uses a Galactic dust distribution model based on the 240$\mu$ COBE data (Drimmel & Spergel, in preparation) to calculate absorbtions. This model is a refined version of the earlier work of Spergel et al. (1997), adding nonaxisymmetric structure to the dust distribution model, the most important of which is dust associated with the local Orion arm. The added structure in the dust distribution results in additional reddening in the directions tangent to the local arm, thus improving the modeled longitude distributions (see Drimmel et al. 1999 as compared to Drimmel et al. 1997). This same model is used to estimate reddenings and absorptions to the nonLC stars. Observational errors are modeled for generating observed quantities. The simulated catalogues in Smart et al. (1998) used parallax errors, $\sigma_{\rm t}$, equal to 1mas, and proper motion errors, $\sigma_{\mu}$, were described by simple functions of galactic latitude, as seen in the Hipparcos catalogue ([@mignard97]). These errors are actually better described as varying with ecliptic latitude and apparent magnitude. We have thus improved our model for $\sigma_{\rm t}$ and $\sigma_{\mu}$ by interpolating from Tables 3.2.4-6 given in the first volume of the Hipparcos catalogue ([@HIPCAT]). We have also added a 0.01 magnitude error to the apparent magnitudes. For our purposes here the Hipparcos errors in position ($l,b$) are negligible. However, the improvement of the model of photometric distance error makes the greatest difference in the simulated observable kinematics of the synthetic catalogues. Previously our error model involved adding a relative gaussian error of 25 percent of the distance. However, if the errors are normally distributed in absolute magnitude, it is the errors in the distance modulus that are normally distributed. We now model the error in photometric distance by assuming a normal distribution of errors in the absolute magnitude. This error is not only the result of the intrinsic scatter of absolute magnitudes, termed “cosmic error”, but also due to errors introduced by uncertainties in the determination of the absorption, intrinsic scatter in the colors, and misclassification errors. Not all these errors are guaranteed to be gaussian. The photometric error of our observed sample of stars is evaluated using the photometric parallax $\pi_{\rm P}$ verses trigonometric parallax $\pi_{\rm t}$ distribution for the sample of 3840 OB stars brighter than 7.5. Fig. \[pipi\_all\] shows the $\pi_{\rm P}$–$\pi_{\rm t}$ distributions for the north and the south. The degraded photometric parallaxes of the north, due to the presence nonLC stars, are responsible for the larger scatter as compared to the south. To illustrate this effect Fig. \[pipi\_nord\] shows the $\pi_{\rm P}$–$\pi_{\rm t}$ diagrams for the north decomposed into the stars with and without luminosity class. It can be seen that the northern stars which possess full initial classification (Fig. 5 left) have approximately the same scatter as the southern stars (Fig. 4 right), and we will assume the same error model can be used for both. One difference between the northern and southern stars with full initial classification is a systematic bias of $\pi_{\rm P}$ in the south for stars with $\pi_{\rm t}$ centered at 7.5mas. This bias is due to the presence of stars in the direction of $\rho$ Ophiuchus whose absorptions are systematically overestimated. Assuming that the observational errors in the Hipparcos trigonometric parallax are well described, we can estimate the photometric error using the synthetic catalogue generator, comparing observed and generated $\pi_{\rm P}$–$\pi_{\rm t}$ distributions. For the northern stars with luminosity classification an error model which approximately reproduces the observed distribution of trigonometric and photometric parallaxes is $$\sigma_M = .5 + {\rm max}(0,(m-6.5)/6) - M/12, \label{photerr}$$ where the apparent magnitude term can be interpreted as being the result of a higher occurrence of classification error at fainter magnitudes, and the absolute magnitude term can be interpreted as an increasing cosmic error and/or uncertainty for the more luminous stars. To characterize the distribution a robust linear fit is made to the observed $\pi_{\rm P}$–$\pi_{\rm t}$ distribution in the north (Fig. \[pipi\_nord\]), giving a slope of 0.77 and a mean absolute deviation in $\pi_{\rm t}$ of 1.28, after 17 stars are excluded whose absolute deviations in $\pi_{\rm t}$ are greater than 6 mean absolute deviations (presumably misclassified stars). Meanwhile the modeled distribution (Fig. \[pipi\_mod\]) produces a slope of 0.80, with 1.06 being the mean absolute deviation. For comparison, Fig. \[sigm\_1\] shows the $\pi_{\rm P}$–$\pi_{\rm t}$ distribution for a model generated with a purely gaussian magnitude error of $\sigma_M = 1$ magnitude, showing that such a high error does not reproduce well the observed $\pi_{\rm P}$–$\pi_{\rm t}$ distribution. For the nonLC stars the signatures due to misclassification are prominent, its nongaussian nature leaving clear artifacts in the distribution, especially at $\pi_{\rm t} > 2\mas$ and $\pi_{\rm P} < 3\mas$ (Fig. \[pipi\_nord\] right). These are intrinsically dim stars incorrectly assigned a high luminosity. In contrast, intrinsically bright stars misclassified as low luminosity stars do not produce clear signatures, but do cause stars of low $\pi_{\rm t}$ to be systematically biased toward higher $\pi_{\rm P}$. We model the misclassification errors as offsets to the true absolute magnitudes, with a binned gaussian probability of misclassification. =7.0cm However, before applying such errors the distribution of nonLC stars must be modeled. The synthetic catalogue generator does not at this time generate colors, so spectral types were simply correlated with the main sequence absolute magnitudes to infer a model of the distribution of nonLC stars in the north. Fig. \[nonlcmod\] shows the model of the relative fraction of nonLC stars in the north. Once a star is randomly determined to be nonLC, a binning error is calculated using $$p={\rm int}(1.5 N + 3 {\rm log} d_0 + 1.4), \label{offstep}$$ where $N$ is a gaussian deviate of unit variance and $d_0$ the true distance, $(3 {\rm log} d_0 + 1.4)$ being a bias reflecting that nearby stars are more likely to be given an over-luminous classification, and distant stars a under-luminous classification. The resulting index $p$ is then used to indicate the appropriate offset, as given in Table \[offsets\]. In addition to this offset, gaussian noise of $\sigma_M = 0.3$ magnitude is added to the absolute magnitudes. Much of the added complexity of misclassification in the nonLC stars will not be needed in what follows, as for stars with $\pi \leq 2\mas$ only the positive offsets will contribute significantly to the error. Improved distances ================== In order to improve upon the distances, a distance estimate is used that corresponds to the weighted mean parallax of the trigonometric and photometric parallaxes: $$\frac{1}{d} = \pi_{\rm w} = \frac{\pi_{\rm t}/\sigma_{\rm t}^2 + \pi_{\rm P}/\sigma_{\rm P}^2} {1/\sigma_{\rm t}^2 + 1/\sigma_{\rm P}^2}, \label{distance}$$ where $\sigma_{\rm P}=\pi_{\rm P} \sigma_M / 2.17$ ([@Smith85]), and $\sigma_{\rm t}$ is interpolated from Tables 3.2.4-6 in volume one of the Hipparcos catalogue. For this distance estimate stars with initial full spectral classification are assumed to have a $\sigma_M$ as specified by Eq. (\[photerr\]), while nonLC stars are assigned $\sigma_M=3$ magnitudes. This prescription is used for both synthetic and observed catalogues. \[offsets\] [llll]{}\ $p$ & offset & Magnitude range & Misclass.\ \[+5pt\]\ $-$4 & $-6.8 - M$ & & V $\rightarrow$ Ia\ $-$3 & $-5 - 2M/2.45$ & & V $\rightarrow$ Ib\ $-$2 & $-1$ & $M > -1.2$ & V $\rightarrow$ III\ & $-(5.9+M)/2.45$ & $M \leq -1.2$ & III $\rightarrow$ II\ $-$1 & $-3 - M/2.45$ & & V $\rightarrow$ II\ 1 & 1. & & III $\rightarrow$ V\ $>$1 & 1. & $M > -3.1$ & III $\rightarrow$ V\ & 2.5 & $-5.2<M\leq -3.1$ & II $\rightarrow$ V\ & 3.5 & $M\leq-5.2$ & I $\rightarrow$ V\ \ To test whether this new distance estimator is more or less biased than the photometric distance alone, the synthetic catalogue generated in the previous section for the northern sky is used to compare the estimated distance with the true distance. Figs. \[newdist\] and \[photdist\] show the result of this Monte Carlo test; for stars with an initial luminosity class, the variance and skew of the deviations from a robust linear fit to the photometric distances, are 2.02 and 1.04, while with the new distance estimate they are 0.41 and -0.07 respectively, showing that the uncertainties are effectively reduced. The slopes and intercepts of both distance estimates show no significant systematic biases. For the nonLC stars the new distance estimator reduces the scatter produced by misclassifications, though the distances can still have significant error. Using the typical errors $\sigma_M=.7$ and $\sigma_{\rm t}=1.2\mas$, the error of $\pi_{\rm w}$ for a star at $\pi_{\rm t}=\pi_{\rm P}=1\mas$ is about .3mas, for which the distance estimate would be $1^{+.45}_{-.24}\kpc$. We also point out that for stars in our $\pi_{\rm t} \leq 2\mas$ sample, the photometric distance will dominate in Eq. (\[photdist\]), as it is only at a distance of $d = \sigma_M/(2.17\sigma_{\rm t})$ that $\sigma_{\rm P} = \sigma_{\rm t}$. Warped spatial distribution =========================== [llllll]{}\ Bin (kpc) & N & Data & No warp & Warp & $1/2$ Warp\ \ 6.5$< r <$7.5 & 140 & $-$0.39 & $-$0.24 $\pm$ 0.94 & 0.34 $\pm$ 0.91 & 0.29 $\pm$ 0.73\ 7.5$< r <$8.5 & 500 & 1.67 & $-$0.34 $\pm$ 0.44 & 1.08 $\pm$ 0.49 & 0.25 $\pm$ 0.44\ 8.5$< r <$9.5 & 161 & 2.05 & 0.07 $\pm$ 0.93 & 2.54 $\pm$ 0.86 & 1.45 $\pm$ 0.80\ \ \[slopes\] As the Sun is near the line of nodes of the Galactic warp the vertical deviation from the Galactic plane is small, but a local slope with respect to the Galactic plane is produced in the stellar distribution. Using the subsample of the 894 bright ($m \leq 7.5$ magnitudes), distant ($\pi_{\rm t} < 2\mas$) OB stars, the slopes of the stellar distribution are found in the heliocentric coordinates $z$ and $y$, $y$ being taken in the direction of rotation, for bins in galactocentric radius $r$, ($r_\odot \equiv 8\kpc$). The slopes were found using a standard robust linear fitting routine which minimizes the absolute deviations in $y$. In Table \[slopes\] we express these slopes as a tilt angle $\theta$, corresponding to the tilt of a galactocentric ring whose axis of tilt goes through the Sun, consistent with the assumption that the line of nodes of the Galactic warp goes through the Sun. It is well known that OB stars are preferentially found in associations, resulting in a “clumpiness” of the distribution on a larger scale than would be found in a kinematically relaxed population. Nevertheless, we do not expect the presence of associations to affect our results above, as we are describing the overall distribution of OB stars on a scale significantly larger than the size of a typical association. To test this presumption stars which were found to be within the spatial limits of a given association were identified. For this purpose the positions ($l,b,d$) and limits ($\pm \Delta l, \pm \Delta b$) for 38 associations were taken from the summary list in Lang (1992), while the line-of-sight extent of each association ($\Delta d = d {\rm max}(\Delta l, \Delta b)$) was doubled to take into account possible errors in the distance. Only four associations have ten or more stars from our complete subsample, these being the Cygnus 7 (10 stars), Carina (13 stars), Cepheus 2 (19 stars), and Orion (17 stars) associations, where a total of 102 stars are found inside all associations. Removing each of these sets of stars in turn, the tilts were redetermined. It was found that only one of the associations (Cepheus 2) influenced the measured slopes by more than 0.5 deg. A warp is effected in the modeled distribution by describing the vertical distribution as a function of $z' = z - Z_{\rm w}(r,\phi)$, where the function $Z_{\rm w}(r,\phi)$ describes the spatial form of the warp in galactocentric cylindrical coordinates, $\phi$ being taken in the direction of rotation. From radio observations the warp is seen to be symmetric out to about $r = 16\kpc$, and can generally be described by $$Z_{\rm w} = h(r) \sin (\phi - \phi_{\rm w} + \omega_{\rm p} t) , \label{zwarp}$$ $\phi_{\rm w}$ being the phase of the warp, and $\omega_{\rm p}$ the precession rate of the warp in the direction [opposite]{} to Galactic rotation, and with $$h(r)=\left\{\begin{array}{ll} (r-r_{\rm w})^2/r_{\rm h}, & ~r > r_{\rm w} \\ 0, & ~r \le r_{\rm w} \end{array} \right. \label{height}$$ being a height function parameterized by $r_{\rm h}$ and the radius at which the warp starts, $r_{\rm w}$. The radio observations show that the Sun lies close to the line of nodes, that is, $\|\phi_{\rm w}\| < 10$ degrees. As the systematic velocity is not sensitive to $\phi_{\rm w}$ in this range ([@SMA96A]) we have assumed $\phi_{\rm w} = 0$ in this study. The assumption of a long-lived warp is effectively implemented by taking $\omega_{\rm p}$ to be a constant with respect to $r$ and time $t$. Table \[slopes\] shows the mean tilts obtained from thirty synthetic catalogues with and without a warp, one warped distribution being generated with the parameters $r_{\rm h}=15\kpc$ and $r_{\rm w}=6.5\kpc$, while a second model with half this amplitude ($r_{\rm h}=30\kpc$) is also shown for comparison. The first set of parameters produce tilts that are consistent with the observed tilts, and these are adopted as our estimated warp parameters in what follows. As noted in our earlier work, the observed tilts indicate a Galactic warp starting within the Solar Circle. The use of a robust linear fitting routine does not provide formal errors on individual fits, hence no error is given on the tilts of the data. In the case of the synthetic catalogues we have the freedom to directly evaluate the uncertainty by generating multiple catalogues, as we have done. Inasmuch as the synthetic catalogues are accurate statistical representations of the observed data, the uncertainties of their tilts will be representative of the uncertainties in the tilts of the observed catalogue. However, the synthetic catalogues do not include the clumpiness due to the presence of OB associations, which will increase the uncertainties. The 0.5 deg influence of the Cepheus 2 association on the tilts can be taken as an estimate of the probable additional random error that should be added to the uncertainties quoted in Table \[slopes\]. In any case, the observed tilts are significant and inconsistent with the no-warp hypothesis. Having estimated the parameters that describe the spatial form of the warp in the OB stars, we are now prepared to consider the kinematic signature accompanying such a warp. Kinematic signature of the warp =============================== The presence of a long-lived warp in the stellar disk will result in systematic motions perpendicular to the Galactic plane; for an observer in the plane of the Galaxy such motions will be primarily observable in the galactic latitude component of the proper motion. Taking ${\bf x} = (\cos l \cos b ,\sin l \cos b, \sin b)$ as the unit pointing vector to a star, the unit vector tangential to the line-of-sight (LOS) in the direction of increasing galactic latitude is $${\bf b} = \frac{ {\bf x} \times ({\bf k} \times {\bf x}) } {\|{\bf x} \times ({\bf k} \times {\bf x})\|} = \frac{{\bf k} - {\bf x} \sin b}{\cos b}, \label{bvector}$$ where ${\bf k} = (0,0,1)$, the unit vector perpendicular to the Galactic plane. The component of a star’s relative motion in the direction of ${\bf b}$ is $$( {\bf v} - {\bf v_\odot} ) \cdot {\bf b} = 4.74 d \mu_b, \label{bmotion}$$ where $\bf{v_\odot}$ is the solar motion, $\bf{v}$ the star’s velocity with respect to the local standard of rest (LSR), $d$ and $\mu_b$ being the distance in kiloparsecs and the galactic latitude proper motion in ${\rm mas\,yr}^{-1}$ respectively. Taking ${\bf v_\odot} = (U,V,W)_\odot$ and ${\bf v} = (U,V,W)$, defined in the same right hand coordinate system as $\bf{x}$, one can write from Eqs. (\[bvector\]) and (\[bmotion\]): $$W = \frac{4.74 d \mu_b}{\cos b} + W_\odot + (S-S_\odot) \tan b, \label{zmotion}$$ where $S$ is the component of the star’s velocity parallel to the Galactic plane and in the plane that contains the LOS and is at right angles to the Galactic plane, that is $S=U \cos l + V \sin l$, and similarly for $S_\odot$. Since distant stars are being considered, we do not wish to assume that the stars are in the solar neighborhood (Oort’s approximation), and $S$ will therefore contain a contribution from both differential Galactic rotation and the star’s peculiar motion. Neglecting the peculiar velocity contribution for the moment, we can write $$S = \left( \overline{v}_\phi \frac{r_\odot}{r} - v_{\rm LSR} \right) \sin l, \label{difrot}$$ where $\overline{v}_\phi$ is the average rotational velocity of the stellar population, and $v_{\rm LSR} = 220\kms$. Because the OB stars do not follow Stromberg’s asymmetric drift equation ([@DB98b]), we simply take $\overline{v}_\phi = v_{\rm LSR} - 3.0(r-r_\odot)$, found to be consistent for this stellar population ([@DRI97A]). =10.0cm Finally, adopting $(U,V,W)_\odot = (9,5,7)\kms$ ([@DRI97A]), with positive $U$ being radially inwards, we can calculate an observed $W$ for each star using Eq. (\[zmotion\]). (Our adopted value for the Solar Motion is comparable with other recent determinations (i.e. [@DB98b]), but it should be noted from Eq. (\[zmotion\]) that $W_\odot$ enters only as an offset term.) The stars are binned in galactocentric radius to find the observed systematic vertical velocity $\langle W \rangle$ as a function of $r$; this is shown in Fig.\[vzrfig\]. The agreement of the complete portion of our sample with the entire sample is consistent with the assumption that stars fainter than 7.5 form a kinematically unbiased sample. By ignoring the peculiar stellar velocities parallel to the Galactic plane, $S_$, in Eq. (\[difrot\]), we have effectively ignored a $\langle S_* \tan b \rangle$ contribution in each bin. This is done from necessity; we cannot calculate $S_$ for stars from the Hipparcos catalogue as we do not have their complete space motion. However, from Monte Carlo tests we find that $\langle S_ \tan b \rangle$ is typically less than $0.05\kms$. We now compare the observed vertical systematic velocities $\langle W \rangle$ with those expected from a long-lived warp. As the vertical velocity $W$ of each star is with respect to the LSR, $W = W_* + \overline{v}_z(r,\phi) - \overline{v}_z(r_\odot,0)$, $W_*$ being the star’s peculiar vertical motion, and $\overline{v}_z(r,\phi)$ the systematic vertical velocity at $(r,\phi)$ resulting from the warp. Upon averaging $\langle W \rangle = \overline{v}_z(r,\phi) - \overline{v}_z(r_\odot,0)$, the difference in the systematic vertical motion at the star’s and the Sun’s position. =10.0cm To find the systematic vertical velocity $\overline{v}_z(r,\phi)$ due to a warp as described by Eq. (\[zwarp\]), we consider Jeans continuity equation of stellar dynamics. For the case of no radial motions we have $$\frac{\partial \nu}{\partial t} + \frac{\partial (\nu \overline{v}_z)}{\partial z} + \frac{1}{r} \frac{\partial (\nu \overline{v}_\phi)}{\partial \phi} = 0. \label{continuity}$$ If the stellar density $\nu$ is purely a function of $z'=z-Z_{\rm w}$ and $\partial \overline{v}_z / \partial z = 0$, the systematic vertical motion can be shown to be $$\overline{v}_z(r,\phi) = \left( \frac{\overline{v}_\phi}{r} + \omega_{\rm p} \right) h(r) \cos \phi ~; \label{sysvel}$$ The systematic motion induced by a warp is simply the result of the stars rotating with respect to the warp structure. In our synthetic catalogues this systematic velocity is added to stars if they are in the warp (i.e. $r>r_{\rm w}$). Fig. \[vzrfig\] shows the systematic $\langle W \rangle$ for a typical synthetic catalogue generated with a non-precessing warp having the parameters obtained from the sloped stellar distribution. This demonstrates that a non-precessing warp, consistent with the sloped distribution of our sample, should be easily detectable. The $\langle W \rangle$ of the synthetic catalogue lies systematically below the theoretical curve primarily due to a bias introduced by the error in the photometric distances. Effect of bias, amplitude, and precession ========================================= The bias introduced by distance errors can be understood after some consideration. Stars whose distance is overestimated are likely to have larger distance errors than those whose distance are underestimated, as the distance error is normal in the distance modulus. Together with the fact that in a magnitude limited catalogue there are more stars closer to the Sun than further away, the skew in the distance errors means that stars with larger observed distances, $d_{\rm o}$, are more likely to be stars whose true distance, $d$, is smaller. The effect of the misplacement of a star to larger distances is that its measured relative vertical motion $W_{\rm o}(d_{\rm o})$ will be smaller than its true relative vertical motion, $W(d)$. This can be seen from Eq. (\[zmotion\]), for if differential Galactic rotation and vertical peculiar motions are neglected, we can write $W(d)=( 4.74 \mu_b / \cos b) d + W_\odot$, and similarly for $W_{\rm o}(d_{\rm o})$. Eliminating the common factor $( 4.74 \mu_b / \cos b)$ it can be shown that $W_{\rm o}(d_{\rm o}) - W(d) = ( W(d) - W_\odot) (d_{\rm o}/d - 1)$. Hence, as long as $W(d) < W_\odot$ and $d_{\rm o}/d > 1$, $W_{\rm o}(d_{\rm o})$ will be less than $W(d)$. As we look out from the Galactic center $r$ is proportional to $d$, so if $d_{\rm o} > d$ due to error, $r_{\rm o} > r$, and from the above $W_{\rm o}(r_{\rm o}) < W(r)$. Finally, the kinematic signature of a warp increases outward, that is $W(r) < W(r_{\rm o})$, allowing us to write $W_{\rm o}(r_{\rm o}) < W(r_{\rm o})$; the average relative systematic velocity measured at a given $r_{\rm o}$ is smaller than it’s true systematic relative velocity at $r_{\rm o}$. In addition to the above bias there are two other possible reasons why the observed systematic motions may be lower than that predicted for a long-lived warp. The first, trivially, is that the actual amplitude of the warp may be smaller than we have estimated from the observed slope of the spatial distribution. This may indeed be the case as radio data yield a warp amplitude between .3 and .4kpc at a galactic radius of 10kpc ([@BUR94]), while our warp parameters produce an amplitude of .8kpc. In Fig. \[h\_allonone24s1\] the signature expected from our standard warp ($r_{\rm h} = 15\kpc$) is shown together with the signature of a warp with half amplitude ($r_{\rm h} = 30\kpc$). Lastly, the warp may be precessing in the direction of Galactic rotation. Fig. \[h\_allonone24s1\] shows the expected signal of a warp of half amplitude precessing at $-13\kmskpc$. While these three effects, taken together, have significantly reduced the expected velocity signature, we still have not recovered the negative velocities seen in our Hipparcos sample. Could higher precession rates, or higher errors in the photometric distances produce the velocity signature that we detect? The fourth model in Fig. \[h\_allonone24s1\] shows the effect of a precession rate of $-25\kmskpc$ on a warp of half amplitude with elevated photometric error ($\sigma_M = 1. + {\rm max}(0,(m-6.5)/6) - M/12$). In this extreme case negative vertical motions are finally achieved. Discussion ========== We have confirmed with a larger and more complete sample of distant OB stars that the observed kinematics and structure perpendicular to the Galactic plane are together inconsistent with the hypothesis of a long-lived non-precessing warp. However, with a revised model for the errors, negative systematic velocities can be produced, but only with excessively large precession rates, a smaller warp and large photometric errors. A precession rate even as high as $-20\kmskpc$ is questionable on physical grounds, for it would require that the warp was precessing faster than Galactic rotation at a galactocentric radius of only 11kpc! Could the photometric error parameter even be as high as one magnitude? The $\pi_{\rm P}$–$\pi_{\rm t}$ distribution of models generated with such errors (see Fig. \[sigm\_1\]), compared to the observed $\pi_{\rm P}$–$\pi_{\rm t}$ distribution (Fig. \[pipi\_nord\]) would suggest not. If we do not accept such large photometric errors and precession rates we are left with the same conclusions that we arrived at in our previous work; either the Galactic warp is not long-lived, or there are other systematic motions present in the Galactic disk. Galactic companions could be responsible for a short-lived warp, or at least modify the velocities, but the regularity of the Galactic warp out to at least 16kpc from the Galactic center makes this seem unlikely. Nevertheless, numerical simulations will have to be carried out to explore this possibility. Alternatively, the presence of other systematic vertical oscillations in the Galactic disk could mask the kinematic signature of the warp, but should also be evidenced by accompanying deviations from the Galactic plane. The warp amplitudes as inferred from the OB stars found here (.4 to .8kpc at $r=10\kpc$) are larger than those found in the HII gas (.3 to .4kpc at $r=10\kpc$, [@BUR94]) and the dust (.3 kpc at $r=10\kpc$, [@FRE98]), and may suggest the presence of such structures. In addition Dehnen (1998) also finds that the kinematics of nearby stars are not consistent with a Galactic warp that begins within the Solar Circle and could thus produce a local tilt in the stellar distribution. Indeed, the observed slopes and anomalous kinematic signature may not be due to a warp at all, but to some other as yet unidentified phenomenon. The possibility of other systematic motions being present in the disk complicates any evaluation of the kinematics of the Galactic warp. Until a deeper and more accurate astrometric survey is completed the possibility of other kinematic effects cannot be excluded nor, if present, be disentangled from the expected kinematic signature of the warp in the stellar distribution. Such a survey will eventually be given by a future astrometric mission like GAIA, which will also, via parallaxes, largely dispense with the bias introduced by the photometric distance errors. Acknowledgments =============== Thanks are extended to James Binney and Stefano Casertano for useful discussions, and the reviewer who contributed to the quality of this contribution. R. Drimmel and R. L. Smart acknowledge support from the University of Turin. R. L. Smart also acknowledges support from the British Royal Society. Binney J., 1992, , 51 A., 1993, In: J. P. Cassinelli, E. B. Churchwell (eds.) [ASP Conf. Ser. 35: Massive Stars: Their Lives in the Interstellar Medium]{} W. B., [Hartmann]{} D., 1994, , 189 W., 1998, , 2384 W., [Binney]{} J. J., 1998, , 387 Djorgovski S., Sosin C., 1989, , L13 Drimmel R., Smart R. L., Lattanzi M. G., 1997, In: M. A. C. Perryman (ed.) [Hipparcos Venice ’97]{}, p. 607, ESA SP-402 Drimmel R., Smart R. L., Lattanzi M. G., 1999, In: R. Dvorak (ed.) [Modern Astrometry and Astrodynamics]{}, Kluwer , 1997, Freudenreich H., 1998, , 495 Freudenreich H., Berriman G., Dwek E., et al., 1994, , L69 N., [Smith-Moore]{} M., 1988, K. R., 1992, F., 1997, In: M. A. C. Perryman (ed.) [Hipparcos Venice ’97]{}, p. 5, ESA SP-402 T., 1982, In: [Landolt-Bornstein: Numerical Data and Functional Relationships in Science and Technology]{} Smart R. L., Lattanzi M. G., 1996, , 104 Smart R. L., Drimmel R., Lattanzi M. G., 1997, In: M. A. C. Perryman (ed.) [Hipparcos Venice ’97]{}, p. 715, ESA SP-402 Smart R. L., Drimmel R., Lattanzi M. G., Binney J. J., 1998, , 471 Smart R. L., Drimmel R., Lattanzi M. G., 1999, In: R. Dvorak (ed.) [Modern Astrometry and Astrodynamics]{}, Kluwer H., 1985, , 413 Spergel D. 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--- author: - 'Felix M. Haehl' bibliography: - 'sphereical.bib' title: Comments on universal properties of entanglement entropy and bulk reconstruction --- Introduction {#sec:introduction} ============ It is by now well known that some conformal field theories (CFTs) admit a dual description in terms of semiclassical gravity in one higher dimension [@Aharony:1999ti]. But it is so far not clear what are the precise CFT data that one needs to know in order to reconstruct various specific features of the bulk theory. A full knowledge of the CFT should be exactly equivalent to full knowledge of bulk quantum gravity, the challenge being to find the precise dictionary. We wish to focus on the question which subset of CFT parameters one needs to have access to in order to reconstruct just a classical bulk geometry and its linearized dynamics. Once this question is answered in terms of fundamental CFT parameters, the practical question is which CFT observables are sensitive to these parameters in a useful way. One hint is the fact that a local bulk metric should be represented in the boundary CFT in a non-local fashion. An important CFT feature which does have non-local properties is the entangled structure of the underlying quantum state. It has therefore been proposed that knowing enough about the entanglement structure of the CFT state is sufficient to determine the gravitational dynamics of the bulk. In particular, entanglement entropy of spatial regions in the CFT appears to be an illuminating quantity to study in this context, c.f., [@VanRaamsdonk:2009ar; @VanRaamsdonk:2010pw; @Lashkari:2013koa; @Blanco:2013joa; @Faulkner:2013ica]. The basic rationale for this idea is that entanglement entropy of a given spatial region $\mathcal{A}$ can be computed in semiclassical holography as the area of a certain extremal surface [@Ryu:2006bv; @Ryu:2006ef; @Hubeny:2007xt] which, of course, encodes information about the metric. By varying this entanglement entropy (for instance by changing $\mathcal{A}$, or by varying the underlying quantum state, or by deforming the CFT spectrum), the corresponding change of the minimal surface encodes information about the dynamics of the bulk metric. Making this idea precise and general would be significant progress towards our understanding of the holographic nature of quantum states and gravity. We wish to carefully distinguish differently strong versions of the expectation that entanglement encodes bulk geometry. The weakest version of the [“entanglement is geometry”]{} statement simply says that the most efficient way of calculating entanglement entropy is by using geometric methods. For holographic CFTs the paradigmatic manifestation of this statement is the Ryu-Takayanagi conjecture [@Ryu:2006bv; @Ryu:2006ef]. But even for non-holographic CFTs there is evidence that geometric concepts sometimes provide the most natural and efficient way of computing entanglement entropy. The first such statement is the realization that vacuum entanglement entropy of spherical regions can be conformally mapped to thermal entropy which sometimes has a natural interpretation in terms of black hole thermodynamics [@Casini:2011kv]. Refining this idea, Faulkner [@Faulkner:2014jva] revisited the problem of computing the response of vacuum entanglement entropy of spheres to a deformation of the CFT by a relevant scalar operator [@deHaro:2000xn; @Hung:2011ta; @Liu:2012eea; @Nishioka:2014kpa; @Rosenhaus:2014woa; @Rosenhaus:2014zza]. For this computation, nothing is assumed about the existence or the nature of a holographic dual of the CFT. As a CFT calculation this is a formidable problem. Intriguingly, the most convenient way of repackaging this calculation turns out to be in terms of an auxiliary gravitational problem, where the entanglement entropy is represented as a minimal surface area in cosmological Einstein gravity responding to a scalar field perturbation. This is a powerful result, in particular in face of the fact that entanglement entropy is notoriously hard to compute otherwise. One of the goals of this paper is to elucidate from a gravitational point of view how this universality comes about and to reconcile these statements with AdS/CFT duality (where bulk descriptions generically have a structure which is not necessarily given by just Einstein gravity). A stronger form of the “entanglement is geometry” conjecture is the idea that in holographic CFTs the detailed bulk dynamics can be extracted entirely from knowledge of entanglement entropy for a suitable set of regions $\mathcal{A}$. Even in the semiclassical regime, this is a strong assertion because the detailed bulk dynamics may involve not only Einstein gravity, but also higher curvature corrections [@Fursaev:2013fta; @Hung:2011xb; @deBoer:2011wk; @Dong:2013qoa; @Camps:2013zua; @Bhattacharyya:2013gra; @Bhattacharyya:2014yga]. The imprint of higher curvature corrections on equations of motion should then be reconstructible from CFT entanglement entropies. See [@Lashkari:2013koa; @Faulkner:2013ica; @Swingle:2014uza] for recent progress towards deriving linearized Einstein equations from a first law of entanglement [@Bhattacharya:2012mi; @Blanco:2013joa; @Wong:2013gua; @Allahbakhshi:2013rda]. It will be a topic of this paper to illuminate the way in which CFT entanglement does or does not discriminate between different higher curvature interactions in the bulk. Let us compare these two versions of the “entanglement is geometry” statement. If spherical region entanglement entropy in deformations of any arbitrary CFT can be computed using linearized Einstein gravity [@Faulkner:2014jva], it follows that this quantity is certainly not sufficient to distinguish even CFTs with a semiclassical gravity dual from those without; let alone distinguish a bulk theory governed by Einstein gravity from any other higher derivative theory with the same spectrum of low energy excitations. Therefore, for computing this particular type of entanglement entropy, geometry is always an excellent tool; but in order to make sure that this geometry is actually the lowest order manifestation of a semiclassical AdS/CFT duality, more input is required. Of course, the bulk reconstruction from CFT data proceeds gradually: the more CFT parameters the entanglement entropies considered are sensitive to, the more details of the bulk dynamics are a-priori expected to be derivable from it. In this note we wish to investigate how to find entanglement entropies which one can expect to be sufficient to reconstruct various bulk features such as linearized dynamics, higher curvature couplings etc. In the context of spherical region entanglement entropy and scalar operator deformations, we will give a detailed analysis both from a CFT point of view and from the bulk perspective. As we will see, at the first relevant order in the perturbation, the entanglement entropy is only sensitive to few correlation functions which have a universal form (up to normalization) thanks to conformal invariance. From the bulk perspective the analogous statement is that the backreaction of a maximally symmetric geometry to the presence of a scalar probe is universal for any acceptable theory of gravity.[^1] A careful comparison reveals that this entanglement data is not sufficient to conclude anything about the presence of higher curvature terms in the bulk Lagrangian. For the purpose of bulk reconstruction we will outline how to gain sensitivity to such features, but will also demonstrate explicitly that the existence of some higher curvature interactions is much easier to conclude than the precise structure of the latter. The structure of this note is as follows. In §\[sec:universality\] we first review the context of our discussion and then give CFT and gravity arguments to demonstrate universal properties of entanglement entropy of spherically symmetric regions. In §\[sec:exampleGB\] we illustrate our abstract discussion with the example of curvature squared theories of gravity and Gauss-Bonnet theory in particular. Finally, §\[sec:discussion\] contains a discussion and relates to recent developments in this context. We generalize our discussion to a wider class of higher curvature theories in §\[app:generalization\], allowing also for fourth order equations of motion. In §\[app:Wald\] we perform the most general background field expansion for gravity at second order in the metric perturbation, thus uncovering the universality of Gauss-Bonnet theory within this class of theories. Some facts about the first law of entanglement are reviewed in §\[sec:1stLaw\]. Various useful formulae can be found in §\[sec:perturbation\]. Universality of entanglement entropy of spherical regions {#sec:universality} ========================================================= In this section we analyze universal features both of CFTs and of gravitational theories, which explain universal properties of entanglement entropy of ball shaped spatial regions in CFTs deformed by a scalar operator. We will then compare these using holography and draw conclusions for bulk reconstruction from entanglement. Setup {#sec:setup} ----- Consider a holographic CFT${}_d$ in the vacuum state and on Minkowski spacetime. Assume further that the theory has a classical gravity dual described by Einstein theory and consider a ball shaped spatial region $\mathcal{A}$ of radius $R$. The reduced density matrix associated with $\mathcal{A}$ is obtained by tracing out the degrees of freedom outside of $\mathcal{A}$: $$\label{eq:rhoDef} \rho_\mathcal{A} = \text{Tr}_{\mathcal{A}^c}\, \|0\rangle\langle 0\| \,.$$ Without loss of generality, we will assume $\rho_{\mathcal{A}}$ is normalized, i.e., $\text{Tr} \,\rho_{\mathcal{A}} = 1$. The entanglement entropy of $\mathcal{A}$ is defined as the von Neumann entropy of $\rho_\mathcal{A}$ and can be computed in the bulk as the area of the minimal surface ending on the boundary of the ball [@Ryu:2006bv]: $${{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A}) = -\text{Tr}( \rho_\mathcal{A} \, \log \rho_\mathcal{A}) = \frac{\text{Vol}(\mathcal{E}_{min})}{4 {G_{_N}}} \,, $$ where $\mathcal{E}_{min}$ is the corresponding bulk minimal surface and ${G_{_N}}$ denotes Newton’s constant in $d+1$ dimensions. Since the region $\mathcal{A}$ is spherical, its causal development can be mapped by a conformal transformation to a direct product of time with a maximally symmetric hyperbolic space [@Casini:2011kv]. This transformation further maps the CFT vacuum state to a thermal state. The extremal surface is no longer anchored at the boundary but instead wraps the horizon of a hyperbolic black hole. The computation of entanglement entropy of $\mathcal{A}$ thus reduces to a calculation of the horizon entropy of a hyperbolic black hole at temperature $T = 1/(2\pi R)$: $$\label{eq:EinsteinEntropy} {{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A}) = S_{thermal}(\mathbb{H}_{d-1}) = \frac{\text{Vol}(\mathbb{H}_{d-1})}{4{G_{_N}}}\,,$$ where $\mathbb{H}_{d-1}$ is the hyperbolic horizon slice. For notational simplicity, we will from now on work with Planck units, i.e., ${\ell_\text{P}}^{d-1} = 8\pi {G_{_N}}$. Now let us ask what changes if the bulk gravitational theory is not just Einstein gravity, but also contains some higher curvature corrections. For concreteness, consider a $d+1$-dimensional bulk theory of the form $$\label{eq:HigherDerDef} \begin{split} I = \frac{1}{2{\ell_\text{P}}^{d-1}} \int \sqrt{-g} \; \mathcal{L} \equiv \frac{1}{2{\ell_\text{P}}^{d-1}} \int \sqrt{-g} \, \left(R + \frac{d(d-1)}{\ell^2}+ \mathcal{L}_{h.d.} \right) \,, \end{split}$$ where $\mathcal{L}_{h.d.}\equiv \mathcal{L}_{h.d.}(g_{ab},R_{abcd},\nabla_e R_{abcd},\ldots)$ encodes higher curvature corrections. The cosmological length scale $\ell$ is related to the AdS scale ${\ell_\text{AdS}}$; e.g., in Einstein gravity, $\ell={\ell_\text{AdS}}$. The thermal entropy (and thus the entanglement entropy) is now given by the Wald entropy [@Iyer:1994ys] of the hyperbolic horizon [@Myers:2010tj; @Myers:2010xs; @Hung:2011xb]: $$\label{eq:WaldEntropy} \begin{split} {{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A}) &= S_{_{Wald}}(\mathbb{H}_{d-1}) = \frac{-2\pi}{2{\ell_\text{P}}^{d-1}} \, \int_{\mathbb{H}_{d-1}} \sqrt{\gamma} \; \left(-2+\frac{\delta \mathcal{L}_{h.d.}}{\delta R_{abcd}} \, n_{ab}\, n_{cd}\right) \\ &= \frac{2\pi}{\pi^{d/2}} \Gamma\left(d/2\right) \, a_d^* \; \frac{\text{Vol}(\mathbb{H}_{d-1})}{{\ell_\text{AdS}}^{d-1}} \,, \end{split}$$ where $n_{ab}$ denotes the binormal to the hyperbolic horizon $\mathbb{H}_{d-1}$ normalized as $n^{ab}n_{ab} = -2$ and $\gamma$ is the induced metric. The constant $a_d^$ depends on the details of the gravitational theory. For example, in Einstein gravity $\mathcal{L}_{h.d.}=0$ and we find by comparison with $$\label{eq:adEH} (a_d^)_{Einstein} = \frac{\pi^{d/2}}{\Gamma(d/2)} \frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}}\,.$$ In fact, since the Wald entropy density is constant on maximally symmetric AdS${}_{d+1}$, we can immediately read off the general expression for the constant $a_d^$: $$\label{eq:adHD} a_d^ = \frac{\pi^{d/2}}{\Gamma(d/2)} \frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}} \left(1- \frac{1}{2}\frac{\delta \mathcal{L}_{h.d.}}{\delta R_{abcd}} \, n_{ab}\, n_{cd}\bigg{\|}_{AdS}\right) = -\frac{\pi^{d/2}{\ell_\text{AdS}}^{d+1}}{d\, \Gamma(d/2)} \, \frac{1}{2{\ell_\text{P}}^{d-1}} \, \mathcal{L}\|_{AdS}\,.$$ Here, the second step is a simple consequence of the existence of an AdS${}_{d+1}$ background solution and the maximal symmetry of the geometry (see, e.g., [@Myers:2010tj] for a detailed derivation).[^2] The important feature of is that the entanglement entropy is always proportional to the horizon area $\text{Vol}(\mathbb{H}_{d-1})$ (or equivalently to $\text{Vol}(\mathcal{E}_{min})$), irrespective of the details of the gravitational theory. In particular, just knowing ${{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A})$ does not give any insight as to what higher derivative corrections the bulk dual of the given CFT may entail: any higher derivative corrections to Einstein gravity will only change the overall normalization but not the volume scaling in . But any such change of $a_d^$ due to higher curvature corrections can be equivalently interpreted as simply a renormalization of the ratio ${\ell_\text{AdS}}/{\ell_\text{P}}$ within Einstein gravity. The precise renormalization factor is given by the bracket in . The effect of higher derivative terms is hence not visible for its only imprint is to renormalize a dimensionless parameter of the theory. Note that the above expressions are divergent and the matchings performed should hence be done more carefully. For example, the volume factor in should be regulated such that its leading divergent piece scales with the area of the sphere $\mathcal{A}$. This divergence depends on the cut-off, but in a consistent way such that the CFT short distance cut-off (regulating the UV divergence of entanglement entropy near the boundary of $\mathcal{A}$) matches the bulk cut-off (regulating the divergent area of $\mathbb{H}_{d-1}$) [@Ryu:2006ef]. More importantly, the expression also contains a universal piece which is constant in even dimensions and logarithmically divergent in odd dimensions. For this reason, the above matching conditions strictly speaking refer to the universal piece, which takes the form [@Ryu:2006ef; @Myers:2010xs] $$S_{EE,\text{univ}}^{(0)}(\mathcal{A}) = \begin{cases} -(-1)^{d/2} \, 4 \, a_d^ \, \log (2R/\delta) & (d\text{ even}) \\ (-1)^{(d-1)/2} \, 2\pi \, a_d^* & (d \text{ odd}) \end{cases}$$ where $\delta$ is the UV cut-off scale in the CFT. It is only for spherical regions $\mathcal{A}$ that our arguments above give the correct result without explicitly dealing with these additional subtleties, see e.g. [@Hung:2011xb]. From a CFT point of view, $a_d^$ is a fundamental parameter. As we have just seen, it characterizes the normalization of the universal part of vacuum entanglement entropy. It is a remarkable fact about entanglement entropy that in even dimensions $a_d^$ coincides with the coefficient $a$ of the Euler density in the conformal trace anomaly [@Schwimmer:2008yh; @Myers:2010xs; @Myers:2010tj]. In odd dimensions, $a_d^$ is proportional to $\log Z_{S^d}$, the sphere partition function of the CFT [@Casini:2011kv]. For a detailed reconstruction of bulk dynamics from entanglement, one would like to probe the particular structure of the gravitational action. In this note we wish to study deformations of the above setup and ask what CFT quantities one can use in order to gain sensitivity to features which are peculiar to higher curvature terms in the gravity dual. Clearly the vacuum entanglement entropy of balls $\mathcal{A}$ will not be enough by the above argument. Consider therefore a deformation of the CFT by some relevant scalar operator $\mathcal{O}$, uniformly coupled to the CFT with a coupling $\lambda$. Given the arguments above, it is clear what bulk computation we should do in order to learn about the change in $S_{EE}(\mathcal{A})$: add to the gravitational problem a scalar field dual to $\mathcal{O}$ with suitable boundary conditions, solve the scalar equation of motion in the AdS${}_{d+1}$ background and compute the backreaction on the minimal surface area perturbatively in $\lambda$. As we will see, the linearized answer is still universal in a way very similar to how was universal for the unperturbed problem. Namely, its functional form is completely fixed such that it only depends on a few parameters which encode the influence of higher curvature couplings. These parameters can again be absorbed in a renormalization of dimensionless quantities such that the answer is indistinguishable from what one would have obtained in Einstein gravity minimally coupled to a scalar field. This is the main universality statement studied in this section: [For holographically computing entanglement entropy of spherical regions in CFTs deformed by a relevant scalar operator, any physically acceptable theory of a spin-2 and a spin-0 field with appropriately tuned values of the couplings gives the same result as cosmological Einstein gravity minimally coupled to a scalar. In particular, this quantity does not distinguish between different higher curvature theories of gravity.* ]{} In the rest of this section, we will explain these statements from the CFT and from the gravity side. CFT explanation for universality {#sec:CFTExplanation} -------------------------------- We start by reviewing from a CFT point of view the ingredients involved in computing entanglement entropy of spherical regions in perturbed states. Our discussion closely follows [@Faulkner:2014jva; @Rosenhaus:2014woa; @Rosenhaus:2014zza]. Consider a CFT on $d$-dimensional Euclidean flat space and perform the following deformation of the CFT action ${{I}^{\text{\tiny (0)}}}_{CFT}$ by a relevant scalar operator $\mathcal{O}$ of dimension $\Delta<d$: $$I_{CFT} = {{I}^{\text{\tiny (0)}}}_{CFT} + \lambda \int d^dx \; \mathcal{O}(x) \,.$$ We can compute the perturbative changes of the entanglement entropy $S_{EE}(\mathcal{A})$ by writing the reduced density matrix as a path integral. This path integral lives on a manifold which is the Euclidean spacetime with a cut along $\partial \mathcal{A}$. The matrix element corresponding to some field configurations $\phi_+$ and $\phi_-$ on the two sides of the cut is $$\begin{split} \langle \phi_- \| \rho_\mathcal{A}\| \phi_+ \rangle &= \int_{\small\substack{\phi(\partial\mathcal{A}_+) = \phi_+ \\ \phi(\partial\mathcal{A}_-) = \phi_-}} \,[\mathcal{D}\phi] \; e^{-I_{CFT}}\\ &= \int_{\small\substack{\phi(\partial\mathcal{A}_+) = \phi_+ \\ \phi(\partial\mathcal{A}_-) = \phi_-}} \,[\mathcal{D}\phi] \; e^{-{{I}^{\text{\tiny (0)}}}_{CFT}} \left( 1 - \lambda \int d^dx\, \mathcal{O}(x) + \frac{\lambda^2}{2} \iint d^dx\,d^dy\, \mathcal{O}(x) \mathcal{O}(y) + \ldots \right) \,. \end{split}$$ Now define a modular Hamiltonian $H_\mathcal{A}$ via $\rho_\mathcal{A} = e^{-H_\mathcal{A}}$. We can then use the identity $S_{EE}(\mathcal{A}) = \text{Tr}(\rho_\mathcal{A} \, H_\mathcal{A})$ to compute the change in the entanglement to any desired order in $\lambda$:[^3] $$\label{eq:PertSEE} \begin{split} S_{EE}(\mathcal{A}) &= {{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A}) + \lambda \, \left[\frac{\partial S_{EE}(\mathcal{A})}{\partial \lambda}\right]_{\lambda = 0} + \frac{\lambda^2}{2} \, \left[\frac{\partial^2 S_{EE}(\mathcal{A})}{\partial \lambda^2} \right]_{\lambda=0} + \ldots \\ &= {{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A}) + \lambda \left[ -\langle \mathcal{O} H_\mathcal{A} \rangle + \langle \frac{\partial H_\mathcal{A}}{\partial \lambda} \rangle \right]_{\lambda=0} + \frac{\lambda^2}{2} \left[\langle \mathcal{O} \mathcal{O} H_\mathcal{A} \rangle - \langle \mathcal{O} \frac{\partial H_\mathcal{A}}{\partial \lambda} \rangle + \langle \frac{\partial^2 H_\mathcal{A}}{\partial^2 \lambda} \rangle\right]_{\lambda=0} + \ldots \\ &= {{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A}) - \lambda \langle \mathcal{O} {{H}_\mathcal{A}^{\text{\tiny (0)}}}\rangle + \frac{\lambda^2}{2} \left( \langle \mathcal{O} \mathcal{O} {{H}_\mathcal{A}^{\text{\tiny (0)}}}\rangle - \langle \mathcal{O} \frac{\partial H_\mathcal{A}}{\partial \lambda} \rangle\right) + \ldots \end{split}$$ where ${{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A})$ is universal up to an overall normalization $a_d^$ as explained in §\[sec:setup\], and similarly ${{H}_\mathcal{A}^{\text{\tiny (0)}}}$ is the unperturbed modular Hamiltonian. In the last step we used the fact that $\rho_\mathcal{A}$ is normalized which implies $\langle \frac{\partial H_\mathcal{A}}{\partial \lambda} \rangle = 0$. We can now use that $\mathcal{A}$ is a spherical region, which brings about another important simplification: the modular Hamiltonian can be traded for an integrated stress tensor, since for spherical $\mathcal{A}$ with radius $R$ centered at the origin, one finds $$\label{eq:ModHamDef} H_\mathcal{A} = 2\pi \int_{\mathcal{A}} d^{d-1}x \; \frac{R^2-\vec{x}\,{}^2}{2R} \, T_{00}(x) \,,$$ for $\mathcal{A}$ lying on the $x^0 = 0$ slice. In our problem, correlation functions involving the modular Hamiltonian can therefore be regarded as correlation functions involving the conformal stress tensor. But for stress tensor correlation functions, we can use well-known features of CFTs: on general grounds, two-point correlations between a stress tensor and a primary operator are constrained by conformal invariance and conformal Ward identities [@Erdmenger:1996yc]. In fact, independent of the details of our problem, conformal invariance of a theory in a flat spacetime enforces $\langle \mathcal{O}T_{\mu\nu} \rangle = 0$. Therefore the ${\mathscr{O}}(\lambda)$ perturbation in vanishes.[^4] Furthermore, the perturbation of $T_{\mu\nu}$ is uniform in $\mathcal{O}$ and we have $\frac{\partial T_{\mu\nu}}{\partial\lambda} = - \delta_{\mu\nu}\,\mathcal{O}$. To see this, recall that for a Euclidean flat boundary, we have $$\label{eq:CFTstress} T_{\mu\nu}(x) = \frac{2}{\sqrt{g}} \frac{\delta I_{CFT}}{\delta g^{\mu\nu}(x)} = T_{\mu\nu}^{\text{\tiny (0)}}(x) - \delta_{\mu\nu} \, \lambda\, \mathcal{O}(x) \,.$$ The entanglement entropy in the deformed theory, , hence takes a generic form [@Rosenhaus:2014zza]: $$\label{eq:PertSEE2} \begin{split} S_{EE}(\mathcal{A}) &= {{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A}) + \frac{\lambda^2}{2} \left( \langle \mathcal{O} \mathcal{O} {{H}_\mathcal{A}^{\text{\tiny (0)}}}\rangle - \langle \mathcal{O} \mathcal{O} \rangle\right) + \ldots \end{split}$$ As we can see, this is only sensitive to the correlators $\langle \mathcal{O} \mathcal{O} T_{\mu\nu} \rangle$ and $\langle \mathcal{O} \mathcal{O} \rangle$. These, however, are known to be universal functions in any conformal field theory which only depend on one overall normalization constant: $$\label{eq:OOnormalization} \begin{split} \langle \mathcal{O}(x_1)\mathcal{O}(x_2) \rangle = \frac{ {C_{\mathcal{O}}}}{(x_1-x_2)^{2\Delta}} \end{split}$$ and a similar universal function for $\langle \mathcal{O}(x_1) \mathcal{O}(x_2) T_{\mu\nu}(x_3) \rangle$ whose normalization is also determined solely by ${C_{\mathcal{O}}}$ [@Osborn:1993cr; @Erdmenger:1996yc]. The entanglement entropy of $\mathcal{A}$ is hence completely universal up to ${\mathscr{O}}(\lambda^2)$, only depending on the fundamental constants $\{a_d^\,,\,{C_{\mathcal{O}}}\}$ and the operator dimension $\Delta$. The divergent structure of the expression has been computed in [@Rosenhaus:2014zza] just using standard Euclidean CFT techniques. However, we must now face an additional subtlety: the analysis of [@Faulkner:2014jva], which treats the problem at hand using the replica trick and carefully analyzes the analytic structure of the resulting partition functions, leads to a different result which involves a non-trivial finite piece. Nevertheless the basic lesson of this more involved analysis using the replica trick turns out to be still the same; namely the ${\mathscr{O}}(\lambda^2)$ perturbation of entanglement entropy is determined solely by correlators $\langle\mathcal{O}\mathcal{O}\rangle$ and $\langle\mathcal{O}\mathcal{O}T_{\mu\nu}\rangle$. To be precise, the finite contribution to the correction at ${\mathscr{O}}(\lambda^2)$ reads [@Liu:2012eea; @Nishioka:2014kpa; @Faulkner:2014jva]:[^5] $$\label{eq:SEECFT} S_{EE}(\mathcal{A}) = {{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A}) - \lambda^2 R^{2(d-\Delta)}\, \frac{\pi^{\frac{d+1}{2}}(d-\Delta) \Gamma(1+\tfrac{d}{2} - \Delta)}{2\,\Gamma(\tfrac{3}{2} + d - \Delta)} + \ldots \,,$$ where $R$ is the radius of the ball $\mathcal{A}$. In this expression the normalization ${C_{\mathcal{O}}}$ has been fixed in a way that will prove convenient when comparing with holographic calculations. However, we will not need the explicit form of $S_{EE}(\mathcal{A})$ for the following analysis. Gravitational explanation for universality {#sec:GravExplanation} ------------------------------------------ Having seen that the second order computation of perturbed entanglement entropy of spherical regions is only sensitive to $\langle \mathcal{O}\mathcal{O}\rangle$ and $\langle \mathcal{O}\mathcal{O}T_{\mu\nu}\rangle$ correlators (which are universal up to one normalization constant), we now want to explain what this means in gravity. ### Holographic parameter matching {#sec:HoloMatching} A major hint for understanding the entanglement universality from a holographic point of view is the result of [@Faulkner:2014jva], where it is shown that for the CFT deformation characterized by a coupling $\lambda$ and an operator dimension $\Delta$ as described in §\[sec:CFTExplanation\], one can calculate the entanglement entropy at ${\mathscr{O}}(\lambda^2)$ by solving an auxiliary gravitational problem described by the action $$\label{eq:GravAux} \begin{split} I_{aux} = \frac{1}{2{\ell_\text{P}}^{d-1}}\int \sqrt{-g} \, \left[R + \frac{d(d-1)}{\ell^2} - \frac{{\lambda_\phi}}{2}\left( (\nabla \phi)^2 + m^2 \, \phi^2 \right)\right] \,. \end{split}$$ The AdS${}_{d+1}$ background geometry ${\bar g}_{ab}$ has a scale which coincides with the cosmological scale: ${\ell_\text{AdS}}^2 = \ell^2$. Further, ${\lambda_\phi}$ parametrizes the normalization of the scalar action. The prescription is to solve first the scalar wave equation in the background ${\bar g}_{ab}$ and then compute its backreaction on the metric at next order in perturbation theory to get the change of the area of the Ryu-Takayanagi minimal surface. For this task we are therefore not interested in the full action , but only in the linearized equations of motion due to a scalar probe in AdS${}_{d+1}$ background. In the following all objects with a bar (e.g., $\bar{\nabla}_a$) refer to the background metric ${\bar g}_{ab}$. We will now comment on the number of free parameters in this problem and how to determine them, given that they should induce the CFT parameters at the boundary in a way consistent with AdS/CFT. A priori there are three dimensionless quantities $\{({\ell_\text{AdS}}/{\ell_\text{P}})\,,\,{\ell_\text{AdS}}^2m^2,{\lambda_\phi}\}$ in the action . The scalar wave equation in AdS, $({\bar{\Box}}-m^2)\phi= 0$, has two asymptotic solutions with respective falloff behavior $z^{\Delta_+}$ and $z^{\Delta_-}$, where the boundary is at $z=0$ (in Poincaré coordinates such as ) and $$\Delta_\pm \equiv \frac{d}{2} \pm \sqrt{\frac{d^2}{4} + {\ell_\text{AdS}}^2 \, m^2} \,.$$ The boundary condition for the scalar is required to be such that it sources an operator with given dimension $\Delta$ and coupling $\lambda$. By standard AdS/CFT techniques [@Aharony:1999ti], we therefore assume[^6] $\Delta \geq d/2$, tune the scalar mass such that $\Delta_+ = \Delta$ and fix the dominant asymptotics as $\phi(z\rightarrow 0) \sim \lambda \, z^{\Delta_-}$. The mass of the scalar is also immediately fixed by demanding that it sources an operator with dimension $\Delta$; the usual prescription for this is to take $-d^2/4 < {\ell_\text{AdS}}^2\,m^2<0$ such that $$\label{eq:MassRelation} {\ell_\text{AdS}}^2 \, m^2 = \Delta (\Delta-d) \,.$$ The remaining two parameters of the theory also need to be fixed by matching with the CFT. On the one hand, ${\ell_\text{AdS}}/{\ell_\text{P}}$ makes an appearance in the normalization of the universal scaling of entanglement in the CFT in terms of the generalization of $a$-central charge, c.f., and can hence be fixed by demanding . On the other hand, ${\lambda_\phi}$ parametrizes rescalings of the bulk field $\phi$ in the same sense in which the dual boundary two-point function $\langle\mathcal{O}\mathcal{O}\rangle$ depends on an overall normalization ${C_{\mathcal{O}}}$ as in . Since $\phi$ has already been equipped with boundary conditions which lead to the correct coupling $\lambda$ for $\mathcal{O}$, we can follow standard techniques to work out the normalization ${\lambda_\phi}$ in terms of ${C_{\mathcal{O}}}$. This involves computing the bulk to boundary propagator for $\phi$ and matching its boundary two-point function with the CFT correlator $\langle\mathcal{O}\mathcal{O}\rangle$. The result is [@Aharony:1999ti; @Hung:2011ta] $$\label{eq:cOOmatch} {C_{\mathcal{O}}}= \frac{\Delta-d/2}{\pi^{d/2}} \frac{\Gamma(\Delta)}{\Gamma(\Delta-d/2)} \frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}}\;{\lambda_\phi}\,.$$ We stress that, as shown in [@Faulkner:2014jva], the auxiliary gravitational problem just described can be used for computing perturbed entanglement entropy of ball shaped regions in [any]{} CFT. Given the universality of the correlators $\langle \mathcal{O}\mathcal{O}\rangle$ and $\langle \mathcal{O}\mathcal{O}T_{\mu\nu}\rangle$ in all CFTs (including those which are dual to cosmological Einstein gravity), this result is not very surprising. It is, however, quite remarkable that the most natural way of rewriting the CFT problem is in terms of the auxiliary gravitational system . As we shall see, there are many other (higher curvature) models of gravity which could equally well serve as the auxiliary system for this problem. Turning this logic around, all these different gravitational theories are indistinguishable if only perturbed entanglement data of spheres is measured. We now turn to illuminating this point from the gravity side. ### Construction of the most general gravitational model {#sec:GravConstruction} We now want to make more precise the statement that all physically acceptable gravity models are equivalent for the computation at hand. Our approach is to construct a large class of auxiliary gravitational systems which make the same predictions as in this respect. Let us take the CFT data as given and try to construct a holographic description in terms of semiclassical gravity ’bottom up’. The first step of such a program could be the question: what are the minimal bulk ingredients if they are supposed to compute entanglement entropy of spheres in vacuum and after turning on the perturbation? We will now answer this question by building general linearized bulk actions which satisfy the following conditions: - The field content is a symmetric spin-2 tensor (the metric) and a scalar field, which source the boundary operators $T_{\mu\nu}$ and $\mathcal{O}$, respectively. The model gives dynamics to these fields and defines a coupling between them which allows to compute the linearized backreaction $h_{ab}$ on an AdS${}_{d+1}$ background ${\bar g}_{ab}$ after perturbing the latter with the scalar probe $\phi$. - There are no ghost degrees of freedom; in particular the linearized equations of motion on AdS${}_{d+1}$ are second order in derivatives. (See below for a justification of this assumption and §\[app:generalization\] for generalizations in the case where the higher curvature terms are perturbatively small and this assumption can be dropped.) - The action is diffeomorphism invariant (c.f., [@Ogievetsky:1965; @PhysRevD.33.3613]). We will find that these conditions specify the linearized action uniquely up to normalization constants; see for the final result. Let us briefly explain why the equations of motion should be second order. It is well known that equations of motion which are higher than second order lead to pathological behaviour which we want to exclude from our discussion of theories which are dual to unitary CFTs. In particular, higher order equations of motion generically give rise to further propagating degrees of freedom with negative energy [@Stelle:1977ry; @Stelle:1977aa; @Barth:1983aa; @Biswas:2011ar; @Conroy:2015wfa].[^7] The presence of ghosts in such theories thus leads to classical instabilities [@Boulware:1985nm] (which would lead to negative norm states and therefore non-unitary quantum theories upon quantization). These features would also be visible in the dual CFT as operators with complex conformal dimension or negative norm. Indeed, via holography the problem of perturbed entanglement entropy is described in the CFT by correlation functions of the stress tensor $T_{\mu\nu}$ and a relevant scalar operator $\mathcal{O}$. This setup should be described in gravity by nothing more than a spin-2 graviton which sources $T_{\mu\nu}$ and a scalar field to source $\mathcal{O}$.[^8] If the graviton equations of motion were higher than second order in derivatives, then the additional unwanted ghost modes would couple to other (non-unitary) operators. Therefore, whatever the full gravitational theory is, after linearization of the action around AdS${}_{d+1}$ the excitation $h_{ab}$ is bound to appear in such a way as to yield second order equations of motion. Despite these arguments in favour of two-derivative equations of motion, higher derivative models with higher order equations of motion are regularly studied as toy models for semiclassical corrections to Einstein gravity induced by some consistent truncation of full unitary quantum gravity (which is assumed to be UV complete). Such models can be justified as sensible toy models in a perturbative framework as long as the ghost modes are very heavy and do not go on-shell (which turns out to be the case whenever the higher derivative couplings are small). We explore this setup in §\[app:generalization\], where we show that in this context our arguments concerning entanglement entropy still hold. The reason for the latter is that at low energies the ghosts can be ignored and the sector of physical modes is exactly the same as for ghost-free models which we are now going to analyze. Let us now construct the minimal ingredients for a linearized action for $h_{ab}$ and $\phi$, which leads to second order equations of motion. We will argue that the most general action of this type is the same as the linearization of Einstein-scalar theory up to various normalizations. For illustration, we can hence start by expanding the metric around an AdS${}_{d+1}$ background solution ${\bar g}_{ab}$ as $g_{ab} = {\bar g}_{ab} + h_{ab}$ and linearizing the Einstein action . Including some convenient normalization factors, this procedure leads to[^9] $$\label{eq:GeneralAction2ndOrder} \begin{split} I = \frac{1}{2{\ell_\text{P}}^{d-1}}\int\sqrt{-{\bar g}} \; \Bigg\{ & -\frac{2d}{{\ell_\text{AdS}}^2}\frac{\Gamma(d/2)}{\pi^{d/2}}\frac{{\ell_\text{P}}^{d-1}}{{\ell_\text{AdS}}^{d-1}} \, a_d^* + \frac{{\lambda_h}}{2}\, h_{ab}\, {\mathcal{G}}^{abcd}\,h_{cd} + {\lambda_\phi}\,\left[ -\frac{1}{2} \left( ({\bar{\nabla}}\phi)^2 + m^2 \phi^2 \right)\right] \\ & + \frac{{\lambda_\phi}}{2}\, h^{ab} \left( - \frac{1}{2} \left[({\bar{\nabla}}\phi)^2 + m^2 \phi^2\right]{\bar g}_{ab} + {\bar{\nabla}}_a \phi \,{\bar{\nabla}}_b \phi \right)+ {\mathscr{O}}(h^3,h^2\phi^2)\Bigg\} \,, \end{split}$$ where ${\mathcal{G}}_{abcd}$ is the graviton kinetic operator $$\label{eq:EHlichnerowicz} \begin{split} {\mathcal{G}}_{abcd}&= \frac{1}{2}\left({{\bar g}}_{a(c} {{\bar g}}_{d)b}- {{\bar g}}_{ab} {{\bar g}}_{cd}\right) \, {\bar{\Box}}- {\bar{\nabla}}_{(c} {{\bar g}}_{d)(a}{\bar{\nabla}}_{b)}+ \frac{1}{2} \left( {{\bar g}}_{ab} {\bar{\nabla}}_c {\bar{\nabla}}_d + {{\bar g}}_{cd} {\bar{\nabla}}_a {\bar{\nabla}}_b \right) \\ &\quad + \frac{d}{{\ell_\text{AdS}}^2}\left(\frac{1}{2}\,{{\bar g}}_{ab} {{\bar g}}_{cd} - {{\bar g}}_{a(c} {{\bar g}}_{d)b}\right) \,. \end{split}$$ A few explanations are in order concerning the way we wrote :\ ([i$\,$]{}) In writing we introduced a normalization ${\lambda_h}$ in the graviton kinetic term for later convenience. By expanding the Einstein-scalar action , one finds simply $({\lambda_h})_{Einstein} = 1$, but we will shortly encounter other (higher derivative) theories with other values of ${\lambda_h}$.\ ([ii$\,$]{}) Similarly, we wrote a constant term proportional to $a_d^$. This is also using the benefit of hindsight (in particular ) to already implement the correct generalization for higher derivative theories. For the present case of Einstein gravity, $(a_d^)_{Einstein}$ is just given by . For more general theories, notice that the constant term in is determined by the requirement that switching off $h_{ab}$ and $\phi$ in the action is equivalent to evaluating the full non-linear Lagrangian on the AdS background. By means of we know that the constant thus obtained needs to be $a_d^$ (with appropriate pre-factors).\ ([iii$\,$]{}) Note also that we did not write the ${\mathscr{O}}(h)$ term of the action. This term would just define the equations of motion for the AdS${}_{d+1}$ background geometry ${\bar g}_{ab}$. By assuming an expansion of the form $g_{ab} = {\bar g}_{ab} + h_{ab}$, such terms vanish automatically.\ The graviton-scalar action was obtained by linearizing the Einstein-scalar system around a given AdS background with scale ${\ell_\text{AdS}}$. However, we will now argue that [any]{} theory which satisfies the basic conditions listed at the beginning of this subsection, takes this form. As we will see, the only freedom is parameterized by the normalizations $\{a_d^, {\lambda_h}, {\lambda_\phi}, m^2\}$. Of these, $a_d^$ has already been fixed manifestly by using . Also ${\lambda_\phi}$ and ${\ell_\text{AdS}}^2\,m^2$ could be fixed in terms of field theory quantities via and . The graviton normalization, ${\lambda_h}$, will be discussed shortly. This will then demonstrate the universality of the dynamics of perturbing an AdS geometry with a scalar field up to normalizations.* #### Graviton propagator on AdS${}_{d+1}$: Clearly, the field content in the bulk should involve a spin-0 field $\phi$ to probe the geometry, whose boundary condition $\phi(z\rightarrow 0) \sim \lambda \, z^{d-\Delta}$ sources the relevant deformation $\mathcal{O}$. But before specifying the matter probe in this much detail, let us discuss the general gravitational response to any small perturbation. This is described by a spin-2 field propagating on the background ${\bar g}_{ab}$. We will write this as $g_{ab} = {\bar g}_{ab} + h_{ab}$, where $h_{ab}$ is the linearized perturbation. We now want to establish the most general physically acceptable equations of motion for $h_{ab}$, as described above. It is straightforward to write the most general (up to total derivatives) action at ${\mathscr{O}}(h^2)$ of the symmetric spin-2 field which involves at most two derivatives. We can parametrize it as (see, e.g., [@Alvarez:2006uu]) $$\label{eq:2ndOrderLin} \begin{split} I_{(h^2)} =\frac{1}{2{\ell_\text{P}}^{d-1}}\int \sqrt{-{\bar g}} \; \bigg\{ -\frac{{\lambda_h}}{4} {\bar{\nabla}}_a h_{bc} {\bar{\nabla}}^a h^{bc} & + \frac{b_1}{2} {\bar{\nabla}}_b h^{ac} {\bar{\nabla}}_a h^b_c - \frac{b_2}{2} {\bar{\nabla}}^a h {\bar{\nabla}}^b h_{ab} \\ & + \frac{b_3}{4} {\bar{\nabla}}_a h {\bar{\nabla}}^a h + \frac{M^2}{2} \left( h^2 - b_4 \,h_{ab} h^{ab} \right) \bigg\} \,, \end{split}$$ where $h \equiv h^{ab} {\bar g}_{ab}$. All raising and lowering of indices as well as covariant derivatives with a bar refer to ${\bar g}_{ab}$.[^10] We can already see that compared to the infinite number of possible higher derivative theories of gravity, the above linearized action is quite restrictive, having only six free parameters $\{{\lambda_h},M^2,b_1,b_2,b_3,b_4\}$. In fact the action will be even more constrained once we impose that the full theory should exhibit diffeomorphism invariance. It is a classic result that demanding a diffeomorphism invariant theory beyond the linearized level singles out a unique graviton propagator [@VanNieuwenhuizen:1973fi]. In our explicit notation, we can give a simple justification of this statement by considering a spin-2 gauge transformation (aka linearized diffeomorphism) $h_{ab} \rightarrow h_{ab} + 2\partial_{(a} \xi_{b)}$. The ${\mathscr{O}}(h^2)$ action transforms nontrivially: $$\begin{split} \delta I_{(h^2)} = \frac{1}{2{\ell_\text{P}}^{d-1}}\int \sqrt{-{\bar g}} \; \xi^a \bigg\{& (b_1-{\lambda_h}) {\bar{\nabla}}^b {\bar{\Box}}h_{ab} + (b_1-b_2) {\bar{\nabla}}_a {\bar{\nabla}}_c {\bar{\nabla}}_d h^{cd} + (b_3-b_2) {\bar{\nabla}}_a {\bar{\Box}}h \\ & - \left(2M^2 - \frac{d\,b_2}{{\ell_\text{AdS}}^2} \right) {\bar{\nabla}}_a h + \left(2b_4\, M^2 - \frac{2\,d\,b_1}{{\ell_\text{AdS}}^2}\right) {\bar{\nabla}}_b h^b_a \bigg\}\,, \end{split}$$ where we used partial integration and . In order for the action to be diffeomorphism invariant, the above expression has to vanish for arbitrary $\xi^a$. Since all terms are independent, this condition forces upon us five relations among the six parameters: $$b_1 = b_2 = b_3 = {\lambda_h}\,,\qquad M^2 = \frac{d}{2{\ell_\text{AdS}}^2}\,{\lambda_h}\,,\qquad b_4 = 2\,.$$ Any other choice of parameters would not correspond to the linearization of a diffeomorphism invariant theory. The only free parameter which is left undetermined is the overall scale ${\lambda_h}$ of the graviton action. In a full non-linear theory of gravity this is, of course, fixed by the normalization of the action and we will compute some examples in §\[sec:exampleGB\] to illustrate this. For instance, we have already seen that the graviton action is exactly the same as the second order term in a perturbative expansion of the Einstein-Hilbert action with cosmological constant scale $\ell = {\ell_\text{AdS}}$ and overall normalization ${\lambda_h}= 1$. This construction shows that by just looking at the linearized graviton propagator on AdS of any reasonable diffeomorphism invariant theory of gravity, we will never be able to distinguish the theory from cosmological Einstein gravity. The underlying reason for this is the universality of spin-2 propagation on maximally symmetric backgrounds. #### Coupling to scalar matter: Having constructed the most general quadratic action for the metric perturbation, we should now in a similar fashion argue that couples $h_{ab}$ to the scalar field in the most general way at this order. There are several possible Yukawa couplings and self-interaction terms. However, for consistency of our perturbative scheme, the only terms we are interested in at this order are $\phi\phi$ self-interactions and $h\phi\phi$ couplings. We will now describe the most general actions of these kinds: - [Scalar field kinetic and mass term:]{} At lowest order in perturbation theory (the probe limit), we are only concerned with couplings of $\phi \phi$ to the background geometry. Up to total derivatives, the most general such action with at most two time derivatives may be parametrized in terms of a mass parameter $m^2$ and an overall dimensionless normalization ${\lambda_\phi}$: $$\label{eq:ScalarAction2} I_{(\phi^2)} =\frac{1}{2{\ell_\text{P}}^{d-1}} \int \sqrt{-{\bar g}}\; {\lambda_\phi}\, \left\{ -\frac{1}{2} \left( ({\bar{\nabla}}\phi)^2 + m^2 \phi^2 \right)\right\}\,.$$ This action defines the background dynamics of $\phi$ as $({\bar{\Box}}- m^2) \phi = 0$. - [Backreaction on the geometry:*]{} There are several possible couplings of the type $h\phi\phi$, which are inequivalent upon integration by parts. We can parametrize them as follows: $$\label{eq:hphiphi} \begin{split} I_{(h\phi^2)} = \frac{1}{2{\ell_\text{P}}^{d-1}}\int \sqrt{-{\bar g}} \; \bigg\{ -\frac{h}{4} \big( c_1 ({\bar{\nabla}}\phi)^2 &+ c_2 \, m^2 \phi^2 \big) + \frac{c_3}{2}\, h^{ab} {\bar{\nabla}}_a \phi {\bar{\nabla}}_b \phi \\ & + c_4\, h^{ab} \phi {\bar{\nabla}}_a {\bar{\nabla}}_b \phi + c_5\, h \phi {\bar{\Box}}\phi \bigg\} \,, \end{split}$$ Clearly the coefficients $c_2$ and $c_5$ are on-shell equivalent. We will thus set $c_5 = 0$ in what follows. Now demanding diffeomorphism invariance, an argument completely analogous to the one we gave for implies the following conditions for the remaining parameters:[^11] $$\label{eq:constraints1} c_1 = c_2 = c_3 \,, \quad c_4 = 0 \,.$$ which leaves us with only one free parameter $c_1$. However, it is immediately clear that the overall normalization has to be the same as in , i.e., $$c_1 = {\lambda_\phi}\,.$$ If this normalization was any different, $h_{ab}$ would not be the response to a scalar with mass $m^2$ and normalization ${\lambda_\phi}$. Taking into account these constraints, we can rewrite as $$\label{eq:Tphi} \begin{split} I_{(h\phi^2)} &= \frac{1}{2{\ell_\text{P}}^{d-1}}\int \sqrt{-{\bar g}} \; \frac{{\lambda_\phi}}{2} \, h^{ab}\, T_{ab}^{(\phi)} \,,\\ \text{with}\quad T_{ab}^{(\phi)} &\equiv - \frac{1}{2} \left[({\bar{\nabla}}\phi)^2 + m^2 \phi^2\right]{\bar g}_{ab} + {\bar{\nabla}}_a \phi \,{\bar{\nabla}}_b \phi \,, \end{split}$$ Summarizing these two points, we recognize the most general scalar-graviton couplings which are consistent with the perturbative scheme and diffeomorphism invariance as being exactly the same as the expansion of the following action up to ${\mathscr{O}}(h)$: $$\label{eq:Matteraction} \begin{split} I_{matter} &= I_{(\phi^2)} + I_{(h\phi^2)} + {\mathscr{O}}(h^2) \\ &=\frac{1}{2{\ell_\text{P}}^{d-1}} \int \sqrt{-g} \; {\lambda_\phi}\, \left[ -\frac{1}{2} \left( (\nabla\phi)^2 + m^2 \phi^2 \right)\right] +{{\mathscr{O}}(h^2)} \,, \end{split}$$ where $g_{ab} = {\bar g}_{ab} + h_{ab}$. Of course, this result is not too surprising as this is the standard action for a massive scalar minimally coupled to gravity. #### Most general second order action: We can summarize the above discussion as follows. The most general way of coupling $h_{ab}$ to a scalar $\phi$ in a way that is consistent with our perturbative scheme and respects diffeomorphism invariance and some very basic constraints coming from the particular CFT computation we are interested in, is given by . This action has only four dimensionless parameters $\{{\ell_\text{AdS}}/{\ell_\text{P}}\,,\, m^2{\ell_\text{AdS}}^2\,, {\lambda_\phi}\,,{\lambda_h}\}$. We already explained in §\[sec:HoloMatching\] how for our problem the first three of these have to be determined holographically in terms of the dimensionless CFT quantities $\{a_d^\,,\,\Delta\,,\,{C_{\mathcal{O}}}\}$. We will argue shortly that the fourth parameter ${\lambda_h}$ does not enter the computation of interest and therefore its value is irrelevant for computing perturbed entanglement entropy. In addition, there is, of course, the boundary coupling $\lambda$ which enters the gravitational setup by prescribing the boundary condition for $\phi$. As a consequence, any covariant theory of gravity whose linearization takes the form , provides a good auxiliary system for computing $S_{EE}(\mathcal{A})$ at second order in perturbation theory. For instance, the action with ${\lambda_h}= 1$ is precisely the same as the linearization of the cosmological Einstein-scalar action via $g_{ab} = {\bar g}_{ab} + h_{ab}$. There are, of course, many other non-linear actions whose linearization on maximally symmetric backgrounds also takes the form with a different value of ${\lambda_h}$, e.g., those of Lovelock type [@Lovelock:1971yv] or quasi-topological gravity [@Myers:2010ru]. Indeed, even if the CFT in question is actually dual to some particular higher derivative theory; for the purpose of calculating the perturbed entanglement entropy, one can always work with the Einstein action after suitably matching the parameters $\{{\ell_\text{AdS}}/{\ell_\text{P}}\,,\, m^2{\ell_\text{AdS}}^2 \,, {\lambda_\phi}\}$ with the CFT boundary conditions. The different linearized actions will then only differ by their value of ${\lambda_h}$. We will see examples of this in §\[sec:exampleGB\]. Since the normalization ${\lambda_h}$ is the only ingredient in which would genuinely distinguish bulk theories with higher derivative couplings from those without, we now need to understand why the linearized computation of perturbed entanglement entropy of a ball is not sensitive to this. #### Holographic interpretation of ${\lambda_h}$: Before we continue with the gravitational analysis, it is instructive to elaborate on the meaning of ${\lambda_h}$, which has a very natural interpretation from the CFT point of view. The graviton normalization ${\lambda_h}$ corresponds to the normalization of the CFT stress tensor two-point function, usually denoted by ${C_{_T}}$. It sets the strength of the leading divergence in the two-point function of CFT stress tensors: $$\label{eq:TTcorr} \langle T_{\mu\nu}(x) T_{\rho\sigma}(0) \rangle = \frac{{C_{_T}}}{x^{2d}} \, \mathcal{I}_{\mu\nu,\rho\sigma}(x) \,,$$ with some well known universal function $\mathcal{I}_{\mu\nu,\rho\sigma}(x)$ [@Osborn:1993cr; @Erdmenger:1996yc]. In $d=2$, the normalization agrees with the usual $c$-central charge, ${C_{_T}}=c$. Similarly, in $d=4$ one finds ${C_{_T}}= (40/\pi^2) c$. Once a holographic description of the CFT has been given, usual AdS/CFT methods can be used to compute the left hand side of as a bulk graviton correlator. Since ${\lambda_h}$ sets the normalization of the graviton propagator in the presence of matter sources (c.f., ), it is rather clear that ${C_{_T}}$ will be proportional to ${\lambda_h}$. Doing the holographic calculation in detail leads to the standard result [@Buchel:2009sk] $$\label{eq:cTmatch} {C_{_T}}= \frac{f_d}{2}\, \frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}} \,{\lambda_h}\,,\quad \text{where}\quad f_d \equiv \frac{2d(d+1)}{d-1} \frac{\Gamma(d)}{\pi^{d/2}\Gamma(d/2)} \,.$$ It is now obvious that once ${\ell_\text{AdS}}/{\ell_\text{P}}$ has been fixed such as to reproduce the other central charge $a_d^$ via matching , there is no free parameter left in which could absorb the normalization ${\lambda_h}$. In our setup, gravity theories with different ${\lambda_h}$ will hence describe CFTs with different ${C_{_T}}$. Conversely, a CFT measurement of ${C_{_T}}$ would allow to for a distinction between different bulk models. However, we will now argue that the perturbed entanglement entropy of spheres is agnostic about this. ### Second order Wald entropy is independent of graviton normalization {#sec:lhindependence} As we have seen, any gravity theory whose action linearized around AdS${}_{d+1}$ takes the form can be used to compute second order perturbations of entanglement entropy of spheres. However, we identified one parameter ${\lambda_h}$ in the action which sets the graviton normalization and turns out to be sensitive to more detailed bulk structures such as higher curvature terms. We will now argue that this parameter does not enter the second order entanglement entropy.[^12] This is the gravity version of the statement that is independent of ${C_{_T}}$. We recall that due to the spherical symmetry of $\mathcal{A}$, extrinsic curvatures on the minimal surface $\mathcal{E}_{min}$ vanish and the entanglement entropy functional to be evaluated on $\mathcal{E}_{min}$ is just the Wald functional. In vacuum this reduces to just the area of $\mathcal{E}_{min}$ multiplied by a normalization which is basically determined by $a_d^$ as in . The perturbation can then be computed in holography as $$\label{eq:DeltaWald} \delta S_{EE}(\mathcal{A}) = \delta S_{Wald} (\mathcal{E}_{min}) = \delta \left( -2\pi \int_{\mathcal{E}_{min}} \sqrt{\gamma} \; \frac{\delta \mathcal{L}}{\delta R_{abcd}} \,n_{ab} \, n_{cd} \right)\,.$$ Clearly $\delta S_{Wald}(\mathcal{E}_{min})$ can be expanded perturbatively by considering different powers of $h_{ab}$ and $\phi$. From the structure of the gravitational backreaction it follows that there is no contribution at ${\mathscr{O}}(\lambda)$. At ${\mathscr{O}}(\lambda^2)$ (i.e., at first order in $h_{ab}$) the variation in acts on the objects inside the integral, but it does not change the surface $\mathcal{E}_{min}$ itself.[^13] Since the surface $\mathcal{E}_{min}$ is maximally symmetric, we anticipate that this variation of the Wald functional is just a variation of the area form (up to normalization). That is, the evaluation of should yield precisely the same answer as in Einstein gravity up to normalization. Let us make this slightly more explicit to figure out the correct normalization. To evaluate at ${\mathscr{O}}(\lambda^2)$, we should expand the integrand linearly in the perturbation: $$\label{eq:DeltaWald2} \delta S_{Wald}(\mathcal{E}_{min}) = -2\pi \int_{\mathcal{E}_{min}} \left[\sqrt{\gamma} \; \frac{\delta \mathcal{L}}{\delta R_{abcd}} \,n_{ab} \, n_{cd} \right]_{{\mathscr{O}}(h)} \,.$$ The obvious way to evaluate this is to start with the full non-linear Lagrangian, expand the Wald entropy density around the AdS background, and then identify the ${\mathscr{O}}(h)$ term in . However, even without knowledge of the full theory it is clear that ultimately the integrand can only be sensitive to the part of $\mathcal{L}$ which is at most quadratic in $h_{ab}$, i.e., we should be able to predict the answer just based on the most general second order Lagrangian . Moreover, only the purely gravitational sector of is relevant for Wald entropy at ${\mathscr{O}}(\lambda^2)$. This is quite clear from the observations that a minimally coupled scalar field in gravity does not couple to curvature and the action agrees with that of a minimally coupled scalar at the order we are interested in. From these observations, we are led to conclude that the integrand in formally has the same structure in any allowed theory of gravity up to the normalization ${\lambda_h}$. Therefore, without loss of generality, we can compute the integral as if it came from Einstein’s theory and multiply by ${\lambda_h}$. But in that case, the answer for $\delta S_{Wald}(\mathcal{E}_{min})$ is well known: it is just the variation of the area functional. Therefore, can be rewritten in general as a variation of area, multiplied by the normalization ${\lambda_h}$ that encodes the higher derivative dependence: $$\label{eq:deltaSWald4} \begin{split} \delta &S_{Wald}(\mathcal{E}_{min}) = 2\pi\left(\frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}}\, {\lambda_h}\right) \frac{\int_{\mathcal{E}_{min}} \sqrt{\bar{\gamma}} \; (\tfrac{1}{2}\bar\gamma^{\alpha\beta} h_{\alpha\beta})}{{\ell_\text{AdS}}^{d-1}} + {\mathscr{O}}(h^2) \,. \end{split}$$ where $\bar\gamma_{\alpha\beta}$ is the background metric induced on $\mathcal{E}_{min}$ by ${\bar g}_{ab}$ and $h_{\alpha\beta}$ is the induced metric perturbation.[^14] Note that the arguments just given were heuristic and we are discarding boundary terms which would be important to regulate the divergences. A detailed derivation of can be found in §\[app:Wald\]. The integral can now be used to derive the independence of $\delta S_{Wald}(\mathcal{E}_{min})$ on ${\lambda_h}$. We just need to find the on-shell dependence of $h_{\alpha\beta}$ on ${\lambda_h}$ and combine it with the prefactor in . To this end, consider the linearized equations of motion implied by the general action : $$\label{eq:generalEOM0} \begin{split} {\lambda_h}\, {\mathcal{G}}_{ab}{}^{cd} \, h_{cd} = - \frac{{\lambda_\phi}}{2} \, T_{ab}^{(\phi)} \,. \end{split}$$ From this equation it is clear how to go from the universality of graviton propagation to the universality of perturbed Wald entropy: apart from the overall ${\lambda_h}$ every dimensionless parameter in has been fixed by the boundary input $\{a_d^,\Delta,{C_{\mathcal{O}}}\}$. The combination ${\lambda_h}h_{cd}$ therefore satisfies a universal equation of motion which is independent of the non-linear details of the bulk theory. Since the perturbed Wald entropy depends on precisely this particular combination it is clear that $\delta S_{Wald}(\mathcal{E}_{min})$ will be the same in all theories we consider. Said differently, the function $h_{ab}$ which satisfies the backreaction equation is proportional to $1/{\lambda_h}$ such that the ${\lambda_h}$ dependence in cancels. We conclude that $\delta S_{EE}(\mathcal{A}) = \delta S_{Wald}(\mathcal{E}_{min})$ is independent of ${\lambda_h}$. This is consistent with the field theory result being independent of ${C_{_T}}$. This concludes our analysis of the role which various dimensionless parameters play. Having verified the cancellation of ${\lambda_h}$, the explicit solution of and evaluation of can be done without loss of generality at ${\lambda_h}= 1$, i.e., Einstein gravity. We refrain from repeating this calculation. See, for example, [@Liu:2012eea; @Nishioka:2014kpa] for an explicit evaluation of in the case of a scalar field source. The result of their computation perfectly matches the CFT answer . Example: Gauss-Bonnet theory {#sec:exampleGB} ============================ In this section we wish to illustrate the general argument, using the example of gravity theories with Lagrangians quadratic in curvature. This discussion will make contact with a number of previous higher derivative computations of entanglement entropy. In particular we will show that various results for perturbations of entanglement entropy of spherical regions dual to higher derivative theories can be equally well reproduced from a calculation in cosmological Einstein gravity by renormalizing the discrete set of free parameters which play a role at second order in perturbation theory, i.e., $\{{\ell_\text{AdS}}/{\ell_\text{P}}\,,\, m^2{\ell_\text{AdS}}^2 \,, {\lambda_\phi}\}$. We will also see the appearance of the additional parameter ${\lambda_h}$ which drops out of the formula for entropy. Two gravity models {#sec:models} ------------------ The basic gravitational model that we want to take as the reference system is just the auxiliary Einstein-Hilbert theory described by $$\label{eq:EHmod} \begin{split} I = \frac{1}{2{\ell_\text{P}}^{d-1}}\int \sqrt{-g} \, \left[R + \frac{d(d-1)}{\ell^2} - \frac{{\lambda_\phi}}{2}\left( (\nabla \phi)^2 + m^2 \, \phi^2 \right)\right] \,, \end{split}$$ with asymptotic boundary condition ${\phi}(z\rightarrow 0) \simeq \lambda\,z^{d-\Delta}$ such that ${\phi}$ sources a scalar operator with dimension $\Delta$ and coupling $\lambda$ in the CFT. We parametrize the lowest order backreaction of the scalar probe on the geometry as $g_{ab} = {\bar g}_{ab} + h_{ab}$, where ${\bar g}_{ab}$ is AdS${}_{d+1}$ with scale ${\ell_\text{AdS}}^2 = \ell^2$. The independent dimensionless parameters in the gravitational problem can therefore be taken as $\{({\ell_\text{AdS}}/{\ell_\text{P}})\,,\,m^2{\ell_\text{AdS}}^2\,,\,{\lambda_\phi}\}$, while ${\lambda_h}= 1$ for Einstein gravity. In this section we wish to reproduce predictions of four-derivative theories by using the above Einstein theory with renormalized parameters. The only four-derivative theory which passes the consistency condition of yielding second order equations of motion is Gauss-Bonnet theory, which we will discuss here. Nevertheless, in §\[app:example\] we explore the most general four-derivative theory of gravity and come to very similar conclusions in the case where the higher derivative terms are small perturbative corrections to . With this motivation in mind, let us take the following action as the theory to compare with: $$\label{eq:HigherDerPert} \begin{split} \tilde{I} &= \frac{1}{2{{{\ell}}_\text{P}}^{d-1}}\int \sqrt{-g} \, \left[R + \frac{d(d-1)}{{\tilde{\ell}}^2} + {\tilde{\ell}}^2\,\mathcal{L}_{h.d.} - \frac{{\tilde\lambda_\phi}}{2} \left( (\nabla \phi)^2 + \tilde{m}^2 \, \phi^2 \right)\right] \\ &\quad \text{where } \mathcal{L}_{h.d.} = \gamma \left(R^2 - 4 R_{ab} R^{ab} + R_{abcd} R^{abcd}\right) \end{split}$$ again with a fixed boundary condition ${\phi}(z\rightarrow 0) \simeq \lambda\,z^{d-\Delta}$. The AdS${}_{d+1}$ background solution of this theory will have some other scale ${\tilde{\ell}_\text{AdS}}$ which is a function of ${\tilde{\ell}}$ and $\gamma$. In the context of both theories $I$ and $\tilde{I}$, we will use ${\bar g}_{ab}$ to refer to the AdS${}_{d+1}$ background. From the context it should be clear whether ${\bar g}_{ab}$ is the background with ${\ell_\text{AdS}}$ or with ${\tilde{\ell}_\text{AdS}}$. Our basic aim is to show that for the purpose of carrying out the perturbed entanglement entropy computation of spherical regions, the actions and are indistinguishable if the dimensionless parameters $\{({\ell_\text{AdS}}/{\ell_\text{P}})\,,\,m^2 {\ell_\text{AdS}}^2\,,\,{\lambda_\phi}\}$ are suitably expressed as functions of $\{({\tilde{\ell}_\text{AdS}}/{{{\ell}}_\text{P}})\,,\,\tilde{m}^2 {\tilde{\ell}_\text{AdS}}^2\,,\,{\tilde\lambda_\phi}\,,\,\gamma\}$ such that they are consistent with the CFT boundary conditions. In order to show this, we will now derive the relevant ingredients for such a computation order by order in $\lambda$ from both $I$ and $\tilde I$. We will demonstrate that the parameters of Einstein gravity can always be chosen such that the two perturbative expansions match. The higher derivative coupling $\gamma$ can be absorbed entirely into renormalizations of the other dimensionless quantities. Parameter matching {#sec:matching} ------------------ We will now determine the relation between the dimensionless parameters of the models $I$ and $\tilde I$ such that both make identical predictions for vacuum and perturbed entanglement entropy of spheres. #### Finding the AdS${}_{d+1}$ background solution: Before turning on the scalar perturbation, we need to determine the maximally symmetric AdS${}_{d+1}$ background solutions of the two theories and . It is well known that such solutions exist in both theories and are ghost-free [@Boulware:1985aa] if we choose the respective AdS scales $$\label{eq:BGsols} \begin{split} {\ell_\text{AdS}}^2 &= \ell^2 \,,\\ {\tilde{\ell}_\text{AdS}}^2 &= \frac{{\tilde{\ell}}^2}{2} \left( 1+ \sqrt{1- 4(d-2)(d-3) \, \gamma }\,\right) \,. \end{split}$$ We will henceforth expand around the AdS${}_{d+1}$ background and thus treat ${\ell_\text{AdS}}$ and $\ell$ as equivalent (and similarly with tildes). #### Matching of Wald entropy (or $a_d^$): Let us remind the reader of the result of §\[sec:introduction\] that Wald entropy of maximally symmetric spaces is always proportional to area. We can thus compute the unperturbed entanglement entropy of $\mathcal{A}$ as the Wald entropy of hyperbolic black holes corresponding to the two AdS radii of Einstein and Gauss-Bonnet gravity, respectively. By integrating for the two gravity models, one finds $$\begin{split} S_{Wald} &= \frac{2\pi}{\pi^{d/2}} \Gamma\left(d/2\right) \, a_d^* \; \frac{\text{Vol}(\mathbb{H}_{d-1})}{{\ell_\text{AdS}}^{d-1}}\,, \qquad a_d^* = \frac{\pi^{d/2}}{\Gamma(d/2)} \frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}} \,,\\ \tilde{S}_{Wald} &= \frac{2\pi}{\pi^{d/2}} \Gamma\left(d/2\right) \, \tilde{a}_d^* \; \frac{\text{Vol}(\mathbb{H}_{d-1})}{{\tilde{\ell}_\text{AdS}}^{d-1}} \,, \qquad \tilde{a}_d^* = \frac{\pi^{d/2}}{\Gamma(d/2)} \frac{{\tilde{\ell}_\text{AdS}}^{d-1}}{{{{\ell}}_\text{P}}^{d-1}} \left( 1 - 2(d-1)(d-2) \,\gamma\, \frac{{\tilde{\ell}}^2}{{\tilde{\ell}_\text{AdS}}^2}\right) \,. \end{split}$$ Given the entanglement entropy ${{S}^{\text{\tiny (0)}}}_{EE}(\mathcal{A})$ with fixed normalization in the CFT, it can be reproduced by the computation of Wald entropy in either of the gravity models by requiring $a_d^* = \tilde{a}_d^$, i.e., $$\label{eq:lPrenorm} \frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}} =\frac{{\tilde{\ell}_\text{AdS}}^{d-1}}{{{{\ell}}_\text{P}}^{d-1}} \left( 1 - 2(d-1)(d-2) \,\gamma\, \frac{{\tilde{\ell}}^2}{{\tilde{\ell}_\text{AdS}}^2}\right) \,.$$ This determines the dimensionless ratio ${\ell_\text{AdS}}/{\ell_\text{P}}$ in Einstein gravity in terms of $\{{\tilde{\ell}_\text{AdS}}/{{{\ell}}_\text{P}}\,,\,\gamma\}$ in Gauss-Bonnet theory such that both theories make the same predictions for Wald entropy of the maximally symmetric horizon. From the CFT point of view, this matching is the condition that $a_d^$ is a fixed fundamental parameter of the CFT which should be reproduced by either candidate of a gravity dual. #### Matching of scalar coupling (or ${C_{\mathcal{O}}}$): Perturbing the CFT with $\mathcal{O}$ means introducing a massive scalar in the gravitational action. The mass of the scalar is constrained by the requirement that the boundary value of the scalar sources the operator $\mathcal{O}$ with given dimension $\Delta$. This fixes $$\label{eq:MassRel} {\ell_\text{AdS}}^2\,m^2 = \Delta (\Delta-d) = {\tilde{\ell}_\text{AdS}}^2\,\tilde{m}^2 \,.$$ The dynamics of the scalar field is derived at lowest order from the propagation on AdS${}_{d+1}$ vacuum: $$\label{eq:ScalarSol} \left({\bar{\Box}}- m^2\right)\phi = 0 \,,\qquad \phi(z \rightarrow 0) \sim \lambda \, z^{d-\Delta} \,,$$ and similarly with tildes. Once the boundary condition is imposed in this way, standard holographic techniques lead to the matching between the normalization ${C_{\mathcal{O}}}$ of the relevant CFT correlators and the normalization of the scalar field action via : the two gravity theories $I$ and $\tilde I$ lead to the same ${C_{\mathcal{O}}}$ provided that $${\lambda_\phi}\, \frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}}= {\tilde\lambda_\phi}\,\frac{{\tilde{\ell}_\text{AdS}}^{d-1}}{{{{\ell}}_\text{P}}^{d-1}} \,,$$ so the ratio ${\lambda_\phi}/{\tilde\lambda_\phi}$ should be fixed to be the same as in . #### Computation of perturbed entanglement entropy: We finally want to check explicitly the arguments of §\[sec:lhindependence\] and verify that after the matching conditions above have been implemented, second order entanglement entropy is automatically the same for $I$ and $\tilde I$. We will also verify that the difference between ${\lambda_h}=1$ and ${\tilde{\lambda}_h}$ cannot be absorbed in any further free parameters. This will lead to the conclusion that the central charge ${C_{_T}}$ is genuinely different in the CFTs dual to $I$ compared to those dual to $\tilde I$ after the above matchings have been performed. Our perturbative scheme in the CFT demands that we consider the linearized equations of motion in the bulk. The linearized equations of motion can be obtained from the actions , by linearizing around ${\bar g}_{\mu\nu} = g_{\mu\nu} - h_{\mu\nu}$. We find for $I$ and $\tilde I$, respectively [@Deser:2002jk]: $$\label{eq:GravEOM} \begin{split} {\lambda_h}\,{\mathcal{G}}_{ab}{}^{cd} h_{cd} &= -\frac{{\lambda_\phi}}{2}\, T_{ab}^{(\phi)} \,, \\ {\tilde{\lambda}_h}\, \tilde{{\mathcal{G}}}_{ab}{}^{cd} h_{cd} &= -\frac{{\tilde\lambda_\phi}}{2} \, \, \tilde{T}_{ab}^{(\phi)} \,,\qquad \end{split}$$ where ${T}_{ab}^{(\phi)}$ and $\tilde{T}_{ab}^{(\phi)}$ are the source terms for scalar matter with mass $m^2$ and $\tilde{m}^2$, respectively propagating on AdS${}_{d+1}$ with scale ${\ell_\text{AdS}}$ and ${\tilde{\ell}_\text{AdS}}$. The kinetic operators ${\mathcal{G}}_{abcd}$ and $\tilde{\mathcal{G}}_{abcd}$ are defined as in , but, of course, also refer to backgrounds with respective scales ${\ell_\text{AdS}}$ and ${\tilde{\ell}_\text{AdS}}$. The coefficients ${\lambda_h}$ and ${\tilde{\lambda}_h}$ in can be computed for $I$ and $\tilde I$ theories and turn out to be $$\label{eq:ratioR} {\lambda_h}= 1 \,,\qquad {\tilde{\lambda}_h}= \left( 1 - 2(d-2)(d-3) \, \gamma \, \frac{{\tilde{\ell}}^2}{{\tilde{\ell}_\text{AdS}}^2}\right)\,.$$ Clearly there is no free parameter left in which could be renormalized to absorb these normalizations. From it is hence clear that the ${C_{_T}}$ charges of the dual CFTs will differ by the ratio ${\lambda_h}/{\tilde{\lambda}_h}$. At this stage, any CFT observable which did depend on ${C_{_T}}$ would therefore genuinely distinguish between the two gravity models. As shown before, $\delta S_{EE}(\mathcal{A})$ does not depend on ${C_{_T}}$, though. Our matching prescriptions , and the graviton kinetic term scaling have, of course, appeared in the literature before. For example, see [@deBoer:2011wk] for a computation of perturbed sphere entanglement entropy in Gauss-Bonnet theory. One can easily verify that their computation gives an answer which takes the form of perturbed entanglement entropy in Einstein gravity with dimensionless parameters rescaled as predicted by our general analysis.[^15] At this order the entanglement entropy computation for spherical regions is therefore not sensitive to the Gauss-Bonnet coupling. Discussion {#sec:discussion} ========== Let us summarize our arguments and comment on implications for the program of bulk reconstruction. #### Summary: We considered the second order perturbation of CFT vacuum entanglement entropy of spheres due to scalar operator deformations. This quantity exhibits a universal functional form which only depends on a small number of fundamental CFT parameters to set various normalizations. We explored a similar universality statement about gravity, which explains why the corresponding holographic computation can always be recast in terms of one universal gravitational system, viz., linearized Einstein gravity coupled to a free massive scalar field. We argued that any ghost-free gravitational theory other than Einstein gravity can be used to make identical predictions due to the uniqueness of diffeomorphism invariant spin-2 propagation on AdS. For instance, we demonstrated how in Gauss-Bonnet theory both the vacuum and the perturbed sphere entanglement entropy (obtained by computing Wald entropy of a minimal surface) can be reproduced from just Einstein gravity by suitably renormalizing the dimensionless parameters of the theory. In this sense, knowledge of this entanglement data does not allow to distinguish between Gauss-Bonnet theory and Einstein gravity. More precisely, the perturbed sphere entanglement only depends on the dimension $\Delta$ of the deforming operator $\mathcal{O}$, the generalized $a$-central charge $a_d^$ (in vacuum) and the normalization ${C_{\mathcal{O}}}$ of the $\langle\mathcal{O}\mathcal{O}\rangle$ correlator (at second order in the perturbation). We saw that any viable gravity model has enough dimensionless parameters to account for these dependencies (the scalar mass ${\ell_\text{AdS}}^2m^2$, the normalization of the gravitational action, ${\ell_\text{AdS}}/{\ell_\text{P}}$, and the normalization of the scalar field, ${\lambda_\phi}$, respectively). In particular, no higher curvature terms in the bulk are required in order to obtain a sufficient number of couplings for consistently performing the holographic matching. In this problem, the only signature of higher curvature couplings would be a renormalization of the parameters of cosmological Einstein gravity. This implies that perturbed entanglement entropy of spheres can be computed using any ghost-free gravitational model as an auxiliary system, even for CFTs without any simple gravity dual at all [@Faulkner:2014jva]. In face of this, it is clear that for the derivation of bulk dynamics from entanglement in the sense of AdS/CFT it is desirable to reconstruct bulk features which are not entirely universal in the above sense. The obvious question arises which quantities one should consider instead of or in addition to entanglement entropy of spheres in order to achieve a dependence on bulk features which are not the same for all CFTs. We will come back to this question below. #### Generalizations: There are some modifications of our analysis which would be interesting to explore. For instance, one could compute the perturbation to higher order in $\lambda$. By extending the perturbation theory of §\[sec:CFTExplanation\], it is clear that entanglement entropy will then be sensitive to higher point correlation functions which would probe bulk dynamics beyond the linear approximation. While this would be very interesting to see explicitly, the major complication in the bulk would be a backreaction not just on the Wald entropy density, but on the minimal surface itself. Once the minimal surface has non-vanishing extrinsic curvatures, ambiguities in the entanglement entropy functional need to be addressed [@Dong:2013qoa; @Camps:2013zua; @Bhattacharyya:2013gra; @Bhattacharyya:2014yga]. By resolving this issue, one may hope to derive bulk dynamics beyond linear order. We mainly focused on perturbations of entanglement due to deformations of the CFT by a relevant scalar operator. As another generalization, one can imagine deformations by operators of other spin. As long as the deformation does not involve a stress tensor, our conclusions are expected to hold since the new operator comes at least with an overall normalization which is all that is required to account for the renormalization of ${\ell_\text{AdS}}/{\ell_\text{P}}$. Consider, however, the case where the CFT is deformed by a stress tensor insertion. In that case, the first interesting perturbation of the entanglement entropy occurs at ${\mathscr{O}}(\lambda)$ instead of ${\mathscr{O}}(\lambda^2)$, where one gets a contribution $\sim \langle T T \rangle$ with normalization set by ${C_{_T}}$. In the bulk the problem of gravitational backreaction changes in a similar way and the extremal surface will feel a perturbation at ${\mathscr{O}}(\lambda)$. It would be interesting to explore this case in detail and consider the dependence on ${C_{_T}}$ to draw similar conclusions about the effect of higher curvature couplings. The dependence on ${C_{_T}}$ is expected to lead to a way of distinguishing Einstein gravity duals from higher curvature ones. In fact it has been argued before (as we review in §\[sec:1stLaw\]) that generic perturbations of spherical region entanglement entropy can be used to derive linearized bulk equations including their normalization set by ${C_{_T}}$ [@Faulkner:2013ica]. Further generalizations could involve perturbations of the CFT other than operator deformations. For instance, a small geometric perturbation of the spatial sphere should lead to similar conclusions as those just mentioned. Similarly, $1/N$ corrections have recently been explored in order to improve on the entanglement first law [@Kelly:2015mna]. It would be interesting to pursue this further and explore ways to reconstruct bulk dynamics at a level where one can distinguish between different higher curvature theories. Instead of considering individual approaches in detail, let us now outline in general how a good choice of CFT observables can be made which are sensitive to interesting bulk features in a controlled way. #### How to “bootstrap” bulk theories: As we have just seen, there are, of course, simple ways to circumvent the problem of perturbed sphere entanglement being insensitive to the desired detailed bulk information. Roughly speaking, one needs to consider more detailed CFT structures in order to distinguish between different bulk models. In general one could imagine bootstrapping the bulk theory using the CFT input. Clearly it is hard to derive bulk dynamics directly from fundamental CFT parameters such as two-point functions and OPE coefficients. But entanglement entropy seems to provide a physically meaningful quantity with well-understood universal properties on both sides of the duality, which depends on these fundamental building blocks of the CFT in a useful way. At lowest order, the vacuum entanglement entropy of spheres can be used to measure the generalized $a$-charge $a_d^$. In our particular context we used this to fix the normalization ${\ell_\text{AdS}}/{\ell_\text{P}}$ of any possible bulk dual (whose dynamics is uniquely fixed in the same way as the functional form of entanglement entropy is universal). As we have seen, a small perturbation of this entanglement entropy does not affect the dynamics enough to determine if there are any higher curvature terms present in the bulk theory. To answer this question, one would have to measure some other entanglement entropy which is sensitive to further details of the bulk dynamics. One natural CFT parameter that we have only encountered briefly so far is the stress tensor central charge. Let us elaborate how any quantity which is sensitive to ${C_{_T}}$ will also be sensitive to bulk higher derivative couplings and hence will not be reproducible by just Einstein gravity once ${\ell_\text{AdS}}/{\ell_\text{P}}$ has been fixed by matching vacuum entanglement entropy of spheres as outlined in this note. In our context, the ratio $a_d^/{C_{_T}}$ computed holographically is independent of ${\ell_\text{AdS}}/{\ell_\text{P}}$ and genuinely takes different values in the presence of higher curvature terms. In fact it is clear from and that higher curvature terms in the bulk are necessary whenever $a_d^ / {C_{_T}}= 2 \pi^{d/2} / \Gamma(d/2)f_d$ does [not]{} hold. This is, of course, just a generalization of the well-known statement that higher curvature terms in the bulk theory are required whenever the $a$- and $c$-central charges parametrizing the boundary conformal anomaly differ [@Henningson:1998gx; @Blau:1999vz; @Nojiri:1999mh]. Any set of CFT observables which depends on both $a_d^$ and ${C_{_T}}$ can hence be used to detect if higher curvature terms in the bulk are necessary. The problem of genuine bulk reconstruction is therefore not at all the restriction to quantities which are universal in the sense that they only depend on an overall normalization. But it is now clear that one has to consider [enough]{} such quantities in order to see a necessity to account for their values in terms of higher curvature interactions. There are many examples of entanglement-related CFT quantities which one may envisage to compute in this context. For instance, the entanglement entropies of other surfaces such as infinite strips or cylinders are sensitive to both $a_d^$ and ${C_{_T}}$ [@Solodukhin:2008dh; @Hung:2011xb; @deBoer:2011wk]. Further, the presence of corners and kinks in the spatial region of interest leads to divergences which exhibit a universal dependence on the opening angle and (at least in holographic CFTs) depend on $a_d^$ and ${C_{_T}}$ in an interesting way [@Hirata:2006jx; @Fradkin:2006; @Casini:2006hu; @Myers:2012vs; @Bueno:2015rda; @Bueno:2015xda; @Alishahiha:2015goa; @Miao:2015dua]. Other interesting quantities with universal features and dependence on ${C_{_T}}$ are Rényi entropies of spherical regions (e.g., in the entanglement entropy limit $q\rightarrow 1$) [@Hung:2011nu; @Galante:2013wta; @Perlmutter:2013gua]. All these quantities have in common a contribution with a universal functional form and a dependence on few CFT parameters (in particular $a_d^$ and ${C_{_T}}$). They could hence be used to distinguish between CFTs with a dual description in terms of Einstein gravity and CFTs which require higher curvature terms in the bulk. However, note that even if some CFT measurement of $a_d^$ and ${C_{_T}}$ leads to the conclusion that there can exist a semiclassical bulk theory only if contains higher curvature interactions are present, it is usually not possible to determine precisely what higher curvature terms are required. Indeed, the signature of, say, a Gauss-Bonnet term on $a_d^/{C_{_T}}$ can equally well be reproduced by some other higher derivative term. For illustration of this statement see also §\[app:Wald\], where it is shown how any higher curvature theory simplifies to the background field expansion of Gauss-Bonnet form if one is only concerned with linearized equations of motion. In this sense Gauss-Bonnet theory is the most general theory which can account for arbitrary values of $a_d^$ and ${C_{_T}}$ in the CFT. In order to further bootstrap the precise bulk dynamics, one needs again access to more CFT data (e.g., entanglement data which depends on further 2-point functions or OPE coefficients). The reason is, again, that gravitons propagating on AdS have very universal dynamics. This is a manifestation of the familiar problem that entanglement entropies which are computable tend to be very universal and contain only discrete information about the actual bulk dynamics. It would be interesting to use this philosophy to derive more detailed (non-linear) bulk dynamics for some simple CFT with known holographic dual from first principles. Generalizing this approach may then ultimately lead to a genuine reconstruction of the bulk geometry and its dynamics by using a suitable set of entanglement entropies. I am very grateful to Jyotirmoy Bhattacharya and Mukund Rangamani for collaboration during early stages of this project, numerous insightful discussions and comments on a draft. I also wish to thank Rob Myers for useful comments. I am supported by a Durham Doctoral Fellowship. Generalization to bulk theories with higher order equations of motion {#app:generalization} ===================================================================== In the main text, we considered the case where the equations of motion for the metric perturbation in the bulk were strictly second order. The motivation for doing so was to exclude ghost modes which would occur if the graviton equations were higher than second order. However, if the higher curvature contributions to gravity are parametrically small, they can be interpreted, e.g., as toy models for perturbative string theory corrections to Einstein gravity. The ghost modes are then an artefact of the low energy truncation and can be ignored, which allows for a considerable increases of the number of acceptable gravitational models covered by our analysis. In order to illustrate this, we first generalize in §\[app:example\] the Gauss-Bonnet theory of §\[sec:exampleGB\] to the case of all Lagrangians which are quadratic in curvature. In §\[app:generalArgument\] we then give a more abstract argument to show that this example actually illustrates the general case. Example: general curvature squared theories {#app:example} ------------------------------------------- In this section we generalize the analysis of §\[sec:exampleGB\] to theories where the gravitational sector is described by $$\label{eq:HigherDerPert2} \begin{split} \tilde{I} &= \frac{1}{2{{{\ell}}_\text{P}}^{d-1}}\int \sqrt{-g} \, \left[R + \frac{d(d-1)}{{\tilde{\ell}}^2} + {\tilde{\ell}}^2\,\mathcal{L}_{h.d.} - \frac{{\tilde\lambda_\phi}}{2} \left( (\nabla \phi)^2 + \tilde{m}^2 \, \phi^2 \right)\right] \\ &\quad \text{where } \mathcal{L}_{h.d.} = \alpha \, R^2 + \beta\, R_{ab} R^{ab} +\gamma \left(R^2 - 4 R_{ab} R^{ab} + R_{abcd} R^{abcd}\right) \end{split}$$ We do this to explicitly demonstrate that such theories – despite the presence of massive ghost modes – are acceptable in a perturbative sense. Indeed, we will see how the ghost degrees of freedom are suppressed and hence can be neglected for the computations we want to perform. We want to model the theory with the cosmological Einstein-scalar theory . That is, we will give the matching conditions which express the parameters of the latter theory, $\{({\ell_\text{AdS}}/{\ell_\text{P}})\,,m^2{\ell_\text{AdS}}^2\,,{\lambda_\phi}\}$, in terms of $\{({\tilde{\ell}_\text{AdS}}/{{{\ell}}_\text{P}})\,,\,\tilde{m}^2 {\tilde{\ell}_\text{AdS}}^2\,,\,{\tilde{\lambda}_h}\,,\,{\tilde\lambda_\phi},\alpha,\beta,\gamma\}$ such that the theories make the same predictions for entanglement entropy of spheres in vacuum even after deforming the theory with a scalar operator. The main new ingredient is the presence of four-derivative terms in the equations of motion. Before discussing the linearized equations of motion, let us give the analogue of the matching conditions , . The stable AdS${}_{d+1}$ background solutions in the two theories have the following scales in terms of the cosmological constant: $$\label{eq:BGsols2} \begin{split} {\ell_\text{AdS}}^2 &= \ell^2 \,,\\ {\tilde{\ell}_\text{AdS}}^2 &= \frac{{\tilde{\ell}}^2}{2} \left( 1+ \sqrt{1- \frac{4(d-3)}{(d-1)} \, \big(d(d+1)\,\alpha+d\,\beta + (d-1)(d-2)\, \gamma\big) }\,\right) \,. \end{split}$$ The generalization of the $a_d^$ matching condition then becomes $$\label{eq:lPrenorm2} \frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}} =\frac{{\tilde{\ell}_\text{AdS}}^{d-1}}{{{{\ell}}_\text{P}}^{d-1}} \left( 1 - 2\big(d (d+1)\alpha+d\,\beta + (d-1)(d-2) \,\gamma\big) \frac{{\tilde{\ell}}^2}{{\tilde{\ell}_\text{AdS}}^2}\right) \,.$$ Exactly as in §\[sec:matching\], the previous equation also immediately fixes the ratio of scalar couplings, i.e., ${\tilde\lambda_\phi}/{\lambda_\phi}$ is given by the value bracket on the right hand side of . This guarantees that holographic computations of $\langle \mathcal{O}\mathcal{O}\rangle$ yield a unique normalization ${C_{\mathcal{O}}}$. Consider now the linearized graviton equations in the two theories. For this, we find it convenient to gauge fix the metric. In the transverse traceless gauge (${\bar{\nabla}}_a h^{ab} = 0 = h$) the universal linearized equations of motion now generalize to the following two equations for Einstein and curvature squared theories, respectively [@Deser:2002jk]: $$\label{eq:GravEOM2} \begin{split} {\lambda_h}\left( {\bar{\Box}}+ \frac{2}{{\ell_\text{AdS}}^2}\right) h_{ab} &= -{\lambda_\phi}\, T_{ab}^{(\phi)} \,, \\ {\tilde{\lambda}_h}\,\left( {\bar{\Box}}+ \frac{2}{{\tilde{\ell}_\text{AdS}}^2}\right) h_{ab} +\beta \, \ell^2 \, \left( {\bar{\Box}}+ \frac{2}{{\tilde{\ell}_\text{AdS}}^2}\right)^2 h_{ab} &= -{\tilde\lambda_\phi}\, \, \tilde{T}_{ab}^{(\phi)} \,,\qquad \end{split}$$ where the coefficients ${\lambda_h}$ and ${\tilde{\lambda}_h}$ in are now given as $$\label{eq:ratioR2} {\lambda_h}= 1 \,,\qquad {\tilde{\lambda}_h}= \left( 1 - 2\big(d(d+1)\alpha+d\,\beta + (d-2)(d-3) \, \gamma \big) \frac{{\tilde{\ell}}^2}{{\tilde{\ell}_\text{AdS}}^2}\right)\,.$$ We now want to argue that the second term in the second line of can be neglected for our computation of entanglement entropy. Indeed, this term is obviously suppressed if $\beta$ is treated as a small parameter and in the physical on-shell limit $({\bar{\Box}}+ 2/{\tilde{\ell}_\text{AdS}}^2) h_{ab} \rightarrow 0$ (see also [@Bueno:2015xda]). Further, we can also rewrite the second line of in the following factorized form: $$\label{eq:mostGeneralEOM} \begin{split} \beta \, \ell^2 \left({\bar{\Box}}+ \frac{2}{{\tilde{\ell}_\text{AdS}}^2}\right) \left( {\bar{\Box}}+ \frac{2}{{\tilde{\ell}_\text{AdS}}^2} + \frac{{\tilde{\lambda}_h}}{\beta \, \ell^2} \right) h_{ab} = -{\tilde\lambda_\phi}\, T^{(\phi)}_{ab} \,. \end{split}$$ From this expression one can anticipate the two poles of the graviton propagator. The Fierz-Pauli type mass term $\sim 2/{\tilde{\ell}_\text{AdS}}^2$ corresponds to the physical graviton mode. The other mass term $\sim (2/{\tilde{\ell}_\text{AdS}}^2 + {\tilde{\lambda}_h}/\beta \ell^2)$ schematically leads to the ghost which is very heavy for small $\beta$ and hence does not go on-shell at low energies. It can be seen as an artefact of truncating some well-behaved theory at four-derivative order. General argument {#app:generalArgument} ---------------- Despite the previous subsection appearing to be restricted to the specific case of curvature squared theories, the conclusion is actually more general. If higher curvature terms, which appear in the gravitational action with some dimensionless coupling $\alpha$, lead to ghosts modes, then the latter generically have a mass which scales with $1/\alpha$. If these modes are interpreted as due to consistently truncating perturbative quantum gravity (i.e., $\|\alpha\|\ll 1$), the ghosts are unphysical and indeed never go on-shell at low energies. In such a perturbative framework the presence of ghosts is hence not due to an instability of the theory (the underlying full theory is assumed to be unitary), but it is just a rather harmless consequence of neglecting certain sectors of the theory which are not being probed at low energies. To illustrate these statements, consider, for example, a gravity sector of the form $\mathcal{L}_{grav} = \mathcal{L}_{grav}(g_{ab},R_{abcd})$ such that the equations of motion are fourth order in derivatives. The most general four-derivative gravitational equations of motion linearized around AdS are precisely the same as those captured by the curvature squared theories in . This can be shown using the method of background field expansion; see, e.g., [@Amsel:2012se], where it was indeed argued that the linearized action of any higher derivative theory of this type is the same as that of the most general curvature squared theory.[^16] Concretely, given some such higher curvature theory, the action expanded to second order around AdS takes the form with parameters $\{\alpha,\beta,\gamma\}$ determined as linear combinations of the original higher derivative couplings. We can therefore ignore the ghosts of these theories as long as the higher derivative couplings are viewed as a semiclassical truncation stemming from some unitary UV-complete theory. Perturbations of Wald entropy: background field expansion {#app:Wald} ========================================================= This appendix complements the analysis of §\[sec:lhindependence\] by computing the expansion of the Wald entropy density at ${\mathscr{O}}(h_{ab})$, derived from the general second order Lagrangian . This analysis will also serve to illuminate how the universality of graviton propagation on AdS simplifies the method of background field expansion, thus demonstrating a more general universality statement about quantities which depend on both $a_d^$ and ${C_{_T}}$. Indeed, we will show in what sense Einstein-Gauss-Bonnet theory is the most general theory at ${\mathscr{O}}(h^2)$. Any other physically acceptable higher curvature theory has the same background field expansion at this order. It is straightforward that we can focus on the pure gravity part of the general Lagrangian since the minimally coupled matter sector does not couple to curvature. We wish to give a similar argument for the constant term. Ignoring the matter sector puts us in the realm of [@Sen:2014nfa] (see also [@Faulkner:2013ica]), where a detailed analysis of perturbed Wald entropy on a maximally symmetric AdS background has been performed. The best strategy to compute the perturbative expansion of Wald entropy is to rewrite the expansion in $h_{ab}$ as a background field expansion of curvatures $R_{abcd}$ around $-\frac{1}{{\ell_\text{AdS}}^2}(g_{ac}g_{bd}-g_{ad}g_{bc})$. More explicitly, instead of writing explicit metric perturbations $h_{ab}$ we write the Lagrangian solely in terms of $$\begin{split} \Delta R_{abcd} &= R_{abcd} - \left[- \frac{1}{{\ell_\text{AdS}}^2} (g_{ac}g_{bd} - g_{ad} g_{bc})\right] \\ & = {{}^{\text{\tiny (1)}}\!R}_{abcd} + \frac{1}{{\ell_\text{AdS}}^2} \left( {\bar g}_{ac} h_{bd} + {\bar g}_{bd} h_{ac} - {\bar g}_{ad}h_{bc} - {\bar g}_{bc} h_{ad}\right) + \mathcal{O}(h^2)\,, \end{split}$$ where $g_{ab}={\bar g}_{ab} + h_{ab}$ is the full metric and indices of expressions like this are raised with the full metric. This way, the pure gravity part of our general quadratic Lagrangian can be interpreted as an expansion in $\Delta R_{abcd}$ and its contractions and derivatives. To wit, we can always write the pure gravitational couplings of as a linear combination of all possible combinations of $\Delta R_{abcd}$ which contain terms relevant at ${\mathscr{O}}(h^2)$. Clearly such combinations contain at most two factors of $\Delta R_{abcd}$, but they may contain any number of derivatives in general: $$\label{eq:Rexpansion} \begin{split} I_{grav} &= \frac{1}{2{\ell_\text{P}}^{d-1}} \int \sqrt{-{\bar g}} \; \left(-\frac{2d}{{\ell_\text{AdS}}^2}\frac{\Gamma(d/2)}{\pi^{d/2}}\frac{{\ell_\text{P}}^{d-1}}{{\ell_\text{AdS}}^{d-1}} \, a_d^ + \frac{{\lambda_h}}{2}\, h_{ab}\, {\mathcal{G}}^{abcd}\,h_{cd} + {\mathscr{O}}(h^3)\right) \\ &= \frac{1}{2{\ell_\text{P}}^{d-1}} \int \sqrt{-g} \; \left( \mathcal{L}^{(I)}_{\Delta R} + \mathcal{L}^{(II)}_{\Delta R} \right) \,,\\ \text{with } & \; \begin{cases} \mathcal{L}^{(I)}_{\Delta R} \equiv a_0 + a_2\, \Delta R \\ \mathcal{L}^{(II)}_{\Delta R} \equiv b_1 \,{\ell_\text{AdS}}^2\, (\Delta R)^2 + b_2 \, {\ell_\text{AdS}}^2\,\Delta R^{ab} \Delta R_{ab} + b_3 \,{\ell_\text{AdS}}^2\, \Delta R^{abcd} \Delta R_{abcd} + \ldots + {\mathscr{O}}(h^3) \end{cases} \end{split}$$ Let us explain this way of parameterizing the action piece by piece. First observe that from the form of it follows that $a_0$ is just $$\label{eq:a0match} a_0 =-\frac{2d}{{\ell_\text{AdS}}^2}\frac{\Gamma(d/2)}{\pi^{d/2}}\frac{{\ell_\text{P}}^{d-1}}{{\ell_\text{AdS}}^{d-1}} \, a_d^* \,.$$ Similarly, one can easily verify that the vanishing of the linearized Lagrangian (i.e. the equation of motion for the background solution ${\bar g}_{ab}$) demands $a_0 = -(2d/{\ell_\text{AdS}}^2)a_2$. This fixes the form of $\mathcal{L}^{(I)}_{\Delta R}$ and shows that the latter is all that would ever show up in the case of Einstein gravity (with an overall pre-factor $a_2$). Therefore, $\mathcal{L}^{(II)}_{\Delta R}$ is the term which parameterizes contributions to the graviton propagator which only come from higher curvature terms. More tensor structures involving derivatives and contractions of $\Delta R_{abcd}\Delta R_{efgh}$ could be written in in order to match the expansion of an arbitrary theory of gravity at higher orders in $h_{ab}$. However, this is not needed to reproduce the first line, i.e., the universal parts up to ${\mathscr{O}}(h^2)$. To wit, the two lines of match if we choose $$b_1 = -\frac{b_2}{4} = b_3 = \frac{1}{4(d-2)} \,\left({\lambda_h}-\frac{\Gamma(d/2)}{\pi^{d/2}}\frac{{\ell_\text{P}}^{d-1}}{{\ell_\text{AdS}}^{d-1}} \, a_d^\right) \,.$$ Note that this choice is schematically of the Gauss-Bonnet form. We can interpret this as the statement that a Gauss-Bonnet term is all that is needed to reproduce the linearized graviton action with arbitrary graviton normalization. This is a generalization of the results of the main text: in CFT language, a quantity which is sensitive to both $a_d^$ and ${C_{_T}}$ may undeniably detect the presence of higher curvature terms, but it does not distinguish between Gauss-Bonnet theory and any other higher curvature theory in the same way that quantities only knowing about $a_d^$ can always be computed using Einstein gravity. The reason is that linearized actions up to ${\mathscr{O}}(h^2)$ only require a small number of terms in the background field expansion. As a consistency check, we can make the relation between parameters $\{a_0,a_2,b_1,b_2,b_3\}$ and ${C_{_T}}$ manifest by referring to the results of [@Sen:2014nfa] (see also [@Faulkner:2013ica]). There it was shown that an action of the form leads to ${\lambda_h}= a_2 + 4(d-2) b_3$ which is proportional to ${C_{_T}}$ via . This is in perfect agreement with the above statements. Since is manifestly written as both a perturbative expansion in the metric fluctuation and as a functional of curvatures, we can easily compute the associated perturbative expansion of Wald entropy density by taking derivatives with respect to $R_{abcd}$. Note that the way the expansion is written makes it explicit that any variation with respect to $R_{abcd}$ reduces the order in $h_{ab}$ by precisely one and the number of derivatives by two. The unperturbed Wald entropy is hence only sensitive to $\mathcal{L}^{(I)}_{\Delta R}$ and is readily checked to agree with : $$\label{eq:Wald00} \begin{split} {{S}^{\text{\tiny (0)}}}_{Wald}(\mathcal{E}_{min}) = -2\pi \, \frac{1}{2{\ell_\text{P}}^{d-1}} \int_{\mathcal{E}_{min}} \sqrt{\bar\gamma} \, \left[\frac{\delta \mathcal{L}^{(I)}_{\Delta R}}{\delta R_{abcd}}\, n_{ab} n_{cd}\right]_{\text{AdS}} = \frac{2\pi}{\pi^{d/2}} \Gamma\left(d/2\right) \, a_d^ \, \frac{\int_{\mathcal{E}_{min}} \sqrt{\bar{\gamma}}}{{\ell_\text{AdS}}^{d-1}}\,, \end{split}$$ where the subscript ‘AdS’ refers to evaluation on the background ${\bar g}_{ab}$. Similarly, the perturbation of Wald entropy is now computable via the two contributions $$\label{eq:Pert0} \begin{split} \delta S_{Wald}(\mathcal{E}_{min}) = -2\pi \, \frac{1}{2{\ell_\text{P}}^{d-1}} \int_{\mathcal{E}_{min}} \left[\sqrt{\gamma} \;\frac{\delta (\mathcal{L}^{(I)}_{\Delta R}+ \mathcal{L}^{(II)}_{\Delta R})}{\delta R_{abcd}}\, n_{ab} n_{cd}\right]_{{\mathscr{O}}(h)} \,, \end{split}$$ where ‘${\mathscr{O}}(h)$’ picks out the term proportional to $h_{ab}$. We find the following contribution from $\mathcal{L}^{(I)}_{\Delta R}$: $$\label{eq:PertI} \begin{split} \delta S_{Wald,\,(I)}(\mathcal{E}_{min}) = -2\pi \, \frac{1}{2{\ell_\text{P}}^{d-1}} \int_{\mathcal{E}_{min}} \left[\sqrt{{\gamma}}\;(-2a_2)\right]_{{\mathscr{O}}(h)} = \frac{2\pi}{\pi^{d/2}} \Gamma\left(d/2\right) \, a_d^* \, \frac{\int_{\mathcal{E}_{min}} \sqrt{\bar{\gamma}} \; (\tfrac{1}{2}\bar\gamma^{\alpha\beta} h_{\alpha\beta})}{{\ell_\text{AdS}}^{d-1}} \,, \end{split}$$ which could also be obtained by simply perturbing the volume form in . Similarly, one can compute the higher curvature contributions from $\mathcal{L}^{(II)}_{\Delta R}$ at ${\mathscr{O}}(h)$ as follows:[^17] $$\label{eq:PertII} \begin{split} \delta &S_{Wald,\,(II)}(\mathcal{E}_{min}) =\\ &\quad= -2\pi \, \frac{1}{2{\ell_\text{P}}^{d-1}} \int_{\mathcal{E}_{min}} \sqrt{\bar{\gamma}} \; 2b_1\,{\ell_\text{AdS}}^2\bigg[ \Delta R \, g^{c[a} g^{b]d} - 4 \Delta R_{ef} \, g^{e[b} g^{a][c} g^{d]f} + \Delta R^{abcd} \bigg] n_{ab}n_{cd} \\ &\quad= 4\pi \,b_1\, {\ell_\text{AdS}}^2\, \frac{1}{{\ell_\text{P}}^{d-1}} \int_{\mathcal{E}_{min}} \sqrt{\bar{\gamma}} \,\left[ R + 2 \, R^{ab} \, n_{ac} n_b{}^c - \frac{1}{2} \, R^{abcd} \, n_{ab} n_{cd} \right]_{{\mathscr{O}}(h)} + {\mathscr{O}}(h^2) \\ &\quad= 4\pi \,b_1\, {\ell_\text{AdS}}^2\, \frac{1}{{\ell_\text{P}}^{d-1}} \int_{\mathcal{E}_{min}} \sqrt{\bar{\gamma}} \; \mathcal{{{}^{\text{\tiny (1)}}\!R}}+ {\mathscr{O}}(h^2) \,, \end{split}$$ where ${\mathcal {{}^{\text{\tiny (1)}}\!R}}$ is the linearized intrinsic Ricci scalar on $\mathcal{E}_{min}$ in units of ${\ell_\text{AdS}}$ (i.e., is essentially the first order perturbation of Wald entropy in pure Gauss-Bonnet theory). In the last step we used the fact that extrinsic curvatures vanish on $\mathcal{E}_{min}$, so the projected Ricci scalar in the penultimate line coincides with the intrinsic one. We can further simplify the expression using $$\mathcal{ {{}^{\text{\tiny (1)}}\!R}} = - \bar{\mathcal{R}}^{\alpha\beta} \, \delta \gamma_{\alpha\beta} + \bar\nabla_\alpha (\ldots) = \frac{(d-2)}{{\ell_\text{AdS}}^2} \, \bar\gamma^{\alpha\beta} h_{\alpha\beta}+ \bar\nabla_\alpha (\ldots) \,,$$ where $\bar{\mathcal{R}}^{\alpha\beta}$ is the unperturbed intrinsic Ricci scalar and $\bar \nabla_\alpha$ is the intrinsic covariant derivative compatible with $\bar\gamma_{\alpha\beta}$ and hence only yields boundary terms which we are not keeping track of.[^18] Hence becomes $$\label{eq:PertFinalII} \begin{split} \delta &S_{Wald,\,(II)}(\mathcal{E}_{min}) = 2\pi \left({\lambda_h}-\frac{\Gamma(d/2)}{\pi^{d/2}}\frac{{\ell_\text{P}}^{d-1}}{{\ell_\text{AdS}}^{d-1}} \, a_d^*\right) \frac{\int_{\mathcal{E}_{min}} \sqrt{\bar{\gamma}} \; (\tfrac{1}{2}\bar\gamma^{\alpha\beta} h_{\alpha\beta})}{{\ell_\text{P}}^{d-1}} \end{split}$$ Finally adding the two contributions to perturbed Wald entropy, and , we conclude that reads $$\begin{split} \delta &S_{Wald}(\mathcal{E}_{min}) = 2\pi\left(\frac{{\ell_\text{AdS}}^{d-1}}{{\ell_\text{P}}^{d-1}}\, {\lambda_h}\right) \frac{\int_{\mathcal{E}_{min}} \sqrt{\bar{\gamma}} \; (\tfrac{1}{2}\bar\gamma^{\alpha\beta} h_{\alpha\beta})}{{\ell_\text{AdS}}^{d-1}} \,. \end{split}$$ First law of entanglement {#sec:1stLaw} ========================= In the main text we have focused on perturbations of entanglement entropy due to scalar deformations. Let us now drop such particular assumptions and consider a slightly more general setup which has been argued to serve as a quantitative derivation of bulk dynamics from entanglement. For theories in which the Ryu-Takayanagi proposal is valid, a first law of entanglement entropy has been shown to be equivalent to linearized bulk equations of motion [@Lashkari:2013koa; @Blanco:2013joa; @Wong:2013gua; @Allahbakhshi:2013rda; @Faulkner:2013ica; @Swingle:2014uza]. The entanglement first law $$\label{eq:EE1stLaw} \delta S_{EE}(\mathcal{A}) = \text{Tr}(\delta \rho_\mathcal{A}\, H_\mathcal{A}) \equiv \delta \langle H_\mathcal{A} \rangle$$ is purely a CFT statement which says that any change of $S_{EE}(\mathcal{A})$ due to some small perturbation $\delta \rho_\mathcal{A}$ of the density matrix is given by the change of modular energy associated with the spatial sphere $\mathcal{A}$. (This statement can easily be derived from the formalism of §\[sec:CFTExplanation\].) For CFTs with a semiclassical gravity dual, this statement has a natural analog in gravity: the left hand side of corresponds to a perturbation of Wald entropy as in . Similarly the right hand side of translates to a change of gravitational energy $E_{grav}(\mathcal{A})$ of the Rindler patch of AdS${}_{d+1}$ associated with the causal diamond of $\mathcal{A}$. That is, the CFT entanglement first law is equivalent to the gravitational first law $$\label{eq:Grav1stLaw} \delta S_{Wald}(\mathcal{E}_{min}) = \delta E_{grav}(\mathcal{A}) \,.$$ The gravitational energy $E_{grav}(\mathcal{A})$ is defined as an integral of the holographic stress tensor $T_{ab}^{grav}$ over a spatial bulk surface of bulk codimension two, which lies inside the Rindler patch of $\mathcal{A}$ and ends on $\partial \mathcal{A}$. Without loss of generality, we push this bulk surface to the boundary, so the gravitational energy can be computed as a boundary integral over $\mathcal{A}$: $$\label{eq:ModEgrav} E_{grav}(\mathcal{A}) = \int_{\mathcal{A}} d\Sigma^\mu \; T^{grav}_{\mu\nu} \, \zeta^\nu_\mathcal{A}\,,$$ where $d\Sigma^\mu$ is the volume form on the $(d-1)$-dimensional surface $\mathcal{A}$ and $\zeta^\nu_\mathcal{A}$ is the conformal Killing vector generating the Rindler flow inside the causal development of $\mathcal{A}$. In holographic CFTs the CFT stress tensor $T_{\mu\nu}$ of agrees with the holographic stress tensor $T_{\mu\nu}^{grav}$. One can easily verify that the gravitational energy then agrees with the definition of modular energy in the CFT, . The holographic stress tensor is determined by the dominant asymptotics of the graviton fluctuation on AdS${}_{d+1}$. Let us work in Poincaré coordinates $$\label{eq:coords} ds^2 = \frac{{\ell_\text{AdS}}^2}{z^2} \left( dz^2 + \eta_{\mu\nu} \, dx^\mu dx^\nu \right) + h_{ab}\, dx^a dx^b\,.$$ Then the relevant graviton falloff near the boundary is given by the following piece of the Fefferman-Graham expansion: $$h_{ab}(x^\rho,z) \, dx^a dx^b = {\ell_\text{AdS}}^2 \, z^{d-2} \, h_{\mu\nu}^{(d)}(x^\rho) \, dx^\mu dx^\nu +\ldots \,.$$ The power of $z$ is chosen such that the induced boundary stress tensor $\delta T_{\mu\nu}^{grav}$ is finite (i.e., $z$-independent). It was shown in [@Faulkner:2013ica; @Sen:2014nfa] that for a wide class of theories the holographic stress tensor associated with this bulk perturbation evaluates to $$\label{eq:Tgrav} \delta T_{\mu\nu}^{grav} = \frac{d\,{C_{_T}}}{f_d} \; h^{(d)}_{\mu\nu} \,,$$ where $f_d$ is the number defined in . Note that this form of $\delta T_{\mu\nu}^{grav}$ is consistent with our considerations in §\[sec:lhindependence\]. Indeed, $\delta S_{Wald}(\mathcal{E}_{min})$ in the present context can be obtained from the gravitational energy integral over the combination : $$\delta S_{Wald}(\mathcal{E}_{min}) = \frac{d\,{C_{_T}}}{f_d}\int_{\mathcal{A}} d\Sigma^\mu \; h^{(d)}_{\mu\nu} \, \zeta^\nu_\mathcal{A}\,.$$ This has the same structure as , i.e., an integral over a quantity (${C_{_T}}$ or ${\lambda_h}$ times a metric perturbation) which satisfies a universal equation of motion. The linearized equations of motion can be derived from the first law statements above. Indeed, assuming the Ryu-Takayanagi conjecture, it was shown in [@Faulkner:2013ica; @Sen:2014nfa] that for small spherical regions $\mathcal{A}$ is equivalent to linearized gravitational equations of motion for any higher derivative Lagrangian $\mathcal{L}_{grav} = \mathcal{L}_{grav}(g_{ab},R_{abcd},\nabla_eR_{abcd},\ldots)$. As argued in §\[sec:GravExplanation\], on an AdS background these equations are the same for any theory of gravity with second order equations of motion, up to overall normalizations which encode the higher curvature contributions. Depending on the precise setup, even these normalizations may be absorbed into renormalizations of dimensionless quantities. In particular the normalization ${\lambda_h}$ inside ${C_{_T}}$ always multiplies $({\ell_\text{AdS}}/{\ell_\text{P}})^{d-1}$, c.f., . Unless ${\ell_\text{AdS}}/{\ell_\text{P}}$ has already been fixed (e.g., by a condition such as ), this ratio can serve to account for any ${\lambda_h}$ dependence. Second order gravitational perturbation theory {#sec:perturbation} ============================================== In this appendix we list some relevant formulae for gravitational perturbation theory up to second order in the graviton. We take the ansatz $$g_{ab} = {\bar g}_{ab} + h_{ab}$$ and are interested in perturbations up to ${\mathscr{O}}(h^2)$. The background solution ${\bar g}_{ab}$ is AdS${}_{d+1}$ with curvature $-1/{\ell_\text{AdS}}^{2}$, so the following simplifications apply for the background curvature tensors: $$\label{eq:AdSsimplify} \begin{split} {\bar R}_{abcd} &= -\frac{1}{{\ell_\text{AdS}}^2} ({\bar g}_{ac} {\bar g}_{bd} - {\bar g}_{ad} {\bar g}_{bc}) \,, \qquad {\bar R}_{ab} = - \frac{d}{{\ell_\text{AdS}}^2} \, {\bar g}_{ab} \,, \qquad {\bar R}= - \frac{d(d+1)}{{\ell_\text{AdS}}^2}\,. \end{split}$$ Let us evaluate and define various quantities, where superscript numbers refer to the order in perturbation theory: $$\begin{split} g^{ab} &= {\bar g}{}^{ab} - h^{ab} + h^{ac}h_c^b + {\mathscr{O}}(h^3) \,,\\ \Gamma^\rho_{ab} &= {\bar{\Gamma}}^\rho_{ab} + {{}^{\text{\tiny (1)}}{\Gamma}}^c_{ab} - h^c_d\; {{}^{\text{\tiny (1)}}{\Gamma}}^d_{ab} + {\mathscr{O}}(h^3) \,,\\ R^a{}_{bcd} &= {\bar R}^a{}_{bcd} +{{}^{\text{\tiny (1)}}\!R}^a{}_{bcd} \\ &\qquad\qquad- \left[h^a_e \; {{}^{\text{\tiny (1)}}\!R}^{e}{}_{bcd} + {\bar g}{}^{ae} {\bar g}_{fg} \left( {{}^{\text{\tiny (1)}}{\Gamma}}^g_{ce} \; {{}^{\text{\tiny (1)}}{\Gamma}}^f_{db} - {{}^{\text{\tiny (1)}}{\Gamma}}^g_{de} \; {{}^{\text{\tiny (1)}}{\Gamma}}^f_{cb} \right)\right]+ {\mathscr{O}}(h^3)\,, \\ R_{ab} &= {\bar R}_{ab} + {{}^{\text{\tiny (1)}}\!R}_{ab} - \left[h^c_d \; {{}^{\text{\tiny (1)}}\!R}^d{}_{acb} + {\bar g}{}^{ce} {\bar g}_{fg} \left( {{}^{\text{\tiny (1)}}{\Gamma}}^g_{ce} \;{{}^{\text{\tiny (1)}}{\Gamma}}^f_{ba} - {{}^{\text{\tiny (1)}}{\Gamma}}^g_{ce} \; {{}^{\text{\tiny (1)}}{\Gamma}}^f_{ca} \right)\right]+ {\mathscr{O}}(h^3)\,, \\ R &= {\bar R}+ {{}^{\text{\tiny (1)}}\!R}+ \bigg\{ h^{ac} h_{c}^b\,{\bar R}_{ab} - h^{ab} \; {{}^{\text{\tiny (1)}}\!R}_{ab} \\ &\qquad\qquad\qquad\; - {\bar g}{}^{ab} \left[h^c_d \; {{}^{\text{\tiny (1)}}\!R}^d{}_{acb} + {\bar g}{}^{ce} {\bar g}_{fg} \left( {{}^{\text{\tiny (1)}}{\Gamma}}^g_{ce} \; {{}^{\text{\tiny (1)}}{\Gamma}}^f_{ba} - {{}^{\text{\tiny (1)}}{\Gamma}}^g_{be} \; {{}^{\text{\tiny (1)}}{\Gamma}}^f_{ca} \right)\right] \bigg\} + {\mathscr{O}}(h^3)\,. \end{split}$$ In these expressions the linearized objects are given in terms of $h_{ab}$ by $$\label{eq:1stOrderQ} \begin{split} {{}^{\text{\tiny (1)}}{\Gamma}}^c_{ab} &\equiv \frac{1}{2} {\bar g}{}^{cd} \left( {\bar{\nabla}}_a h_{bd} + {\bar{\nabla}}_b h_{ad} - {\bar{\nabla}}_d h_{ab} \right) \,, \\ {{}^{\text{\tiny (1)}}\!R}^a{}_{bcd} &= {\bar{\nabla}}_{[c}{\bar{\nabla}}_{d]} h^a_b + {\bar{\nabla}}_{[c} {\bar{\nabla}}_b h^a_{d]} - {\bar{\nabla}}_{[c} {\bar{\nabla}}^a h_{d]b} \,, \\ {{}^{\text{\tiny (1)}}\!R}_{ab} &= {\bar{\nabla}}_c {\bar{\nabla}}_{(a} h^c_{b)} - \frac{1}{2} {\bar{\Box}}h_{ab} - \frac{1}{2} {\bar{\nabla}}_a {\bar{\nabla}}_b h \,, \\ {{}^{\text{\tiny (1)}}\!R}&= {{}^{\text{\tiny (1)}}\!R}_{ab} \,{\bar g}{}^{ab} - {\bar R}^{ab}\, h_{ab} = {\bar{\nabla}}_a {\bar{\nabla}}_b h^{ab} - {\bar{\Box}}h - {\bar R}^{ab} \, h_{ab} \,, \end{split}$$ where $h \equiv h^a_a = {\bar g}{}^{ab} h_{ab}$. For the expansion of actions it is also useful to note the following identity: $$\sqrt{-g} = \sqrt{-{\bar g}} \left( 1 + \frac{1}{2} h + \frac{1}{8} (h^2 - 2 h^a_b \, h^b_a ) + {\mathscr{O}}(h^3)\right) \,.$$ [^1]: By an “acceptable theory of gravity” we mean any semiclassical gravitational theory with the correct spectrum of excitations (in particular without ghosts). See however §\[app:generalization\]. [^2]: The first term in the bracket of comes from the Einstein contribution in the action . In a theory $\mathcal{L} = d(d-1)/\ell^2 + \mathcal{L}_{h.d.}$ without pure Ricci scalar, this term would disappear. [^3]: Operators without argument are understood to be integrated, e.g., $\mathcal{O} \equiv \int d^dx \; \mathcal{O}(x)$. [^4]: We will use indices $a,b,\ldots$ for bulk coordinates, $\mu,\nu,\ldots$ for boundary directions, and $\alpha,\beta,\ldots$ for intrinsic coordinates on the minimal surface $\mathcal{E}_{min}$. [^5]: This is assuming that $\mathcal{O}$ is a relevant operator with $\Delta \neq d/2$. See [@Liu:2012eea; @Nishioka:2014kpa] for a discussion of the case $\Delta = d/2$ in holographic theories. [^6]: The assumption $\Delta \geq d/2$ corresponds to performing so-called standard quantization. Note that $(d-2)/2 \leq \Delta \leq d/2$ would still be compatible with field theory unitarity. The latter regime can be treated by choosing $\Delta_- = \Delta$. See [@Hung:2011ta] for a discussion of this case. [^7]: \[footnote:fR\] There are special circumstances where the additional degrees of freedom can be dealt with by using appropriate field redefinitions. For instance, $f(R)$ theories of gravity naively have higher order equations of motion, but are known to be equivalent under field redefinitions to Einstein gravity coupled to a (positive energy) scalar field such that the boundary theory is unitary [@Sotiriou:2008rp]. The graviton in this case couples not only to the boundary stress tensor, but in addition to some scalar operator. The CFT therefore has some more specific features than we assumed in the beginning. For validating our conclusions in the most general circumstances, it would be interesting to explore the most general requirements for a bulk theory to admit a unitary dual CFT. A first step in this direction might be a gauge-fixed analysis where the scalar part of the graviton is set to zero, c.f., the transverse traceless gauge employed in §\[app:generalization\]. This would at least take care of $f(R)$ theories. [^8]: Of course, it would be possible for $\mathcal{O}$ to be some other scalar operator, e.g., one corresponding to the square of a scalar field. Since such scenarios would not change our analysis qualitatively, we will assume that $\mathcal{O}$ is simply sourced by $\phi$ with appropriate boundary condition. [^9]: To verify this by hand, the identities in §\[sec:perturbation\] are helpful. [^10]: Note that the coefficient ${\lambda_h}$ of the first term in cannot be vanishing as this would mean that no spin-2 degrees of freedom propagate at all. By this we ensure that no gauge choice can eliminate the spin-2 mode. For instance, we could choose to work in a transverse traceless gauge (${\bar{\nabla}}_a h^{ab} = 0 = h$), but the first term would still survive as it is the source of the spin-2 component. On a similar note, we need ${\lambda_h}> 0$ in order for the spin-2 graviton not to be ghost. [^11]: Alternatively, one can determine the coefficients by demanding conservation of the matter stress tensor, ${\bar{\nabla}}^b T_{ab}^{(\phi)} = 0$. [^12]: See §\[sec:1stLaw\] for related observations based on the entanglement first law. [^13]: Variations of the bifurcation surface only affect the Wald entropy at the next order, which is beyond our discussion. To see this, note that the surface $\mathcal{E}_{min}$ extremizes the Wald entropy functional upon variation with respect to the metric. First order variations of the metric (which are ${\mathscr{O}}(\lambda^2)$ in our conventions) hence leave $\mathcal{E}_{min}$ unchanged. [^14]: Explicitly, if $\xi^\alpha(x)$ denotes the coordinates of $\mathcal{E}_{min}$ embedded in AdS${}_{d+1}$, we have $$\bar\gamma_{\alpha\beta} = \frac{\partial x^a}{\partial \xi^\alpha} \frac{\partial x^b}{\partial \xi^\beta} \, {\bar g}_{ab} \,,\qquad h_{\alpha\beta} = \frac{\partial x^a}{\partial \xi^\alpha} \frac{\partial x^b}{\partial \xi^\beta} \, h_{ab} \,.$$ For more conventions see §\[app:Wald\]. [^15]: See, e.g., Eq. (7.7) of [@deBoer:2011wk] with the understanding that their Gauss-Bonnet coupling $\lambda$ equals $2\gamma$ in our notation. [^16]: In this context, see also §\[app:Wald\]. [^17]: Our conventions are as follows: the embedded version of the intrinsic metric $\gamma_{\alpha\beta}$ is $\gamma_{ab} = g_{ab} + t_at_b-n_an_b$ where $t^a$ and $n^a$ are normal to the minimal surface such that $t^2 = -1$, $n^2 = 1$, $t^a n_a = 0$ and the binormal $n^{ab} \equiv n^{a} t^{b} - t^a n^b$. [^18]: Boundary terms are important to get a finite result as in . Since we are only interested in identifying the fate of the dimensionless parameters, we are being implicit about these subtleties and assume that there always exist suitable Gibbons-Hawking-York boundary terms.
--- abstract: 'We report on a measurement of the fraction of events with a $W$ or $Z$ boson which are produced diffractively in ${\bar p}p$ collisions at $\sqrt{s}=1.96$ TeV, using data from 0.6 fb$^{-1}$ of integrated luminosity collected with the CDF II detector equipped with a Roman-pot spectrometer that detects the $\bar p$ from $\bar{p}+p\rightarrow {\bar p}+[X+W/Z]$. We find that $(1.00\pm 0.11)\%$ of $W$s and $(0.88\pm 0.22)\%$ of $Z$s are produced diffractively in a region of antiproton [or proton]{} fractional momentum loss $\xi$ of $0.03<\xi<0.10$ and 4-momentum transferred squared $t$ of $-1<t<0$ (GeV/$c$)$^2$, where we account for the events in which the proton scatters diffractively while the antiproton dissociates, $\bar{p}+p\rightarrow [X+W/Z]+p$, by doubling the measured proton dissociation fraction. We also report on searches for $W$ and $Z$ production in double Pomeron exchange, $p+\bar{p}\rightarrow p+[X+W/Z]+\bar{p}$, and on exclusive $Z$ production, $\bar{p}+p\rightarrow {\bar p}+Z+p$. No signal is seen above background for these processes, and comparisons are made with expectations.' title: Diffractive $W$ and $Z$ Production at the Fermilab Tevatron --- Introduction ============ Approximately one quarter of the inelastic $\bar{p}p$ collisions at the Tevatron are diffractive interactions, in which a strongly-interacting color singlet quark/gluon combination with the quantum numbers of the vacuum (the [Pomeron]{}, ${I$\!\!$P}$) is presumed to be exchanged [@Collins]-[@Donnachie]. As no radiation is expected from such an exchange, a pseudorapidity region devoid of particles, called a [rapidity gap]{} [@rapidity], is produced. Diffractive processes are classified by the topology of the final state as single diffraction (SD), double Pomeron exchange (DPE), and double diffraction dissociation. In SD, the $\bar{p}(p)$ remains intact escaping the collision with momentum close to that of the original beam momentum and separated by a rapidity gap from the products of the ${I$\!\!$P}$-$p(\bar{p})$ collision, usually referred to as a [forward gap]{}; in DPE both the $\bar p$ and the $p$ escape, resulting in two forward rapidity gaps; and in double diffraction dissociation a central gap is formed while both the $p$ and $\bar p$ dissociate. A special case of rapidity gap events is [exclusive]{} production where a particle state is centrally produced, such as a dijet system or a $Z$ boson. Diffraction has traditionally been described using Regge theory (see Refs. [@Collins]-[@Donnachie]). In [hard]{} diffraction, such as jet or $W$ production, in addition to the colorless exchange there is a hard scale which allows one to explore both the mechanism for diffraction and the partonic structure of the Pomeron. Whereas diffractive dijets can be produced via quarks or gluons, to leading order a diffractive $W$ is produced via a quark in the Pomeron. Production by gluons is suppressed by a factor of $\alpha_s$, and can be distinguished from quark production by an additional associated jet, as shown in Fig. \[fig:WZ\_fig1\]. Combining cross section measurements of diffractive dijet production and diffractive $W$ production can be used to determine the quark/gluon content of the Pomeron [@run1dijet1997]. In Tevatron Run I, CDF measured the fraction of events with dijets [@run1dijet1997]-[@run1dijet2002], $W$ bosons [@run1], $b$ quarks [@cdfdiffrb], or $J/\psi$s [@jpsi] which are produced diffractively, and found in all cases a fraction of approximately $1\%$ (including both ${I$\!\!$P}$-$\bar p$ and ${I$\!\!$P}$-$p$ production). ![image](WZ_fig1.eps){width="50.00000%"} In the Run I measurements, with the exception of [@run1dijet2000; @run1dijet2002] which used a Roman-pot spectrometer (RPS), both the CDF and D0 collaborations used the rapidity gap signature for identifying diffractive events and extracting the diffractive fraction. The interpretation of the results obtained by this method is complicated by the issue of gap survival probability, the likelihood that a rapidity gap produced in a diffractive interaction will not be filled by the products of additional parton-parton interactions in the same $\bar pp$ collision. In addition, there are experimental problems associated with the definition of the gap size, $\Delta\eta$, e.g., penetration of the gap by low transverse momentum ($p_T$) particles originating at the interaction point (IP) from the diffractively dissociated (anti)proton. To ameliorate this problem, CDF allowed up to two particles in the nominal gap region and introduced the term “gap acceptance” for the fraction of events selected by this criterion. The gap survival and gap acceptance probabilities both require model dependent Monte Carlo simulations followed by a detector simulation. At $\sqrt s=1.8$ TeV, CDF observed a diffractive $W$ signal with a probability $1.1\times 10^{-4}$ of being caused by a fluctuation from nondiffractive (ND) events and measured the fraction of diffractive $W$ events to be $[1.15\pm 0.51\mbox{(stat.)}\pm 0.20\mbox{(syst.)}]\%$ [@run1]. D0 studied both diffractive $W$ and $Z$ production in Run I and found a diffractive fraction, uncorrected for gap survival, of $\left (0.89^{+0.19}_{-0.17}\right )\%$ for $W$s and $\left (1.44^{+0.61}_{-0.52}\right )\%$ for $Z$s [@d0]. However, the gap survival estimated by D0 using Monte Carlo simulations was $(21\pm4)\%$, which would yield $W$ and $Z$ fractions approximately four times larger than those of CDF. An observation of an anomalously high diffractive $W/Z$ production rate could be evidence for beyond-standard-model theories, such as that of Ref. [@AlanWhite] in which the Pomeron couples strongly to the electroweak sector through a pair of [sextet]{} quarks. The Run I CDF and D0 measurements using rapidity gaps rely on model dependent corrections for the gap acceptance and background, making their interpretation difficult. The analysis presented here using the RPS makes no gap requirements and consequently is model-independent. \[sec:fwdde\]Detector ===================== The CDF II detector is a multi-purpose detector described in detail in Ref. [@cdf]. It consists of a central detector and a forward detector system designed for diffractive physics studies (see Fig. \[fig:WZ\_fig2\]). ![image](WZ_fig2.eps){width="80.00000%"} The central detector comprises a precision tracking system ($\|\eta\|\lesssim2$), central and plug calorimeters ($\|\eta\|<1.1 $ and $1.1<\|\eta\|<3.6$, respectively), with electromagnetic followed by hadronic sections, and muon spectrometers outside the central calorimeters ($\|\eta\|\lesssim 1.0$). The tracking system is coaxial with the beam pipe and consists of silicon strip detectors surrounded by the Central Outer Tracker (COT), a cylindrical wire drift chamber inside a 1.4 T solenoidal magnetic field. Proportional strip and wire chambers embedded inside the electromagnetic calorimeter provide an accurate position measurement of the source of electromagnetic showers. Wire chambers and scintillator counters outside the calorimeters make up the muon detectors. The forward detectors (see Ref. [@DPEdijet]) consist of the Cherenkov Luminosity Counters (CLC) ($3.7<\|\eta\|<4.7$), which record charged particles coming from the IP and are used primarily for monitoring the luminosity; two MiniPlug calorimeters (MPCAL) covering the pseudorapidity region $3.5<\|\eta\|<5.1$; beam shower counters (BSC) within $5.4<\|\eta\|<7.4$ surrounding the beam pipe in several locations to detect charged particles and photons through conversion to $e^+e^-$ pairs in a 1 radiation length Pb plate placed in front of the first counter; the RPS, consisting of three Roman-pot (RP) detectors approximately 56 m from the nominal IP and 20 m from a string of Tevatron dipole magnets, used to detect and measure the momentum of diffracted antiprotons with fractional momentum loss $\xi$ in the region of $0.03\lesssim\xi\lesssim0.10$. Each RPS detector consists of a trigger scintillation counter and clad scintillating fibers for tracking, covering 20 mm in the $x$ direction to measure the distance from the beam in the plane of the Tevatron ring, and 20 mm in the $y$ direction perpendicular to the plane of the Tevatron. The fibers in both $x$ and $y$ in each RP were arranged in two layers, spaced by 1/3 of a fiber width to give three times better position resolution than a single layer [@ken_thesis; @run1dijet2000]. For RPS detector arrangement and track reconstruction details see Appendix C in Ref. [@ken_thesis]. The resolutions in $\xi$ and $t$ (the 4-momentum transferred squared) for tracks recorded in the RPS were $\delta\xi=0.001$ and $\delta t=0.07$ (GeV/$c$)$^2$. The RPS acceptance is concentrated in the region of $0.03\lesssim\xi\lesssim 0.10$ and $-1<t<0$ (GeV/$c$)$^2$, as shown in Fig. \[fig:WZ\_fig3\]. The RPS was installed for use in the Tevatron Run Ic (1995–96) and was operated in Run II from 2002 to February 2006. A large fraction of events in which all three RPS trigger counters were hit are due to background which we refer to as [splash]{} events. These events are characterized by a large signal measured in the trigger counters as well as hits in almost all of the fibers. One example of a splash process is a diffractive $\bar p$ which is outside the RPS acceptance showering in material near the RPS stations. Another example is beam halo induced interactions in nearby material. The kinematics of the diffractive antiproton are reconstructed from the track position and angle in the RPS, and the beam position and angle at the IP using a model of the Tevatron beam optics between the Roman pots and the IP. ![RPS acceptance as a function of $\xi$ and $t$ obtained from simulation using the transport parameters between the nominal interaction point and the Roman pots.[]{data-label="fig:WZ_fig3"}](WZ_fig3.eps){width="40.00000%"} \[sec:datasets\]$W$/$Z$ event selection ======================================= The event selection begins by requiring a central ($\|\eta\|\!<\!1.1$) high-$p_T$ electron or muon consistent with events with a $W$ or $Z$ decaying leptonically. The triggering $e$($\mu$) is required to have a reconstructed $E^e_T(p^\mu_T)>20$ GeV (GeV/$c$); for $Z$ candidates, the second lepton is subjected to looser identification (ID) requirements, and electrons are also accepted if detected in the plug calorimeter within $1.2\!<\|\eta\|\!<2.8$. Our data set with the forward detectors fully operational corresponds to an integrated luminosity of $\sim 0.6$ fb$^{-1}$ for both the electron and muon samples. The $W$ candidate selection criteria require an $e$ or $\mu$ which passes tight ID requirements [@Wcrosssect] and has $E_T^e(p_T^\mu)>25$ GeV (GeV/$c$), missing transverse energy [@missingET] (corrected for event vertex $z$-position, $p_T^\mu$ of muons that traverse the detector, and mismeasured hadronic jets) of $\met\,>25$ GeV, and $W$ transverse mass, $M_T^W=\sqrt{2(p_T^{l}p_T^{\nu}-\overrightarrow{p}_T^l\cdot\overrightarrow{p}_T^{\nu})}/c$, in the region $40<M_{T}^{W}<120$ GeV/$c^2$. The $Z\rightarrow ee$ selection criteria include the same requirements on the first electron plus a central $e$ with looser ID requirements, pertaining mainly to COT track quality and allowed range of ratio of energies deposited in the hadronic and electromagnetic calorimeters, or an $e$ in the plug calorimeter with $E_T^e>25$ GeV. Similarly, the $Z\rightarrow \mu\mu$ selection criteria include the same requirements on the first $\mu$ plus a loose-ID $\mu$ with $p_T^\mu>25$ GeV/$c$. For both the $Z\rightarrow ee$ and $Z\rightarrow \mu\mu$ channels, we also require $66<M_Z<116$ GeV/$c^2$. The events are required to have a primary interaction vertex within 60 cm (in $z$) of the nominal IP. Events with multiple vertices are not explicitly rejected in the event selection; instead, the fraction of events expected to have a single interaction is calculated. The number of events passing the $W$ and $Z$ candidate selection requirements is 308 915 $W\rightarrow e\nu$, 259 465 $W\rightarrow \mu\nu$, 31 197 $Z\rightarrow ee$, and 15 603 $Z\rightarrow \mu\mu$. The probability $P_n$ of a beam crossing producing $n$ inelastic interactions in addition to the hard interaction that produces the $W$/$Z$ is obtained from Poisson statistics: $P_n={\bar n}^n e^{-\bar n}/n!$, where ${\bar n}={\cal L}\cdot\sigma_{\rm inel}/f^{\rm cross}_{\rm eff}$ is the mean number of interactions, which depends on the instantaneous luminosity ${\cal L}$, the inelastic cross section $\sigma_{\rm inel}$, and the effective beam crossing frequency (disregarding the transition regions of empty beam bunches) of $f^{\rm cross}_{\rm eff}=1.7$ MHz. For the value of $\sigma_{\rm inel}$ at $\sqrt s=1.96$ TeV we use $59.3\pm 2.3$ mb, obtained by extrapolation from the CDF measurements of the elastic and total cross sections at $\sqrt s=1.8$ TeV [@CDF_elastic; @CDF_total]. The fraction of $W$ and $Z$ events from a single interaction is determined independently for each one of three datasets comprising our event sample, collected at different average instantaneous luminosities. This fraction is $f_{\rm 1-int}=(47.4\pm 1.3)\%$ for the Aug’02–Dec’03 dataset, $(25.1\pm 1.2)\%$ for Dec’04–Jul’05, and $(20.1\pm 1.0)\%$ for Sept’05–Feb’06, where the uncertainty is based on that in $\sigma_{\rm inel}$, which is common in all data sets. The weighted average over all datasets is $\langle f_{\rm 1-int}\rangle =(25.6\pm 1.2)\%$. Taking into account bunch-to-bunch variation in luminosity, because our W/Z was more likely to be produced by a bunch with higher luminosity, we find that the single-interaction probability becomes systematically lower. A correction factor of $0.97\pm 0.01$ is applied to $\langle f_{\rm 1-int}\rangle$ to account for this effect. \[sec:dataanal\]Diffractive event selection =========================================== Diffractive events are first selected by requiring that all three of the RPS trigger counters have energy deposited within a specified run-dependent range. Although a minimum energy is required in order to select events where the RPS detectors are hit, we also require a reconstructed RPS track. The upper bound on the energy imposed in order to remove background splash events is another important selection requirement. As splash events are caused by secondaries from an interaction in material near the RPS, rather than by a diffractive antiproton within the RPS acceptance, these events tend to have large energy deposited in the RPS trigger counters, as well as a large fraction of RPS tracking fibers hit. Next, a reconstructed RPS track is also required. In this step, we accept events where the $\xi$ and $t$ reconstructed from the track are within the range of $0.03<\xi<0.10$ and $\|t\|<1$ (GeV/$c$)$^2$. In an event with multiple ${\bar p}p$ interactions, the diffracted antiproton may originate from a different interaction than the one producing the $W$ or $Z$. This [overlap background]{} is dominated by events in which a ND $W/Z$ is superimposed on an inclusive SD interaction with the $\bar p$ detected in the RPS. The main tool for removing overlap backgrounds is the value of $\xi_{\bar p}$ reconstructed using the calorimeters, $$\label{eq:xical}\xi^{\rm cal}_{\bar p}=\sum_{i=1}^{N_{\rm towers}}\frac{E^i_{\rm T}}{\sqrt{s}}e^{-\eta^i},$$ where $\eta^i$ is the $\eta$–value of the center of a tower and the sum is carried over all calorimeter towers with $E_T$ greater than a calorimeter-dependent threshold chosen to reject noise (see Ref. [@DPEdijet]). The $\xi^{\rm cal}_{\bar p}$ values were calibrated by comparing diffractive dijet data, collected concurrently with the diffractive $W/Z$ data, with Monte Carlo generated events. Calibrated $\xi^{\rm cal}_{\bar p}$ values were found to be in good agreement with values measured by the RPS. The resolution in $\xi^{\rm cal}_{\bar p}$ is dominated by that of the energy measurement by MPCAL, which is $\sim 30$% resulting in a root mean square deviation $\Delta\log\xi=0.1$. In a diffractive Z event with no additional interactions, $\xi^{\rm cal}_{\bar p}\approx \xi^{\rm RPS}$ and should fall within the RPS acceptance region of $0.03<\xi<0.10$ or $-1.5<\log{\xi}<-1.0$. In an event with multiple interactions, $\xi^{\rm cal}_{\bar p}$ would be larger due to energy from the additional interaction. Therefore, we expect all diffractive Z interactions with no additional interaction to have $\xi^{\rm cal}_{\bar p}<0.10$ and we can remove most of the overlap background with this requirement. Some events with $\xi^{\rm cal}_{\bar p}<0.10$ may still have multiple interactions. We use the distribution of $\xi^{\rm cal}_{\bar p}$ from the nondiffractive Z sample (before the requirement of a RPS track) to estimate the overlap background at small $\xi^{\rm cal}_{\bar p}$. One does not expect to find perfect agreement between the ND and SD distributions at $\xi^{\rm cal}_{\bar p}>0.1$ since the SD candidates in that region always contain at least one other interaction, while a fraction of the ND events may be due to a single interaction. By normalizing the ND to the SD distribution in the region of $-1.0<\log{\xi}<-0.4$ we obtain a reasonable estimate of the overlap background in the region $\xi^{cal}_{\bar p}<0.1$ within the assigned conservative uncertainty. Figure \[fig:WZ\_fig4\] shows the distributions of $\xi^{\rm cal}_{\bar p}$ for $W$ and $Z$ candidate events with a RPS track. For $Z$s, the distribution for ND events is also shown, normalized to the distribution from events with a RPS track in the region of $-1.0<\log_{10}\xi^{\rm cal}_{\bar p}<-0.4$. The excess of RPS events over ND ones for $\log_{10}\xi^{\rm cal}_{\bar p}<-1.0$ ($\xi_{\bar p}^{cal}<0.10$) contains the SD signal. In determining the fraction of $Z$ events which are diffractive in Sec. \[WZfraction\], we require $\xi^{cal}_{\bar p}<0.10$ (number of events $N^Z_{\xi<0.10}$) and subtract a background determined from the number of events expected in the normalized ND distribution with $\xi^{cal}_{\bar p}<0.10$ ($N^Z_{bgnd}$). In diffractive $W$ events, since the neutrino is not directly detected by the CDF II detector, $E_T^\nu$ is not included in the $\sum E_T$ of calorimeter towers used to determine $\xi_{\bar p}^{\rm cal}$ through Eq. (\[eq:xical\]). Requiring $\xi_{\bar p}^{\rm cal}<\xi_{\bar p}^{\rm RPS}$ removes events with multiple interactions in a similar way as the requirement on $\xi_{\bar p}^{cal}<0.10$. In addition, knowing the kinematics of the diffracted antiproton allows us to reconstruct the longitudinal momentum of the neutrino and thereby the $W$ mass, $M_{\rm W}$, as described in Sec. \[sec:Wkin\]. Requiring $M_{\rm W}$ to be within the range $50<M_{\rm W}<120$ GeV/$c^2$ removes almost all the remaining multiple-interaction or misreconstructed events. The $\xi_{\bar p}^{\rm cal}$ distribution for diffractive $W$ candidates is shown in Fig. \[fig:WZ\_fig4\], and the number of $W$ and $Z$ events passing the diffractive event selection criteria is listed in Table \[t:diffrWZ\]. ![image](WZ_fig4.eps){width="70.00000%"} ---------------------- ------------------------- --------------------------- ---------------------------- [$W\rightarrow e\nu$]{} [$W\rightarrow \mu\nu$]{} $W\rightarrow l(e/\mu)\nu$ RPS-trigger-counters 6663 5657 12 320 RPS-track 5124 4201 9325 $50<M_W<120$ 192 160 352 [$Z\rightarrow ee$]{} [$Z\rightarrow \mu\mu$]{} $Z\rightarrow ll$ RPS-trigger-counters 650 341 991 RPS-track 494 253 747 $\xi^{\rm cal}<0.10$ 24 12 36 ---------------------- ------------------------- --------------------------- ---------------------------- : \[t:diffrWZ\] $W$ and $Z$ events passing successive selection requirements. Results ======= $W$ mass from diffractive events\[sec:Wkin\] -------------------------------------------- In nondiffractive $W$ production, the neutrino transverse energy $E^\nu_T$ is inferred from the $\met$ but the neutrino longitudinal momentum $p^\nu_z$ is unknown. However, in diffractive $W$ production the missing $p^\nu_z$ yields a difference between the $\xi_{\bar p}^{\rm cal}$, calculated from the energy deposited in the calorimeters using Eq. (\[eq:xical\]), and the $\xi^{\rm RPS}_{\bar p}$ determined from the RPS track. This difference allows one to determine $p^\nu_z$, and thereby the full $W$ kinematics through Eqs. (\[eq:etanu\]–\[eq:pEw\]): $$\xi^{\rm RPS}-\xi_{\bar p}^{\rm cal}=\sum_{i=1}^{\rm all\;towers}\frac{\met^i}{\sqrt{s}}e^{-\eta^\nu}\label{eq:etanu},$$ $$p_z^\nu=\met/\tan{\left[2\tan^{-1}{(e^{-\eta^\nu})}\right]}\label{eq:pznu},$$ $$M_{\rm w}^2=2\left [E_e\sqrt{\met^2+p_z^{\nu\,2}}-p_x^e p_x^\nu-p_y^e p_y^\nu-p_z^e p_z^\nu\right]\label{eq:mw},$$ $$p_z^{\rm w}=p_z^e+p_z^\nu,\; E^{\rm w}=E^e+\sqrt{\met^2+p_z^{\nu\,2}}.\label{eq:pEw}$$ Using the full $W$ kinematics results in a Gaussian $M_{\rm W}$ distribution permitting a more accurate determination of $M_{\rm W}$ from a given number of events. For the diffractive sample of 352 events listed in Table \[t:diffrWZ\], this method yields the mass distribution shown in Fig. \[fig:WZ\_fig5\] from which we obtain $M_{\rm W}^{\rm diff}=(80.9\pm0.7)$ GeV/$c^2$, in agreement with the world average of $M_W^\text{PDG}=(80.398\pm0.025)$ GeV/$c^2$ [@PDG]. ![Reconstructed $M_{\rm W}^{\rm diff}$ with a Gaussian fit.[]{data-label="fig:WZ_fig5"}](WZ_fig5.eps){width="45.00000%"} \[WZfraction\]Diffractive $W/Z$ fraction ---------------------------------------- The RPS acceptance is calculated using: $$A_{\rm RPS}=\frac{N}{\sum_{i=1}^{\rm N} A^i_{\rm RPS}(\xi_i,t_i)^{-1}},$$ where the sum is over $N$ diffractive $W$ events with a RPS track and $A^i_{\rm RPS}(\xi,t)$ is the acceptance for an event at ($\xi_i,t_i$) shown in Fig. \[fig:WZ\_fig3\]. For our diffractive $W$ event samples, we measure $A_{\rm RPS}=[88\pm 12(\rm stat.)]\%$ (Aug’02–Dec’03) and $A_{\rm RPS}=[75\pm 5(\rm stat.)]\%$ (Dec’04–Feb’06) for events within $0.03<\xi<0.10$ and $\|t\|<1$ (GeV/$c$)$^{2}$. These values are consistent with those determined with better statistical precision from the data of our exclusive dijet production paper [@DPEdijet], which are mainly due to variations in beam angle dispersion at the collision point and RPS alignment over the period of data taking. As the dijet data were taken concurrently with the diffractive $W$/$Z$ data, we use the acceptances of the dijet paper with an increased systematic uncertainty to account for differences in the $\xi$ and $t$ distributions expected between $W$/$Z$ and dijet production. The values being used are: $$A_{\rm RPS}= [83\pm 5(\rm syst.)]\%\mbox{~(Aug'02--Dec'03),}$$ $$A_{\rm RPS}= [78\pm 5(\rm syst.)]\%\mbox{~(Dec'04--Feb'06).}$$ The requirement on RPS trigger counter energy was made to remove splash events due to beam-related background and events with diffractive antiprotons outside the RPS acceptance. However, it also removes some diffractive signal events. The efficiency for this selection requirement to retain good diffractive events is determined as the fraction of all events with a reconstructed RPS track which pass the requirement on trigger counter energy. This efficiency, $\epsilon_{\rm RPStrig}$, varies with dataset within the range $68\%-80\%$, as the selection requirements were chosen independently for different running periods to account for ageing of the counters. The average RPS track reconstruction efficiency is [@DPEdijet] $$\epsilon_{\rm RPStrk}=[87\pm 1(\mbox{stat.})\pm 6(\mbox{syst.})]\%,$$ where the systematic uncertainty was chosen to bring consistency over all data sets. ### Double Pomeron exchange\[DPE/SD\] Double Pomeron exchange events are the sub-sample of the $W/Z$ SD events in which both the $\bar p$ and $p$ remain intact. As for the SD events, defined by the requirement of $\xi_{\bar p}^{\rm cal}\leqslant 0.1$, a DPE interaction should have in addition $\xi^{\rm cal}_{p}\leqslant 0.1$, where $\xi^{\rm cal}_{p}=\sum_{i=1}^{\rm all-towers}(E^i_T/ \sqrt{s})\,e^{+\eta^i}$. In this region of $\xi^{\rm cal}_{p}$ there are 45 $W$ and two $Z$ events in our candidate $W/Z$ data. Monte Carlo studies show that these are consistent with the expected numbers of SD events in which the gap on the proton side is due to multiplicity fluctuations, without any DPE contribution. Using fits of Monte Carlo templates to the data conducted according to Ref. [@Barlow], profile likelihood limits were set at the 95% confidence level of 1.5% and 7.8% on the fraction of diffractive $W$ and $Z$ events produced by DPE, respectively. These limits are consistent with the expectation of no observable signal due to the reduced collision s-value from 0.1s ($\sqrt s\sim 600$ GeV) in SD to $(0.1)^2s$ ($\sqrt s\sim 200$ GeV) in DPE. ### Search for exclusive $Z$ production\[exclZsearch\] We have examined whether any of the $Z$ candidate events in the DPE event sample are produced exclusively through the process ${\bar p}+p\rightarrow {\bar p}+Z+p$. In the Standard Model, this process is predicted to proceed by photoproduction, where a virtual photon radiated from the $\bar p$ ($p$) fluctuates into a $q\bar q$ loop which scatters elastically on the $p$ ($\bar p$) by two-gluon exchange forming a $Z$. Since $W$s cannot be produced exclusively, because there must be at least another charged particle in the event, we use $W$s as a control sample. A limit on exclusive $Z$ production has recently been published by the CDF collaboration. At a 95% confidence level, the exclusive $Z$ production cross section was found to be $\sigma_{\rm excl}^Z<0.96$ pb, a factor of ${\cal{O}}(10^{-3})$ of predictions based on the Standard Model (see Ref. [@exclZ]-[@albrow_forshaw]). The search method relied on strict excusivity requirements to ensure that nothing is present in the detector except for the two leptons from $Z\rightarrow l^{+l}l^{-}$. The present search is based on our data sample of $0.6$ fb$^{-1}$ integrated luminosity, which is a $\sim 30\%$ sub-sample of the events used in the above search. However, we use a different method which is more transparent to the background introduced by detector noise that can spoil the exclusivity requirement, and expand our event selection beyond the $Z$-mass window to include all $ee$($\mu\mu$) pairs with an electron (muon) $E_T(p_T)>25$ GeV(GeV/$c$). The method consists of comparing the total energy in the calorimeter to the dilepton mass. At first, we determine the dilepton mass $M_{ll}$ from the $e$ or $\mu$ momentum using calorimeter ($e$) or track ($\mu$) information. For $W$s, we evaluate $M_W$ using Eq. (\[eq:mw\]). Then, we determine the system mass $M_X$ from calorimeter towers, $M_X=\sqrt{(\Sigma E)^2-(\Sigma {\vec p})^2}$, with the caveat that we substitute the muon track information for the corresponding calorimeter tower, and in the case of $W$ events include the $\nu$ momentum. Thus, if nothing is present in the calorimeter besides the dilepton pair (or the $W$), the event will satisfy the condition $M_X=M_{ll}$ (or $M_X=M_W$). ![System mass $M_X$ vs $M_W$ (left) for $W$ events with a RPS track and $\xi^{\rm cal}<\xi^{\rm RPS}$, and mass $M_{ll}$ (right) for dilepton events with a RPS track and lepton $p_T>25$ GeV/$c$. Exclusive $W/Z$ candidates are expected to fall on the $M_X=M_W\,(M_{ll})$ line.[]{data-label="fig:WZ_fig6"}](WZ_fig6a.eps "fig:"){width="28.00000%"}![System mass $M_X$ vs $M_W$ (left) for $W$ events with a RPS track and $\xi^{\rm cal}<\xi^{\rm RPS}$, and mass $M_{ll}$ (right) for dilepton events with a RPS track and lepton $p_T>25$ GeV/$c$. Exclusive $W/Z$ candidates are expected to fall on the $M_X=M_W\,(M_{ll})$ line.[]{data-label="fig:WZ_fig6"}](WZ_fig6b.eps "fig:"){width="28.00000%"} Figure \[fig:WZ\_fig6\] shows $M_X$ versus $M_W$ ($M_{ll}$) for $W$ (dilepton) events with a RPS track. Note that $M_X$ depends on the calorimeter thresholds: increasing the thresholds could cause more events to move onto the $M_X=M_W\,(M_{ll})$ line, but these events would not be truly exclusive. The diffractive $W$ sample acts as a control sample for this type of background. Calorimeter noise could also move truly exclusive events off the $M_X=M_W\,(M_{ll})$ line. We looked at “empty crossings" collected using a beam-crossing trigger and selecting events with no reconstructed tracks or hits in the Cherenkov luminosity counters. The energy and momentum of the “noise" in the calorimeter for these events was added to the candidate exclusive events and the difference in reconstructed $M_X$ determined. The largest mean deviation found in the different running periods was 0.5 GeV, which is represented by the width of the $M_X=M_W\,(M_{ll})$ line in Fig. \[fig:WZ\_fig6\]. No candidates for exclusive production are observed within this band, neither in the control sample of the $W$ events (left) nor in the $Z$ event sample (right). This result is compatible with the upper bound on exclusive $Z$ production set in Ref. [@exclZ], as expected. ### Diffractive fractions The diffractive fractions $R_{\rm W}$ and $R_{\rm Z}$ for $0.03\!<\!\xi\!<\!0.10$ and $\|t\|\!<\!1$ (GeV/$c$)$^2$ are obtained after RPS acceptance and background corrections. The corrections include division by the product of the RPS acceptance, $A_{\rm RPS}$, the efficiency of the selection requirement on RPS trigger counter energy, $\epsilon_{\rm RPStrig}$, the RPS tracking efficiency, $\epsilon_{\rm RPStrk}$, and the fraction of ND events which are expected to have a single interaction, $N^{\rm 1-int}_{ND}=N_{ND}\cdot \langle f_{\rm 1-int}\rangle$. To account for SD events in which the proton, instead of the antiproton, remains intact we multiply $N^W_{SD}$ or $N^Z_{SD}\equiv N^Z_{\xi<0.10}-N^Z_{\rm bgnd}$ by a factor of two. $$R_W(R_Z)=\frac { 2\cdot N_{SD}^W(N_{SD}^Z) } { A_{\rm RPS}\cdot \epsilon_{\rm RPStrig}\cdot \epsilon_{\rm RPStrk} \cdot N^{\rm 1-int}_{ND} }$$ The resulting diffractive fractions are: $$\begin{aligned} R_{\rm W}=[1.00 \pm 0.05\,(\mbox{stat.}) \pm 0.10(\,\mbox{syst.})]\%,\\ R_{\rm Z}=[0.88 \pm 0.21(\,\mbox{stat.}) \pm 0.08(\,\mbox{syst.})]\%,\end{aligned}$$ where the systematic uncertainties are obtained from the contributions of the RPS acceptance and the trigger and tracking efficiencies. Conclusions =========== We have measured the fraction of events with a $W$ or $Z$ boson which are produced diffractively in ${\bar p}p$ collisions at $\sqrt{s}=1.96$ TeV using data from 0.6 fb$^{-1}$ of integrated luminosity collected with the CDF II detector incorporating a Roman-pot spectrometer to detect diffracted antiprotons. Within a region of antiproton [or proton]{} fractional momentum loss $\xi$ of $0.03<\xi<0.10$ and 4-momentum transferred squared $t$ of $-1<t<0$ (GeV/$c$)$^2$, we find that $(1.00\pm 0.11)\%$ of $W$s and $(0.88\pm 0.22)\%$ of $Z$s are produced diffractively, where the events in which the proton scatters diffractively and the antiproton dissociates are accounted for by doubling the measured proton dissociation fraction. We have also conducted a search for $W$ and $Z$ events produced by double-Pomeron exchange, $\bar p+p\rightarrow \bar p+[X+W/Z]+p$, and set confidence level upper limits of 1.5% and 7.7% on the fraction of DPE/SD events, respectively. Finally, we searched for exclusive $Z$ production, $\bar p+p\rightarrow \bar p+Z+p$. No exclusive $Z$ candidates were found within the DPE event sample, compatible with the CDF Run II published limit. Acknowledgments =============== We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium für Bildung und Forschung, Germany; the World Class University Program, the National Research Foundation of Korea; the Science and Technology Facilities Council and the Royal Society, UK; the Institut National de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation for Basic Research; the Ministerio de Ciencia e Innovación, and Programa Consolider-Ingenio 2010, Spain; the Slovak R&D Agency; and the Academy of Finland. [99]{} P. D. B. Collins, [An Introduction to Regge Theory and High Energy Physics]{}, Cambridge University Press, Cambridge, (1977). V. Barone and E. Predazzi, [High-Energy Particle Diffraction]{}, Springer Press, Berlin, (2002). S. Donnachie, G. Dosch, P. Landshoff, and O. Nachtmann, [Pomeron Physics and QCD]{}, Cambridge University Press, Cambridge, (2002). 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--- author: - \| [Qiuyi Dai]{}\ [Department of Mathematics, Hunan Normal University]{}\ [Changsha Hunan 410081, P.R.China]{}\ [E-mail: daiqiuyi@yahoo.com.cn]{}\ [Yuxia Fu]{}\ [Department of Applied Mathematics, Hunan University]{}\ [Changsha Hunan 410082, P.R.China]{}\ [E-mail: fu\_yuxia@yahoo.com.cn]{}\ title: 'Faber-Krahn inequality for Robin problem involving p-Laplacian ' --- [AMS subject classification:]{} 35P15, 35P30, 35J65, 35J70 [Key words:]{} Robin problem, p-Laplacian, Faber-Krahn type inequality Preface {#preface .unnumbered} ======= The result of this paper was announced at first on a conference held at the Wuhan Institute of Physics and Mathematics in May 2007. The final version of this paper was finished in September 2008 when the first author worked as a research fellow at The Australian National University. As soon as we completed our paper, we sent a copy of our preprint to D.Daners (see item 6 in the reference of [@sBDan]) since our result is related to a previous paper [@sDan] of him. Five months later, D.Bucur and D.Daners give an alternative proof of our result in February 2009. Though the paper has been published (see [@sBDan]), their proof depends completely on Proposition 2.2, Corollary 2.3 and Proposition 2.7 of this paper which, to our knowledge, can not been found in other materials. For completeness of their proof and the reader’s convenience, our paper will be published here. 1 Introduction {#introduction .unnumbered} ============== Let $\Omega \subset R^N(N\geq 2)$ be an open bounded smooth domain, we consider the following eigenvalue problem $$\left\{\begin{array}{ll}\label{eq1} -{\rm{div}}(\|\nabla u\|^{p-2}\nabla u)=\lambda \|u\|^{p-2}u & \mbox{in}\ \ \Omega,\\ \|\nabla u\|^{p-2}\frac{\partial u}{\partial \nu}+\beta \|u\|^{p-2}u=0 & \mbox{on}\ \ \partial\Omega,\\ \end{array} \right.$$ where $1<p<+\infty$, $\nu$ is the outward unit normal of $\partial\Omega$ and $\beta$ is a non-negative constant. The p-Laplacian ${\rm{div}}(\|\nabla u\|^{p-2}\nabla u)$ arises in many applications such as non-Newtonian fluids, quasi-regular and quasi-conformal mapping theory and Finsler geometry etc. An important special case of the p-Laplacian is the well known Laplacian $\Delta u={\rm{div}}(\nabla u)$ which corresponds to $p=2$. Problem (\[eq1\]) is called Dirichlet when $\beta=+\infty$, Neumann when $\beta=0$, and Robin when $0<\beta<+\infty$. The main purpose of this paper is to prove a Faber-Krahn type inequality for the Robin problem of the p-Laplacian. This inequality says that amongst all the domains of fixed volume, the ball has the smallest first eigenvalue. The study of this kinds of inequalities can be traced back to 1877 [@sRay]. Let $B$ denote a ball in $R^N$, and $\lambda^D_1(\Omega)$ denote the first eigenvalue of the following eigenvalue problem $$\label{eq2} \left\{ \begin{array}{ll} -\Delta\psi=\lambda\psi &x\in\Omega,\\ \psi=0 &x\in\partial\Omega. \end{array} \right.$$ Rayleigh [@sRay] conjectured that $$\label{eq3} \lambda^D_1(\Omega)\geq\lambda^D_1(B)\ \ \ \mbox{for}\ \ \Omega\subset R^N\ \ \ \mbox{with}\ \|\Omega\|=\|B\|,$$ and the equality hold if and only if $\Omega=B$. This conjecture was proved independently by Faber [@sFab] and Krahn [@sKra; @sKra1] in the 1920’s by making use of Schwartz symmetrization. Since then, the inequality (\[eq3\]) was known as Faber-Krahn inequality. In 1999, a proof of Faber-Krahn type inequality for the Dirichlet problem of the p-Laplacian was given by T. Bhattacherya [@sBha]. Recently, Faber-krahn type inequality was generalized to Robin problem of the Laplacian by M.H.Bossel [@sBos] for dimension $N=2$, and by D.Daners [@sDan] for dimension $N\geq 3$ but left the equality case open. A little bit later, D.Daners and J.Kennedy complete the proof of equality case in [@sDanK]. Note that the generalization of the Faber-Krahn inequality from Dirichlet problem to Robin problem is not trivial as, unlike in the Dirichlet problem, the first eigenvalue of Robin problem is not monotone as the domain expands (see [@sGio]). For more information of the Faber-Krahn type inequality on manifold , we refer to [@sIssa]. Since the level surface of the first eigenfunction of Robin problem intersects with the boundary $\partial\Omega$, the Schwartz symmetrization of the first eigenfunction generally does not decrease its Dirichlet integral and hence the Schwartz symmetrization method does not apply to the proof of Faber-Krahn inequality for Robin problem. Therefore, new approach must be employed in the proof of the Faber-Krahn inequality for Robin problem. The two crucial tools used by D.Daners [@sDan] to prove the Faber-Krahn inequality for Robin problem of the Laplacian are the Bessel functions and a new formula for the first eigenvalue by making use of level sets of the corresponding eigenfunction. To prove the Faber-Krahn type inequality for Robin problem of the p-Laplacian with $p\neq 2$, we mainly face two difficulties. One is the lack of Bessel functions and the other is the degeneracy of the operator. The tools we use to overcome these difficulties are some new abstract propositions of the first eigenfunction and some approximation procedure. The main results of this paper can be stated as the following 0.1in [[Theorem 1.1.]{} Let $1<p<+\infty$ and $\lambda_1(\Omega)$ be the first eigenvalue of problem (1.1) with $0<\beta<+\infty$. If $B$ is an open ball such that $\|B\|=\|\Omega\|$, then $\lambda_1(B)\leq\lambda_1(\Omega)$.]{} 0.1in [Remark.]{} Theorem 1.1 is proved under the assumption that $\Omega$ is smooth. However, by an approximation method similar to that used in [@sDan], we can prove that Theorem 1.1 is still true for the domains of Lipschitz type. 0.1in [[Theorem 1.2.]{} Let $1<p<+\infty$ and $B$ be a ball satisfying $\|B\|=\|\Omega\|$. If $\lambda_1(\Omega)=\lambda_1(B)$, then, up to a translation, we have $\Omega=B$.]{} 0.11in We also point out here that a symmetry result due to Gidas, Ni and Nirenberg [@sGNN] plays a crucial role in the proof of Theorem 1.2 when $p=2$ (see D.Daners and J.Kennedy [@sDanK]). However, this kind of result is not available for p-Laplace equation when $p>2$ and $p\neq N$ (see however [@sKeP] for the case $p=N$). Fortunately, we can prove a symmetry result needed in the proof of Theorem 1.2 in the special case of eigenvalue problem, though we can not prove more general symmetry result as in [@sGNN]. The contents of the rest of this paper are as follows: $\S 2$The First Eigenvalue and Eigenfunction. $\S 3$ Level Sets Formula of $\lambda_1(\Omega)$. $\S 4$ The lower bound of $\lambda_1(\Omega)$. $\S 5$ Proof of Theorem 1.1. $\S 6$ Proof of Theorem 1.2. 2 The First Eigenvalue and Eigenfunction {#the-first-eigenvalue-and-eigenfunction .unnumbered} ======================================== In this section, we give definition and some properties of the first eigenvalue and its corresponding eigenfunction of problem (\[eq1\]). We focus on the case $0<\beta<+\infty$, since the case $\beta=+\infty$ has been resolved and the case $\beta=0$ is trivial. Let $\mathcal{K}=\{u\in W^{1,p}(\Omega);\\|u\\|_{L^p(\Omega)}=1\}$. Let $$\label{eq4} \lambda_1(\Omega)=\inf\{\int_{\Omega}\|\nabla u\|^p+\beta\int_{\partial\Omega}\|u\|^p;\ u\in\mathcal{K}\}$$ be the first eigenvalue and $\psi$ the corresponding eigenfunction of problem (\[eq1\]). 0.2cm [[Proposition 2.1.]{} Let $\lambda_1(\Omega)$ be defined as in (\[eq4\]). Then $\lambda_1(\Omega)>0$ can be achieved by some positive function $\psi$.]{} 0.2cm [Proof.]{} Define functional $\Phi(u)$ on $\mathcal{K}$ by $$\Phi(u)=\int_{\Omega}\|\nabla u\|^p dx +\beta\int_{\partial \Omega}\|u\|^p dx.$$ It is obvious that $\Phi(u)$ is a convex functional. By Theorem 1.3 in Chapter 5 of [@sGuoD], $\Phi(u)$ is weakly lower semi-continuous on $\mathcal{K}$. Let $\{u_j\}_{j=1}^{\infty}$ be a minimum sequence of $\lambda_1$ on $\mathcal{K}$, that is, $\int_{\Omega}\|u_j\|^p dx=1$ and $$\int_{\Omega}\|\nabla u_j\|^p dx +\beta\int_{\partial \Omega}\|u_j\|^p dx\rightarrow \lambda_1(\Omega),\ \ \rm{as}\ \ j\rightarrow +\infty.$$ Since $\{u_j\}$ is bounded in $W^{1,p}(\Omega)$ and the embedding $W^{1,p}(\Omega)\hookrightarrow L^p(\Omega)$ is compact, there exists $u\in W^{1,p}(\Omega)$ such that $$u_j\rightharpoonup u \ \ \rm{weakly\ in}\ W^{1,p}(\Omega),$$ $$u_j\rightharpoonup u \ \ \rm{strongly\ in}\ L^p(\Omega).$$ Hence, by the weakly lower semi-continuity of $\Phi(u)$, we have $$\Phi(u)\leq \underline{\lim}_{j\rightarrow+\infty} \Phi(u_j)=\lambda_1(\Omega).$$ On the other hand, we have $\lambda_1(\Omega)\leq\Phi(u)$ due to $u\in\mathcal{K}$ and the definition of $\lambda_1(\Omega)$. Thus $\lambda_1(\Omega)=\Phi(u)$. Let $\psi=\|u\|$ , it is easy to check that $\lambda_1(\Omega)=\Phi(\psi)$. Moreover, $\psi$ is positive in $\bar{\Omega}$ by the strong maximum principle (see Lemma 2.6 below) . Thus, we complete the proof of proposition 2.1. 0.2cm [[Proposition 2.2.]{} Let $\lambda_1(\Omega)$ be the first eigenvalue of problem (\[eq1\]). Then $\lambda_1(\Omega)$ is simple in the sense that if $\psi_1>0$ and $\psi_2>0$ are two eigenfunctions corresponding to $\lambda_1(\Omega)$, then $\psi_2=C\psi_1$ and $C$ is a constant.]{} [Proof.]{} Suppose that $\psi_1$ and $\psi_2$ are two eigenfunctions corresponding to $\lambda_1(\Omega)$, and $\psi_1,\ \psi_2>0$. Then $\psi_i,\, i=1,2$ satisfy $$\label{eq5} \left\{\begin{array}{ll} -{\rm{div}}(\|\nabla\psi_i\|^{p-2}\nabla\psi_i)=\lambda \|\psi_i\|^{p-2}\psi_i & \mbox{in}\ \Omega,\\ \|\nabla\psi_i\|^{p-2}\frac{\partial\psi_i}{\partial \nu}+\beta \|\psi_i\|^{p-2}\psi_i=0 & \mbox{on}\ \partial\Omega.\\ \end{array} \right.$$ Let $$\eta_1=\psi_1-\psi_2^p\psi_1^{1-p}=\frac{\psi_1^p-\psi_2^p}{\psi_1^{p-1}}, \ \ \eta_2=\psi_2-\psi_1^p\psi_2^{1-p}=\frac{\psi_2^p-\psi_1^p}{\psi_2^{p-1}}.$$ Multiplying equation (\[eq5\]) by $\eta_i\ (i=1,2)$ and integrating by parts, we obtain $$\int_{\Omega}\|\nabla\psi_i\|^{p-2}\nabla\psi_i\cdot\nabla\eta_i+ \beta\int_{\partial\Omega}\psi_i^{p-1}\eta_i-\lambda_1\int_{\Omega}\psi_i^{p-1}\eta_i=0,\ \ (i=1,2).$$ It follows $$\label{eq6} \begin{array}{ll} &\int_{\Omega}(1+(p-1)(\frac{\psi_2}{\psi_1})^p)\|\nabla\psi_1\|^p+ \int_{\Omega}(1+(p-1)(\frac{\psi_1}{\psi_2})^p)\|\nabla\psi_2\|^p\\[3mm] &-\int_{\Omega}(p(\frac{\psi_2}{\psi_1})^{p-1}\|\nabla\psi_1\|^{p-2} +p(\frac{\psi_1}{\psi_2})^{p-1}\|\nabla\psi_2\|^{p-2})\nabla\psi_1\cdot\nabla\psi_2=0. \end{array}$$ Noticing that $\nabla(\ln\psi_i)=\frac{\nabla\psi_i}{\psi_i}$, (\[eq6\]) can be rewritten as $$\label{eq7} \begin{array}{ll} &\int_{\Omega}(\psi_1^p+(p-1)\psi_2^p)\|\nabla\ln\psi_1\|^p+(\psi_2^p+(p-1)\psi_1^p)\|\nabla \ln\psi_2\|^p\ \ \ \ \ \\[3mm] &=p\int_{\Omega}(\psi_2^p\|\nabla \ln\psi_1\|^{p-2}+\psi_1^p\|\nabla \ln\psi_2\|^{p-2})\nabla\ln\psi_1\cdot\nabla\ln\psi_2 \end{array}$$ Hence $$\label{eq8} \begin{array}{ll} \int_{\Omega}(\psi_1^p-\psi_2^p)(\|\nabla \ln\psi_1\|^p-\|\nabla \ln\psi_2\|^p)\\ = p\int_{\Omega}\psi_2^p\|\nabla \ln\psi_1\|^{p-2}(\nabla \ln\psi_1)\cdot(\nabla \ln\psi_2-\nabla \ln\psi_1)\\ -p\int_{\Omega}\psi_1^p\|\nabla \ln\psi_2\|^{p-2}(\nabla \ln\psi_2)\cdot(\nabla \ln\psi_2-\nabla \ln\psi_1). \end{array}$$ Observing that (see [@sLind]) $$\label{eq9} \|\xi_2\|^p-\|\xi_1\|^p\geq p \|\xi_1\|^{p-2}\xi_1\cdot(\xi_2-\xi_1)+C(p)\frac{\|\xi_2-\xi_1\|^p}{2^p-1},\ \ \forall\ \xi_1,\xi_2\in R^n,$$ we obtain $$\label{eq10} \begin{array}{ll} \|\nabla \ln\psi_1\|^p-\|\nabla \ln\psi_2\|^p\\ \geq p \|\nabla \ln\psi_2\|^{p-2}(\nabla \ln\psi_2)\cdot(\nabla \ln\psi_1-\nabla \ln\psi_2)+C_1(p)\frac{\|\nabla \ln\psi_1-\nabla \ln\psi_2\|^p}{2^p-1}, \end{array}$$ and $$\label{eq11} \begin{array}{ll} \|\nabla \ln\psi_2\|^p-\|\nabla \ln\psi_1\|^p\\ \geq p \|\nabla \ln\psi_1\|^{p-2}(\nabla \ln\psi_1)\cdot(\nabla \ln\psi_2-\nabla \ln\psi_1)+C_2(p)\frac{\|\nabla \ln\psi_2-\nabla \ln\psi_1\|^p}{2^p-1}, \end{array}$$ where $C(p)$, $C_1(p)$ and $C_2(p)$ are positive constants depend only on $p$. From (\[eq8\]), (\[eq10\]) and (\[eq11\]), we deduce $$-\frac{C_1(p)+C_2(p)}{2^p-1}\int_{\Omega}(\frac{1}{\psi_2^p}+\frac{1}{\psi_1^p}) \|\nabla\ln\psi_1-\nabla\ln\psi_2\|^p\geq 0.$$ This implies that $\nabla (\ln\psi_1-\ln\psi_2)=0$, namely, $\psi_2=C\psi_1$. This completes the proof of Proposition 2.2. 0.2cm [[Corollary 2.3.]{} If $\Omega=B(0)$ is a ball, then the first eigenfunction $\psi$ of problem (\[eq1\]) is radially symmetry, that is, $\psi(x)=\psi(r)$ with $r=\|x\|$.]{} [Proof.]{} The conclusion of Corollary 2.3 comes immediately from the simplicity of $\lambda_1(\Omega)$ and the rotational invariance of problem (\[eq1\]). 0.2cm To state our next proposition of the first eigenfunction, we need the following two lemmas which were proved in the appendix of [@sSaka]. 0.2cm [[Lemma 2.4.]{}(Weak comparison principle) Let $\Omega\in R^N$ be a bounded domain with smooth boundary $\partial\Omega$. Let $u_1, u_2\in W^{1,p}(\Omega)$ satisfy $$-{\rm{div}}(\|\nabla u_1\|^{p-2}\nabla u_1)\leq-{\rm{div}}(\|\nabla u_2\|^{p-2}\nabla u_2)\ \ \ \ \mbox{in}\ \ \Omega$$ in weak sense. Then $u_1\leq u_2$ on $\partial\Omega$ implies $u_1\leq u_2$ in $\Omega$.]{} 0.2cm [[Lemma 2.5.]{}(Hopf’s lemma) Let $\Omega\in R^N$ be a bounded domain with smooth boundary $\partial\Omega$. Let $u\in C^1(\overline{\Omega})$ satisfy $$\left\{\begin{array}{ll}\label{eq12} -{\rm{div}}(\|\nabla u\|^{p-2}\nabla u)\geq 0 & x\in\Omega,\\ u>0 & x\in\Omega. \end{array} \right.$$ If $u=0$ at $x_0\in\partial\Omega$, then $\frac{\partial u}{\partial\nu}(x_0)<0$, where $\nu$ denotes the unit outward vector normal to $\partial\Omega$.]{} 0.2cm We also need the following strong maximum principle which is a special case of Theorem 1.1 in [@sSerrin]. 0.2cm [[Lemma 2.6.]{}(Strong maximum principle) If $u\in C^1(\Omega)$ satisfies the following inequalities in weak sense $$\left\{\begin{array}{ll}\label{eq13} -{\rm{div}}(\|\nabla u\|^{p-2}\nabla u)\geq 0 & x\in\Omega,\\ u\geq 0 & x\in\Omega. \end{array} \right.$$ Then, $u(x_0)=0$ for some $x_0\in\Omega$ implies $u(x)\equiv 0$ in $\Omega$.]{} 0.2cm [[Proposition 2.7.]{} Let $B_R(0)$ be a ball in $R^N$ with radius $R$ and center $0$. If $\psi(x)=\psi(r)$ denotes the first eigenfunction of problem (\[eq1\]) on $B_R(0)$, then $\psi'(r)<0$ for any $0<r\leq R$.]{} 0.2cm [Proof:]{} For any fixed $r_0\in (0,R)$, we have $$\left\{\begin{array}{ll} -{\rm{div}}(\|\nabla\phi(r_0)\|^{p-2}\nabla \phi(r_0))\leq-{\rm{div}}(\|\nabla \phi(x)\|^{p-2}\nabla\phi(x))& x\in B_{r_0}(0)\\ \phi(x)=\phi(r_0) & x\in\partial B_{r_0}(0). \end{array} \right.$$ Hence, by lemma 2.4, we have $$\phi(x)\geq\phi(r_0)\ \ \ \ \ \ x\in B_{r_0}(0).$$ Since $\phi(x)$ is not a constant, it follows from lemma 2.6 that $$\phi(x)>\phi(r_0)\ \ \ \ \ \ x\in B_{r_0}(0).$$ Let $w(x)=\phi(x)-\phi(r_0)=\phi(r)-\phi(r_0)$. Then, $w(x)$ satisfies $$\left\{\begin{array}{ll} -{\rm{div}}(\|\nabla w(x)\|^{p-2}\nabla w(x))=\lambda_1\phi(x)>0 & x\in B_{r_0}(0)\\ w(x)>0 & x\in B_{r_0}(0)\\ w(x)=0 & x\in\partial B_{r_0}(0). \end{array} \right.$$ Consequently, lemma 2.5 implies that $\phi'(r_0)<0$. Noting that $r_0$ is arbitrary, the conclusion of proposition 2.7 then follows. 0.2cm We conclude this section with the following proposition which is essential for the proof of Theorem 1.1. 0.2cm [[Proposition 2.8.]{} Let $B_R(0)$ be a ball in $R^N$ with radius $R$ and center $0$. Let $\psi(x)=\psi(r)$ denote the first eigenfunction of problem (\[eq1\]) on $B_R(0)$. If $g(r)=\|\psi'(r)\|/\psi(r)$, then $g'(r)>0$ for $0<r<R$, and $g(r)\leq\beta^{\frac{1}{p-1}}$ for any $r\in [0,R]$.]{} 0.2cm [Proof:]{} It follows from Proposition 2.7 and the standard regularity theory of elliptic equations that $\psi\in C^{\infty}(B_R(0)\setminus\{0\})$. Consequently, $0<g\in C^{\infty}(0,R)$. Now, we compute $$\label{eq14} g'=(-\frac{\psi'}{\psi})'=-\frac{\psi''}{\psi}+g^2,$$ $$\label{eq15} g''=-\frac{\psi'''}{\psi}+3gg'-g^3.$$ From the equation satisfied by $\psi$, we have $$\label{eq16} -(p-1)\|\psi'\|^{p-2}\psi''-\frac{N-1}{r}\|\psi'\|^{p-2}\psi'=\lambda_1\psi^{p-1}.$$ It follows that $$\label{eq17} -(p-1)\psi''-\frac{N-1}{r}\psi'=\lambda_1\frac{\psi}{g^{p-2}}.$$ Differentiating the above equation, we obtain $$\label{eq18} -(p-1)\frac{\psi'''}{\psi}-\frac{N-1}{r}\frac{\psi''}{\psi}-\frac{N-1}{r^2}g=-\frac{\lambda_1}{g^{p-3}}- \frac{(p-2)\lambda_1g'}{g^{p-1}}.$$ Since $-\frac{\psi''}{\psi}=g'-g^2$, it follows from (\[eq17\]) that $$\label{eq19} \begin{array}{ll} \lambda_1 &=-(p-1)g^{p-2}\frac{\psi''}{\psi}+\frac{N-1}{r}g^{p-1}\\ &=(p-1)g^{p-2}g'-(p-1)g^p+\frac{N-1}{r}g^{p-1}. \end{array}$$ Hence $$-\frac{\lambda_1}{g^{p-3}}=-(p-1)gg'+(p-1)g^3-\frac{N-1}{r}g^2,$$ and $$-\frac{\psi'''}{\psi}=g^3+\frac{(N-1)g}{(p-1)r^2}-fg',$$ where $f=g+\frac{N-1}{(p-1)r}+\frac{(p-2)\lambda_1}{(p-1)g^{p-1}}$. Substituting this equation into (\[eq15\]), we infer that for any $r\in(0,R)$ $$g''(r)+[f(r)-3g(r)]g'(r)=\frac{(N-1)g(r)}{(p-1)r^2}>0.$$ we claim that $g'\neq 0$ in $(0,R)$. For if there exists $r_0\in(0,R)$ such that $g'(r_0)=0$, then $g''(r_0)>0$. Hence $r_0$ is a minimum point of $g$. Since $g\geq 0$ and $g(0)=0$, it follows from the continuity of $g$ that $g(r_0)=0$. This contradicts with the fact that $g>0$ in $(0,R)$. Consequently, $g'$ has definite sign in $(0,R)$. This implies immediately that $g'>0$ in $(0,R)$. For if $g'\leq 0$ in $(0,R)$, then we have $g(r)\leq g(0)=0$ in $(0,R)$, a contradiction. Finally, the a priori estimate $g(r)\leq\beta^{\frac{1}{p-1}}$ for any $r\in [0,R]$ follows from the facts that $g'>0$ and $g(R)=\beta^{\frac{1}{p-1}}$. This completes the proof of Proposition 2.8. 3 Level Sets Formula of $\lambda_1(\Omega)$ {#level-sets-formula-of-lambda_1omega .unnumbered} =========================================== For an open set $U\subset\Omega$, we define the interior and exterior boundary of $U$ respectively by $$\partial_I U=\partial U\cap\Omega,\ \ \ \partial_E U=\partial U\cap \partial\Omega.$$ Then $\partial U=\partial_I U\cup\partial_E U$ is a disjoint union. For any $\varphi\in C(\bar\Omega)$ and $\varphi(x)\geq 0$, we define a functional $H_{\Omega}(U,\varphi)$ by $$\label{eq20} H_{\Omega}(U,\varphi)=\frac{1}{\|U\|}(\int_{\partial_I U}\varphi\ d\sigma +\int_{\partial_E U} \beta\ d\sigma -(p-1)\int_U \varphi^{\frac{p}{p-1}}\ dx),$$ where $\sigma$ is the $(N-1)$-dimensional Hausdorff measure defined on $\partial U$ and $\|U\|$ is the Lebesgue measure of $U$. Since $\varphi$ is continuous on $\partial_I U$ and $U$, all integrals in $H_{\Omega}(U,\varphi)$ are well defined. In the following, we reformulate $\lambda_1(\Omega)$ by $H_{\Omega}(U,\varphi)$. To this end, we always denote by $\psi$ the first eigenfunction of (\[eq1\]) and sometimes denote $\psi$ by $\psi_{\Omega}$ when we want to emphasize the dependence of $\psi$ on the domain $\Omega$ . Furthermore, we choose $\psi$ so that $\psi>0$ and $\\|\psi\\|_{L^\infty(\Omega)}=1$. By regularity results of DiBenedetto [@sDiBen] and Tolksdorf [@sTolk; @sTolk1], we know that $\psi$ belongs to $C^{1,\alpha}(\overline{\Omega})$ for some $0<\alpha<1$. Let $$m=\min_{x\in \bar{\Omega}}\psi(x).$$ Then, by Hopf’s boundary point Lemma we have $m>0$. For any $t\in(m,1)$, we denote by $U_t$ the level set of $\psi$, that is $$U_t=\{x\in\Omega;\ \psi(x)>t\},$$ then $U_t$ is open and the interior boundary of $U_t$ is the level surface $$S_t=\partial_I U_t=\{x\in\Omega;\ \psi(x)=t\}.$$ Hence, $S_t=\emptyset$, if $t\not\in\ (m,1]$. Bearing all notations $\psi$, $U_t$ and $S_t$ in mind, we prove 0.2cm [[Proposition 3.1.]{} Let $\lambda_1(\Omega)$ be the first eigenvalue of problem (\[eq1\]), $\psi$ be the corresponding eigenfunction , and $H_{\Omega}(U,\varphi)$ be defined as in (\[eq20\]). Then $$\label{eq21} \lambda_1(\Omega)=H_{\Omega}(U_t,\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}})\ \ \ \ \mbox{for}\ \ t\in(m,1).$$]{} 0.2cm As in [@sDan], second partial derivatives of eigenfunction will be involved in the proof of Proposition 3.1. However, it is well known that, in general, the best possible regularity results of Problem (\[eq1\]) is $C^{1,\alpha}$. Hence, to prove Proposition 3.1, we consider the following regularized problem $$\label{eq22} \left\{ \begin{array}{ll} -{\rm{div}}[(\varepsilon u_{\varepsilon}^2+\|\nabla u_{\varepsilon}\|^2)^{\frac{p-2}{2}}\nabla u_{\varepsilon}]=\lambda^{\varepsilon} \|u_{\varepsilon}\|^{p-2}u_{\varepsilon}-\varepsilon(\varepsilon{u_{\varepsilon}}^2+\|\nabla u_{\varepsilon}\|^2)^{\frac{p-2}{2}}u_{\varepsilon},\ \ &x\in\Omega,\\[3mm] (\varepsilon u_{\varepsilon}^2+\|\nabla u_{\varepsilon}\|^2)^{\frac{p-2}{2}}\frac{\partial u_{\varepsilon}}{\partial\nu}+\beta u_{\varepsilon}^{p-1}=0,&x\in\partial\Omega. \end{array} \right.$$ where $1<p<+\infty$, $\nu$ is the outward unit normal of $\partial\Omega$. Define $\lambda_1^{\varepsilon}(\Omega)$ by $$\lambda_1^{\varepsilon}(\Omega)=\inf\{\int_{\Omega} (\varepsilon u_{\varepsilon}^2+\|\nabla u_{\varepsilon}\|^2)^{\frac{p}{2}}+\beta\int_{\partial \Omega}u_{\varepsilon}^p;\ u_{\varepsilon}\in\mathcal{K}\}.$$ Then, we have 0.2cm [[Lemma 3.2.]{} For any $\varepsilon>0$, $\lambda_1^{\varepsilon}(\Omega)$ is attained by a positive function $\psi_{\varepsilon}\in\mathcal{K}$. Moreover, up to a subsequence, we have $\lim\limits_{\varepsilon\rightarrow 0} \lambda_1^{\varepsilon}(\Omega) =\lambda_1(\Omega)$, and $$\lim\limits_{\varepsilon\rightarrow 0}\psi_{\varepsilon}=\widehat{\psi} \ \ \ \ \mbox{in}\ \ C^1(\overline{\Omega})$$ where $\lambda_1(\Omega)$ is the first eigenvalue of problem (\[eq1\]) and $\widehat{\psi}$ is a corresponding eigenfunction with $\|\|\widehat{\psi}\|\|_{L^p(\Omega)}=1$. ]{} 0.2cm [Proof.]{} The conclusion that $\lambda_1^{\varepsilon}(\Omega)$ is attained by a positive function $\psi_{\varepsilon}$ can be proved in the same way as that of Proposition 2.1. To prove the second part of Lemma 3.2, We first note that $\psi_{\varepsilon}$ is bounded in $W^{1,p}(\Omega)$ when $\varepsilon$ is small enough. Hence, up to a subsequence, we may assume that $$\psi_{\varepsilon}\rightharpoonup\tilde{\psi}\ \ \rm{weakly\ in\ } W^{1,p}(\Omega),\ \ \rm{as}\ \varepsilon\rightarrow 0.$$ Since $W^{1,p}(\Omega)\hookrightarrow L^p(\Omega)$ is compact, we also have $\tilde{\psi}\in\mathcal{K}$. Because convex functional is weakly lower semi-continuous, we have $$\label{eq23} \int_{\Omega}\|\nabla\tilde{\psi}\|^p+\beta\int_{\partial\Omega}\|\tilde{\psi}\|^p \leq\underline{\lim}_{\varepsilon\rightarrow 0}(\int_{\Omega}\|\nabla\psi_{\varepsilon}\|^p+\beta\int_{\partial\Omega}\|\psi_{\varepsilon}\|^p)$$ On the other hand, if $\widehat{\psi}$ is the first eigenfunction of problem (\[eq1\]) with $\|\|\widehat{\psi}\|\|_{L^p(\Omega)}=1$, then by the definition of $\lambda_1(\Omega)$ and $\lambda_1^\varepsilon(\Omega)$, we have $$\label{eq24} \begin{array}{ll} \lambda_1(\Omega)&=\int_{\Omega}\|\nabla\widehat{\psi}\|^p+\beta\int_{\partial\Omega}\|\widehat{\psi}\|^p \leq\int_{\Omega}\|\nabla\psi_{\varepsilon}\|^p+\beta\int_{\partial\Omega}\|\psi_{\varepsilon}\|^p\\ &\leq\int_{\Omega}(\varepsilon\psi_{\varepsilon}^2+\|\nabla\psi_{\varepsilon}\|^2)^{\frac{p}{2}} +\beta\int_{\partial\Omega}\psi_{\varepsilon}^p\\[3mm] &=\lambda_{\varepsilon}(\Omega)\leq \int_{\Omega}(\varepsilon\widehat{\psi}^2+\|\nabla\widehat{\psi\|}^2)^{\frac{p}{2}}+\beta\int_{\partial\Omega} \widehat{\psi}^p \end{array}$$ Let $\varepsilon\rightarrow 0$ on the both side of the above inequality, we obtain $$\label{eq25} \lim_{\varepsilon\rightarrow 0}\lambda_{\varepsilon}(\Omega)=\lim_{\varepsilon\rightarrow 0}\int_{\Omega}\|\nabla\psi_{\varepsilon}\|^p+\beta\int_{\partial\Omega}\|\psi_{\varepsilon}\|^p =\int_{\Omega}\|\nabla\widehat{\psi}\|^p+\beta\int_{\partial\Omega}\|\widehat{\psi}\|^p=\lambda_1(\Omega)$$ From (\[eq23\]) and (\[eq25\]), we infer that $$\int_{\Omega}\|\nabla\tilde{\psi}\|^p+\beta\int_{\partial\Omega}\|\tilde{\psi}\|^p \leq\int_{\Omega}\|\nabla\widehat{\psi}\|^p+\beta\int_{\partial\Omega}\|\widehat{\psi}\|^p=\lambda_1(\Omega).$$ Hence, $\tilde{\psi}$ is a minimizer of $\lambda_1(\Omega)$. This implies that $\tilde{\psi}=\widehat{\psi}$ due to the simplicity of $\lambda_1(\Omega)$ and $\|\|\tilde{\psi}\|\|_{L^p(\Omega)}=\|\|\widehat{\psi}\|\|_{L^p(\Omega)}=1$. Consequently, $\psi_{\varepsilon}\rightharpoonup\widehat{\psi}$ weakly in $W^{1,p}(\Omega)$ as $\varepsilon\rightarrow 0$. Finally, by the regularity theory of Tolksdorf [@sTolk; @sTolk1] and DiBenedetto [@sDiBen], we know that for $\varepsilon\in(0,1)$,there exists $\alpha\in(0,1)$ and a positive constant $C$ independent of $\varepsilon$ such that $\|\|\psi_{\varepsilon}\|\|_{C^{1,\alpha}}(\overline{\Omega})\leq C$. Hence, up to a subsequence, $\psi_\varepsilon$ converges to $\widehat{\psi}$ in $C^1(\overline{\Omega})$. This completes the proof of Lemma 3.2. 0.2cm [Proof of Proposition 3.1.]{} For any fixed $t\in(m,1)$, let $\nu$ denote the outward unit vector normal to $\partial U_t$. If we denote by $\psi_\varepsilon$ the solution of Problem (\[eq22\]) obtained in Lemma 3.2, then by the standard regularity theory of elliptic equations we know that $\psi_\varepsilon\in C^{\infty}(\Omega)$. Hence, by divergence Theorem, we have $$\label{eq26} \begin{array}{ll} -\int_{\partial U_t} \frac{(\varepsilon\psi_{\varepsilon}^2+\|\nabla\psi_{\varepsilon}\|^2)^{\frac{p-2}{2}}}{\psi_{\varepsilon}^{p-1}} \frac{\partial\psi_{\varepsilon}}{\partial\nu}d\sigma\\ =-\int_{U_t}{\rm{div}} (\frac{(\varepsilon\psi_{\varepsilon}^2+\|\nabla\psi_{\varepsilon}\|^2)^{\frac{p-2}{2}}\nabla\psi_{\varepsilon}}{\psi_{\varepsilon}^{p-1}})dx\\ =-\int_{U_t}\frac{{\rm{div}}((\varepsilon\psi_{\varepsilon}^2+\|\nabla\psi_{\varepsilon}\|^2)^{\frac{p-2}{2}}\nabla\psi_{\varepsilon})}{\psi_{\varepsilon}^{p-1}} dx +(p-1)\int_{U_t}\frac{(\varepsilon\psi_{\varepsilon}^2+\|\nabla\psi_{\varepsilon}\|^2)^{\frac{p-2}{2}}\|\nabla\psi_{\varepsilon}\|^2}{\psi_{\varepsilon}^p}dx\\ =\lambda_1^{\varepsilon}(\Omega)\|U_t\|-\varepsilon\int_{U_t}\frac{(\varepsilon\psi_\varepsilon^2+\|\nabla\psi_\varepsilon\|^2)^{\frac{p-2}{2}}}{\psi_\varepsilon^{p-2}} +(p-1)\int_{U_t}\frac{(\varepsilon\psi_{\varepsilon}^2+\|\nabla\psi_{\varepsilon}\|^2)^{\frac{p-2}{2}}\|\nabla\psi_{\varepsilon}\|^2}{\psi_{\varepsilon}^p}dx \end{array}$$ Passing to the limit in (\[eq26\]) as $\varepsilon\rightarrow 0$, we obtain $$\label{eq27} -\int_{\partial U_t} \frac{\|\nabla\widehat{\psi}\|^{p-2}}{\widehat{\psi}^{p-1}} \frac{\partial\widehat{\psi}}{\partial\nu}d\sigma =\lambda_1(\Omega)\|U_t\| +(p-1)\int_{U_t}\frac{\|\nabla\widehat{\psi}\|^{p}}{\widehat{\psi}^p}dx$$ Since $\lambda_1(\Omega)$ is simple, we have $$\label{eq50} -\int_{\partial U_t} \frac{\|\nabla\psi\|^{p-2}}{\psi^{p-1}} \frac{\partial\psi}{\partial\nu}d\sigma =\lambda_1(\Omega)\|U_t\| +(p-1)\int_{U_t}\frac{\|\nabla\psi\|^{p}}{\psi^p}dx.$$ By the boundary condition, we have $$\beta=-\frac{\|\nabla\psi\|^{p-2}}{\psi^{p-1}}\frac{\partial\psi} {\partial \nu},\ \ \ x\in \partial_E U_t,$$ due to $\partial_E U_t\subset \partial \Omega$. Noticing further that $\|\nabla\psi\|=-\frac{\partial \psi}{\partial\nu}$ on $S_t$, we obtain from the definitions of $S_t$ and $\partial_EU_t$ that $$\label{eq28} -\int_{\partial U_t}\frac{\ \ \|\nabla\psi\|^{p-2}}{\psi^{p-1}} \frac{\partial\psi}{\partial \nu}d\sigma=\int_{S_t}\frac{\|\nabla\psi\|^{p-2}}{\psi^{p-1}}\|\nabla\psi\| d\sigma+\int_{\partial_E U_t}\beta \ d\sigma .$$ Now, the conclusion of Proposition 3.1 follows from (\[eq50\]) and (\[eq28\]). 4 The lower bound of $\lambda_1(\Omega)$ {#the-lower-bound-of-lambda_1omega .unnumbered} ========================================= In this section, we give a lower bound of $\lambda_1(\Omega)$. Let $$\label{eq29} \mathcal{M}_{\beta}=\{\varphi(x)\in C(\Omega);\varphi(x)\geq0, \overline{\lim_{x\to z}}\varphi(x)\leq \beta,\ \ \forall z\in\partial\Omega\}.$$ Keep in use the same notations $\psi$, $U_t$ and $S_t$ as in the previous section. Since $\psi(x)\in C^1(\overline{\Omega})$, it is easy to see that $(\frac{\|\nabla\psi\|}{\psi})^{p-1}\in\mathcal{M}_{\beta}$ if and only if $\psi$ is a constant on $\partial \Omega$. In fact, if $\psi$ is a constant on $\partial \Omega$ then $\frac{\partial \psi}{\partial \nu}=-\|\nabla\psi\|$ on $\partial \Omega$. Hence $$\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}}=-\frac{\|\nabla\psi\|^{p-2}}{\psi^{p-1}}\frac{\partial \psi}{\partial \nu}=\beta\ {\rm {on}}\ \partial\Omega.\ \ \ $$ This implies that $(\frac{\|\nabla\psi\|}{\psi})^{p-1}\in\mathcal{M}_{\beta}$. On the other hand, if $(\frac{\|\nabla\psi\|}{\psi})^{p-1}\in\mathcal{M}_{\beta}$, then $$\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}}\leq\beta=-\frac{\|\nabla\psi\|^{p-2}}{\psi^{p-1}} \frac{\partial\psi}{\partial \nu}\leq\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}}\ \ \forall\ x\in\partial\Omega.$$ Hence $\frac{\partial\psi}{\partial\nu}=-\|\nabla\psi\|$ for all $x\in\partial\Omega$, which implies that $\psi$ is a constant on $\partial\Omega$. The main results of this section can be stated as 0.2cm [[Theorem 4.1.]{} For every $\varphi\in \mathcal{M}_{\beta}$, there exists a set $I\subset [0,1]$ with positive measure such that $$\label{eq30} \lambda_1(\Omega)\geq H_{\Omega}(U_t,\varphi)\ \ \ \ \mbox{for all}\ \ \ t\in I .$$]{} 0.2 cm [[Theorem 4.2.]{} Let $\varphi(x)\in\mathcal{M}_{\beta}$, and $\psi$ be the first eigenfunction of problem (\[eq1\]). If $\varphi\neq\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}}$, then there exists a set $I\subset[m,1]$ with positive measure such that $$H_{\Omega}(U_t,\varphi)<\lambda_1(\Omega) \ \ \mbox{for all}\ \ t\in I.$$]{} 0.2cm To prove Theorems, we prove some lemmas first. For any given $\varphi$ and $\varphi\geq0$, let $$\label{eq31} \omega(x):=\varphi(x)-\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}},\ \ x\in\Omega.$$ Then we have 0.2cm [[Lemma 4.3.]{} For any $\varphi\in \mathcal{M}_{\beta}$, let $\omega$ be defined as (\[eq31\]). Then for any $\varepsilon>0$ there exists $\delta>0$ such that $\omega(x)\leq\varepsilon$ for all $x\in\Omega$ with ${\rm{dist}}(x,\partial\Omega)<\delta$.]{} 0.2cm [Proof.]{} Since $\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}}$ is continuous on the compact set $\overline{\Omega}$, we have that for any fixed $\varepsilon>0$, there exists $\delta_0>0$ such that $$\label{eq32} \|\frac{\|\nabla\psi(x)\|^{p-1}}{\psi(x)^{p-1}}-\frac{\|\nabla\psi(z)\|^{p-1}} {\psi(z)^{p-1}}\|<\frac{\varepsilon}{2}\ \ \ {\rm{for\ any}}\ \ x,\ z\in\overline{\Omega},\ \|x-z\|<\delta_0.$$ For any fixed $z\in\partial\Omega$ and the above fixed $\varepsilon$, by the assumption that $\overline{\lim}_{x\to z}\varphi\leq\beta$, we can choose $r_z>0$ such that $$\sup_{x\in B(z,r_z)\cap\Omega}\varphi(x)\leq\beta+\frac{\varepsilon}{2},$$ that is $$\label{eq33} \varphi(x)-\beta\leq\frac{\varepsilon}{2},\ \ {\rm{for\ all}}\ x\in B(z,r_z)\cap\Omega,$$ where $B(z,r_z)$ denotes the ball with radius $r_z$ and center $z$ . Since the set $\{B(z,r_z),\ z\in\partial\Omega\}$ of balls form an open cover of the compact set $\partial\Omega$, we can select a finite sub-cover $\{B(z_i,r_i)\}_{i=1}^n$ with $r_i=r_{z_i}$. Let $\delta\leq \min\{r_1,r_2,\cdots,r_n,\delta_0\}$ be so small that $x\in\bigcup\limits_{i=1}^nB(z_i,r_i)$ whenever $x\in\Omega$ satisfying dist$(x,\partial\Omega)<\delta$. Then, for any $x\in\Omega$ with dist$(x,\partial\Omega)<\delta$, there exists $i_0\in\{1,2,\cdots,n\}$ such that $x\in B(z_{i_0},r_{i_0})$. By the boundary condition, we have $$\label{eq60} \beta=-\frac{\|\nabla\psi\|^{p-2}}{\psi^{p-1}}\frac{\partial \psi}{\partial \nu}(z_{i_0})\leq\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}}(z_{i_0}).$$ It follows from (\[eq32\]), (\[eq33\]) and (\[eq60\]) that for any $x\in\Omega$ with dist$(x,\partial\Omega<\delta$, we have $$\label{eq34} \begin{array}{ll} \omega(x)=\varphi(x)-\frac{\|\nabla\psi(x)\|^{p-1}}{\psi(x)^{p-1}} &=\varphi(x)-\beta+\beta-\frac{\|\nabla\psi(x)\|^{p-1}}{\psi(x)^{p-1}}\\ &\leq\frac{\varepsilon}{2}+\frac{\|\nabla\psi(z_{i_0})\|^{p-1}}{\psi(z_{i_0})^{p-1}} -\frac{\|\nabla\psi(x)\|^{p-1}}{\psi(x)^{p-1}}\\ &\leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon. \end{array}$$ This is just the desired conclusion of Lemma 4.3. 0.2cm [[Lemma 4.4.]{} Suppose that $\varphi\in C(\Omega)$ is non-negative such that $\varphi\in L^1(U)$ for every open set $U\subset\Omega$. Let $\omega$ be defined as (\[eq31\]). Set $$F(t):=\int_t^1\frac{1}{\tau}\int_{S_{\tau}}\omega d\sigma d\tau,\ \ {\rm{for}}\ t\in(m,1).$$ Then $F$ is absolutely continuous on $(\varepsilon,1)$ for all $\varepsilon\in(0,1)$ and $$\frac{d}{dt}F(t)=-\frac 1t\int_{S_t}\omega d\sigma,$$ for almost all $t\in(0,1)$. ]{} 0.2cm [Proof.]{} Fix $\varepsilon\in(0,1)$. By the assumption $\varphi\in C(\Omega)\cap L^1(U_{\varepsilon})$ and the co-area formula, we have $$\int_{\varepsilon}^1\frac{1}{\tau}\int_{S_{\tau}}\varphi\ d\sigma d\tau=\int_{U_\varepsilon}\frac{\varphi}{\psi}\|\nabla\psi\|\ dx<\infty$$ and $$\int_{\varepsilon}^1\frac{1}{\tau}\int_{S_{\tau}}\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}}\ d\sigma\ d\tau=\int_{U_\varepsilon}\frac{\|\nabla\psi\|^p}{\psi^p}\ dx<\infty.$$ Let $$f(\tau):=\frac{1}{\tau}\int_{S_{\tau}}\omega\ d\sigma,$$ Then $f(\tau)\in L^1((\varepsilon,1))$, thus $F(t)=\int_t^1f(\tau)\ d\tau$ is absolutely continuous on $(\varepsilon, 1)$ and differentiable almost everywhere. Moreover $$F'(t)=-f(t)=-\frac{1}{t}\int_{S_{t}}\omega\ d\sigma.$$ This completes the proof of Lemma 4.4. 0.2cm To state our next Lemma, we recall more regularity results of the first eigenfunction $\psi$. By the boundary condition and the Hopf’s boundary point Lemma, we know that $\psi(x)>0$ for any $x\in\partial\Omega$. Consequently $\|\nabla\psi\|(x)>0$ for any $x\in\partial\Omega$. Since $\partial\Omega$ is compact and $\|\nabla\psi\|\in C(\overline{\Omega})$, it is easy to prove that there exists positive number $\alpha$ and a neighborhood $N$ of $\partial\Omega$ in $\Omega$ such that $\|\nabla\psi\|(x)\geq\alpha>0$ for any $x\in N$. This implies that p-Laplacian is uniformly elliptic in $N$. Hence, by the interior regularity theorem of elliptic equations, we know that $\psi\in C^{\infty}(N)$. If we let $m=\min\{\psi(x);x\in\overline{\Omega}\}$ and $K=\{x\in\Omega;\psi(x)=m\}$, then by strong maximum principle we know that $K\subset\partial\Omega$. Noticing furthermore that $K$ is compact, there exists $t_0\in(m,1)$ small enough such that $$S_t\subset N\ \ \ \ \mbox{for any}\ \ \ t\leq t_0.$$ An argument similar to that used by Daners in [@sDan] implies the following lemma since all computations in [@sDan] are local. 0.2cm [[Lemma 4.5.]{} Let $U_t$ and $S_t$ be defined as in section 3. Then $U_t$ is a Lipschitz domain, moreover, there exist $t_1\in(m,t_0)$ and a constant $C>0$ independent of $t$ such that $\sigma(S_t)\leq C\sigma(\partial \Omega)$ for all $t\in (m,t_1)$.]{} 0.2cm [Proof of Theorem 4.1.]{} We give a proof by contradiction. Suppose that there exists $\varphi\in\mathcal{M}_{\beta}$ such that $$\label{eq35} \lambda_1(\Omega)<H_{\Omega}(U_t,\varphi)\ \ {\rm{for\ almost\ all}}\ t\in(m,1).$$ Let $\omega$ be defined as (\[eq31\]) and $F(t)$ be defined as in Lemma 4.4, that is $$\label{eq65} \omega(x):=\varphi(x)-\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}},\ \ x\in\Omega.$$ and $$F(t):=\int_t^1\frac{1}{\tau}\int_{S_{\tau}}\omega d\sigma d\tau,\ \ {\rm{for\ all}}\ t\in(m,1).$$ Then by (\[eq35\]), the definition of $H_{\Omega}(U_t,\varphi)$ and Proposition 3.1, we have $$\label{eq36} \int_{S_t}\omega\ d\sigma -(p-1)\int_{U_t}(\varphi^{\frac{p}{p-1}}-\frac{\|\nabla\psi\|^p}{\psi^p})\ dx=\|U_t\|[H_{\Omega}(U_t,\varphi)-\lambda_1(\Omega)]>0.$$ By Taylor’s expansion, there holds $$\label{eq37} \begin{array}{ll} \varphi^{\frac{p}{p-1}}-\frac{\|\nabla\psi\|^p}{\psi^p}= (\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}}+\omega)^{\frac{p}{p-1}} -\frac{\|\nabla\psi\|^p}{\psi^p}\\ =\frac{p}{p-1}(\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}})^{\frac{1}{p-1}}\omega +\frac 12\frac 1{p-1}\xi^{\frac p{p-1}-2}\omega^2, \end{array}$$ where $\xi$ is a nonnegative function with value between $\varphi$ and $\frac{\|\nabla\psi\|^{p-1}}{\psi^{p-1}}$. From (\[eq36\]), (\[eq37\]), the co-area formula and the definition of $F(t)$, we obtain $$\int_{S_t}\omega\ d\sigma>p\int_{U_t}\frac{\|\nabla\psi\|}{\psi}\omega\ dx= p\int_t^1\int_{S_\tau}\frac 1{\tau}\omega\ d\sigma d\tau=pF(t)$$ for almost all $t\in(m,1)$. It follows from Lemma 4.4 and the above inequality that $$\frac d{dt}(t^pF(t))=-t^pf(t)+pt^{p-1}F(t)=t^{p-1}(-\int_{S_t}\omega\ d\sigma+pF(t))<0$$ for almost all $t\in(m,1)$. Hence, the function $t^pF(t)$ is strictly decreasing on $(m,1)$. Since $F(1)=0$ and $F(t)$ is continuous on $(m,1)$, there exists $\eta>0$ and $t_2\in(m,1)$ such that $F(t)>\eta$ for $t\in(m,t_2].$ On the other hand, by Lemma 4.5, there exists $t_3\in (m,t_2]$ and a constant $C>0$ such that $\sigma(S_t)\leq C\sigma(\partial\Omega)$ for $t\in (m,t_3)$. Set $$\varepsilon_0=\frac{\eta}{C\sigma(\partial\Omega)}.$$ For this fixed $\varepsilon_0$, it follows from Lemma 4.3 that there exists $\delta_0>0$ such that $\omega(x)\leq\varepsilon_0$ for any $x\in\Omega$ with dist$(x,\partial\Omega)<\delta_0$. Noticing that $\psi$ attains its strict minimum on $\partial\Omega$, we can choose $0<t_4<t_3$ so small that dist$(x,\partial\Omega)<\delta_0$ for any $x\in S_t$ and $t\in(m,t_4)$. Hence, for any $t\in(m,t_4)$, there holds $$p\eta<pF(t)<\int_{S_t}\omega\ d\sigma\leq\varepsilon_0\sigma(S_t) \leq\varepsilon_0 C\sigma(\partial\Omega)\leq\eta$$ which is a contradiction. Thus we complete the proof of Theorem 4.1. 0.2cm [Proof of Theorem 4.2.]{} We give a proof by contradiction. Assume that $\varphi\neq\frac{\|\nabla \psi\|^{p-1}}{\psi^{p-1}}$ and that $$\label{eq38} H_{\Omega}(U_t,\varphi)\geq\lambda_1(\Omega),\ \ {\rm{for\ almost\ all }}\ t\in(m,1).$$ Similar to the proof of Theorem 4.1, by the definition of $H_{\Omega}(U_t,\varphi)$ and Proposition 3.1, we have $$\label{eq39} \int_{S_t}\omega\ d\sigma\geq pF(t)+\frac 12\frac{p}{p-1}\int_{U_t}\xi^{\frac{2-p}{p-1}}\omega^2dx\ \ \ \mbox{for almost all}\ \ t\in(m,1),$$ and $$\frac d{dt}(t^pF(t))\leq-\frac 12\frac{p}{p-1}\int_{U_t}\xi^{\frac{2-p}{p-1}}\omega^2dx\leq 0, \ \ \mbox{for almost all}\ \ t\in(m,1),$$ where $\xi$ is a nonnegative function with value between $\varphi$ and $\frac{\|\nabla \psi\|^{p-1}}{\psi^{p-1}}$. Hence $t^pF(t)$ is nonincreasing in $(m,1)$. Since $\omega(x)\in C(\Omega)$, $\omega(x)\not\equiv 0$ and $\bigcup\limits_{t\in(m,1)}U_t=\Omega$, there exists $t_0\in(m,1)$ such that $$\label{eq40} \int_{U_{t_0}}\xi^{\frac{2-p}{p-1}}\omega^2dx> 0$$ Moreover, if $t_1,t_2\in(m,1)$ satisfy $t_1<t_2$, then we have $U_{t_2}\subset U_{t_1}$. Hence, the map $$t\mapsto\int_{U_t}\xi^{\frac{2-p}{p-1}}\omega^2dx$$ is non-increasing in $(m,1)$ and $\int_{U_1}\xi^{\frac{2-p}{p-1}}\omega^2dx=0$ due to $U_1=\emptyset$. Let $$t^=\sup\{t\in(m,1),\ \ \int_{U_t}\xi^{\frac{2-p}{p-1}}\omega^2dx>0\}.$$ From (\[eq40\]), we know that $t^\in(m,1]$ and thus $t^pF(t)$ is strictly decreasing on $(m,t^)$ and non-increasing on $[t^,1]$, similar to the proof of Theorem 4.1, there exists $t_3\in(m,t^)$ such that for any $t\in(m, t_3)$, $$p\eta<pF(t)<\int_{S_t}\omega\ d\sigma\leq\varepsilon\sigma(S_t)\leq\eta,$$ which is a contradiction. Hence, we complete the proof of Theorem 4.2. 5 Proof of Theorem 1.1 {#proof-of-theorem-1.1 .unnumbered} ====================== This section devotes to prove Theorem 1.1. To this end, we denote by $\lambda_1(\Omega)$ the first eigenvalue of problem (\[eq1\]) on the domain $\Omega$ and $\psi_{\Omega}$ denotes its corresponding eigenfunction. Furthermore, $B=B_R(0)$ denotes the ball with radius $R$ and center $0$ such that $\|B\|=\|\Omega\|$. Let $U_t$ be the level set and $S_t$ be the level surface of $\psi_{\Omega}$ at level $t$ defined in section 2, and $B_{r(t)}(0)$ be the ball with radius $r(t)$ and center $0$ such that $\|B_{r(t)}(0)\|=\|U_t\|$. Define $$\Phi_B(x)=\frac{\|\nabla\psi_{B}(x)\|^{p-1}}{\psi_{B}^{p-1}(x)}\ \ \ \mbox{for}\ \ \ x\in B_R(0).$$ By Corollary 2.3, $\Phi_B$ is radially symmetry. So, we only need to consider the radial function $$G(r)=\Phi_B(\|x\|)=\frac{\|\psi_B'(r)\|^{p-1}}{\psi_B^{p-1}(r)}=g^{p-1}(r)\ \ \ \mbox{for}\ \ \ r\in(0,R)$$ where $g(r)$ is the function defined in Proposition 2.8. Then by Proposition 2.8, we know that $G(r)$ is strictly increasing in $(0,R)$. Consequently, $G(r)\leq G(R)=\beta$ for any $r\in [0,R]$. we construct our test function as the following. For any $t\in(m,1)$ and $x\in S_t$, we set $$\Phi(x)=G(r(t)).$$ It is obvious that $\Phi$ is well defined since $\Omega$ is a disjoint union of $S_t$, $t\in(m,1]$. Moreover, $\Phi\in\mathcal{M}_\beta(\Omega)$ due to $\Phi$ is continuous and $\Phi(x)\leq\beta$ for all $x\in\overline{\Omega}$. It is also not too difficult to see that $$\label{eq41} \int_{U_t}\|\Phi\|^{\frac{p}{p-1}}dx=\int_{B_{r(t)}}\Phi_B^{\frac{p}{p-1}}dx.$$ Since by the construction the level sets of $\Phi$ and $\Phi_B$ have the same measure. Now, we are in a position to prove Theorem 1.1. 0.2cm [Proof of Theorem 1.1.]{} Since $\Phi\in\mathcal{M}_{\beta}$, we conclude from Theorem 4.1 that there exist a set $I\subset(m,1)$ with positive measure such that $$\label{eq42} \lambda_1(\Omega)\geq H_{\Omega}(U_{t},\Phi)\ \ \ \mbox{for\ all}\ \ t\in I .$$ Noticing that $\sigma(\partial B_{r_t})\leq\sigma(\partial U_t)$ for all $t\in(m,1]$, and $\Phi(x)=G(r(t))\leq\beta$ when $x\in S_t$, we have $$\label{eq43} \begin{array}{ll} \int_{\partial B_{r(t)}}\Phi_B(x)d\sigma=G(r(t))\sigma(\partial B_{r(t)})&\leq G(r(t))\sigma(\partial U_t)\\ &=G(r(t))(\int_{S_t}d\sigma +\int_{\partial_EU_t}\ d\sigma)\\ &\leq\int_{S_t}\Phi d\sigma +\int_{\partial_E U_t}\beta d\sigma. \end{array}$$ Hence, from (\[eq41\]), (\[eq43\]) and the definitions of $H_{B}(B_{r(t)}(0),\Phi_B)$ and $H_{\Omega}(U_t,\Phi)$, we have $$\label{eq44} H_{B}(B_{r(t)}(0),\Phi_B)\leq H_{\Omega}(U_t,\Phi)\ \ \ \ \forall t\in(m,1).$$ Since, by Proposition 3.1, we have $\lambda_1(B)=H_{B}(B_{r(t)}(0),\Phi_B)$ for any $t\in(m,1)$, it follows from (\[eq42\]) and (\[eq44\]) that $$\lambda_1(\Omega)\geq\lambda_1(B).$$ This completes the proof of Theorem 1.1. 6 Proof of Theorem 1.2 {#proof-of-theorem-1.2 .unnumbered} ====================== This section devotes to prove Theorem 1.2. To this end, we keep in use of all notations in section 5, and prove some lemmas first. 0.2cm [[Lemma 6.1.]{} Suppose that $\Omega$ satisfies that $\lambda_1(\Omega)=\lambda_1(B)$ with $\|\Omega\|=\|B\|$. Then $$\Phi=\frac{\|\nabla \psi_{\Omega}\|^{p-1}}{\psi_{\Omega}^{p-1}}\ \ \ \ \mbox{and}\ \ \ H_{\Omega}(U_t,\Phi)=\lambda_1(B),$$ for almost all $ t\in(m,1)$.]{} 0.2cm [Proof.]{} If $\lambda_1(\Omega)=\lambda_1(B)$, then by Proposition 3.1 and (5.4), we have $\lambda_1(\Omega)=\lambda_1(B)=H_B(B(r(t)),G)\leq H_{\Omega}(U_t,\Phi)$ for almost all $t\in(m,1)$. Hence by Theorem 4.2, $\Phi=\frac{\|\nabla \psi_{\Omega}\|^{p-1}}{\psi_{\Omega}^{p-1}}$. Again, by Proposition 3.1, we obtain $H_{\Omega}(U_t,\Phi)=\lambda_1(\Omega)=\lambda_1(B)$, for almost all $t\in(m,1)$. 0.2cm [[Lemma 6.2.]{} Let $\psi_{\Omega}$ be the eigenfunction corresponding to the first eigenvalue $\lambda_1(\Omega)$, and $U_t$ be the level set of $\psi_{\Omega}$. Then $H_{\Omega}(U_t,\Phi)=\lambda_1(B)$ if and only if $U_t$ is a ball and $\sigma(\partial_EU_t)=0$.]{} 0.2cm [Proof.]{} It follows from Proposition 3.1 that $\lambda_1(B)=H_B(B(r(t)),G)$ for all $t\in(m,1)$. By the construction of $G$ and $\Phi$, we know that the level sets of $G$ and $\Phi$ have the same measure. Hence $$\int_{U_t}\|\Phi\|^{\frac p{p-1}} dx=\int_{B_{r(t)}}\|G\|^{\frac p{p-1}} dx,\ \ {\rm{for\ all}}\ t\in(m,1).$$ Using the definitions of $H_{\Omega}(U,\varphi)$ and $\Phi$, we have $$\label{eq45} \begin{array}{ll} H_{\Omega}(U_t,\Phi)&=\frac{1}{\|U_t\|}(\int_{\partial_I U_t}\Phi d\sigma +\int_{\partial_E U_t} \beta d\sigma -(p-1)\int_{U_t} \Phi^{\frac{p}{p-1}} dx),\\[3mm] &=\frac{1}{\|B_{r(t)}\|}[G(r(t))\sigma(S_t)+\beta \sigma(\partial_EU_t)-(p-1)\int_{B_{r(t)}} G^{\frac{p}{p-1}}dx]. \end{array}$$ If $U_t$ is a ball and $\sigma(\partial_EU_t)=0$, then $\sigma(S_t)=\sigma(\partial B_{r(t)})$ and $$\label{eq46} \begin{array}{ll} H_{\Omega}(U_t,\Phi)=\frac{1}{\|B_{r(t)}\|}[G(r(t))\sigma(\partial B_{r(t)}) -(p-1)\int_{B_{r(t)}} G^{\frac{p}{p-1}}dx]\\ =H_B(B_{r(t)},G)=\lambda_1(B). \end{array}$$ Conversely, if $H_{\Omega}(U_t,\Phi)=\lambda_1(B)$, then for this $t$, $$G(r(t))\sigma(S_t)+\beta \sigma(\partial_EU_t)=G(r(t))\sigma(\partial B_{r(t)}).$$ Noticing that $S_t=\partial_I U_t=\partial U_t-\partial_E U_t$, we have $$\sigma(\partial_EU_t)(\beta-G(r(t)))=G(r(t))(\sigma(\partial B_{r(t)})-\sigma(\partial U_t)).$$ This is only possible when $\sigma(\partial_EU_t)=0$ and $\sigma(\partial B_{r(t)})=\sigma(\partial U_t)$, since $0<G(r(t)))<\beta$ for all $t\in(m,1)$ and $\|B_{r(t)}\|=\|U_t\|$ implies $\sigma(\partial B_{r(t)})\leq\sigma(\partial U_t)$. But we know that the ball is the unique minimizer of the isoperimetric inequality. Hence, $U_t=B_{r(t)}+z$ for some $z\in R^N$. This completes the proof of Lemma 6.2. 0.2cm [[Lemma 6.3.]{} Assume that $u(x)\geq 0$ satisfies that $-{\rm{div}}(\|\nabla u\|^{p-2}\nabla u)=\lambda u^{p-1}$ in $\Omega$ for some $\lambda>0$. Suppose further that for some $t>0$ the level set $\{ x\in\Omega, u(x)>t\}=B_{r(t)}(x_0)$ is a ball with radius $r(t)$ and center $x_0$. If $u\in C(\overline{B}_{r(t)}(x_0))$ and $\sigma(\partial_E B_{r(t)}(x_0))=0$, then $u$ is radially symmetric with respect to $x_0$ in $B_{r(t)}(x_0)$.]{} 0.2cm This lemma is crucial to the proof of Theorem 1.2. In the case $p= 2$, the conclusion of the Lemma 6.3 is a famous result due to Gidas, Ni and Nirenberg [@sGNN] (see also Corollary 3.4 in [@sFra]). In the case $1<p<2$, the conclusion of Lemma 6.3 was given in [@sDama]. In the case $p=N$, the conclusion of Lemma 6.3 was proved in [@sKeP]. However, the conclusion of Lemma 6.3 for the case $p>2$ and $p\neq N$ is not available so far. Here, we give a proof of Lemma 6.3 for all $p\in(1,+\infty)$. 0.2cm [Proof of Lemma 6.3.]{} By the assumption, we know that for the same $t>0$ in the above Lemma, $u(x)$ is a solution of the following Dirichlet problem $$\label{eq47} \left\{ \begin{array}{ll} -{\rm{div}}(\|\nabla v\|^{p-2}\nabla v)=\lambda v^{p-1} &\mbox{in}\ B_{r(t)}(x_0),\\ v\geq 0 & \mbox{in}\ B_{r(t)}(x_0),\\ v=t &\mbox{on}\ \partial B_{r(t)}(x_0). \end{array} \right.$$ By Lemma 2.4 and Lemma 2.5, we know that any solution of problem (\[eq47\]) is strictly positive in $\overline{\Omega}$. Since problem (\[eq47\]) is invariant under rotation, we can prove Lemma 6.3 by proving that uniqueness theorem is valid for (\[eq47\]). To this end, we denote $B_{r(t)}(x_0)$ by $\Omega$ for simplicity, and suppose that $v_1>0$ and $v_2>0$ are two solutions of problem (\[eq47\]). Then $v_i(x)\ (i=1,2)$ satisfy $$\label{eq48} \left\{\begin{array}{ll} -{\rm{div}}(\|\nabla v_i\|^{p-2}\nabla v_i)=\lambda \|v_i\|^{p-2}v_i & \mbox{in}\ \Omega,\\ v_i=t & \mbox{on}\ \partial\Omega.\\ \end{array} \right.$$ Let $$\eta_1=v_1-v_2^pv_1^{1-p}=\frac{v_1^p-v_2^p}{v_1^{p-1}}, \ \ \eta_2=v_2-v_1^pv_2^{1-p}=\frac{v_2^p-v_1^p}{v_2^{p-1}}.$$ It is obvious that $\eta_i=0\ (i=1,2)$ on $\partial\Omega$. Multiplying equation (\[eq48\]) by $\eta_i\ (i=1,2)$ and integrating by parts, we obtain $$\int_{\Omega}\|\nabla v_i\|^{p-2}\nabla v_i\cdot\nabla\eta_i -\lambda\int_{\Omega}v_i^{p-1}\eta_i=0,\ \ (i=1,2).$$ By a similar argument to that used in the proof of Proposition 2.2, we infer that $\nabla (\ln v_1-\ln v_2)=0$, namely, $v_2=Cv_1$ for some constant $C$. Since $v_1(x)=v_2(x)=t$ for $x\in\partial\Omega$, we obtain that $C=1$ and $v_1(x)\equiv v_2(x)$ on $\overline{\Omega}$. Hence, the solution of problem (\[eq47\]) is unique, and hence, the symmetry result of Lemma 6.3 follows. 0.2cm [Proof of Theorem 1.2.]{} Let $\Omega$ satisfy $\lambda_1(\Omega)=\lambda_1(B)$ and $\|\Omega\|=\|B\|$, $U_t$ be the level set of eigenfunction $\psi_{\Omega}$ correspond to $\lambda_1(\Omega)$. Then by Lemma 6.1, $H_{\Omega}(U_t,\Phi)=\lambda_1(B)$ for almost all $t\in (m,1)$, and so, $U_t$ is a ball for any $t\in(m,1)$ and $\sigma(\partial_EU_t)=0$ by Lemma 6.2. At this stage, Lemma 6.3 implies that $\psi_{\Omega}$ is radially symmetry inside $U_t$, and all interior level sets $U_{\tau}$ for $\tau\in(t,1)$ are concentric balls. In particular, for all $t\in(m,1]$, the level sets $U_t$ are concentric balls. Therefore, $\Omega=\bigcup_{t\in (m,1)}U_t$ is a ball. 0.5cm [Acknowledgements*]{}. Heartfelt thanks are given to Professor Xu-jia Wang for many invaluable comments. 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--- abstract: 'In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is expressed uniformly in terms of natural parameters of the group. In the case of factorizations of minimal length, we recover a formula due to P. Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex case. For the symmetric group, our formula specializes to a formula of D. M. Jackson.' address: - 'CNRS, LIAFA, Université Paris Diderot – Paris 7' - 'Institut für Algebra, Zahlentheorie, Diskrete Mathematik, Universität Hannover' author: - Guillaume Chapuy - Christian Stump bibliography: - 'ChapuyStump2012.bib' title: \| Counting factorizations of Coxeter elements\ into products of reflections --- [^1] Introduction ============ One of the many equivalent forms of Cayley’s formula [@Cay1889] counting labeled trees asserts that the number of factorizations of the long cycle $(1,2,\dots,N)$ as a product of $(N-1)$ transpositions is $$\begin{aligned} \#\big\{ \tau_1 \tau_2 \cdots \tau_{N-1} = (1,2,\ldots,N) \big\} = N^{N-2}.\end{aligned}$$ A widely considered generalization of this formula comes from the field of map enumeration. Instead of considering minimal factorizations, one can as well consider higher genus factorizations, which are factorizations of the long cycle into $N-1+2g$ transpositions. The term genus and the letter $g$ come from the fact that this $g$ is indeed the genus of a surface associated to a natural embedded graph representing the factorization, see for example [@LandoZvonkine] for further details. The following exponential generating function identity, proved in [@Jac1988 Corollary 4.2], provides a beautiful and compact way to count such factorizations, $$\begin{aligned} \sum_{g\geq 0} \frac{t^{N-1+2g}}{(N-1+2g)!}\#\Big\{ \tau_1 \tau_2\dots\tau_{N-1+2g}=(1,2,\dots,N)\Big\} = \frac{1}{N!}\left(e^{t\frac{N}{2}}-e^{-t\frac{N}{2}}\right)^{N-1}.\end{aligned}$$ Observe that near $t=0$ this formula has the expansion $\frac{t^{N-1}}{(N-1)!} N^{N-2}$, in agreement with Cayley’s formula. Another way to generalize Cayley’s formula is to count factorizations of Coxeter elements in real reflection groups, and, even more generally, in well-generated complex reflection groups. If we replace the symmetric group by any well-generated complex reflection group $W$ of rank $n$ with Coxeter number $h$, the number of factorizations of a fixed Coxeter element $c \in W$ into a product of $n$ reflections is given by the formula $$\begin{aligned} \label{eq:DTZ} \#\big\{ \tau_1 \tau_2 \dots \tau_n = c \big\} = \frac{n!}{\|W\|}h^n.\end{aligned}$$ In the case of real reflection groups, this formula was proved in the early 1970’s in a letter from P. Deligne to E. Looijenga [@Deligne] crediting discussions with J. Tits and D. Zagier. The remaining cases were then proved by D. Bessis in [@Bes2007 Proposition 7.5], using the geometry of braid groups. Formula was as well rediscovered for real reflection groups in the context of the noncrossing partition lattice, see [@Chapoton Proposition 9], [@Armstrong Corollary 3.6.10], and [@Reading Theorem 3.6]. For the symmetric group ${\mathcal{S}}_N$ the rank is given by $n=N-1$ and the Coxeter number is $h=N$, we thus again get back Cayley’s formula. In this paper, we provide a uniform generalization of both results by counting higher genus factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. The main result is the following identity for their exponential generating function. We refer to Section \[sec:background\] for the definitions used here. \[thm:main\] Let $W$ be an irreducible well-generated complex reflection group of rank $n$. Let $c$ be a Coxeter element in $W$, let ${\mathcal{R}}$ be the set of all reflections in $W$, and let ${\mathcal{R}}^$ be the set of all reflecting hyperplanes. Define $$\begin{aligned} {{\operatorname{FAC}}}_W(t) := \sum_{\ell\geq 0} \frac{t^\ell}{\ell!} \#\Big\{ (\tau_1,\tau_2,\dots,\tau_\ell)\in \mathcal{R}^\ell\ ,\ \tau_1 \tau_2 \dots \tau_\ell = c \Big\}\end{aligned}$$ to be the exponential generating function of factorizations of $c$ into a product of reflections. Then ${{\operatorname{FAC}}}_W(t)$ is given by the formula $$\begin{aligned} {{\operatorname{FAC}}}_W(t) = \frac{1}{\|W\|}\left(e^{t\|{\mathcal{R}}\|/n}-e^{-t\|{\mathcal{R}}^\|/n}\right)^{n}. \label{eq:coxetergenus}\end{aligned}$$ The Coxeter number $h$ is given by $(\|{\mathcal{R}}\|+\|{\mathcal{R}}^\|)/n$, see Formula in Section \[sec:background\] below. Therefore the expansion of near $t=0$ gives back Formula . Moreover, in the case of real reflection groups one has $\|{\mathcal{R}}\|=\|{\mathcal{R}}^\|$, so the exponents in the righthand side of are given by $th/2$ and $-th/2$. For real reflection groups, Theorem \[thm:main\] has thus the following form. \[cor:Coxeter\] Let $W$ be an irreducible real reflection group of rank $n$ and Coxeter number $h$. Then ${{\operatorname{FAC}}}_W(t)$ is given by the formula $$\begin{aligned} \label{eq:corollary} {{\operatorname{FAC}}}_W(t) = \frac{1}{\|W\|}\left(e^{th/2}-e^{-th/2}\right)^{n}. \end{aligned}$$ \[rem:automorphism\] There are several conventions in the literature for the definition of Coxeter elements in well-generated complex reflection groups. For the definition we use here, there might be several conjugacy classes ${\mathcal{C}}_\zeta$ of Coxeter elements for the various primitive $h$-th roots of unity $\zeta$. One of these conjugacy classes corresponds to the Coxeter elements in the sense of the more restrictive definition used for example in [@Bes2007]. Notice that, strictly speaking, the definition of ${{\operatorname{FAC}}}_W(t)$ depends a priori on the conjugacy class of the chosen Coxeter element. Therefore, Theorem \[thm:main\] implies as well that the number of factorizations is independent of this choice, which is why we suppress the information of the conjugacy class in the notation ${{\operatorname{FAC}}}_W(t)$. This seems to illustrate a more general phenomenon, namely that enumerative properties of Coxeter elements seem to be valid for the more general definition. For the infinite families of matrix groups defined in Section \[sec:background\], this phenomenon can easily be explained by the existence of automorphisms of the group, inherited from automorphisms of the base field $\mathbb{Q}[\zeta]$, that preserve reflections and send one class of Coxeter elements to another. For exceptional groups, however, we do not know of such a simple explanation[^2]. We refer the reader to Section \[sec:background\] for more details on our definition and for a comparison with other definitions. Finally, we want to emphasize again that Theorem \[thm:main\] is in particular true with the more restrictive definition of Coxeter elements. There is no loss of generality in only considering irreducible groups in Theorem \[thm:main\] and Corollary \[cor:Coxeter\]. This is meant in the following sense. Let $W$ be a well-generated reducible complex reflection group and write $W=W_1\times \dots \times W_k$, where the $W_i$ are the irreducible components. The Coxeter elements in $W$ are, by definition, the product of the Coxeter elements of each $W_i$, and the set of reflections in $W$ is the union of the set of reflections in all $W_i$, and one thus obtains immediately that ${{\operatorname{FAC}}}_W(t)$ can be computed by applying Theorem \[thm:main\] to all its irreducible components, $${{\operatorname{FAC}}}_W(t) = {{\operatorname{FAC}}}_{W_1}(t) \cdots {{\operatorname{FAC}}}_{W_k}(t).$$ Our proof of Theorem \[thm:main\] uses the classification of complex reflection groups. We first treat the case of the infinite families, and then check the finitely many remaining exceptional cases. This is possible since one can encode the a priori infinite sum ${{\operatorname{FAC}}}_W(t)$ as a finite sum over all irreducible characters, see Formula below. Of course, it is natural to hope for a uniform proof of this theorem, i.e., a proof that does not rely on the classification. However, we point out that even in the simpler case of Formula , whose proof goes back to [@Deligne], only a uniform recurrence formula is known that can be used to prove this formula case-by-case. Similarly, multiple results in [@Chapoton; @Armstrong; @Reading; @Bes2007; @BR2007] rely on the classification either by directly using case-by-case analysis, or by relying on the counting formula for noncrossing partitions for which still no uniform proof is known either. In Section \[sec:background\], we recall needed background on complex reflection groups. In Section \[sec:dihedralgroup\], we prove the main theorem for the cyclic groups and the dihedral groups by explicit computations. In Section \[sec:representationtheory\], we recall a general approach to the enumeration of factorizations in groups via representation theory. In Section \[sec:classicaltypes\], we prove the main theorem for the infinite families of well-generated complex reflection groups. In Section \[sec:exceptionaltypes\], we then present computer verifications for the exceptional well-generated complex reflection groups. Since we were not able to provide references for explicit descriptions of the irreducible characters of the infinite families of irreducible well-generated complex reflection groups, we describe them explicitly in Appendix \[app:A\]. In the final Appendix \[app:exceptionaltypes\], we provide parts of the data that we have used in the computer verification of the main theorem for exceptional groups, namely the evaluations of the irreducible characters of exceptional well-generated complex reflection groups at reflections and at one conjugacy class of Coxeter elements. Background on complex reflection groups {#sec:background} ======================================= In this section, we recall some background on complex reflection groups. All these results can be found for example in [@LT2009] to which we also refer for further details. We moreover adapt some notation from [@BR2007]. Let $V = {\mathbb{C}}^n$ be a complex vector space of dimension $n$. A [(complex) reflection]{} is a linear transformation of $V$ that has finite order and whose fix space ${\operatorname{Fix}}(w) := {{\operatorname{ker}}}({{1\!\!1}}- w)$ is a hyperplane in $V$. Such a hyperplane is called a [reflecting hyperplane]{}. A [complex]{} (or [unitary]{}) [reflection group]{} $W$ is a finite subgroup of ${\operatorname{GL}}(V)$ generated by reflections. We denote the set of reflections in $W$ by ${\mathcal{R}}$ and the set of corresponding reflecting hyperplanes by ${\mathcal{R}}^$. The space $V$ and the action of $W$ on $V$ are called [reflection]{} (or [natural]{}) [representation]{} of $W$. We say that $W$ is [irreducible]{} if this representation is irreducible, i.e. if no proper subspace of $V$ is stable under the action of $W$. In this case, the dimension $n$ of $V$ is called the [rank]{} of $W$. Throughout this paper, we assume that $W$ is irreducible. We say that two complex reflection groups are [isomorphic]{} if they are isomorphic as abstract groups and if their reflection representations are isomorphic. G.C. Shephard and J.A. Todd classified complex reflection groups in [@ST1954]. They moreover used this classification to show that when $W$ acts on the symmetric algebra $S = {\operatorname{Sym}}(V^) \cong {\mathbb{C}}[x_1,\dots,x_n]$, its ring of invariants $S^W$ is again a polynomial algebra, generated by homogeneous polynomials $f_1,\dots,f_n$ of uniquely determined degrees $d_1 \leq \dots \leq d_n$. These are called the [degrees]{} of $W$. This result was as well proved by C. Chevalley in [@Che1955]. It turns out that there is an equivalent way to define the degrees using the [coinvariant algebra]{} $S/{\langle \mathbf{f} \rangle}$, where $\langle \mathbf{f} \rangle = \langle f_1,\ldots,f_n \rangle = S_+^W$ is the ideal in $S$ generated by all invariants without constant term. Both Shephard and Todd [@ST1954] and Chevalley [@Che1955] showed that $S/{\langle \mathbf{f} \rangle}$ carries the regular representation of $W$. Thus, $S/{\langle \mathbf{f} \rangle}$ contains exactly $k$ copies of any irreducible $W$-representation $U$ of dimension $k$. In particular $S/{\langle \mathbf{f} \rangle}$ contains $n$ copies of $V$. The multiset of [$U$-exponents]{} $e_1(U),\ldots,e_k(U)$ is given by the degrees of the homogeneous components of $S/{\langle \mathbf{f} \rangle}$ in which these $k$ copies of $U$ occur. It is known, see e.g. [@LT2009 Section 4.1], that the degrees of $W$ are uniquely determined by saying that the $V$-exponents are equal to $d_1-1,\ldots,d_n-1$. This characterization has the advantage that one can as well define the [codegrees]{} $d_1^* \geq \ldots \geq d_n^$ by saying that the $V^$-exponents are given by $d_1^+1,\ldots,d_n^+1$. It is well known that the degrees and the codegrees determine the number of reflections and the number of reflecting hyperplanes in $W$, $$\begin{aligned} \|{\mathcal{R}}\| = (d_1-1) + \ldots + (d_n-1) \qquad \|{\mathcal{R}}^\| = (d^_1+1) + \ldots + (d^_n+1), \label{eq:reflectionshyperplanes}\end{aligned}$$ see [@LT2009 Theorem 4.14(ii) and Appendix C, Section 1.2]. As in [@Mal99 Remark 6.13], we say that $W$ is [well-generated]{} if the following two equivalent properties hold, (i) $W$ is generated by $n$ reflections, (ii) the degrees and the codegrees satisfy $d_i + d_i^ = d_n$. \[eq:degreecodegree\] The equivalence of these two properties was observed in [@OS1980], using the Shephard-Todd classification. For well-generated complex reflection groups, the [Coxeter number]{} $h$ is defined to be the largest degree $d_n$ of $W$. Observe that together with the counting formulas in yields $$\begin{aligned} \|R\|+\|R^\| = nh. \label{eq:nhRR}\end{aligned}$$ An element $c$ in a complex reflection group $W$ is called [regular]{} if it has an eigenvector lying in the complement of the reflecting hyperplanes for $W$ and furthermore $\zeta$-[regular]{} if this eigenvector may be taken to have eigenvalue $\zeta$. In this case, the multiplicative order $d$ of $\zeta$ is called a [regular number]{} for $W$. An integer $d$ is a regular number if and only if it divides as many degrees as codegrees, see e.g. [@LT2009 Theorem 11.28]. If $W$ is well-generated, $d_n = h$ and $d^_n = 0$ are the only degrees and codegrees that are integer multiples of $h$. Thus, $h$ is always a regular number in this situation. For any primitive $h$-th root of unity $\zeta$ there thus exists a regular element $c_\zeta \in W$ with eigenvalue $\zeta$, see [@LT2009 Remark 11.23]. Any such element is called [Coxeter element]{}. For any $\zeta$, all $\zeta$-regular elements are $W$-conjugate, see [@LT2009 Corollary 11.25]. Thus, the class $C_\zeta$ of Coxeter elements for a particular primitive $h$-th root of unity $\zeta$ is closed under conjugation. Nonetheless, observe that according to our definition, there are in general more than a single conjugacy class of Coxeter elements, since the conjugacy classes for the various primitive $h$-th roots of unity may differ. For comparison, we notice that the paper [@Bes2007] defines Coxeter elements as $e^{\frac{2i\pi}{h}}$-regular elements, i.e. elements of the class $C_{\zeta}$ with $\zeta=e^{\frac{2i\pi}{h}}$. Therefore all Coxeter elements in the sense of [@Bes2007] are also Coxeter elements with our definition, but the converse is not true in general. To check that an element of $W$ is a Coxeter element, we will use several times the following necessary and sufficient condition. An element $w \in W$ is a Coxeter element if and only if it has an eigenvalue that is a primitive $h$-th root of unity. This result is well known to specialists, but we have not found it explicitly in the literature in the generality discussed here. For real groups, a proof can be found for example in [@Kan2001 Theorem 32-2C]. For completeness, we provide a proof of the general case that follows exactly the same lines. The arguments are very similar to the ones used in the proof of [@LT2009 Theorem 11.15]. Since $W$ is irreducible and well-generated, we have $$\begin{aligned} h = d_n > d_{n-1},\ldots,d_2,d_1. \label{eq:hismax} \end{aligned}$$ Fix now $\zeta$ to be a primitive $h$-th root of unity. We have already seen that there exists a regular element $c_\zeta \in W$ with eigenvalue $\zeta$, and that its conjugacy class ${\mathcal{C}}_{\zeta}$ is the conjugacy class of all $\zeta$-regular elements in $W$. It follows from [@LT2009 Theorem 11.24(iii)] and that $$\|{\mathcal{C}}_\zeta\| = \|W\|/\|Z(c_\zeta)\| = d_1 \cdots d_{n-1},$$ where $Z(c_\zeta)$ denotes the centralizer of $c_\zeta$, and where we used the fact that $\|W\| = d_1 \cdots d_n$. On the other hand, we have that the Pianzola-Weiss polynomial for $\zeta$ is $$P(T) := \sum_{w \in W} T^{\dim{V(w,\zeta)}} = d_1 \cdots d_{n-1}(T+h-1),$$ where $V(w,\zeta) = \{ v \in V : w(v) = \zeta \cdot v\}$ is the eigenspace for $w$ with eigenvalue $\zeta$, see [@LT2009 Corollary 10.39]. The coefficient $d_1 \cdots d_{n-1}$ of $T$ in $P(T)$ counts the number of elements in $W$ that have an eigenvalue $\zeta$. Since all elements in ${\mathcal{C}}_\zeta$ have $\zeta$ as an eigenvalue, we conclude that all elements in $W$ that have $\zeta$ as an eigenvalue are already contained in ${\mathcal{C}}_\zeta$, and are thus Coxeter elements. The statement follows. The classification of well-generated complex reflection groups {#sec:classification} -------------------------------------------------------------- In the remainder of this section, we recall the classification of irreducible well-generated complex reflection groups from [@ST1954]. A monomial matrix is a square matrix with exactly one non-zero entry in each row and column. Let $G(r,p,n)$ with $p$ dividing $r$ be the group of all monomial $(n \times n)$-matrices with entries being $r$-th roots of unity, such that the product of the non-zero entries is an $(r/p)$-th root of unity. It is a subgroup of $GL({\mathbb{C}}^n)$, and it has order $n! r^n/p$. Notice that the special case $G(1,1,n)$ is the group of all permutation matrices of size $n$. This group leaves invariant the proper subspace $V_0$ of ${\mathbb{C}}^n$ formed by vectors whose coordinates sum to zero. We denote by ${\widehat G(1,1,n)}$ the restriction of $G(1,1,n)$ to this subspace. Hence ${\widehat G(1,1,n)}\subset GL(V_0)$ is an irreducible complex reflection group of rank $n-1$. \[thm:shephardtodd\] Let $W$ be an irreducible complex reflection group. Then $W$ is isomorphic to one of the following complex reflection groups: - the group $G(r,p,n)$ for some integers $r\geq 2$ and $n,p \geq 1$ such that $p$ divides $r$ (if $n=1$, we impose moreover that $p=1$). - the symmetric group ${\widehat G(1,1,n)}$ for some $n\geq 5$; - one of $34$ exceptional groups. Following [@ST1954], we denote the $34$ exceptional [Shephard-Todd classification types]{} by $G_4$ through $G_{37}$; the degrees and codegrees of all irreducible complex reflection groups were already computed therein. For the groups $G(r,p,n)$, the degrees are given by $$\begin{aligned} r,2r,\dots,(n-1)r,rn/p, \label{eq:degrees}\end{aligned}$$ while the codegrees are given by $$\begin{aligned} 0,r,\dots,(n-1)r &\text{ if } p < r, \nonumber \\ 0,r,\dots,(n-2)r,(n-1)r-n &\text{ if } p = r. \label{eq:codegrees}\end{aligned}$$ For the symmetric group ${\mathcal{S}}_n\simeq{\widehat G(1,1,n)}$, the degrees and codegrees are given by $2,3,\dots,n$ and $0,1,\dots,n-2$, respectively. rank Well-generated exceptional classification types --------------- ------------------------------------------------------------------------ \[-10pt\] $2$ $G_4,G_5,G_6,G_8,G_9,G_{10},G_{14},G_{16},G_{17},G_{18},G_{20},G_{21}$ \[-10pt\] $3$ $G_{23}=H_3,G_{24},G_{25},G_{26},G_{27}$ \[-10pt\] $4$ $G_{28}=F_4,G_{29},G_{30}=H_4,G_{32}$ \[-10pt\] $5$ $G_{33}$ \[-10pt\] $6$ $G_{34},G_{35}=E_6$ \[-10pt\] $7$ $G_{36}=E_7$ \[-10pt\] $8$ $G_{37}=E_8$ : All well-generated exceptional classification types.[]{data-label="tab:exceptionaltypes"} This implies that the three infinite families of irreducible well-generated complex reflection groups are given by ${\widehat G(1,1,n)}$, $G(r,1,n)$ and $G(r,r,n)$. Observe that the infinite families of real reflection groups are the particular cases - $G(r,r,2) = I_2(r)$, - ${\widehat G(1,1,n)}= A_{n-1}$, - $G(2,1,n) = B_n$, - $G(2,2,n) = D_n$. Observe also that the group $G(r,1,1)$ is the cyclic group $C_r$ of order $r$. The complete list of well-generated exceptional Shephard-Todd classification types is given in Table \[tab:exceptionaltypes\], their degrees and codegrees can be found in Table \[tab:exceptionaldegreescodegrees\] in Appendix \[app:exceptionaltypes\]. We will later use this classification to prove Theorem \[thm:main\]. In particular, see Appendix \[app:exceptionaltypes\] for the representation theoretic data for the exceptional types. Direct proof for the cyclic groups and dihedral groups {#sec:dihedralgroup} ====================================================== As a warmup exercise, we prove the main theorem for the cyclic group $C_r$ and for the dihedral group $I_2(r)$. Observe that the cyclic groups will as well be covered in Section \[par:Gr1n\], while the proof for the dihedral groups will not be covered in Section \[par:Grrn\] and is thus only given here. [The proof for the cyclic group $G(r,1,1)=C_r$ with $r\geq 2$.]{} The cyclic group $G(r,1,1)=C_r$ formed by the $r$-th roots of unity is a complex reflection group of rank $n=1$. It has one reflecting hyperplane, and since any element of the group different from $1$ is a reflection, it contains $(r-1)$ reflections. Let $\zeta$ be a primitive $r$-th root of unity. Since the Coxeter number is $h=r$, the element $\zeta$ is a Coxeter element. For $\ell \geq0$, let $f_\ell$ be the number of factorizations of $\zeta$ into a product of $\ell$ reflections. Then $(r-1)^\ell-f_{\ell}$ is the number of $\ell$-tuples of reflections $(\zeta_1,\zeta_2,\dots,\zeta_\ell)$ whose product is not equal to $\zeta$. Given such a tuple, there is a unique reflection $\zeta_{\ell+1}$ such that $\zeta_1\zeta_2\dots\zeta_\ell\zeta_{\ell+1} = \zeta$. We thus obtain the recurrence relation $$f_{\ell+1} = (r-1)^\ell - f_\ell.$$ Given that $f_0=0$ and $f_1=1$, this implies that $f_\ell = \frac{1}{r}\big((r-1)^\ell-(-1)^\ell\big)$ for all $\ell \geq 0$. We thereby obtain $${{\operatorname{FAC}}}_{C_r}(t) = \sum_{\ell\geq 0} \frac{t^\ell}{\ell!}f_\ell = \frac{1}{r}\left(e^{(r-1)t} - e^{-t}\right),$$ which yields Formula in this case. [The proof for the dihedral group $G(r,r,2)=I_2(r)$ with $r\geq 2$.]{} Let $\zeta$ be a primitive $r$-th root of unity. The dihedral group $G(r,r,2)=I_2(r)$ consists of the $r$ reflections $s_i = \left(\begin{smallmatrix} 0&\zeta^i\\ \zeta^{-i}&0 \end{smallmatrix}\right)$ and the $r$ elements $t_i = \left(\begin{smallmatrix} \zeta^i&0\\ 0&\zeta^{-i} \end{smallmatrix}\right)$, for $1 \leq i \leq r$ (the elements $t_i$ are the “rotations” in the classical representation of $I_2(r)$ as the symmetry group of the regular $r$-gon). Notice that the hyperplanes ${{\operatorname{ker}}}({{1\!\!1}}-s_i)$ are all distinct, so that there are $r$ reflecting hyperplanes. The Coxeter number is $h=r$, which shows that $c=s_1 s_0 = t_1$ is a Coxeter element. Note that $s_i s_j = c$ if and only if $i = j+1\ ({\operatorname{mod}}r)$. In particular, since for any integer $g \geq 0$ any product of $2g+1$ reflections is again a reflection, there is a unique other reflection such that the complete product of all $2g+2$ reflections equals $c$. Therefore the number of factorizations of $c$ into $2g+2$ reflections is equal to $r^{2g+1}$. We thereby obtain $${{\operatorname{FAC}}}_{I_2(r)}(t) = \sum_{g\geq0}\frac{t^{2g+2}}{(2g+2)!}r^{2g+1} = \frac{e^{rt}+e^{-rt}-2}{2r}= \frac{1}{2r}\Big(e^{rt/2}-e^{-rt/2}\Big)^2,$$ which yields Formula in this case. A representation theoretic approach to the main theorem {#sec:representationtheory} ======================================================= In this section, we recall a classical approach to the enumeration of factorizations in groups via representation theory. This approach is widely used in the literature to enumerate factorizations in the symmetric group, see e.g. [@LandoZvonkine Appendix A] and the references therein. It is based on the Frobenius formula given in Formula below. For completeness, we recall its proof here, following the same lines as in [@LandoZvonkine Section A.1.3]. Let $W$ be an irreducible well-generated complex reflection group. The group algebra ${\mathbb{C}}[W]$ is decomposed into irreducible $W$-modules as $$\begin{aligned} \label{eq:regular} {\mathbb{C}}[W] = \bigoplus_{\lambda \in \Lambda} \dim(\lambda) V^\lambda,\end{aligned}$$ where $V^\lambda$, $\lambda \in \Lambda$ is a complete list of irreducible representations of $W$. We let ${\mathcal{R}}\subset W$ be the union of the conjugacy classes of all reflections. We use the same letter, in non-calligraphic style, to denote the corresponding sum in the group algebra, $$R := \sum_{\tau \in {\mathcal{R}}} \tau.$$ Moreover, we extend class functions linearly from $W$ to ${\mathbb{C}}[W]$. In particular if $\chi :W\rightarrow {\mathbb{C}}$ is a class function, we use the notation $$\chi(R) := \sum_{\tau \in {\mathcal{R}}} \chi(\tau).$$ We consider the action of $W$ on ${\mathbb{C}}[W]$ by left multiplication. Since an element $w \in W$ acts on $\mathbb{C}[W]$ by permuting the canonical basis, the trace of this action is equal to the number of fixed points under this action, which is $\|W\|$ if $w=1$ and $0$ otherwise. Since $R$ is in the center of ${\mathbb{C}}[W]$, it acts as a scalar on every irreducible representation, so the decomposition gives the Frobenius formula $$\begin{aligned} \#\Big\{ (\tau_1,\tau_2,\dots,\tau_\ell)\in {\mathcal{R}}^\ell \ : \ \tau_1 \tau_2 \dots &\tau_\ell = c \Big\} \nonumber \\ &= \frac{1}{\|W\|} {\mathrm{Tr}}_{\mathbb{C}[W]} \Big( {R}^{\ell} c^{-1} \Big) \nonumber \\ &= \frac{1}{\|W\|} \sum_{\lambda \in \Lambda} \dim(\lambda) {\mathrm{Tr}}_{V^\lambda} \Big( {R}^{\ell} c^{-1} \Big) \nonumber\\ &= \frac{1}{\|W\|} \sum_{\lambda \in \Lambda} \dim(\lambda)^{1-\ell} \chi_\lambda({R})^{\ell} \chi_\lambda(c^{-1})\label{eq:frobenius},\end{aligned}$$ where $\chi_\lambda$ is the character of the representation $\lambda$. Using , we can now deduce that ${{\operatorname{FAC}}}_W(t)$ is given by the finite sum $$\begin{aligned} \label{eq:frobeniusZt} {{\operatorname{FAC}}}_W(t) &= \frac{1}{\|W\|} \sum_{\lambda \in \Lambda} \dim(\lambda) \chi_\lambda(c^{-1}) \exp\left(t \cdot \frac{\chi_\lambda({R})}{\dim(\lambda)}\right).\end{aligned}$$ We will use this formula, together with the classification of complex reflection groups and the knowledge of their irreducible representations to prove Theorem \[thm:main\]. In order to simplify notation, we will often make use of the [normalized character]{} defined as the character divided by the dimension of the corresponding representation $\chi_\lambda({{1\!\!1}}) = \dim(\lambda)$. By convention, a normalized character will be denoted with a tilde, such as $$\widetilde\chi_\lambda := \frac{1}{\dim(\lambda)} \cdot \chi_\lambda.$$ Proof of the main theorem for the infinite families {#sec:classicaltypes} =================================================== In this section, we treat the case of the infinite families of well-generated complex reflection groups, namely ${\widehat G(1,1,n)}\simeq {\mathcal{S}}_n$, $G(r,1,n)$ and $G(r,r,n)$. For technical reasons we exclude the case $G(r,r,2)$, which has been treated in Section \[sec:dihedralgroup\]. As mentioned in the introduction the case of ${\widehat G(1,1,n)}\simeq {\mathcal{S}}_n$ is already known, but we discuss it here to set up notations and to recall some needed results. Our approach for this case is similar e.g. to [@SSV] or to [@LandoZvonkine Appendix A.2.4]. The proof for the symmetric group Sn ------------------------------------ The symmetric group ${\mathcal{S}}_n$ is the group of all permutations of $\{1,2,\dots,n\}$. Its irreducible representations are classically indexed by the set of partitions of $n$, see e.g. [@Sagan]. In this section we are going to use extensively two classical results, the [hook-length formula]{} and the [Murnaghan-Nakayama rule]{}. These formulas give the dimensions of these representations and the evaluation of their characters, respectively. We refer the reader to the original papers [@FRT; @Mur37; @Nak40] or to [@Sagan Sections 3.10 and 4.10] for a detailed treatment of these results. We will use them several times without again recalling these references. A [partition]{} of $n$ is a non-increasing sequence $\lambda=[\lambda_1,\lambda_2,\dots,\lambda_r]$ of positive integers summing to $n$. The integer $n$ is called the [size]{} of $\lambda$, denoted by $\|\lambda\|=n$, and we write $\lambda \vdash n$. The integers $\lambda_i$ are called the [parts]{} of $\lambda$. To denote a partition, we often use superscript notation, we write for example $\lambda=[4^2, 3^1, 1^3]$ for the partition $\lambda = [4,4,3,1,1,1]$ of $14$. By convention, the [empty partition]{} $\varnothing$ is the unique partition of $0$. If not otherwise stated, we denote by $\chi_\lambda$ the character of the representation of ${\mathcal{S}}_n$ indexed by the partition $\lambda \vdash n$. We choose the long cycle $c=(1,2,\dots,n)$ as a Coxeter element. Since all Coxeter elements are conjugated to $c$ in ${\mathcal{S}}_n$, it is enough to prove Theorem \[thm:main\] for this particular element. Since $c$ is a long cycle, as is its inverse $c^{-1}$, the Murnaghan-Nakayama rule implies that $\chi_\lambda(c^{-1})=0$ unless $\lambda$ is a [hook]{}, i.e., unless $\lambda$ is of the form ${\mathfrak{h}_{k}^{n}} := [n-k,1^{k}]$ for some $0\leq k < n$. The following lemma is a straightforward consequence of the hook-length formula and of the Murnaghan-Nakayama rule, and will be useful in several places. \[lemma:hooks\] Let $n \geq 1$ and $0\leq k < n$. Then $$\dim({\mathfrak{h}_{k}^{n}}) = \binom{n-1}{k} \quad \text{and} \quad \chi_{{\mathfrak{h}_{k}^{n}}} \big((1,2,\dots,n)\big) =(-1)^k.$$ The reflections of the group ${\widehat G(1,1,n)}\simeq {\mathcal{S}}_n$ are given by the transpositions $(i,j)$ for $1\leq i < j \leq n$. Hence in view of evaluating , we will also need the value of hook characters on transpositions. If $\lambda=[\lambda_1,\lambda_2,\dots,\lambda_r] \vdash n$, we define the multiset of its contents as $${\operatorname{Content}}(\lambda) := {\big\{\!\!\big\{ j-i : 1 \leq i \leq r, 1 \leq j \leq \lambda_i \big\}\!\!\big\}},$$ where we indicate the multiset notation by double braces. Notice that ${\operatorname{Content}}(\lambda)$ has cardinality $n$. We have the following classical lemma relating character evaluations and contents. \[lemma:contents\] Let $\lambda \vdash n$ be a partition and $\tau \in {\mathcal{S}}_n$ be a transposition. Then we have $$\widetilde\chi_\lambda(\tau) = \frac{2}{n(n-1)} \sum_{ x\in {\operatorname{Content}}(\lambda)} x .$$ As in Section \[sec:representationtheory\], we let $R=\sum_{\tau\in{\mathcal{R}}}\tau$ be the sum of all transpositions in the group algebra ${\mathbb{C}}[{\mathcal{S}}_n]$. Write $R=\sum_{i=1}^{n} J_i$ where $J_i = \sum_{i<j} (i,j)$ is the $j$-th Jucys-Murphy element. It is well known – see e.g. [@OkounkovVershik] – that given any symmetric function $f$ in $n$ variables, the element $f(J_1,J_2,\dots,J_n)$ acts on the irreducible module $V^\lambda \subset {\mathbb{C}}[{\mathcal{S}}_n]$ as the scalar $f({\operatorname{Content}}(\lambda))$. The statement follows by taking $f(x_1,x_2,\dots x_n)=x_1+x_2+\dots +x_n$, and observing that $\chi_\lambda(R) = \frac{n(n-1)}{2}\chi_\lambda(\tau)$. In the particular case of hooks, we obtain $$\begin{aligned} \widetilde\chi_{{\mathfrak{h}_{k}^{n}}}(\tau) &= \frac{n-2k-1}{n-1}. \label{eq:hooktranspo}\end{aligned}$$ We now recalled all ingredients needed to provide a proof of Theorem \[thm:main\] for the symmetric group. Using Equation together with Lemma \[lemma:hooks\], Formula can be rewritten as $$\begin{aligned} {{\operatorname{FAC}}}_W(t) &= \frac{1}{\|W\|} \sum_{k =0}^{n-1} \binom{n-1}{k} (-1)^k e^{t\frac{n(n-2k-1)}{2}}\\ &= \frac{1}{\|W\|} e^{t\binom{n}{2}} \left(1-e^{-tn}\right)^{n-1} \\ &= \frac{1}{\|W\|}\left(e^{t\frac{n}{2}}-e^{-t\frac{n}{2}}\right)^{n-1}.\end{aligned}$$ Since ${\widehat G(1,1,n)}$ has rank $n-1$, and $\|\mathcal{R}\|=\|\mathcal{R}^\|=\binom{n}{2}$, this coincides with in this case. The structure of the proof {#par:4step} -------------------------- In the cases of $G(r,1,n)$ and $G(r,r,n)$, the structure of the proof will be similar to what we just did for the symmetric group. It can be decomposed into the following four steps. 1. Provide a list of all irreducible characters not vanishing on the inverse of the Coxeter element (for ${\mathcal{S}}_n$, these were exactly the hook characters). 2. Compute the dimension of these characters and their evaluation on the inverse of the Coxeter element (for ${\mathcal{S}}_n$, this was given by Lemma \[lemma:hooks\]). 3. Compute the evaluation of these characters on the conjugacy classes of reflections (for ${\mathcal{S}}_n$, this was given by Equation ). 4. Use Formula and simplify the sum (for ${\mathcal{S}}_n$, we just applied Newton’s binomial formula) to prove Formula . In Section \[par:Grrn\], we prove the main theorem for the group $G(r,r,n)$ following the four steps above. In order to do that, we list the irreducible characters of $G(r,r,n)$ that do not vanish on a Coxeter element (Lemma \[lemma:step1Grr\]). Note that since $G(r,r,n)$ is a subgroup of $G(r,1,n)$, another approach would be to apply the Frobenius formula directly in the group $G(r,1,n)$[^3]. This other approach would save us the description of the irreducible characters of $G(r,r,n)$ given in Appendix \[sec:irreduciblecharactersrrn\]. We follow the first approach here because we think that Lemma \[lemma:step1Grr\] is interesting in itself. Indeed, together with the results for $G(r,1,n)$ and with the data for irreducible groups given in Appendix \[app:exceptionaltypes\], this provides an exhaustive list of irreducible characters non vanishing on a Coxeter element in well-generated complex reflection groups, that may be useful in other situations. The proof for the group G(r,1,n) with r>=2 {#par:Gr1n} --------------------------------------------- The Coxeter number for $G(r,1,n)$ is $h = nr$. Let $\zeta$ be a primitive $h$-th root of unity, and let $\xi = \zeta^n$, which is a primitive $r$-th root of unity. Recall that $G(r,1,n)$ is the group of all monomial matrices whose non-zero entries are powers of $\xi$. Given $\sigma\in{\mathcal{S}}_n$ and $(i_1,i_2,\dots,i_n)$ with $0 \leq i_1,\ldots,i_n < r$, we denote by $w = \sigma {\wr}(i_1,i_2,\dots,i_n)$ the element of $G(r,1,n)$ with entry $\xi^{i_\ell}$ at position $(\ell,\sigma_\ell)$ for $1\leq \ell \leq n$. Observe that our presentation depends on the choice of $\xi$ and thus on the choice of $\zeta$. We moreover denote by $\|w\| := \sigma$ the projection onto ${\mathcal{S}}_n$, and by $\\|w\\| := i_1+\ldots+i_n\ ({\operatorname{mod}}r)$. Consider the element $c=c_0{\wr}(0,0,\dots,0,1) \in G(r,1,n)$ where $c_0 \in {\mathcal{S}}_n$ is given by the long cycle $(1,2,\dots,n)$. We claim that $c$ is a Coxeter element of $G(r,1,n)$. To see that, according to the discussion preceding Section \[sec:classification\], it is enough to check that $c$ has an eigenvalue equal to $\zeta$. This is indeed the case, since the fact that $\xi = \zeta^n$ implies that the element $c$ has $n$ eigenvalues given by $\big\{ \zeta, \zeta^{r+1},\zeta^{2r+1},\ldots,\zeta^{(n-1)r+1} \big\}$. \[rem:coxeterClasses\] In the rest of this section, we will prove Theorem \[thm:main\] for the group $G(r,1,n)$, for the particular choice of Coxeter element $c=c_0{\wr}(0,0,\dots,0,1)$. Since $c$ belongs to the conjugacy class ${\mathcal{C}}_{\zeta}$ of all Coxeter elements which are $\zeta$-regular, and since the number of factorizations into reflections is the same for elements in the same conjugacy class, this will also prove the theorem for all elements $c'\in{\mathcal{C}}_\zeta$. Finally, since we fixed $\zeta$ as an arbitrary* primitive $h$-th root of unity, this will imply the result for all Coxeter elements of the group $G(r,1,n)$. The following proposition is well known and describes the irreducible characters of $G(r,1,n)$. We refer to Appendix \[sec:irreduciblecharactersr1n\] for its proof and references. \[prop:Gr1n\] The complete and unambiguous list of irreducible characters of $G(r,1,n)$ is obtained as follows. Let $\vec\lambda = (\lambda^{(0)},\lambda^{(1)},\dots,\lambda^{(r-1)})$ be an $r$-tuple of partitions of total size $n$, and let $k_\ell=\|\lambda^{(\ell)}\|$. One may view $$B:=G(r,1,k_0)\times G(r,1,k_1)\times \dots \times G(r,1,k_{r-1})$$ as a subgroup of $G(r,1,n)$, formed by block-diagonal matrices. The character $\chi_{\vec\lambda}$ of $G(r,1,n)$ is then given by $$\begin{aligned} \label{eq:charGr1n} \chi_{\vec\lambda}(w) &= \frac{1}{\|B\|} \sum_{\substack{s \in G(r,1,n) \\ s^{-1}ws \in B}} \prod_{\ell=0}^{r-1} \chi_{\lambda_\ell}(\|w_\ell\|)\cdot \xi^{\ell\cdot \\|w_\ell\\| },\end{aligned}$$ where $s^{-1}ws \in B$ in the sum is denoted by $(w_0,\ldots,w_{r-1})$, and where $\chi_{\lambda^{(\ell)}}$ denotes the ${\mathcal{S}}_{k_\ell}$-character indexed by $\lambda^{(\ell)}$. Having the list of irreducible characters at hand, we can now proceed with the four steps in the proof as explained in Section \[par:4step\]. [Step 1.]{} The following lemma describes those characters not vanishing on the inverse of the Coxeter element. The character $\chi_{\vec\lambda}$ defined in vanishes on $c^{-1}$ unless $$\vec\lambda = {{}_q{\mathfrak{h}_{k}^{n}}} := (0,\dots,0,{\mathfrak{h}_{k}^{n}},0,\dots0) \text{ for } 0 \leq q < r \text{ and } 0 \leq k < n,$$ where as before ${\mathfrak{h}_{k}^{n}}$ is a hook of size $n$, and where the hook appears at position $q$. Let us consider the evaluation of Formula with $w=c^{-1}$. Observe that $\|c^{-1}\|$ is a long cycle. Thus, the cyclic group it generates acts transitively on $\{1,2,\dots,n\}$, which implies that the sum in is empty unless exactly one of the numbers $k_0,\ldots,k_{r-1}$ is equal to $n$, say $k_q$. Moreover, the Murnaghan-Nakayama rule implies that the evaluation $\chi_{\lambda_q}(c_0)$ is zero unless $\lambda^{(k_q)}$ is a hook. We are therefore left with all the characters listed in the lemma. [Step 2.]{} By the Murnaghan-Nakayama rule and the fact that $\\|c^{-1}\\|=r-1$, the evaluation of gives $$\chi_{{{}_q{\mathfrak{h}_{k}^{n}}}}(c^{-1}) = (-1)^k \xi^{-q}.$$ Moreover, their degrees are easily determined by evaluating on the identity element $$\dim({{}_q{\mathfrak{h}_{k}^{n}}}) = \chi_{{\mathfrak{h}_{k}^{n}}}(1) = \binom{n-1}{k} \mbox{ by Lemma~\ref{lemma:hooks}}.$$ [Step 3.]{} The set of reflections of $G(r,1,n)$ is divided into $r$ conjugacy classes, ${\mathcal{R}}= \bigcup_{\ell=0}^{r-1} {\mathcal{R}}_\ell$, where - for $1 \leq \ell <r$, ${\mathcal{R}}_\ell$ is the set of matrices obtained from the identity matrix by replacing one of its entries by $\xi^\ell$. Hence ${\mathcal{R}}_\ell$ has cardinality $n$, for each $\ell$. - ${\mathcal{R}}_0$ is the set of matrices $M_{i,j}^{(k)}$, for $1\leq i<j\leq n$, $1\leq k \leq r$, where $M_{i,j}^{(k)}$ is obtained from the matrix of the transposition $(i,j)$ by replacing its entries in columns $i$ and $j$ by $\xi^k$ and $\xi^{-k}$, respectively. Hence, ${\mathcal{R}}_0$ has cardinality $r\binom{n}{2}$. For $0\leq \ell < r$ we let $R_\ell = \sum_{\tau \in {\mathcal{R}}_\ell} \tau$ be the corresponding element of the group algebra. We need to evaluate the character $\chi_{{{}_q{\mathfrak{h}_{k}^{n}}}}$ on each of the elements $R_\ell$. If $\tau \in {\mathcal{R}}_\ell$ with $\ell > 0$, then $\\|\tau\\| = \ell$ and $\|\tau\| = {{1\!\!1}}$. Therefore, evaluating yields $\chi_{{{}_q{\mathfrak{h}_{k}^{n}}}}(\tau) = \dim({\mathfrak{h}_{k}^{n}})\xi^{q\ell}$, and thus $$\chi_{{{}_q{\mathfrak{h}_{k}^{n}}}}(R_\ell) = n \dim({\mathfrak{h}_{k}^{n}}) \xi^{q\ell} = n \dim({{}_q{\mathfrak{h}_{k}^{n}}}) \xi^{q\ell}.$$ If $\tau \in {\mathcal{R}}_0$, then $\\|\tau\\| = 0$ and $\|\tau\|$ is a transposition, so by the character evaluates as $$\chi_{{{}_q{\mathfrak{h}_{k}^{n}}}}(\tau) = \chi_{{\mathfrak{h}_{k}^{n}}} (\|\tau\|) = \dim({\mathfrak{h}_{k}^{n}}) \frac{n-2k-1}{n-1} = \dim({{}_q{\mathfrak{h}_{k}^{n}}}) \frac{n-2k-1}{n-1}.$$ Adding all contributions and letting $R$ be the sum of all reflections as in Section \[sec:representationtheory\], we obtain the evaluation of the normalized character $$\widetilde\chi_{{{}_q{\mathfrak{h}_{k}^{n}}}} (R) = \sum_{\ell=1}^{r-1} n \xi^{q\ell} + \frac{n r (n-2k-1)}{2}.$$ [Step 4.]{} From Steps 1–3, Formula for $W=G(r,1,n)$ finally rewrites as $$\begin{aligned} {{\operatorname{FAC}}}_W(t) &= \frac{1}{\|W\|} \sum_{q=0}^{r-1}\sum_{k=0}^{n-1} \binom{n-1}{k} (-1)^k \xi^{-q} \exp\left(t \sum_{\ell=1}^{r-1} n \xi^{q\ell} +t r\frac{ n(n-2k-1)}{2} \right) \\ &= \frac{1}{\|W\|} e^{t r\binom{n}{2}} \left( \sum_{q=0}^{r-1} \xi^{-q} e^{ t \cdot \sum_{l=1}^{r-1} n \xi^{ql}} \right) \left(\sum_{k=0}^{n-1} \binom{n-1}{k}(-1)^k e^{-t rnk} \right) \\ &= \frac{1}{\|W\|} e^{t r\binom{n}{2} } \left(1-e^{-t rn}\right)^{n-1} \left( \sum_{q=0}^{r-1} \xi^{-q} e^{ t \cdot \sum_{\ell=1}^{r-1} n \xi^{q\ell}} \right).\end{aligned}$$ To finish the proof of Theorem \[thm:main\] for the group $G(r,1,n)$, we use the following lemma. \[lem:zetasum\] Let $\xi$ be a primitive $r$-th root of unity. Then $$\sum_{q=0}^{r-1} \xi^{-q} e^{ t \cdot \sum_{\ell=1}^{r-1} n \xi^{q\ell}} = e^{t(r-1)n} -e^{-tn}$$ If $1<q<r-1$ then $\xi^{-q}\neq 1$ and $\sum_{\ell=1}^{r-1}\xi^{q\ell} = -1$ by summing the geometric progression (notice that here we use that $r\geq 2$). We therefore obtain $$\sum_{q=0}^{r-1} \xi^{-q} e^{ t \cdot \sum_{\ell=1}^{r-1} n \xi^{q\ell}} = \xi^0 e^{n(r-1)t} + \sum_{q=1}^{r-1} \xi^{-q} e^{-nt} = e^{(n-1)rt} - e^{-nt}.$$ Using Lemma \[lem:zetasum\], our last expression of ${{\operatorname{FAC}}}_W(t)$ rewrites as $$\begin{aligned} {{\operatorname{FAC}}}_W(t) &= \frac{1}{\|W\|} e^{t r\binom{n}{2} } \left(1-e^{-t rn}\right)^{n-1} \left(e^{t(r-1)n} -e^{-t n} \right)\\ &= \frac{1}{\|W\|} e^{t r\binom{n}{2} + t(r-1)n } \left(1-e^{-t rn}\right)^{n} \\ &= \frac{1}{\|W\|}\left(e^{t\left(r\frac{n+1}{2}-1\right)}-e^{-t\left(r\frac{n-1}{2}+1\right)}\right)^{n}. \end{aligned}$$ Since we have for $W = G(r,1,n)$ that $\|{\mathcal{R}}\| = r\binom{n+1}{2}-n$ and $\|{\mathcal{R}}^\| = r\binom{n}{2}+n$, this coincides with in this case. The proof for the group G(r,r,n) with r>=2 and n>2 {#par:Grrn} -------------------------------------------------------- The group $G(r,r,2)$ is the dihedral group $I_2(r)$, which we have already treated in Section \[sec:dihedralgroup\]. We assume here that $n > 2$. Now, the Coxeter number for $G(r,r,n)$ is $h = (n-1)r$. Let $\zeta$ be a primitive $h$-th root of unity, and let $\xi = \zeta^{n-1}$, which is a primitive $r$-th root of unity. The group $G(r,r,n)$ is a subgroup of $G(r,1,n)$, and we reuse the notation defined in the previous section. Consider the element $c=c_0{\wr}(0,0,\dots,0,1,r-1) \in G(r,r,n)$, where $c_0 \in {\mathcal{S}}_n$ is given by the cycle $(1,2,\dots,n-1)(n)$. We claim that $c$ is a Coxeter element of $G(r,r,n)$. To see that, from the discussion preceding Section \[sec:classification\], it is enough to prove that $c$ has an eigenvalue equal to $\zeta$. This is true, since the fact that $\xi = \zeta^{n-1}$ implies that the element $c$ has $n-1$ eigenvalues given by $\big\{ \zeta,\zeta^{r+1},\zeta^{2r+1},\ldots,\zeta^{(n-2)r+1}\big\}$, while the $n$-th eigenvalue is given by $\zeta^{(n-1)(r-1)}$. In the rest of the proof for the group $G(r,r,n)$ we will work with the particular Coxeter element $c=c_0{\wr}(0,0,\dots,0,1,r-1)$. This is enough to prove Theorem \[thm:main\] for all* Coxeter elements of the group $G(r,r,n)$, for the same reasons as described in Remark \[rem:coxeterClasses\]. [Step 1.]{} The description of irreducible characters is now more complicated than for the group $G(r,1,n)$. Therefore we consider here only the irreducible characters that are of interest to us in the proof of Theorem \[thm:main\]. We refer to Appendix \[sec:irreduciblecharactersrrn\] for more information on irreducible characters of $G(r,r,n)$, and for a proof of the following lemma. \[lemma:step1Grr\] Let $\chi$ be an irreducible character of $G(r,r,n)$ not vanishing on $c^{-1}$. Then $\chi$ is equal to the restriction of the character $\chi_{\vec\lambda}$ of $G(r,1,n)$ defined by $\eqref{eq:charGr1n}$, where $\vec\lambda$ is one of the following partition vectors. - $\vec\lambda=({[n],\dots})$ or $\vec\lambda = ({[1^n],\dots})$, where the dots denote a list of $(r-1)$ empty partitions; - $\vec\lambda = ({{\mathfrak{qh}_{k}^{n}},\dots})$ for some $1\leq k \leq n-3$, where ${\mathfrak{qh}_{k}^{n}}$ denotes the [quasi-hook]{} ${\mathfrak{qh}_{k}^{n}}:= [n-k-1,2,1^{k-1}]$, and where the dots denote a list of $(r-1)$ empty partitions; - $\vec\lambda=({{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots})$ for some $1\leq j < r$ and $0 \leq k \leq n-2$. Here ${\mathfrak{h}_{k}^{n-1}}$ is a hook of size $n-1$ and the partition of size $1$ appears in position $j$. As before the dots denote lists of empty partitions, respectively of lengths $j-1$ and $r-j-1$. [Step 2.]{} This step is encapsulated in the following lemma. \[lemma:step2Grr\] The dimension and evaluation on the inverse $c^{-1}$ of the Coxeter element of the characters listed in Lemma \[lemma:step1Grr\] are given as follows. $$\begin{aligned} \dim({[n],\dots}) = 1, &\quad \chi_{{[n],\dots}}(c^{-1}) = 1, \\ \dim({[1^n],\dots}) = 1, &\quad \chi_{{[1^n],\dots}}(c^{-1}) = (-1)^n, \\ \dim({{\mathfrak{qh}_{k}^{n}},\dots}) = \frac{(n-2-k)k}{n-1}\binom{n}{k+1}, &\quad \chi_{{{\mathfrak{qh}_{k}^{n}},\dots}}(c^{-1}) = (-1)^k, \\ \dim({{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots}) = n \cdot \binom{n-2}{k}, &\quad \chi_{{{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots}}(c^{-1}) = (-1)^k \cdot \xi^{-j}. \end{aligned}$$ If $\lambda$ is a partition of $n$, observe that in the sum defining $\chi_{\lambda,\dots}$, all the elements $s\in G(r,1,n)$ contribute. Therefore one has $\chi_{\lambda,\dots}(w) = \chi_\lambda(\|w\|)$ for all $w\in G(r,1,n)$. This settles the first three cases considered in the lemma, using the Murnaghan-Nakayama rule and the hook-length formula (the only non immediate fact here is that $\dim({\mathfrak{qh}_{k}^{n}}) = \frac{(n-2-k)k}{n-1}\binom{n}{k+1}$; it follows by a simple computation using the hook-length formula, that we leave to the reader). In the fourth case we have to be more careful about the elements $s$ contributing to the sum . When evaluating $\chi_{{{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots}}$ on $c^{-1}$, an element $s\in G(r,1,n)$ contributes to the sum if and only if $s$ belongs to the group $B$ of block-diagonal matrices defined in Proposition \[prop:Gr1n\]. From this we obtain $$\chi_{{{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots}}(c^{-1}) = \chi_{{\mathfrak{h}_{k}^{n-1}}}((1,2,\dots,n-1))\cdot \xi^{-j},$$ which equals $(-1)^k \cdot \xi^{-j}$ by Lemma \[lemma:hooks\]. When evaluating the same character on the identity, then all the elements $s\in G(r,1,n)$ contribute, so we obtain $$\chi_{{{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots}}({{1\!\!1}}) = \frac{\|G(r,1,n)\|}{\|B\|} \chi_{{\mathfrak{h}_{k}^{n-1}}}({{1\!\!1}}) = n\dim({\mathfrak{h}_{k}^{n-1}}) = n\binom{n-2}{k},$$ where we again used Lemma \[lemma:hooks\]. [Step 3.]{} The set of reflections of $G(r,r,n)$ is formed of one conjugacy class, namely the class ${\mathcal{R}}_0$ defined in Section \[par:Gr1n\]. The following lemma gives the evaluation of the normalized irreducible characters at the corresponding group algebra element $R_0:=\sum_{\tau\in{\mathcal{R}}_0}\tau$. For the characters listed in Lemma \[lemma:step1Grr\], the normalized character evaluations on the sum of all reflections are given by $$\begin{aligned} \widetilde\chi_{{[n],\dots}}(R_0) &= r\binom{n}{2}, \label{eq:c1} \\ \widetilde\chi_{{[1^n],\dots}}(R_0) &= -r\binom{n}{2}, \label{eq:c2} \\ \widetilde\chi_{{{\mathfrak{qh}_{k}^{n}},\dots}}(R_0) &= \frac{r(n-1)(n-2-2k)}{2}, \label{eq:c3} \\ \widetilde\chi_{{{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots}}(R_0) &= \frac{r(n-1)(n-2k-2)}{2}. \label{eq:c4} \end{aligned}$$ As already observed in the proof of Lemma \[lemma:step2Grr\], for $\lambda$ a partition of $n$ we have $\chi_{\lambda, \dots}(w) = \chi_\lambda (\|w\|)$ for all $w$. Therefore we have $\chi_{\lambda,\dots}(R_0) = r\binom{n}{2} \chi_\lambda(\|\tau\|)$ where $\tau$ is any element of ${\mathcal{R}}_0$. Since $\|\tau\|$ is a transposition, the three first cases , , therefore follow from Lemma \[lemma:contents\] (we leave to the reader the details of the computation of the sum of contents of the quasi-hook). Now let us consider the evaluation $\chi_{{{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots}}(\tau)$ where $\tau \in {\mathcal{R}}_0$. Note that $\|\tau\|$ is a transposition, say $\|\tau\| = (i,j)$. In , an element $s\in G(r,1,n)$ is such that $s^{-1}\tau s \in B$ if and only if $\|s^{-1}(n)\|\not\in \{i,j\}$. There are $r^n(n-2)(n-1)!$ such elements in $G(r,1,n)$. Therefore evaluates as $$\begin{aligned} \chi_{{{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots}}(\tau) &= \frac{r^n(n-2)(n-1)!}{r^n (n-1)!} \chi_{{\mathfrak{h}_{k}^{n-1}}}(\|\tau\|) \\ &= (n-2) \chi_{{\mathfrak{h}_{k}^{n-1}}}(\|\tau\|) \\ &= (n-2k-2)\dim({\mathfrak{h}_{k}^{n-1}}),\end{aligned}$$ where we used in the last equality. then follows from the equality $\dim({\mathfrak{h}_{k}^{n-1}}) = n\dim(\chi_{{{\mathfrak{h}_{k}^{n-1}},\dots,1,\dots}})$. [Step 4.]{} From Steps 1–3, Formula rewrites as $$\begin{aligned} \|W\|\cdot {{\operatorname{FAC}}}_W(t) &= e^{tr\binom{n}{2}} +(-1)^n e^{-tr\binom{n}{2}} \\ & + \sum_{k=1}^{n-3} \frac{(n-2-k)k}{n-1} \binom{n}{k+1} (-1)^k e^{tr\frac{(n-1)(n-2-2k)}{2}} \\ & + \sum_{k=0}^{n-2} n\binom{n-2}{k} (-1)^k e^{tr\frac{(n-1)(n-2-2k)}{2}} \left(\sum_{j=1}^{r-1}\xi^{-j}\right).\end{aligned}$$ To simplify this expression, first note that the range of summation in the first sum may be extended to $k\in[0,n-2]$ since boundary values vanish. Then, note that $\sum_{j=1}^{r-1}\xi^{-j} = -1$. Finally, observe that we have $$n \binom{n-2}{k} - \frac{(n-2-k)k}{n-1} \binom{n}{k+1} = \binom{n}{k+1}.$$ Therefore the two sums can be merged together and we obtain $$\begin{aligned} {{\operatorname{FAC}}}_W(t) &= \frac{1}{\|W\|} \Bigg( e^{t r\binom{n}{2}} +(-1)^n e^{-t r\binom{n}{2}} \\ & \hspace{50pt} + \sum_{k=0}^{n-2} \binom{n}{k+1} (-1)^{k+1} e^{t r\frac{(n-1)(n-2k-2)}{2}} \Bigg).\end{aligned}$$ We can now finish the proof of Theorem \[thm:main\] for the group $G(r,r,n)$. Notice that the two isolated terms in the last equality may be incorporated to the sum by extending the summation range to $k\in[-1,n-1]$. Making the change of index $k\rightarrow k+1$ we obtain $$\begin{aligned} {{\operatorname{FAC}}}_W(t) &= \frac{1}{\|W\|}\sum_{k=0}^{n} \binom{n}{k} (-1)^{k} e^{t r\frac{(n-1)(n-2k)}{2}}\\ &= \frac{1}{\|W\|}e^{t r\frac{(n-1)}{2}} \left(1-e^{-t r(n-1)}\right)^n \\ &= \frac{1}{\|W\|}\left(e^{t r\frac{(n-1)}{2}} - e^{-t r\frac{(n-1)}{2}}\right)^n.\end{aligned}$$ Since we have for $W = G(r,r,n)$ that $\|{\mathcal{R}}\| = \|{\mathcal{R}}^\| = r\binom{n}{2}$, this coincides with in this case. Proof of the main theorem for the exceptional groups {#sec:exceptionaltypes} ==================================================== We now discuss the verification of Theorem \[thm:main\] for the well-generated exceptional Shephard-Todd classification types. Those were listed in Table \[tab:exceptionaltypes\]. In Table \[tab:exceptionaldegreescodegrees\] in Appendix \[app:exceptionaltypes\], we summarize more detailed information concerning reflections, degrees, codegrees, and number of irreducible representations. The order, degrees, and codegrees are taken from [@BMR1995], while the number of irreducible representations were computed using [Chevie]{} [@CHEVIE]. This software package does not only provide the number of irreducible representations but also the complete character tables[^4]. The irreducible characters are indexed in [Chevie]{} by their [degrees]{} given by the dimension of the corresponding irreducible representation, $\deg(\chi) = \chi({{1\!\!1}})$, together with the smallest integer $k$ for which the irreducible representation $V_\chi$ occurs within the $k$-th symmetric power of the reflection representation $V$, $${\operatorname{occ}(\chi)} := \min\big\{ k : V_\chi \text{ is a summand of } {\operatorname{Sym}}^k(V) \big\}.$$ Moreover, in [Chevie]{}, for each character value at a conjugacy class, a class representative within $W$ is given as a permutation of the roots for $W$. Its matrix in ${\operatorname{GL}}(V)$ thus has rows given by the images of the simple roots. Therefore, one can simply find the class representatives of the class ${\mathcal{C}}_\zeta$ given by all Coxeter elements which are $\zeta$-regular, for each primitive $h$-th root of unity $\zeta$ (see the discussion in the paragraph preceding Section \[sec:classification\]). Similarly, one can find the class representatives of the conjugacy classes of reflections. For the latter, we computed as well the class sizes. To be more precise, we used the [Sage]{} [@sage] interface to [GAP3]{} and [Chevie]{}, together with the [Sage]{} patch on reflection groups[^5] to determine the conjugacy classes of Coxeter elements, and the classes of reflections, and to compute the class sizes and to evaluate the irreducible characters at these classes. As a backup check, we rechecked this data for the irreducible characters using the [Sage]{} interface to [GAP4]{} together with the [GAP4]{} implementation of irreducible characters of permutation groups. As mentioned above, we used here the presentation of the irreducible well-generated complex reflection groups as permutation groups acting on roots. In Table \[fig:G4example\], the computed data for the exceptional complex reflection group $G_4$ is shown as an example. The first two columns provide $\deg(\chi)$ and ${\operatorname{occ}(\chi)}$ which uniquely determine the given irreducible representation $\chi$ of $G_4$. We can read from Table \[tab:exceptionaldegreescodegrees\] that the Coxeter number for $G_4$ is $6$. We thus have two primitive $6$-th roots of unity, namely $\zeta = e^{2\pi i/6}$ and $\zeta^{-1} = e^{10\pi i/ 6}$. Columns $3$ and $4$ in Figure \[fig:G4example\] show the evaluation $\chi(c)$ and $\chi(c^{-1})$ of the irreducible characters at representatives of the two conjugacy classes ${\mathcal{C}}_\zeta$ and ${\mathcal{C}}_{\zeta^{-1}}$. The last column finally gives the evaluation $\chi(R) = \sum_{\tau \in {\mathcal{R}}} \chi(\tau)$ of the irreducible characters at the sum of the reflections. Evaluating the finite sum with this data, one easily checks that Theorem \[thm:main\] holds for the group $W=G_4$. $\deg(\chi)$ ${\operatorname{occ}(\chi)}$ $\chi(c)$ $\chi(c^{-1})$ $\chi(R)$ -------------- ------------------------------ ----------------- ----------------- ----------- $1$ $0$ $1$ $1$ $8$ $1$ $4$ $\zeta_{3}^{2}$ $\zeta_{3}$ $-4$ $1$ $8$ $\zeta_{3}$ $\zeta_{3}^{2}$ $-4$ $2$ $1$ $\zeta_{3}$ $\zeta_{3}^{2}$ $4$ $2$ $3$ $\zeta_{3}^{2}$ $\zeta_{3}$ $4$ $2$ $5$ $1$ $1$ $-8$ $3$ $2$ $0$ $0$ $0$ : Character values for $G_{4}$[]{data-label="fig:G4example"} For the interested reader, the complete and long list of the evaluations for all exceptional well-generated complex reflection groups can be found in Tables \[tab:first\]–\[tab:last\] in Appendix \[app:exceptionaltypes\]. To save space, we only show the evaluation of the irreducible characters at a representative $c$ of the conjugacy class ${\mathcal{C}}_\zeta$ for $\zeta = e^{2\pi i/h}$, and omit the other primitive $h$-th roots of unity. Nonetheless, we checked the other classes as well. The procedure described in this section allowed us to compute Formula for all well-generated exceptional Shephard-Todd classification types, and for all conjugacy classes of Coxeter elements, and thus to verify the following theorem which completes the proof of Theorem \[thm:main\]. Theorem \[thm:main\] holds for all well-generated exceptional Shephard-Todd classification types. Irreducible characters of G(r,1,n) and G(r,r,n). {#app:A} ================================================ In this appendix we give the proofs of Proposition \[prop:Gr1n\] and Lemma \[lemma:step1Grr\]. We do not claim much originality. The proof of Proposition \[prop:Gr1n\] is easily found in the literature, see e.g. [@Macdonald Appendix B]. However, for the group $G(r,r,n)$, we were unable to provide references where the characters are described explicitly enough so that Lemma \[lemma:step1Grr\] could be given as an immediate corollary[^6]. Therefore we include its proof here, and since it follows from the same general theory and helps setting up notation, we also include the proof of Proposition \[prop:Gr1n\]. Both proofs are exercises in elementary representation theory, and only use the description of $G(r,p,n)$ as the semi-direct product of ${\mathcal{S}}_n$ by an abelian group, see below. Irreducible characters of semidirect products with abelian groups {#par:Serre} ----------------------------------------------------------------- The semidirect product with an abelian group is a standard situation in representation theory. Here for convenience we recall briefly the general theory, taken verbatim from [@Serre Section 8.2], to which we refer for the proofs. We follow the notation in this reference, except that we denote by $K$ the group denoted by $H$ in [@Serre], and by $\kappa$, instead of $h$, elements of this group. We let $G,A,{K}$ be finite groups such that $A$ is abelian, $G=A \cdot {K}$, and $A \cap {K}=\{1\}$. We let $X=\mathrm{Hom}(A ,{\mathbb{C}}^)$ be the group of complex irreducible characters of $A$. The group $G$ operates on $X$ by $g x(a) := x(g^{-1} a g)$ for $g\in G, x\in X, a\in A$. We fix $(x_\iota)_{\iota \in X/{K}}$ be a complete system of representative of the orbits of ${K}$ in $X$, and for $\iota \in X/{K}$, we let ${K}_\iota$ be the subgroup of ${K}$ formed by elements ${\kappa}$ such that ${\kappa}x_\iota=x_\iota$. We let $G_\iota = A \cdot {K}_\iota$ be the corresponding subgroup of $G$. We extend $x_\iota$ to $G_\iota$ by defining $x_\iota(a\cdot {\kappa}) = x_\iota(a)$ for $a\in A, {\kappa}\in {K}_\iota$. this defines a one-dimensional character of $G_\iota$. Now, let $\rho$ be an irreducible representation of ${K}_\iota$. By composing $\rho$ with the projection $G_\iota \rightarrow {K}_\iota$, we obtain an irreducible representation $\tilde \rho$ of $G_\iota$. By considering the tensor product $x_\iota \otimes \tilde\rho$ we obtain an irreducible representation of $G_\iota$. We let $\theta_{\iota,\rho}:=\mathrm{Ind}_{G_\iota}^G(x_\iota\otimes\tilde\rho)$ be the corresponding induced representation of $G$. \[prop:Serre\] The representations $\theta_{\iota,\rho}$ form a complete list of irreducible representations of $G$. Moreover two such representations $\theta_{\iota,\rho}$ and $\theta_{\iota',\rho'}$ are isomorphic if and only if $\iota=\iota'$ and $\rho$ is isomorphic to $\rho'$. Note that what precedes applies to the group $G=G(r,p,n)$, upon taking ${K}={\mathcal{S}}_n$, and letting $A$ be the group of diagonal matrices of size $n$, whose entries are $r$-th roots of unity $(\xi_1,\xi_2,\dots,\xi_n)$ such that the product $\xi_1\xi_2\dots \xi_n$ is an $(r/p)$-th root of unity. In the remainder of this section we apply the general theory to the two cases that are of interest to us, namely $p=1$ and $p=r$. In what follows $\xi$ is a fixed primitive $r$-th root of unity. Irreducible characters for the group G(r,1,n) {#sec:irreduciblecharactersr1n} --------------------------------------------- In the case of $G(r,1,n)$ we have $A\cong \left(\mathbb{Z}/r\mathbb{Z}\right)^n$, and $X=\mathrm{Hom}(A,{\mathbb{C}}^)$ is formed of the homomorphisms $\{x_\ell,\ell\in A\}$ given by $$x_\ell(a) = \xi^{ \ell_1 a_1 + \ell_2 a_2 + \dots + \ell_n a_n}$$ for $a \in A$. The orbits of $X$ under the action of ${K}={\mathcal{S}}_n$ are indexed by vectors $\iota=(i_0,i_1,\dots ,i_{r-1})$ of sum $n$, think of $i_m$ as the number of coordinates of $\ell$ equal to $m$. For a given vector $\iota$ there is a natural representative $x_\iota = x_{(0^{i_0},1^{i_1},\dots)}$, and we obtain this way a complete system of representatives of $X/{K}$. The subgroup ${K}_\iota$ of ${\mathcal{S}}_n$ formed by elements ${\kappa}$ such that ${\kappa}x_\iota = x_\iota$ is, clearly, the group of block-diagonal matrices $${K}_\iota={\mathcal{S}}_{i_0}\times{\mathcal{S}}_{i_1}\times \dots \times {\mathcal{S}}_{i_{r-1}}.$$ The irreducible representations of this group are given by $\rho = \rho_{\lambda_0} \otimes \rho_{\lambda_1}\otimes \dots \otimes \rho_{\lambda_{r-1}}$ where $\vec{\lambda}$ is a vector of partitions as in Proposition \[prop:Gr1n\] and $\rho_{\lambda_m}$ is the representation of ${\mathcal{S}}_{i_m}$ indexed by $\lambda_m$, for $0\leq m < r$. Note that the group $G_\iota = A. {K}_\iota$ is nothing but the group $B$ of block-diagonal matrices defined in Proposition \[prop:Gr1n\]. With the notation of Paragraph \[par:Serre\], the tensor product $x_\iota \otimes\tilde\rho $ is a representation of $G_\iota$ of character $$\chi_{x_\iota\otimes \tilde\rho }(w_0,w_1\dots,w_{r-1}) = \prod_{\ell=0}^{r-1} \chi_{\lambda_\ell}(\|w_\ell\|)\xi^{\ell \cdot \\|w_\ell\\| }.$$ The representation $\theta_{\iota,\rho}$ is, by definition, the induced representation $\mathrm{Ind}_{G_\iota}^G(\tilde\rho \otimes x_\iota)$, whose character is given by (for general background on characters of induced representations, see e.g. [@Serre Theorem 12]). Therefore, Proposition \[prop:Serre\] becomes Proposition \[prop:Gr1n\] is this setting. Irreducible characters for the group G(r,r,n) {#sec:irreduciblecharactersrrn} --------------------------------------------- We are going to use the general theory of Paragraph \[par:Serre\] to describe the irreducible representations of $G(r,r,n)$ as induced representations, but we will not go as far as in the case of $G(r,1,n)$. More precisely, in the notation of \[par:Serre\], we will describe explicitly the groups $G_\iota$, but we will not describe explicitly their irreducible representations in general. Instead we will keep in mind that, to prove Lemma \[lemma:step1Grr\], we are only interested in characters that do not vanish on the inverse $c^{-1}$ of the Coxeter element, so we will disregard any group $G_\iota$ that contains no conjugate of $c^{-1}$. As a side remark, let us mention that we could have proved Lemma \[lemma:step1Grr\] directly using Proposition \[prop:Gr1n\] and classical criteria regarding restrictions of induced representations ([@Serre Chapter 7]). However the approach presented here is not longer, and it requires no more background than the one presented in Appendix \[par:Serre\]. Recall that in the case of $G(r,r,n)$ the Coxeter element is $c=c_0\wr (0,\dots,0,1,r-1)$ where $c_0=(1,2,\dots,n-1)(n)$ has cycle type $[n-1,1]$. As in Section \[par:Grrn\] we assume that $n > 2$. In the case of $G(r,r,n)$ we have $A\cong \mathrm{Ker}\ \pi$ where $\pi: \left(\mathbb{Z}/r\mathbb{Z}\right)^n \rightarrow \mathbb{Z}/r\mathbb{Z}$ is the sum-of-coordinate mapping. Therefore $X=\mathrm{Hom}(A,{\mathbb{C}}^)$ is formed of the homomorphisms $$x_\ell : a \in A \longmapsto \xi^{ \ell_1 a_1 + \ell_2 a_2 + \dots \ell_n a_n}$$ for $\ell\in (\mathbb{Z}/r\mathbb{Z})^n /\!\! \approx$, where $\ell \approx \ell'$ if and only if $\ell-\ell'$ is of the form $(z,z,\dots,z)$ for some $z\in \mathbb{Z}/r\mathbb{Z}$. The orbits of $X$ under the action of ${K}={\mathcal{S}}_n$ are indexed by vectors $\iota = (i_0,i_1,\dots,i_{r-1})$ of total sum $n$ considered up to circular permutation (think of $i_m$ as the number of coordinates of $\ell$ equal to $m$, and observe that a translation on $\ell$ by a vector of the form $(z,z,\dots,z)$ acts as a circular permutation on $\iota$). For each such orbit we fix an arbitrary representative $x_\iota = x_{0^{i_0}1^{i_1}\dots}$ in order to obtain a complete system of representatives of $X/{K}$. Fix $\iota=(i_0,i_1,\dots, i_{r-1})$ and let $d$ be the smallest positive integer such that $i_{\ell+d}=i_\ell$ for all $\ell$, indices being taken modulo $r$. Note that $d$ is a divisor of $r$, say $r=md$. We now determine the group ${K}_\iota$, in the notation of \[par:Serre\]. By definition, an element ${\kappa}\in \mathfrak{S}_n$ is such that ${\kappa}x_\iota = x_\iota$ if and only if $${\kappa}(0^{i_0},1^{i_1},\dots,(r-1)^{i_{r-1}}) = (0^{i_0},1^{i_1},\dots,(r-1)^{i_{r-1}}) \text{ modulo $\approx$,}$$ where $\approx$ is the equivalence relation defined above, i.e. iff there exists $k_{\kappa}\in \mathbb{Z}/r\mathbb{Z}$ such that $$\begin{aligned} \label{eq:stabH} {\kappa}(0^{i_0},1^{i_1},\dots,(r-1)^{i_{r-1}}) = (k_{\kappa}^{i_0},(k_{\kappa}+1)^{i_1},\dots,(k_{\kappa}+r-1)^{i_{r-1}}).\end{aligned}$$ Now, let $I_\ell$ be the integer interval $I_\ell=\llbracket i_0 + i_1 + \dots +i_{\ell-1}+1, i_0 + i_1 + \dots + i_\ell\rrbracket$ for $0\leq \ell <r$. Note that $\llbracket 1,r \rrbracket = I_0 \uplus I_1 \uplus \dots \uplus I_{r-1}$. Then implies that for each $0\leq \ell <r$, we have: $$\begin{aligned} \label{eq:stabH2} {\kappa}( I_\ell ) = I_{\ell+k_{\kappa}},\end{aligned}$$ indices taken modulo $r$. In particular, we have that $i_{\ell}=i_{\ell+k_{\kappa}}$ for all $\ell$, so that $k_{\kappa}$ is a multiple of $d$. Unless one of the entries of $\iota$ is equal to $n$ or $n-1$, the inverse $c^{-1}$ of the Coxeter element has no conjugates in the group $G_\iota$. First note that $c^{-1}$ has a conjugate in $G_\iota$ if and only if $c_0^{-1}=\|c^{-1}\|$ has a conjugate in ${K}_\iota$. Since $c_0^{-1}=(n-1,n-2,\dots,1)(n)$ has a fixed point, then by , if such a conjugate ${\kappa}$ exists it is such that $k_{\kappa}=0$, keeping previous notation. Hence using again, the conjugate belongs to the group of block-diagonal matrices ${\mathcal{S}}_{i_0}\times{\mathcal{S}}_{i_1}\times\dots \times {\mathcal{S}}_{i_{r-1}}$. But since $c_0^{-1}$ has cycle type $[n-1,1]$, one of the blocks must be of size $n-1$ or $n$, in order to fit the cycle of length $(n-1)$. Recalling the classical expression of induced characters as a sum over conjugates, see e.g. [@Serre Theorem 12], this implies the following lemma. Unless one of the entries of $\iota$ is equal to $n$ or $n-1$, the character of the representation $\theta_{\iota,\rho}$ defined in \[par:Serre\] vanishes on the inverse $c^{-1}$ of the Coxeter element, for any irreducible representation $\rho$ of ${K}_\iota$. Since we only want to prove Lemma \[lemma:step1Grr\], we may thus focus only on the case when $\iota$ has a coordinate equal to $n$ or $n-1$. We treat both cases separately. In the first case, we have that $\iota = (n, 0,\dots,0)$ up to circular permutation. The group ${K}_\iota$ coincides with ${\mathcal{S}}_n$, and $G_\iota$ coincides with $G(r,r,n)$. For any irreducible representation $\rho=\rho_\lambda$ of ${\mathcal{S}}_n$, the character of the irreducible representation $\theta_{\iota,\rho}$ on an element $w\in G(r,r,n)$ is given, simply, by the ${\mathcal{S}}_n$-character $\chi_\lambda(\|w\|)$. It remains to determine which partitions $\lambda$ of $n$ are such that the character $\chi_\lambda$ does not vanish on the permutation $c_0^{-1}$ of type $[n-1,1]$. Clearly, the Murnaghan-Nakayama rule implies than $\lambda$ is a hook or a quasi-hook, since these are the only partitions containing a strip of length $n-1$. A closer look at the case of hooks shows, moreover, that the only hooks for which the character does not vanish are $[n]$ and $[1^n]$. Finally, it is clear that the characters thus defined coincide with the restrictions to $G(r,r,n)$ of the representations $\chi_{[n],\dots}$, $\chi_{[1^n],\dots}$ and $\chi_{{\mathfrak{qh}_{k}^{n}},\dots}$ as stated in Lemma \[lemma:step1Grr\]. In the second case, we have that, up to a circular permutation, $\iota=(n-1,0,\dots,0,1,0,\dots,0)$ where the $1$ appears in position $j$, for $1\leq j <r$. Notice that since $n\neq 2$ we have $n-1\neq1$ so $\iota$ is fixed by no non-trivial circular permutation. In previous notation, this means that $d=r$, hence all ${\kappa}\in {K}_\iota$ are such that $k_{\kappa}=0 (mod \ r)$. Thus by the group ${K}_\iota$ is the group of block-diagonal matrices $${K}_\iota = {\mathcal{S}}_{n-1}\times {\mathcal{S}}_1.$$ The group $G_\iota$ is the subgroup of $G(r,r,n)$ formed by matrices $w$ such that $\|w\|\in {K}_\iota$. Irreducible representations of ${K}_\iota$ are indexed by partitions $\lambda$ of $n-1$, and for each such partition we may thus construct the representation $\theta_{\iota, \rho}$ of $G(r,r,n)$ as in \[par:Serre\]. By definition, its character is $$\chi(w) = \sum_{\substack{s \in G(r,r,n) \\ s^{-1}ws \in G_\iota}} \chi_{\lambda}(\|w_1\|) \xi^{j \\|w_2\\|}$$ where $s^{-1}ws \in B$ in the sum is denoted by $(w_1,w_2)$ where $w_1$ is the principal $(n-1)\times(n-1)$ submatrix and $w_2$ is the bottom-right entry. Since $\|c^{-1}\|=c_0^{-1}$ has cycle type $[n-1,1]$, the only conjugates $s^{-1}c^{-1}s$ of $c^{-1}$ that are in $G_\iota$ are such that $\|w_1\|$ is an $(n-1)$-cycle. Therefore by the Murnaghan-Nakayama rule one has $\chi(c^{-1})=0$ unless $\lambda$ is a hook of size $n-1$. This concludes the proof of Lemma \[lemma:step1Grr\], by observing that the character we just described coincides with the restriction of the $G(r,1,n)$-character $\chi_{\lambda,\dots,1,\dots}$ as in the statement of Lemma \[lemma:step1Grr\]. Character evaluations for the exceptional groups {#app:exceptionaltypes} ================================================ type order degrees codegrees \# irr.rep. -------------------- ------------- ------------------------- ------------------------- ------------- \[-10pt\] $G_4$ $24$ $4,6$ $0,2$ $7$ \[-10pt\] $G_5$ $72$ $6,12$ $0,6$ $21$ \[-10pt\] $G_6$ $48$ $4,12$ $0,8$ $14$ \[-10pt\] $G_8$ $96$ $8,12$ $0,4$ $16$ \[-10pt\] $G_9$ $192$ $8,24$ $0,16$ $32$ \[-10pt\] $G_{10}$ $288$ $12,24$ $0,12$ $48$ \[-10pt\] $G_{14}$ $144$ $6,24$ $0,18$ $24$ \[-10pt\] $G_{16}$ $600$ $20,30$ $0,10$ $45$ \[-10pt\] $G_{17}$ $1200$ $20,60$ $0,40$ $90$ \[-10pt\] $G_{18}$ $1800$ $30,60$ $0,30$ $135$ \[-10pt\] $G_{20}$ $360$ $12,30$ $0,18$ $27$ \[-10pt\] $G_{21}$ $720$ $12,60$ $0,48$ $54$ \[-10pt\] $G_{23}$ $120$ $2,6,10$ $0,4,8$ $10$ \[-10pt\] $G_{24}$ $336$ $4,6,14$ $0,8,10$ $12$ \[-10pt\] $G_{25}$ $648$ $6,9,12$ $0,3,6$ $24$ \[-10pt\] $G_{26}$ $1296$ $6,12,18$ $0,6,12$ $48$ \[-10pt\] $G_{27}$ $2160$ $6,12,30$ $0,18,24$ $34$ \[-10pt\] $G_{28}$ $1152$ $2,6,8,12$ $0,4,6,10$ $25$ \[-10pt\] $G_{29}$ $7680$ $4,8,12,20$ $0,8,12,16$ $37$ \[-10pt\] $G_{30}$ $14400$ $2,12,20,30$ $0,10,18,28$ $34$ \[-10pt\] $G_{32}$ $155520$ $12,18,24,30$ $0,6,12,18$ $102$ \[-10pt\] $G_{33}$ $51840$ $4,6,10,12,18$ $0,6,8,12,14$ $40$ \[-10pt\] $G_{34}$ $39191040$ $6,12,18,24,30,42 $ $0,12,18,24,30,36$ $169$ \[-10pt\] $G_{35}$ $51840$ $2,5,6,8,9,12$ $0,3,4,6,7,10$ $25$ \[-10pt\] $G_{36}$ $2903040$ $2,6,8,10,12,14,18$ $0,4,6,8,10,12,16$ $60$ \[-10pt\] $G_{37}$ $696729600$ $2,8,12,14,18,20,24,30$ $0,6,10,12,16,18,22,28$ $112$ : The order, degrees and codegrees of all well-generated exceptional Shephard-Todd classification types.[]{data-label="tab:exceptionaldegreescodegrees"} \[tab:first\] [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $8$ $1$ & $4$ & $\zeta_{3}^{2}$ & $-4$ $1$ & $8$ & $\zeta_{3}$ & $-4$ $2$ & $1$ & $\zeta_{3}$ & $4$ $2$ & $3$ & $\zeta_{3}^{2}$ & $4$ $2$ & $5$ & $1$ & $-8$ $3$ & $2$ & $0$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $16$ $1$ & $4$ & $\zeta_{3}$ & $4$ $1$ & $4$ & $\zeta_{3}$ & $4$ $1$ & $8$ & $\zeta_{3}^{2}$ & $4$ $1$ & $8$ & $\zeta_{3}^{2}$ & $-8$ $1$ & $8$ & $\zeta_{3}^{2}$ & $4$ $1$ & $12$ & $1$ & $-8$ $1$ & $12$ & $1$ & $-8$ $1$ & $16$ & $\zeta_{3}$ & $-8$ $2$ & $1$ & $0$ & $8$ $2$ & $3$ & $0$ & $8$ $2$ & $3$ & $0$ & $8$ $2$ & $5$ & $0$ & $-4$ $2$ & $5$ & $0$ & $8$ $2$ & $5$ & $0$ & $-4$ $2$ & $7$ & $0$ & $-4$ $2$ & $7$ & $0$ & $-4$ $2$ & $9$ & $0$ & $-16$ $3$ & $2$ & $-\zeta_{3}^{2}$ & $0$ $3$ & $4$ & $-\zeta_{3}$ & $0$ $3$ & $6$ & $-1$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $14$ $1$ & $4$ & $\zeta_{3}$ & $2$ $1$ & $6$ & $-1$ & $2$ $1$ & $8$ & $\zeta_{3}^{2}$ & $2$ $1$ & $10$ & $-\zeta_{3}$ & $-10$ $1$ & $14$ & $-\zeta_{3}^{2}$ & $-10$ $2$ & $1$ & $\zeta_{12}^{11}$ & $4$ $2$ & $3$ & $-\zeta_{12}^{11}$ & $4$ $2$ & $3$ & $-\zeta_{12}^{7}$ & $4$ $2$ & $5$ & $\zeta_{12}^{7}$ & $4$ $2$ & $5$ & $\zeta_{4}$ & $-8$ $2$ & $7$ & $-\zeta_{4}$ & $-8$ $3$ & $2$ & $0$ & $6$ $3$ & $4$ & $0$ & $-6$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $18$ $1$ & $6$ & $-1$ & $-6$ $1$ & $12$ & $1$ & $-6$ $1$ & $18$ & $-1$ & $-6$ $2$ & $1$ & $\zeta_{4}$ & $12$ $2$ & $4$ & $-1$ & $12$ $2$ & $7$ & $-\zeta_{4}$ & $12$ $2$ & $7$ & $-\zeta_{4}$ & $-12$ $2$ & $10$ & $1$ & $-12$ $2$ & $13$ & $\zeta_{4}$ & $-12$ $3$ & $2$ & $0$ & $6$ $3$ & $4$ & $0$ & $6$ $3$ & $6$ & $0$ & $6$ $3$ & $8$ & $0$ & $-18$ $4$ & $3$ & $\zeta_{4}$ & $0$ $4$ & $5$ & $-\zeta_{4}$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $30$ $1$ & $6$ & $\zeta_{4}$ & $6$ $1$ & $12$ & $-1$ & $6$ $1$ & $12$ & $-1$ & $6$ $1$ & $18$ & $-\zeta_{4}$ & $-18$ $1$ & $18$ & $-\zeta_{4}$ & $6$ $1$ & $24$ & $1$ & $-18$ $1$ & $30$ & $\zeta_{4}$ & $-18$ $2$ & $1$ & $-\zeta_{8}^{3}$ & $12$ $2$ & $4$ & $1$ & $12$ $2$ & $5$ & $\zeta_{8}^{3}$ & $12$ $2$ & $7$ & $\zeta_{8}$ & $-12$ $2$ & $7$ & $\zeta_{8}$ & $12$ $2$ & $8$ & $-1$ & $12$ $2$ & $10$ & $\zeta_{4}$ & $-12$ $2$ & $11$ & $-\zeta_{8}$ & $-12$ $2$ & $11$ & $-\zeta_{8}$ & $12$ $2$ & $13$ & $\zeta_{8}^{3}$ & $-12$ $2$ & $14$ & $-\zeta_{4}$ & $-12$ $2$ & $17$ & $-\zeta_{8}^{3}$ & $-12$ $3$ & $2$ & $0$ & $18$ $3$ & $4$ & $0$ & $18$ $3$ & $6$ & $0$ & $-6$ $3$ & $6$ & $0$ & $18$ $3$ & $8$ & $0$ & $-6$ $3$ & $8$ & $0$ & $-6$ $3$ & $10$ & $0$ & $-6$ $3$ & $12$ & $0$ & $-30$ $4$ & $3$ & $\zeta_{8}$ & $0$ $4$ & $5$ & $-\zeta_{8}^{3}$ & $0$ $4$ & $7$ & $-\zeta_{8}$ & $0$ $4$ & $9$ & $\zeta_{8}^{3}$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $34$ $1$ & $6$ & $\zeta_{4}$ & $10$ $1$ & $8$ & $\zeta_{3}$ & $10$ $1$ & $12$ & $-1$ & $10$ $1$ & $14$ & $\zeta_{12}^{7}$ & $-14$ $1$ & $16$ & $\zeta_{3}^{2}$ & $10$ $1$ & $18$ & $-\zeta_{4}$ & $10$ $1$ & $20$ & $-\zeta_{3}$ & $-14$ $1$ & $22$ & $\zeta_{12}^{11}$ & $-14$ $1$ & $26$ & $-\zeta_{12}^{7}$ & $-14$ $1$ & $28$ & $-\zeta_{3}^{2}$ & $-14$ $1$ & $34$ & $-\zeta_{12}^{11}$ & $-14$ $2$ & $1$ & $0$ & $20$ $2$ & $4$ & $0$ & $20$ $2$ & $5$ & $0$ & $20$ $2$ & $7$ & $0$ & $20$ $2$ & $7$ & $0$ & $-4$ $2$ & $8$ & $0$ & $20$ $2$ & $9$ & $0$ & $-4$ $2$ & $10$ & $0$ & $-4$ $2$ & $11$ & $0$ & $20$ $2$ & $11$ & $0$ & $-4$ $2$ & $12$ & $0$ & $-4$ $2$ & $13$ & $0$ & $-4$ $2$ & $14$ & $0$ & $-4$ $2$ & $15$ & $0$ & $-4$ $2$ & $15$ & $0$ & $-28$ $2$ & $17$ & $0$ & $-4$ $2$ & $18$ & $0$ & $-28$ $2$ & $21$ & $0$ & $-28$ $3$ & $2$ & $-\zeta_{12}^{7}$ & $6$ $3$ & $4$ & $-\zeta_{3}^{2}$ & $6$ $3$ & $6$ & $-\zeta_{4}$ & $6$ $3$ & $6$ & $\zeta_{4}$ & $6$ $3$ & $8$ & $-\zeta_{3}$ & $6$ $3$ & $8$ & $\zeta_{3}$ & $-18$ $3$ & $10$ & $\zeta_{12}^{11}$ & $6$ $3$ & $10$ & $-\zeta_{12}^{11}$ & $6$ $3$ & $12$ & $1$ & $-18$ $3$ & $12$ & $-1$ & $6$ $3$ & $14$ & $\zeta_{12}^{7}$ & $6$ $3$ & $16$ & $\zeta_{3}^{2}$ & $-18$ $4$ & $3$ & $0$ & $16$ $4$ & $5$ & $0$ & $-8$ $4$ & $7$ & $0$ & $-8$ $4$ & $9$ & $0$ & $16$ $4$ & $11$ & $0$ & $-8$ $4$ & $13$ & $0$ & $-8$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $28$ $1$ & $8$ & $\zeta_{3}$ & $4$ $1$ & $12$ & $-1$ & $4$ $1$ & $16$ & $\zeta_{3}^{2}$ & $4$ $1$ & $20$ & $-\zeta_{3}$ & $-20$ $1$ & $28$ & $-\zeta_{3}^{2}$ & $-20$ $2$ & $1$ & $\zeta_{24} + \zeta_{24}^{19}$ & $8$ $2$ & $4$ & $0$ & $8$ $2$ & $5$ & $-\zeta_{24}^{11} - \zeta_{24}^{17}$ & $8$ $2$ & $7$ & $-\zeta_{24} - \zeta_{24}^{19}$ & $8$ $2$ & $8$ & $0$ & $8$ $2$ & $9$ & $\zeta_{8} + \zeta_{8}^{3}$ & $-16$ $2$ & $11$ & $\zeta_{24}^{11} + \zeta_{24}^{17}$ & $8$ $2$ & $12$ & $0$ & $-16$ $2$ & $15$ & $-\zeta_{8} - \zeta_{8}^{3}$ & $-16$ $3$ & $2$ & $-\zeta_{3}$ & $12$ $3$ & $4$ & $-\zeta_{3}^{2}$ & $12$ $3$ & $6$ & $-1$ & $12$ $3$ & $6$ & $1$ & $-12$ $3$ & $8$ & $\zeta_{3}$ & $-12$ $3$ & $10$ & $\zeta_{3}^{2}$ & $-12$ $4$ & $3$ & $0$ & $16$ $4$ & $5$ & $0$ & $-8$ $4$ & $7$ & $0$ & $-8$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $48$ $1$ & $12$ & $\zeta_{5}^{2}$ & $-12$ $1$ & $24$ & $\zeta_{5}^{4}$ & $-12$ $1$ & $36$ & $\zeta_{5}$ & $-12$ $1$ & $48$ & $\zeta_{5}^{3}$ & $-12$ $2$ & $1$ & $\zeta_{5}$ & $36$ $2$ & $7$ & $\zeta_{5}^{2}$ & $36$ $2$ & $13$ & $\zeta_{5}^{3}$ & $36$ $2$ & $13$ & $\zeta_{5}^{3}$ & $-24$ $2$ & $19$ & $\zeta_{5}^{4}$ & $36$ $2$ & $19$ & $\zeta_{5}^{4}$ & $-24$ $2$ & $25$ & $1$ & $-24$ $2$ & $25$ & $1$ & $-24$ $2$ & $31$ & $\zeta_{5}$ & $-24$ $2$ & $37$ & $\zeta_{5}^{2}$ & $-24$ $3$ & $2$ & $0$ & $24$ $3$ & $6$ & $0$ & $24$ $3$ & $10$ & $0$ & $24$ $3$ & $10$ & $0$ & $24$ $3$ & $14$ & $0$ & $24$ $3$ & $14$ & $0$ & $-36$ $3$ & $18$ & $0$ & $24$ $3$ & $18$ & $0$ & $-36$ $3$ & $22$ & $0$ & $-36$ $3$ & $26$ & $0$ & $-36$ $4$ & $3$ & $-\zeta_{5}^{3}$ & $12$ $4$ & $6$ & $\zeta_{5}$ & $12$ $4$ & $8$ & $\zeta_{5}^{3}$ & $12$ $4$ & $9$ & $-\zeta_{5}^{4}$ & $12$ $4$ & $11$ & $-\zeta_{5}$ & $12$ $4$ & $12$ & $\zeta_{5}^{2}$ & $12$ $4$ & $14$ & $\zeta_{5}^{4}$ & $12$ $4$ & $15$ & $-1$ & $-48$ $4$ & $17$ & $-\zeta_{5}^{2}$ & $12$ $4$ & $20$ & $1$ & $-48$ $5$ & $4$ & $-\zeta_{5}^{4}$ & $0$ $5$ & $8$ & $-\zeta_{5}^{3}$ & $0$ $5$ & $10$ & $-1$ & $0$ $5$ & $12$ & $-\zeta_{5}^{2}$ & $0$ $5$ & $16$ & $-\zeta_{5}$ & $0$ $6$ & $5$ & $0$ & $48$ $6$ & $7$ & $0$ & $-12$ $6$ & $9$ & $0$ & $-12$ $6$ & $11$ & $0$ & $-12$ $6$ & $13$ & $0$ & $-12$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $78$ $1$ & $12$ & $\zeta_{5}$ & $18$ $1$ & $24$ & $\zeta_{5}^{2}$ & $18$ $1$ & $30$ & $-1$ & $18$ $1$ & $36$ & $\zeta_{5}^{3}$ & $18$ $1$ & $42$ & $-\zeta_{5}$ & $-42$ $1$ & $48$ & $\zeta_{5}^{4}$ & $18$ $1$ & $54$ & $-\zeta_{5}^{2}$ & $-42$ $1$ & $66$ & $-\zeta_{5}^{3}$ & $-42$ $1$ & $78$ & $-\zeta_{5}^{4}$ & $-42$ $2$ & $1$ & $\zeta_{20}^{17}$ & $36$ $2$ & $7$ & $-\zeta_{20}^{9}$ & $36$ $2$ & $11$ & $-\zeta_{20}^{17}$ & $36$ $2$ & $13$ & $\zeta_{20}$ & $36$ $2$ & $13$ & $\zeta_{20}$ & $-24$ $2$ & $17$ & $\zeta_{20}^{9}$ & $36$ $2$ & $19$ & $-\zeta_{20}^{13}$ & $-24$ $2$ & $19$ & $-\zeta_{20}^{13}$ & $36$ $2$ & $23$ & $-\zeta_{20}$ & $36$ $2$ & $23$ & $-\zeta_{20}$ & $-24$ $2$ & $25$ & $\zeta_{4}$ & $-24$ $2$ & $25$ & $\zeta_{4}$ & $-24$ $2$ & $29$ & $\zeta_{20}^{13}$ & $36$ $2$ & $29$ & $\zeta_{20}^{13}$ & $-24$ $2$ & $31$ & $-\zeta_{20}^{17}$ & $-24$ $2$ & $35$ & $-\zeta_{4}$ & $-24$ $2$ & $35$ & $-\zeta_{4}$ & $-24$ $2$ & $37$ & $\zeta_{20}^{9}$ & $-24$ $2$ & $41$ & $\zeta_{20}^{17}$ & $-24$ $2$ & $47$ & $-\zeta_{20}^{9}$ & $-24$ $3$ & $2$ & $0$ & $54$ $3$ & $6$ & $0$ & $54$ $3$ & $10$ & $0$ & $54$ $3$ & $10$ & $0$ & $54$ $3$ & $12$ & $0$ & $-6$ $3$ & $14$ & $0$ & $54$ $3$ & $14$ & $0$ & $-6$ $3$ & $16$ & $0$ & $-6$ $3$ & $18$ & $0$ & $54$ $3$ & $18$ & $0$ & $-6$ $3$ & $20$ & $0$ & $-6$ $3$ & $20$ & $0$ & $-6$ $3$ & $22$ & $0$ & $-6$ $3$ & $24$ & $0$ & $-6$ $3$ & $24$ & $0$ & $-66$ $3$ & $26$ & $0$ & $-6$ $3$ & $28$ & $0$ & $-6$ $3$ & $28$ & $0$ & $-66$ $3$ & $32$ & $0$ & $-66$ $3$ & $36$ & $0$ & $-66$ $4$ & $3$ & $\zeta_{20}$ & $12$ $4$ & $6$ & $-\zeta_{5}^{3}$ & $12$ $4$ & $8$ & $\zeta_{5}^{4}$ & $12$ $4$ & $9$ & $-\zeta_{20}^{13}$ & $12$ $4$ & $11$ & $\zeta_{20}^{17}$ & $12$ $4$ & $12$ & $\zeta_{5}$ & $12$ $4$ & $13$ & $-\zeta_{20}$ & $12$ $4$ & $14$ & $-\zeta_{5}^{2}$ & $12$ $4$ & $15$ & $\zeta_{4}$ & $-48$ $4$ & $16$ & $\zeta_{5}^{3}$ & $12$ $4$ & $17$ & $-\zeta_{20}^{9}$ & $12$ $4$ & $18$ & $-\zeta_{5}^{4}$ & $12$ $4$ & $19$ & $\zeta_{20}^{13}$ & $12$ $4$ & $20$ & $1$ & $-48$ $4$ & $21$ & $-\zeta_{20}^{17}$ & $12$ $4$ & $22$ & $-\zeta_{5}$ & $12$ $4$ & $24$ & $\zeta_{5}^{2}$ & $12$ $4$ & $25$ & $-\zeta_{4}$ & $-48$ $4$ & $27$ & $\zeta_{20}^{9}$ & $12$ $4$ & $30$ & $-1$ & $-48$ $5$ & $4$ & $-\zeta_{5}^{2}$ & $30$ $5$ & $8$ & $-\zeta_{5}^{4}$ & $30$ $5$ & $10$ & $1$ & $-30$ $5$ & $12$ & $-\zeta_{5}$ & $30$ $5$ & $14$ & $\zeta_{5}^{2}$ & $-30$ $5$ & $16$ & $-\zeta_{5}^{3}$ & $30$ $5$ & $18$ & $\zeta_{5}^{4}$ & $-30$ $5$ & $20$ & $-1$ & $30$ $5$ & $22$ & $\zeta_{5}$ & $-30$ $5$ & $26$ & $\zeta_{5}^{3}$ & $-30$ $6$ & $5$ & $0$ & $48$ $6$ & $7$ & $0$ & $-12$ $6$ & $9$ & $0$ & $-12$ $6$ & $11$ & $0$ & $-12$ $6$ & $13$ & $0$ & $-12$ $6$ & $15$ & $0$ & $48$ $6$ & $17$ & $0$ & $-12$ $6$ & $19$ & $0$ & $-12$ $6$ & $21$ & $0$ & $-12$ $6$ & $23$ & $0$ & $-12$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $88$ $1$ & $12$ & $\zeta_{5}$ & $28$ $1$ & $20$ & $\zeta_{3}$ & $28$ $1$ & $24$ & $\zeta_{5}^{2}$ & $28$ $1$ & $32$ & $\zeta_{15}^{8}$ & $-32$ $1$ & $36$ & $\zeta_{5}^{3}$ & $28$ $1$ & $40$ & $\zeta_{3}^{2}$ & $28$ $1$ & $44$ & $\zeta_{15}^{11}$ & $-32$ $1$ & $48$ & $\zeta_{5}^{4}$ & $28$ $1$ & $52$ & $\zeta_{15}^{13}$ & $-32$ $1$ & $56$ & $\zeta_{15}^{14}$ & $-32$ $1$ & $64$ & $\zeta_{15}$ & $-32$ $1$ & $68$ & $\zeta_{15}^{2}$ & $-32$ $1$ & $76$ & $\zeta_{15}^{4}$ & $-32$ $1$ & $88$ & $\zeta_{15}^{7}$ & $-32$ $2$ & $1$ & $0$ & $56$ $2$ & $7$ & $0$ & $56$ $2$ & $11$ & $0$ & $56$ $2$ & $13$ & $0$ & $56$ $2$ & $13$ & $0$ & $-4$ $2$ & $17$ & $0$ & $56$ $2$ & $19$ & $0$ & $56$ $2$ & $19$ & $0$ & $-4$ $2$ & $21$ & $0$ & $-4$ $2$ & $23$ & $0$ & $56$ $2$ & $23$ & $0$ & $-4$ $2$ & $25$ & $0$ & $-4$ $2$ & $25$ & $0$ & $-4$ $2$ & $27$ & $0$ & $-4$ $2$ & $29$ & $0$ & $56$ $2$ & $29$ & $0$ & $-4$ $2$ & $31$ & $0$ & $-4$ $2$ & $33$ & $0$ & $-4$ $2$ & $33$ & $0$ & $-64$ $2$ & $35$ & $0$ & $-4$ $2$ & $35$ & $0$ & $-4$ $2$ & $37$ & $0$ & $-4$ $2$ & $39$ & $0$ & $-4$ $2$ & $39$ & $0$ & $-64$ $2$ & $41$ & $0$ & $-4$ $2$ & $45$ & $0$ & $-64$ $2$ & $45$ & $0$ & $-64$ $2$ & $47$ & $0$ & $-4$ $2$ & $51$ & $0$ & $-64$ $2$ & $57$ & $0$ & $-64$ $3$ & $2$ & $-\zeta_{15}^{8}$ & $24$ $3$ & $6$ & $-\zeta_{5}^{3}$ & $24$ $3$ & $10$ & $-\zeta_{3}^{2}$ & $24$ $3$ & $10$ & $-\zeta_{3}^{2}$ & $24$ $3$ & $12$ & $-\zeta_{5}$ & $24$ $3$ & $14$ & $-\zeta_{15}^{11}$ & $24$ $3$ & $14$ & $-\zeta_{15}^{11}$ & $-36$ $3$ & $16$ & $-\zeta_{15}^{4}$ & $24$ $3$ & $18$ & $-\zeta_{5}^{4}$ & $24$ $3$ & $18$ & $-\zeta_{5}^{4}$ & $-36$ $3$ & $20$ & $-\zeta_{3}$ & $24$ $3$ & $20$ & $-\zeta_{3}$ & $24$ $3$ & $22$ & $-\zeta_{15}^{13}$ & $24$ $3$ & $22$ & $-\zeta_{15}^{13}$ & $-36$ $3$ & $24$ & $-\zeta_{5}^{2}$ & $24$ $3$ & $24$ & $-\zeta_{5}^{2}$ & $-36$ $3$ & $26$ & $-\zeta_{15}^{14}$ & $24$ $3$ & $26$ & $-\zeta_{15}^{14}$ & $-36$ $3$ & $28$ & $-\zeta_{15}^{7}$ & $24$ $3$ & $28$ & $-\zeta_{15}^{7}$ & $-36$ $3$ & $30$ & $-1$ & $24$ $3$ & $30$ & $-1$ & $24$ $3$ & $32$ & $-\zeta_{15}^{8}$ & $-36$ $3$ & $34$ & $-\zeta_{15}$ & $24$ $3$ & $34$ & $-\zeta_{15}$ & $-36$ $3$ & $36$ & $-\zeta_{5}^{3}$ & $-36$ $3$ & $38$ & $-\zeta_{15}^{2}$ & $24$ $3$ & $38$ & $-\zeta_{15}^{2}$ & $-36$ $3$ & $42$ & $-\zeta_{5}$ & $-36$ $3$ & $46$ & $-\zeta_{15}^{4}$ & $-36$ $4$ & $3$ & $0$ & $52$ $4$ & $6$ & $0$ & $52$ $4$ & $8$ & $0$ & $-8$ $4$ & $9$ & $0$ & $52$ $4$ & $11$ & $0$ & $-8$ $4$ & $12$ & $0$ & $52$ $4$ & $13$ & $0$ & $-8$ $4$ & $14$ & $0$ & $-8$ $4$ & $15$ & $0$ & $-8$ $4$ & $16$ & $0$ & $-8$ $4$ & $17$ & $0$ & $-8$ $4$ & $18$ & $0$ & $52$ $4$ & $19$ & $0$ & $-8$ $4$ & $20$ & $0$ & $-68$ $4$ & $21$ & $0$ & $52$ $4$ & $22$ & $0$ & $-8$ $4$ & $23$ & $0$ & $-8$ $4$ & $24$ & $0$ & $52$ $4$ & $25$ & $0$ & $-68$ $4$ & $26$ & $0$ & $-8$ $4$ & $27$ & $0$ & $52$ $4$ & $28$ & $0$ & $-8$ $4$ & $29$ & $0$ & $-8$ $4$ & $30$ & $0$ & $-8$ $4$ & $31$ & $0$ & $-8$ $4$ & $32$ & $0$ & $-8$ $4$ & $34$ & $0$ & $-8$ $4$ & $35$ & $0$ & $-68$ $4$ & $37$ & $0$ & $-8$ $4$ & $40$ & $0$ & $-68$ $5$ & $4$ & $\zeta_{15}$ & $20$ $5$ & $8$ & $\zeta_{15}^{2}$ & $20$ $5$ & $10$ & $\zeta_{3}^{2}$ & $20$ $5$ & $12$ & $\zeta_{5}$ & $-40$ $5$ & $14$ & $\zeta_{15}^{11}$ & $20$ $5$ & $16$ & $\zeta_{15}^{4}$ & $20$ $5$ & $18$ & $\zeta_{5}^{4}$ & $-40$ $5$ & $20$ & $\zeta_{3}$ & $20$ $5$ & $22$ & $\zeta_{15}^{13}$ & $20$ $5$ & $24$ & $\zeta_{5}^{2}$ & $-40$ $5$ & $26$ & $\zeta_{15}^{14}$ & $20$ $5$ & $28$ & $\zeta_{15}^{7}$ & $20$ $5$ & $30$ & $1$ & $-40$ $5$ & $32$ & $\zeta_{15}^{8}$ & $20$ $5$ & $36$ & $\zeta_{5}^{3}$ & $-40$ $6$ & $5$ & $0$ & $48$ $6$ & $7$ & $0$ & $-12$ $6$ & $9$ & $0$ & $-12$ $6$ & $11$ & $0$ & $-12$ $6$ & $13$ & $0$ & $-12$ $6$ & $15$ & $0$ & $48$ $6$ & $17$ & $0$ & $-12$ $6$ & $19$ & $0$ & $-12$ $6$ & $21$ & $0$ & $-12$ $6$ & $23$ & $0$ & $-12$ $6$ & $25$ & $0$ & $48$ $6$ & $27$ & $0$ & $-12$ $6$ & $29$ & $0$ & $-12$ $6$ & $31$ & $0$ & $-12$ $6$ & $33$ & $0$ & $-12$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $40$ $1$ & $20$ & $\zeta_{3}^{2}$ & $-20$ $1$ & $40$ & $\zeta_{3}$ & $-20$ $2$ & $1$ & $-\zeta_{15}^{2} - \zeta_{15}^{8}$ & $20$ $2$ & $7$ & $-\zeta_{15}^{11} - \zeta_{15}^{14}$ & $20$ $2$ & $11$ & $-\zeta_{15}^{7} - \zeta_{15}^{13}$ & $20$ $2$ & $17$ & $-\zeta_{15} - \zeta_{15}^{4}$ & $20$ $2$ & $21$ & $-\zeta_{5} - \zeta_{5}^{4}$ & $-40$ $2$ & $27$ & $-\zeta_{5}^{2} - \zeta_{5}^{3}$ & $-40$ $3$ & $2$ & $-\zeta_{15}^{7} - \zeta_{15}^{13}$ & $0$ $3$ & $6$ & $-\zeta_{5}^{2} - \zeta_{5}^{3}$ & $0$ $3$ & $10$ & $-\zeta_{15}^{11} - \zeta_{15}^{14}$ & $0$ $3$ & $10$ & $-\zeta_{15}^{2} - \zeta_{15}^{8}$ & $0$ $3$ & $12$ & $-\zeta_{5} - \zeta_{5}^{4}$ & $0$ $3$ & $14$ & $-\zeta_{15} - \zeta_{15}^{4}$ & $0$ $4$ & $3$ & $1$ & $40$ $4$ & $6$ & $-1$ & $40$ $4$ & $8$ & $-\zeta_{3}^{2}$ & $-20$ $4$ & $11$ & $\zeta_{3}^{2}$ & $-20$ $4$ & $13$ & $\zeta_{3}$ & $-20$ $4$ & $16$ & $-\zeta_{3}$ & $-20$ $5$ & $4$ & $0$ & $20$ $5$ & $8$ & $0$ & $20$ $5$ & $12$ & $0$ & $-40$ $6$ & $5$ & $-\zeta_{3}^{2}$ & $0$ $6$ & $7$ & $-\zeta_{3}$ & $0$ $6$ & $9$ & $-1$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $70$ $1$ & $20$ & $\zeta_{3}$ & $10$ $1$ & $30$ & $-1$ & $10$ $1$ & $40$ & $\zeta_{3}^{2}$ & $10$ $1$ & $50$ & $-\zeta_{3}$ & $-50$ $1$ & $70$ & $-\zeta_{3}^{2}$ & $-50$ $2$ & $1$ & $-\zeta_{60}^{19} - \zeta_{60}^{31}$ & $20$ $2$ & $7$ & $\zeta_{60}^{7} + \zeta_{60}^{43}$ & $20$ $2$ & $11$ & $\zeta_{60}^{11} + \zeta_{60}^{59}$ & $20$ $2$ & $13$ & $-\zeta_{60}^{7} - \zeta_{60}^{43}$ & $20$ $2$ & $17$ & $-\zeta_{60}^{23} - \zeta_{60}^{47}$ & $20$ $2$ & $19$ & $\zeta_{60}^{19} + \zeta_{60}^{31}$ & $20$ $2$ & $21$ & $-\zeta_{20}^{13} - \zeta_{20}^{17}$ & $-40$ $2$ & $23$ & $\zeta_{60}^{23} + \zeta_{60}^{47}$ & $20$ $2$ & $27$ & $\zeta_{20} + \zeta_{20}^{9}$ & $-40$ $2$ & $29$ & $-\zeta_{60}^{11} - \zeta_{60}^{59}$ & $20$ $2$ & $33$ & $-\zeta_{20} - \zeta_{20}^{9}$ & $-40$ $2$ & $39$ & $\zeta_{20}^{13} + \zeta_{20}^{17}$ & $-40$ $3$ & $2$ & $\zeta_{15}^{11} + \zeta_{15}^{14}$ & $30$ $3$ & $6$ & $\zeta_{5} + \zeta_{5}^{4}$ & $30$ $3$ & $10$ & $\zeta_{15}^{7} + \zeta_{15}^{13}$ & $30$ $3$ & $10$ & $\zeta_{15} + \zeta_{15}^{4}$ & $30$ $3$ & $12$ & $-\zeta_{5}^{2} - \zeta_{5}^{3}$ & $-30$ $3$ & $14$ & $\zeta_{15}^{2} + \zeta_{15}^{8}$ & $30$ $3$ & $16$ & $-\zeta_{15}^{7} - \zeta_{15}^{13}$ & $-30$ $3$ & $18$ & $\zeta_{5}^{2} + \zeta_{5}^{3}$ & $30$ $3$ & $20$ & $-\zeta_{15}^{11} - \zeta_{15}^{14}$ & $-30$ $3$ & $20$ & $-\zeta_{15}^{2} - \zeta_{15}^{8}$ & $-30$ $3$ & $24$ & $-\zeta_{5} - \zeta_{5}^{4}$ & $-30$ $3$ & $28$ & $-\zeta_{15} - \zeta_{15}^{4}$ & $-30$ $4$ & $3$ & $-\zeta_{4}$ & $40$ $4$ & $6$ & $1$ & $40$ $4$ & $8$ & $-\zeta_{3}$ & $-20$ $4$ & $9$ & $\zeta_{4}$ & $40$ $4$ & $11$ & $-\zeta_{12}^{7}$ & $-20$ $4$ & $12$ & $-1$ & $40$ $4$ & $13$ & $\zeta_{12}^{11}$ & $-20$ $4$ & $14$ & $\zeta_{3}$ & $-20$ $4$ & $16$ & $-\zeta_{3}^{2}$ & $-20$ $4$ & $17$ & $\zeta_{12}^{7}$ & $-20$ $4$ & $19$ & $-\zeta_{12}^{11}$ & $-20$ $4$ & $22$ & $\zeta_{3}^{2}$ & $-20$ $5$ & $4$ & $0$ & $50$ $5$ & $8$ & $0$ & $50$ $5$ & $10$ & $0$ & $-10$ $5$ & $12$ & $0$ & $-10$ $5$ & $14$ & $0$ & $-10$ $5$ & $18$ & $0$ & $-70$ $6$ & $5$ & $-\zeta_{12}^{7}$ & $0$ $6$ & $7$ & $\zeta_{12}^{11}$ & $0$ $6$ & $9$ & $-\zeta_{4}$ & $0$ $6$ & $11$ & $\zeta_{12}^{7}$ & $0$ $6$ & $13$ & $-\zeta_{12}^{11}$ & $0$ $6$ & $15$ & $\zeta_{4}$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $15$ $1$ & $15$ & $-1$ & $-15$ $3$ & $1$ & $\zeta_{5} + \zeta_{5}^{4}$ & $15$ $3$ & $3$ & $\zeta_{5}^{2} + \zeta_{5}^{3}$ & $15$ $3$ & $6$ & $-\zeta_{5} - \zeta_{5}^{4}$ & $-15$ $3$ & $8$ & $-\zeta_{5}^{2} - \zeta_{5}^{3}$ & $-15$ $4$ & $3$ & $1$ & $0$ $4$ & $4$ & $-1$ & $0$ $5$ & $2$ & $0$ & $15$ $5$ & $5$ & $0$ & $-15$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $21$ $1$ & $21$ & $-1$ & $-21$ $3$ & $1$ & $-\zeta_{7} - \zeta_{7}^{2} - \zeta_{7}^{4}$ & $21$ $3$ & $3$ & $-\zeta_{7}^{3} - \zeta_{7}^{5} - \zeta_{7}^{6}$ & $21$ $3$ & $8$ & $\zeta_{7} + \zeta_{7}^{2} + \zeta_{7}^{4}$ & $-21$ $3$ & $10$ & $\zeta_{7}^{3} + \zeta_{7}^{5} + \zeta_{7}^{6}$ & $-21$ $6$ & $2$ & $-1$ & $42$ $6$ & $9$ & $1$ & $-42$ $7$ & $3$ & $0$ & $21$ $7$ & $6$ & $0$ & $-21$ $8$ & $4$ & $1$ & $0$ $8$ & $5$ & $-1$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $24$ $1$ & $12$ & $1$ & $-12$ $1$ & $24$ & $1$ & $-12$ $2$ & $3$ & $0$ & $12$ $2$ & $9$ & $0$ & $12$ $2$ & $15$ & $0$ & $-24$ $3$ & $1$ & $\zeta_{3}$ & $36$ $3$ & $5$ & $\zeta_{3}^{2}$ & $36$ $3$ & $5$ & $\zeta_{3}^{2}$ & $0$ $3$ & $6$ & $-1$ & $0$ $3$ & $13$ & $\zeta_{3}$ & $-36$ $3$ & $13$ & $\zeta_{3}$ & $0$ $3$ & $17$ & $\zeta_{3}^{2}$ & $-36$ $6$ & $2$ & $0$ & $36$ $6$ & $4$ & $0$ & $36$ $6$ & $4$ & $0$ & $0$ $6$ & $8$ & $0$ & $-36$ $6$ & $8$ & $0$ & $0$ $6$ & $10$ & $0$ & $-36$ $8$ & $3$ & $0$ & $48$ $8$ & $6$ & $0$ & $-24$ $8$ & $9$ & $0$ & $-24$ $9$ & $5$ & $-\zeta_{3}^{2}$ & $0$ $9$ & $7$ & $-\zeta_{3}$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $33$ $1$ & $9$ & $-1$ & $15$ $1$ & $12$ & $\zeta_{3}^{2}$ & $-3$ $1$ & $21$ & $-\zeta_{3}^{2}$ & $-21$ $1$ & $24$ & $\zeta_{3}$ & $-3$ $1$ & $33$ & $-\zeta_{3}$ & $-21$ $2$ & $3$ & $\zeta_{3}$ & $30$ $2$ & $9$ & $\zeta_{3}^{2}$ & $30$ $2$ & $12$ & $-\zeta_{3}$ & $-6$ $2$ & $15$ & $1$ & $-6$ $2$ & $18$ & $-\zeta_{3}^{2}$ & $-6$ $2$ & $24$ & $-1$ & $-42$ $3$ & $1$ & $0$ & $45$ $3$ & $4$ & $0$ & $27$ $3$ & $5$ & $0$ & $45$ $3$ & $5$ & $0$ & $9$ $3$ & $6$ & $0$ & $27$ $3$ & $8$ & $0$ & $27$ $3$ & $8$ & $0$ & $-9$ $3$ & $13$ & $0$ & $-27$ $3$ & $13$ & $0$ & $9$ $3$ & $15$ & $0$ & $-27$ $3$ & $16$ & $0$ & $-45$ $3$ & $16$ & $0$ & $-9$ $3$ & $17$ & $0$ & $-27$ $3$ & $20$ & $0$ & $-45$ $6$ & $2$ & $0$ & $54$ $6$ & $4$ & $0$ & $54$ $6$ & $4$ & $0$ & $18$ $6$ & $5$ & $0$ & $18$ $6$ & $7$ & $0$ & $18$ $6$ & $7$ & $0$ & $-18$ $6$ & $8$ & $0$ & $-18$ $6$ & $8$ & $0$ & $18$ $6$ & $10$ & $0$ & $-18$ $6$ & $11$ & $0$ & $-54$ $6$ & $11$ & $0$ & $-18$ $6$ & $13$ & $0$ & $-54$ $8$ & $3$ & $1$ & $48$ $8$ & $6$ & $-1$ & $48$ $8$ & $6$ & $-\zeta_{3}^{2}$ & $-24$ $8$ & $9$ & $\zeta_{3}^{2}$ & $-24$ $8$ & $9$ & $\zeta_{3}$ & $-24$ $8$ & $12$ & $-\zeta_{3}$ & $-24$ $9$ & $5$ & $0$ & $27$ $9$ & $7$ & $0$ & $27$ $9$ & $8$ & $0$ & $-27$ $9$ & $10$ & $0$ & $-27$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $45$ $1$ & $45$ & $-1$ & $-45$ $3$ & $1$ & $\zeta_{15}^{11} + \zeta_{15}^{14}$ & $45$ $3$ & $5$ & $\zeta_{15}^{7} + \zeta_{15}^{13}$ & $45$ $3$ & $5$ & $\zeta_{15} + \zeta_{15}^{4}$ & $45$ $3$ & $7$ & $\zeta_{15}^{2} + \zeta_{15}^{8}$ & $45$ $3$ & $16$ & $-\zeta_{15}^{11} - \zeta_{15}^{14}$ & $-45$ $3$ & $20$ & $-\zeta_{15}^{7} - \zeta_{15}^{13}$ & $-45$ $3$ & $20$ & $-\zeta_{15} - \zeta_{15}^{4}$ & $-45$ $3$ & $22$ & $-\zeta_{15}^{2} - \zeta_{15}^{8}$ & $-45$ $5$ & $6$ & $0$ & $45$ $5$ & $6$ & $0$ & $45$ $5$ & $15$ & $0$ & $-45$ $5$ & $15$ & $0$ & $-45$ $6$ & $2$ & $\zeta_{3}^{2}$ & $90$ $6$ & $4$ & $\zeta_{3}$ & $90$ $6$ & $17$ & $-\zeta_{3}^{2}$ & $-90$ $6$ & $19$ & $-\zeta_{3}$ & $-90$ $8$ & $6$ & $-\zeta_{5}^{2} - \zeta_{5}^{3}$ & $0$ $8$ & $9$ & $\zeta_{5}^{2} + \zeta_{5}^{3}$ & $0$ $8$ & $9$ & $\zeta_{5} + \zeta_{5}^{4}$ & $0$ $8$ & $12$ & $-\zeta_{5} - \zeta_{5}^{4}$ & $0$ $9$ & $4$ & $-\zeta_{3}$ & $45$ $9$ & $6$ & $-1$ & $45$ $9$ & $8$ & $-\zeta_{3}^{2}$ & $45$ $9$ & $9$ & $1$ & $-45$ $9$ & $11$ & $\zeta_{3}^{2}$ & $-45$ $9$ & $13$ & $\zeta_{3}$ & $-45$ $10$ & $3$ & $0$ & $90$ $10$ & $12$ & $0$ & $-90$ $15$ & $5$ & $0$ & $45$ $15$ & $7$ & $0$ & $45$ $15$ & $8$ & $0$ & $-45$ $15$ & $10$ & $0$ & $-45$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $24$ $1$ & $12$ & $1$ & $0$ $1$ & $12$ & $1$ & $0$ $1$ & $24$ & $1$ & $-24$ $2$ & $4$ & $-1$ & $24$ $2$ & $4$ & $-1$ & $24$ $2$ & $16$ & $-1$ & $-24$ $2$ & $16$ & $-1$ & $-24$ $4$ & $1$ & $0$ & $48$ $4$ & $7$ & $0$ & $0$ $4$ & $7$ & $0$ & $0$ $4$ & $8$ & $1$ & $0$ $4$ & $13$ & $0$ & $-48$ $6$ & $6$ & $-1$ & $0$ $6$ & $6$ & $-1$ & $0$ $8$ & $3$ & $0$ & $48$ $8$ & $3$ & $0$ & $48$ $8$ & $9$ & $0$ & $-48$ $8$ & $9$ & $0$ & $-48$ $9$ & $2$ & $0$ & $72$ $9$ & $6$ & $0$ & $0$ $9$ & $6$ & $0$ & $0$ $9$ & $10$ & $0$ & $-72$ $12$ & $4$ & $1$ & $0$ $16$ & $5$ & $0$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $40$ $1$ & $40$ & $1$ & $-40$ $4$ & $1$ & $-\zeta_{4}$ & $80$ $4$ & $3$ & $\zeta_{4}$ & $80$ $4$ & $4$ & $-1$ & $80$ $4$ & $21$ & $-\zeta_{4}$ & $-80$ $4$ & $23$ & $\zeta_{4}$ & $-80$ $4$ & $24$ & $-1$ & $-80$ $5$ & $8$ & $0$ & $40$ $5$ & $16$ & $0$ & $-40$ $6$ & $10$ & $-1$ & $0$ $6$ & $10$ & $-1$ & $0$ $6$ & $10$ & $-1$ & $0$ $6$ & $10$ & $-1$ & $0$ $6$ & $12$ & $1$ & $0$ $10$ & $2$ & $0$ & $160$ $10$ & $6$ & $0$ & $80$ $10$ & $14$ & $0$ & $-80$ $10$ & $18$ & $0$ & $-160$ $15$ & $4$ & $0$ & $120$ $15$ & $4$ & $0$ & $120$ $15$ & $12$ & $0$ & $-120$ $15$ & $12$ & $0$ & $-120$ $16$ & $3$ & $-\zeta_{4}$ & $160$ $16$ & $5$ & $\zeta_{4}$ & $160$ $16$ & $13$ & $\zeta_{4}$ & $-160$ $16$ & $15$ & $-\zeta_{4}$ & $-160$ $20$ & $5$ & $0$ & $80$ $20$ & $6$ & $0$ & $80$ $20$ & $7$ & $0$ & $80$ $20$ & $9$ & $0$ & $-80$ $20$ & $10$ & $0$ & $-80$ $20$ & $11$ & $0$ & $-80$ $24$ & $6$ & $1$ & $0$ $24$ & $7$ & $\zeta_{4}$ & $0$ $24$ & $9$ & $-\zeta_{4}$ & $0$ $30$ & $8$ & $0$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $60$ $1$ & $60$ & $1$ & $-60$ $4$ & $1$ & $\zeta_{5} + \zeta_{5}^{4}$ & $120$ $4$ & $7$ & $\zeta_{5}^{2} + \zeta_{5}^{3}$ & $120$ $4$ & $31$ & $\zeta_{5} + \zeta_{5}^{4}$ & $-120$ $4$ & $37$ & $\zeta_{5}^{2} + \zeta_{5}^{3}$ & $-120$ $6$ & $12$ & $-\zeta_{5} - \zeta_{5}^{4}$ & $0$ $6$ & $20$ & $-\zeta_{5}^{2} - \zeta_{5}^{3}$ & $0$ $8$ & $12$ & $0$ & $0$ $8$ & $13$ & $-1$ & $0$ $9$ & $2$ & $0$ & $180$ $9$ & $6$ & $0$ & $180$ $9$ & $22$ & $0$ & $-180$ $9$ & $26$ & $0$ & $-180$ $10$ & $12$ & $-1$ & $0$ $16$ & $3$ & $1$ & $240$ $16$ & $6$ & $-1$ & $240$ $16$ & $11$ & $\zeta_{5}^{2} + \zeta_{5}^{3}$ & $0$ $16$ & $13$ & $\zeta_{5} + \zeta_{5}^{4}$ & $0$ $16$ & $18$ & $-1$ & $-240$ $16$ & $21$ & $1$ & $-240$ $18$ & $10$ & $0$ & $0$ $24$ & $6$ & $-\zeta_{5}^{2} - \zeta_{5}^{3}$ & $0$ $24$ & $7$ & $1$ & $0$ $24$ & $11$ & $1$ & $0$ $24$ & $12$ & $-\zeta_{5} - \zeta_{5}^{4}$ & $0$ $25$ & $4$ & $0$ & $300$ $25$ & $16$ & $0$ & $-300$ $30$ & $10$ & $\zeta_{5} + \zeta_{5}^{4}$ & $0$ $30$ & $10$ & $\zeta_{5}^{2} + \zeta_{5}^{3}$ & $0$ $36$ & $5$ & $0$ & $360$ $36$ & $15$ & $0$ & $-360$ $40$ & $8$ & $1$ & $0$ $48$ & $9$ & $-1$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $80$ $1$ & $40$ & $\zeta_{3}$ & $-40$ $1$ & $80$ & $\zeta_{3}^{2}$ & $-40$ $4$ & $1$ & $\zeta_{3}$ & $200$ $4$ & $11$ & $\zeta_{3}^{2}$ & $200$ $4$ & $21$ & $1$ & $-40$ $4$ & $41$ & $\zeta_{3}^{2}$ & $-160$ $4$ & $51$ & $1$ & $-40$ $4$ & $61$ & $\zeta_{3}$ & $-160$ $5$ & $4$ & $0$ & $160$ $5$ & $12$ & $0$ & $40$ $5$ & $20$ & $0$ & $160$ $5$ & $36$ & $0$ & $40$ $5$ & $44$ & $0$ & $-200$ $5$ & $52$ & $0$ & $-200$ $6$ & $8$ & $\zeta_{3}^{2}$ & $120$ $6$ & $28$ & $\zeta_{3}$ & $120$ $6$ & $48$ & $1$ & $-240$ $10$ & $2$ & $0$ & $320$ $10$ & $10$ & $0$ & $320$ $10$ & $14$ & $0$ & $-40$ $10$ & $30$ & $0$ & $-280$ $10$ & $34$ & $0$ & $-40$ $10$ & $42$ & $0$ & $-280$ $15$ & $6$ & $0$ & $480$ $15$ & $8$ & $0$ & $120$ $15$ & $16$ & $0$ & $120$ $15$ & $22$ & $0$ & $-240$ $15$ & $24$ & $0$ & $-240$ $15$ & $38$ & $0$ & $-240$ $20$ & $3$ & $0$ & $520$ $20$ & $5$ & $0$ & $280$ $20$ & $7$ & $0$ & $280$ $20$ & $9$ & $0$ & $520$ $20$ & $9$ & $0$ & $160$ $20$ & $12$ & $0$ & $160$ $20$ & $13$ & $0$ & $-80$ $20$ & $16$ & $0$ & $-80$ $20$ & $17$ & $0$ & $280$ $20$ & $19$ & $0$ & $280$ $20$ & $20$ & $0$ & $-80$ $20$ & $21$ & $0$ & $160$ $20$ & $25$ & $0$ & $-440$ $20$ & $29$ & $0$ & $-80$ $20$ & $29$ & $0$ & $-440$ $20$ & $31$ & $0$ & $-440$ $20$ & $33$ & $0$ & $-560$ $20$ & $35$ & $0$ & $-440$ $24$ & $6$ & $-1$ & $480$ $24$ & $16$ & $-\zeta_{3}$ & $-240$ $24$ & $26$ & $-\zeta_{3}^{2}$ & $-240$ $30$ & $4$ & $0$ & $600$ $30$ & $8$ & $0$ & $600$ $30$ & $12$ & $0$ & $-120$ $30$ & $12$ & $0$ & $240$ $30$ & $16$ & $0$ & $-120$ $30$ & $20$ & $0$ & $-120$ $30$ & $20$ & $0$ & $-480$ $30$ & $24$ & $0$ & $-120$ $30$ & $28$ & $0$ & $-480$ $36$ & $5$ & $-\zeta_{3}^{2}$ & $720$ $36$ & $7$ & $-\zeta_{3}$ & $720$ $36$ & $15$ & $-1$ & $-360$ $36$ & $17$ & $-\zeta_{3}^{2}$ & $-360$ $36$ & $25$ & $-\zeta_{3}$ & $-360$ $36$ & $27$ & $-1$ & $-360$ $40$ & $8$ & $0$ & $560$ $40$ & $10$ & $0$ & $560$ $40$ & $14$ & $0$ & $-160$ $40$ & $18$ & $0$ & $-400$ $40$ & $22$ & $0$ & $-160$ $40$ & $24$ & $0$ & $-400$ $45$ & $6$ & $0$ & $360$ $45$ & $10$ & $0$ & $360$ $45$ & $14$ & $0$ & $360$ $45$ & $18$ & $0$ & $360$ $45$ & $22$ & $0$ & $-720$ $45$ & $26$ & $0$ & $-720$ $60$ & $7$ & $0$ & $480$ $60$ & $9$ & $0$ & $120$ $60$ & $11$ & $0$ & $480$ $60$ & $11$ & $0$ & $120$ $60$ & $12$ & $0$ & $480$ $60$ & $13$ & $0$ & $120$ $60$ & $15$ & $0$ & $120$ $60$ & $15$ & $0$ & $-240$ $60$ & $16$ & $0$ & $-240$ $60$ & $17$ & $0$ & $-600$ $60$ & $19$ & $0$ & $-600$ $60$ & $20$ & $0$ & $-240$ $64$ & $8$ & $-\zeta_{3}^{2}$ & $320$ $64$ & $11$ & $\zeta_{3}^{2}$ & $320$ $64$ & $13$ & $\zeta_{3}$ & $320$ $64$ & $16$ & $-\zeta_{3}$ & $320$ $64$ & $18$ & $-1$ & $-640$ $64$ & $21$ & $1$ & $-640$ $80$ & $9$ & $0$ & $640$ $80$ & $13$ & $0$ & $-320$ $80$ & $17$ & $0$ & $-320$ $81$ & $10$ & $\zeta_{3}$ & $0$ $81$ & $12$ & $1$ & $0$ $81$ & $14$ & $\zeta_{3}^{2}$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $45$ $1$ & $45$ & $-1$ & $-45$ $5$ & $1$ & $\zeta_{3}^{2}$ & $135$ $5$ & $3$ & $\zeta_{3}$ & $135$ $5$ & $28$ & $-\zeta_{3}^{2}$ & $-135$ $5$ & $30$ & $-\zeta_{3}$ & $-135$ $6$ & $5$ & $0$ & $90$ $6$ & $20$ & $0$ & $-90$ $10$ & $8$ & $\zeta_{3}^{2}$ & $90$ $10$ & $8$ & $\zeta_{3}$ & $90$ $10$ & $17$ & $-\zeta_{3}^{2}$ & $-90$ $10$ & $17$ & $-\zeta_{3}$ & $-90$ $15$ & $2$ & $0$ & $315$ $15$ & $9$ & $0$ & $45$ $15$ & $12$ & $0$ & $-45$ $15$ & $23$ & $0$ & $-315$ $20$ & $6$ & $-1$ & $180$ $20$ & $15$ & $1$ & $-180$ $24$ & $4$ & $0$ & $360$ $24$ & $19$ & $0$ & $-360$ $30$ & $3$ & $0$ & $450$ $30$ & $4$ & $0$ & $270$ $30$ & $6$ & $0$ & $270$ $30$ & $13$ & $0$ & $-270$ $30$ & $15$ & $0$ & $-270$ $30$ & $18$ & $0$ & $-450$ $40$ & $5$ & $-\zeta_{3}^{2}$ & $360$ $40$ & $5$ & $-\zeta_{3}$ & $360$ $40$ & $14$ & $\zeta_{3}^{2}$ & $-360$ $40$ & $14$ & $\zeta_{3}$ & $-360$ $45$ & $7$ & $0$ & $135$ $45$ & $9$ & $0$ & $135$ $45$ & $10$ & $0$ & $-135$ $45$ & $12$ & $0$ & $-135$ $60$ & $7$ & $0$ & $180$ $60$ & $10$ & $0$ & $-180$ $64$ & $8$ & $1$ & $0$ $64$ & $9$ & $-1$ & $0$ $81$ & $6$ & $0$ & $405$ $81$ & $11$ & $0$ & $-405$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $126$ $1$ & $126$ & $1$ & $-126$ $6$ & $1$ & $\zeta_{3}^{2}$ & $504$ $6$ & $5$ & $\zeta_{3}$ & $504$ $6$ & $85$ & $\zeta_{3}^{2}$ & $-504$ $6$ & $89$ & $\zeta_{3}$ & $-504$ $15$ & $14$ & $\zeta_{3}$ & $630$ $15$ & $16$ & $\zeta_{3}^{2}$ & $630$ $15$ & $56$ & $\zeta_{3}$ & $-630$ $15$ & $58$ & $\zeta_{3}^{2}$ & $-630$ $20$ & $33$ & $1$ & $0$ $20$ & $33$ & $1$ & $0$ $21$ & $2$ & $0$ & $1386$ $21$ & $4$ & $0$ & $1386$ $21$ & $6$ & $0$ & $1134$ $21$ & $60$ & $0$ & $-1134$ $21$ & $68$ & $0$ & $-1386$ $21$ & $70$ & $0$ & $-1386$ $35$ & $6$ & $0$ & $1890$ $35$ & $18$ & $0$ & $630$ $35$ & $36$ & $0$ & $-630$ $35$ & $60$ & $0$ & $-1890$ $56$ & $3$ & $0$ & $3024$ $56$ & $9$ & $0$ & $2016$ $56$ & $45$ & $0$ & $-2016$ $56$ & $57$ & $0$ & $-3024$ $70$ & $9$ & $0$ & $2520$ $70$ & $9$ & $0$ & $2520$ $70$ & $9$ & $0$ & $2520$ $70$ & $45$ & $0$ & $-2520$ $70$ & $45$ & $0$ & $-2520$ $70$ & $45$ & $0$ & $-2520$ $84$ & $13$ & $0$ & $2016$ $84$ & $17$ & $0$ & $2016$ $84$ & $37$ & $0$ & $-2016$ $84$ & $41$ & $0$ & $-2016$ $90$ & $6$ & $-1$ & $3780$ $90$ & $48$ & $-1$ & $-3780$ $105$ & $4$ & $0$ & $4410$ $105$ & $8$ & $0$ & $4410$ $105$ & $8$ & $0$ & $3150$ $105$ & $10$ & $0$ & $3150$ $105$ & $20$ & $0$ & $630$ $105$ & $22$ & $0$ & $630$ $105$ & $26$ & $0$ & $-630$ $105$ & $28$ & $0$ & $-630$ $105$ & $38$ & $0$ & $-3150$ $105$ & $40$ & $0$ & $-3150$ $105$ & $46$ & $0$ & $-4410$ $105$ & $50$ & $0$ & $-4410$ $120$ & $5$ & $-\zeta_{3}$ & $5040$ $120$ & $7$ & $-\zeta_{3}^{2}$ & $5040$ $120$ & $21$ & $-1$ & $0$ $120$ & $21$ & $-1$ & $0$ $120$ & $47$ & $-\zeta_{3}$ & $-5040$ $120$ & $49$ & $-\zeta_{3}^{2}$ & $-5040$ $126$ & $5$ & $0$ & $4536$ $126$ & $7$ & $0$ & $4536$ $126$ & $41$ & $0$ & $-4536$ $126$ & $43$ & $0$ & $-4536$ $140$ & $12$ & $0$ & $2520$ $140$ & $21$ & $0$ & $0$ $140$ & $30$ & $0$ & $-2520$ $189$ & $18$ & $0$ & $1134$ $189$ & $24$ & $0$ & $-1134$ $210$ & $8$ & $0$ & $6300$ $210$ & $10$ & $0$ & $6300$ $210$ & $12$ & $0$ & $3780$ $210$ & $13$ & $0$ & $2520$ $210$ & $17$ & $0$ & $2520$ $210$ & $25$ & $0$ & $-2520$ $210$ & $29$ & $0$ & $-2520$ $210$ & $30$ & $0$ & $-3780$ $210$ & $38$ & $0$ & $-6300$ $210$ & $40$ & $0$ & $-6300$ $280$ & $12$ & $0$ & $5040$ $280$ & $12$ & $0$ & $5040$ $280$ & $30$ & $0$ & $-5040$ $280$ & $30$ & $0$ & $-5040$ $315$ & $6$ & $0$ & $9450$ $315$ & $10$ & $0$ & $5670$ $315$ & $14$ & $0$ & $5670$ $315$ & $18$ & $0$ & $1890$ $315$ & $24$ & $0$ & $-1890$ $315$ & $28$ & $0$ & $-5670$ $315$ & $32$ & $0$ & $-5670$ $315$ & $36$ & $0$ & $-9450$ $336$ & $8$ & $0$ & $8064$ $336$ & $10$ & $0$ & $8064$ $336$ & $17$ & $0$ & $2016$ $336$ & $19$ & $0$ & $2016$ $336$ & $23$ & $0$ & $-2016$ $336$ & $25$ & $0$ & $-2016$ $336$ & $32$ & $0$ & $-8064$ $336$ & $34$ & $0$ & $-8064$ $384$ & $8$ & $-\zeta_{3}$ & $8064$ $384$ & $10$ & $-\zeta_{3}^{2}$ & $8064$ $384$ & $11$ & $\zeta_{3}$ & $8064$ $384$ & $13$ & $\zeta_{3}^{2}$ & $8064$ $384$ & $29$ & $\zeta_{3}$ & $-8064$ $384$ & $31$ & $\zeta_{3}^{2}$ & $-8064$ $384$ & $32$ & $-\zeta_{3}$ & $-8064$ $384$ & $34$ & $-\zeta_{3}^{2}$ & $-8064$ $420$ & $7$ & $0$ & $10080$ $420$ & $11$ & $0$ & $10080$ $420$ & $12$ & $0$ & $7560$ $420$ & $14$ & $0$ & $2520$ $420$ & $16$ & $0$ & $2520$ $420$ & $20$ & $0$ & $-2520$ $420$ & $21$ & $0$ & $0$ $420$ & $22$ & $0$ & $-2520$ $420$ & $30$ & $0$ & $-7560$ $420$ & $31$ & $0$ & $-10080$ $420$ & $35$ & $0$ & $-10080$ $504$ & $9$ & $0$ & $12096$ $504$ & $15$ & $0$ & $3024$ $504$ & $21$ & $0$ & $-3024$ $504$ & $33$ & $0$ & $-12096$ $540$ & $17$ & $-\zeta_{3}$ & $0$ $540$ & $19$ & $-\zeta_{3}^{2}$ & $0$ $540$ & $21$ & $-1$ & $0$ $540$ & $21$ & $-1$ & $0$ $560$ & $9$ & $0$ & $10080$ $560$ & $18$ & $0$ & $0$ $560$ & $18$ & $0$ & $0$ $560$ & $18$ & $0$ & $0$ $560$ & $27$ & $0$ & $-10080$ $630$ & $11$ & $0$ & $7560$ $630$ & $13$ & $0$ & $7560$ $630$ & $14$ & $0$ & $3780$ $630$ & $15$ & $0$ & $7560$ $630$ & $16$ & $0$ & $3780$ $630$ & $20$ & $0$ & $-3780$ $630$ & $22$ & $0$ & $-3780$ $630$ & $23$ & $0$ & $-7560$ $630$ & $25$ & $0$ & $-7560$ $630$ & $27$ & $0$ & $-7560$ $720$ & $16$ & $-\zeta_{3}^{2}$ & $0$ $720$ & $20$ & $-\zeta_{3}$ & $0$ $729$ & $10$ & $\zeta_{3}^{2}$ & $10206$ $729$ & $12$ & $1$ & $10206$ $729$ & $14$ & $\zeta_{3}$ & $10206$ $729$ & $24$ & $1$ & $-10206$ $729$ & $26$ & $\zeta_{3}$ & $-10206$ $729$ & $28$ & $\zeta_{3}^{2}$ & $-10206$ $756$ & $14$ & $0$ & $4536$ $756$ & $16$ & $0$ & $4536$ $756$ & $20$ & $0$ & $-4536$ $756$ & $22$ & $0$ & $-4536$ $840$ & $11$ & $0$ & $10080$ $840$ & $13$ & $0$ & $10080$ $840$ & $13$ & $0$ & $5040$ $840$ & $17$ & $0$ & $5040$ $840$ & $19$ & $0$ & $-5040$ $840$ & $23$ & $0$ & $-5040$ $840$ & $23$ & $0$ & $-10080$ $840$ & $25$ & $0$ & $-10080$ $896$ & $12$ & $0$ & $8064$ $896$ & $15$ & $0$ & $8064$ $896$ & $21$ & $0$ & $-8064$ $896$ & $24$ & $0$ & $-8064$ $945$ & $14$ & $0$ & $5670$ $945$ & $16$ & $0$ & $5670$ $945$ & $20$ & $0$ & $-5670$ $945$ & $22$ & $0$ & $-5670$ $1260$ & $17$ & $0$ & $0$ $1260$ & $19$ & $0$ & $0$ $1280$ & $15$ & $1$ & $0$ $1280$ & $18$ & $-1$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $36$ $1$ & $36$ & $1$ & $-36$ $6$ & $1$ & $-1$ & $144$ $6$ & $25$ & $-1$ & $-144$ $10$ & $9$ & $-1$ & $0$ $15$ & $4$ & $-1$ & $180$ $15$ & $5$ & $0$ & $180$ $15$ & $16$ & $-1$ & $-180$ $15$ & $17$ & $0$ & $-180$ $20$ & $2$ & $0$ & $360$ $20$ & $10$ & $1$ & $0$ $20$ & $20$ & $0$ & $-360$ $24$ & $6$ & $0$ & $144$ $24$ & $12$ & $0$ & $-144$ $30$ & $3$ & $1$ & $360$ $30$ & $15$ & $1$ & $-360$ $60$ & $5$ & $0$ & $360$ $60$ & $8$ & $1$ & $0$ $60$ & $11$ & $0$ & $-360$ $64$ & $4$ & $0$ & $576$ $64$ & $13$ & $0$ & $-576$ $80$ & $7$ & $0$ & $0$ $81$ & $6$ & $0$ & $324$ $81$ & $10$ & $0$ & $-324$ $90$ & $8$ & $-1$ & $0$ [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $63$ $1$ & $63$ & $-1$ & $-63$ $7$ & $1$ & $-1$ & $315$ $7$ & $46$ & $1$ & $-315$ $15$ & $7$ & $0$ & $315$ $15$ & $28$ & $0$ & $-315$ $21$ & $3$ & $0$ & $693$ $21$ & $6$ & $0$ & $567$ $21$ & $33$ & $0$ & $-567$ $21$ & $36$ & $0$ & $-693$ $27$ & $2$ & $0$ & $945$ $27$ & $37$ & $0$ & $-945$ $35$ & $4$ & $-1$ & $945$ $35$ & $13$ & $1$ & $315$ $35$ & $22$ & $-1$ & $-315$ $35$ & $31$ & $1$ & $-945$ $56$ & $3$ & $1$ & $1512$ $56$ & $30$ & $-1$ & $-1512$ $70$ & $9$ & $-1$ & $630$ $70$ & $18$ & $1$ & $-630$ $84$ & $12$ & $0$ & $252$ $84$ & $15$ & $0$ & $-252$ $105$ & $5$ & $0$ & $2205$ $105$ & $6$ & $0$ & $1575$ $105$ & $12$ & $0$ & $315$ $105$ & $15$ & $0$ & $-315$ $105$ & $21$ & $0$ & $-1575$ $105$ & $26$ & $0$ & $-2205$ $120$ & $4$ & $0$ & $2520$ $120$ & $25$ & $0$ & $-2520$ $168$ & $6$ & $0$ & $2520$ $168$ & $21$ & $0$ & $-2520$ $189$ & $5$ & $0$ & $3213$ $189$ & $7$ & $0$ & $2457$ $189$ & $10$ & $0$ & $1323$ $189$ & $17$ & $0$ & $-1323$ $189$ & $20$ & $0$ & $-2457$ $189$ & $22$ & $0$ & $-3213$ $210$ & $6$ & $0$ & $3150$ $210$ & $10$ & $0$ & $630$ $210$ & $13$ & $0$ & $-630$ $210$ & $21$ & $0$ & $-3150$ $216$ & $9$ & $0$ & $1512$ $216$ & $16$ & $0$ & $-1512$ $280$ & $8$ & $1$ & $2520$ $280$ & $9$ & $-1$ & $2520$ $280$ & $17$ & $-1$ & $-2520$ $280$ & $18$ & $1$ & $-2520$ $315$ & $7$ & $0$ & $2835$ $315$ & $16$ & $0$ & $-2835$ $336$ & $11$ & $0$ & $1008$ $336$ & $14$ & $0$ & $-1008$ $378$ & $9$ & $0$ & $1890$ $378$ & $14$ & $0$ & $-1890$ $405$ & $8$ & $0$ & $2835$ $405$ & $15$ & $0$ & $-2835$ $420$ & $10$ & $0$ & $1260$ $420$ & $13$ & $0$ & $-1260$ $512$ & $11$ & $1$ & $0$ $512$ & $12$ & $-1$ & $0$ \[tab:last\] [ v[22px]{} \| v[22px]{} \|\| v[48px]{} \| v[30px]{} ]{} $\deg(\chi)$ & ${\operatorname{occ}(\chi)}$ & $\chi(c)$ & $\chi(R)$ $1$ & $0$ & $1$ & $120$ $1$ & $120$ & $1$ & $-120$ $8$ & $1$ & $-1$ & $720$ $8$ & $91$ & $-1$ & $-720$ $28$ & $8$ & $0$ & $1680$ $28$ & $68$ & $0$ & $-1680$ $35$ & $2$ & $0$ & $2520$ $35$ & $74$ & $0$ & $-2520$ $50$ & $8$ & $0$ & $2400$ $50$ & $56$ & $0$ & $-2400$ $56$ & $19$ & $1$ & $1680$ $56$ & $49$ & $1$ & $-1680$ $70$ & $32$ & $-1$ & $0$ $84$ & $4$ & $-1$ & $5040$ $84$ & $64$ & $-1$ & $-5040$ $112$ & $3$ & $1$ & $6720$ $112$ & $63$ & $1$ & $-6720$ $160$ & $7$ & $0$ & $7680$ $160$ & $55$ & $0$ & $-7680$ $168$ & $24$ & $0$ & $0$ $175$ & $12$ & $0$ & $4200$ $175$ & $36$ & $0$ & $-4200$ $210$ & $4$ & $0$ & $10080$ $210$ & $52$ & $0$ & $-10080$ $300$ & $8$ & $0$ & $10800$ $300$ & $44$ & $0$ & $-10800$ $350$ & $14$ & $0$ & $8400$ $350$ & $38$ & $0$ & $-8400$ $400$ & $7$ & $0$ & $14400$ $400$ & $43$ & $0$ & $-14400$ $420$ & $20$ & $-1$ & $0$ $448$ & $9$ & $-1$ & $13440$ $448$ & $25$ & $-1$ & $0$ $448$ & $39$ & $-1$ & $-13440$ $525$ & $12$ & $0$ & $12600$ $525$ & $36$ & $0$ & $-12600$ $560$ & $5$ & $0$ & $23520$ $560$ & $47$ & $0$ & $-23520$ $567$ & $6$ & $0$ & $22680$ $567$ & $46$ & $0$ & $-22680$ $700$ & $6$ & $0$ & $25200$ $700$ & $16$ & $0$ & $8400$ $700$ & $28$ & $0$ & $-8400$ $700$ & $42$ & $0$ & $-25200$ $840$ & $13$ & $0$ & $15120$ $840$ & $14$ & $0$ & $10080$ $840$ & $26$ & $0$ & $-10080$ $840$ & $31$ & $0$ & $-15120$ $972$ & $12$ & $0$ & $19440$ $972$ & $32$ & $0$ & $-19440$ $1008$ & $9$ & $-1$ & $30240$ $1008$ & $39$ & $-1$ & $-30240$ $1050$ & $10$ & $0$ & $25200$ $1050$ & $34$ & $0$ & $-25200$ $1134$ & $20$ & $1$ & $0$ $1296$ & $13$ & $0$ & $25920$ $1296$ & $33$ & $0$ & $-25920$ $1344$ & $8$ & $1$ & $40320$ $1344$ & $19$ & $0$ & $0$ $1344$ & $38$ & $1$ & $-40320$ $1400$ & $7$ & $0$ & $42000$ $1400$ & $8$ & $0$ & $33600$ $1400$ & $11$ & $0$ & $25200$ $1400$ & $20$ & $0$ & $0$ $1400$ & $29$ & $0$ & $-25200$ $1400$ & $32$ & $0$ & $-33600$ $1400$ & $37$ & $0$ & $-42000$ $1575$ & $10$ & $0$ & $37800$ $1575$ & $34$ & $0$ & $-37800$ $1680$ & $22$ & $1$ & $0$ $2016$ & $19$ & $1$ & $0$ $2100$ & $16$ & $0$ & $25200$ $2100$ & $20$ & $0$ & $0$ $2100$ & $28$ & $0$ & $-25200$ $2240$ & $10$ & $0$ & $40320$ $2240$ & $28$ & $0$ & $-40320$ $2268$ & $10$ & $0$ & $45360$ $2268$ & $30$ & $0$ & $-45360$ $2400$ & $17$ & $0$ & $14400$ $2400$ & $23$ & $0$ & $-14400$ $2688$ & $20$ & $0$ & $0$ $2800$ & $13$ & $0$ & $33600$ $2800$ & $25$ & $0$ & $-33600$ $2835$ & $14$ & $0$ & $22680$ $2835$ & $22$ & $0$ & $-22680$ $3150$ & $18$ & $0$ & $0$ $3200$ & $16$ & $0$ & $19200$ $3200$ & $22$ & $0$ & $-19200$ $3240$ & $9$ & $0$ & $71280$ $3240$ & $31$ & $0$ & $-71280$ $3360$ & $13$ & $0$ & $40320$ $3360$ & $25$ & $0$ & $-40320$ $4096$ & $11$ & $1$ & $61440$ $4096$ & $12$ & $-1$ & $61440$ $4096$ & $26$ & $-1$ & $-61440$ $4096$ & $27$ & $1$ & $-61440$ $4200$ & $12$ & $0$ & $50400$ $4200$ & $15$ & $0$ & $25200$ $4200$ & $18$ & $0$ & $0$ $4200$ & $21$ & $0$ & $-25200$ $4200$ & $24$ & $0$ & $-50400$ $4480$ & $16$ & $1$ & $0$ $4536$ & $13$ & $0$ & $45360$ $4536$ & $18$ & $1$ & $0$ $4536$ & $23$ & $0$ & $-45360$ $5600$ & $15$ & $0$ & $33600$ $5600$ & $19$ & $0$ & $0$ $5600$ & $21$ & $0$ & $-33600$ $5670$ & $18$ & $-1$ & $0$ $6075$ & $14$ & $0$ & $48600$ $6075$ & $22$ & $0$ & $-48600$ $7168$ & $17$ & $-1$ & $0$ [^1]: 2000 Mathematics Subject Classification 20F55 (primary), 05E10 (secondary).\ We first considered the question addressed in this paper during the Formal Power Series and Algebraic Combinatorics conference in Nagoya, Japan, in August 2012. We are grateful to the organizers for this wonderful meeting, and C.S. acknowledges the financial support.\ G.C. acknowledges partial support from the ERC grant StG 208471 – ExploreMaps, the grant grant ANR-08-JCJC-0011 – IComb and the grant ANR-12-JS02-001-01 – Cartaplus.\ C.S. acknowledges support from the German Research Foundation DFG, grant STU 563/2-1 “Coxeter-Catalan combinatorics”.\ We thank Christian Krattenthaler, Vic Reiner, and Vivien Ripoll for very stimulating discussions (in particular for their contributions to Remark \[rem:automorphism\]), Eduard Looijenga for providing us with a copy of [@Deligne], and Gunter Malle and Nathan Reading for valuable comments on previous versions of this paper. [^2]: In the same vein, we observe that the various choices of Coxeter elements lead to isomorphic non-crossing partition lattices (see e.g. [@BR2007] for the definition). For the infinite families, this follows from the same argument as above, whereas we have used a computer to verify this fact for the exceptional groups. [^3]: The only property we need for that is that any conjugate in $G(r,1,n)$ of a reflection in $G(r,r,n)$ is a reflection in $G(r,r,n)$, which is true. Note that a Coxeter element $c$ of $G(r,r,n)$ is not a Coxeter element of the group $G(r,1,n)$, but this does not prevent one to apply the Frobenius formula. [^4]: see <http://www.math.jussieu.fr/~jmichel/gap3/htm/chap073.htm> for examples and a detailed description. [^5]: see <http://trac.sagemath.org/sage_trac/ticket/11187> for further information. [^6]: In [@Stembridge], a description of the irreducible representations of $G(r,r,n)$ is given in a way that is essentially equivalent to ours.
--- abstract: 'We study sets of bounded remainder for the two-dimensional continuous irrational rotation $(\{x_1+t\}, \{x_2+t\alpha \})_{t {\geqslant}0}$ in the unit square. In particular, we show that for almost all $\alpha$ and every starting point $(x_1, x_2)$, every polygon $S$ with no edge of slope $\alpha$ is a set of bounded remainder. Moreover, every convex set $S$ whose boundary is twice continuously differentiable with positive curvature at every point is a bounded remainder set for almost all $\alpha$ and every starting point $(x_1, x_2)$. Finally we show that these assertions are, in some sense, best possible.' address: - 'Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstr. 69, A-4040 Linz, Austria.' - 'Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstr. 69, A-4040 Linz, Austria.' author: - Sigrid Grepstad and Gerhard Larcher date: 'February 29, 2016' title: 'Sets of bounded remainder for the continuous irrational rotation on $[0,1]^2$' --- Introduction ============ In this paper we will be concerned with bounded remainder sets for the two-dimensional irrational rotation on the unit square $I^2 = [0,1)^2$. Let ${\mathbf{x}}= (x_1, x_2) \in I^2$, and let $\alpha \in {\mathbb{R}}\setminus {\mathbb{Q}}$. We say that the function $X : [0, \infty) \mapsto I^2$ defined by $$X(t) = \left( \{x_1+t\}, \{x_2+\alpha t \}\right)$$ is the two-dimensional continuous irrational rotation with slope $\alpha$ and starting point ${\mathbf{x}}$. \[def:cbrs\] Let $S \subset I^2$ be an arbitrary measurable subset of the unit square with Lebesgue measure $\lambda (S)$. We say that $S$ is a bounded remainder set for the continuous irrational rotation with slope $\alpha>0$ and starting point ${\mathbf{x}}=(x_1, x_2) \in I^2$ if the distributional error $$\label{eq:discrepancy} \Delta_T(S, \alpha, {\mathbf{x}}) = \int_0^T \chi_S \left( \{x_1+t\}, \{x_2+\alpha t \} \right) \, dt - T \lambda (S)$$ is uniformly bounded for all $T > 0$. Here, $\chi_S$ denotes the characteristic function for the set $S$. Bounded remainder sets have been extensively studied for the discrete analogue of continuous irrational rotation, that is, for Kronecker sequences $(\{n\alpha_1\}, \{n\alpha_2\}, \ldots , \{n\alpha_s\})_{n=1, 2, \ldots}$ in $[0,1)^s$, where $\alpha_1, \ldots , \alpha_s$ are given reals. In this context, a bounded remainder set $S \subseteq [0,1)^s$ is a measurable set for which the difference $$\left\| \sum_{n=1}^N \chi_S (\{x_1+ n\alpha_1\}, \ldots , \{x_s+n\alpha_s\}) - N \lambda (S) \right\|$$ is uniformly bounded for all integers $N{\geqslant}1$ and almost every point $(x_1, \ldots , x_s) \in [0,1)^s$. In the simplest case when $s=1$ and $S$ is just an interval, bounded remainder sets for the Kronecker sequences were explicitly characterized by Hecke [@hecke], Ostrowski [@ostrowski1; @ostrowski2] and Kesten [@kesten]. In the general multi-dimensional case, a characterization of bounded remainder sets in terms of equidecomposability to certain parallelepipeds was recently given in [@grlev]. Without going into further detail on the known results for the Kronecker sequences, let us simply emphasize that in the discrete case, a given set $S \subset [0,1)^s$ is a bounded remainder set for only “very few” choices of $(\alpha_1, \ldots , \alpha_s)$. Likewise, given a vector $(\alpha_1, \ldots, \alpha_s)$, the class of sets $S$ which are of bounded remainder with respect to this vector is, in some sense, small. Once we consider bounded remainder sets for the continuous irrational rotation, the situation turns out to be quite different. In light of recent work by Jószef Beck, this is not entirely unexpected. Beck studied distributional properties of the continuous irrational rotation in [@beck1; @beck2; @beck3], and showed in particular that: Let $S \subseteq I^2$ be an arbitrary Lebesgue measurable set in the unit square with positive measure. Then for every $\varepsilon >0$, almost all $\alpha>0$ and every starting point $\mathbf{x}=(x_1, x_2)\in I^2$, we have $$\int_0^T \chi_S \left( \{x_1+t\}, \{x_2+\alpha t\} \right) \, dt - T \lambda (S) = o \left( (\log T)^{3+\varepsilon} \right) .$$ As pointed out by Beck, the polylogarithmic error term is shockingly small compared to the linear term $T \lambda (S)$. Moreover, it holds for all measurable sets $S$. It is thus natural to ask if imposing certain regularity conditions on $S$ could give an even lower bound on the error term. The aim of this paper is to show that the estimate of Beck can be significantly improved for a large collection of sets $S$. We show that: \[thm:mainpoly\] For almost all $\alpha>0$ and every $\mathbf{x} \in I^2$, every polygon $S \subset I^2$ with no edge of slope $\alpha$ is a bounded remainder set for the continuous irrational rotation with slope $\alpha$ and starting point $\mathbf{x}$. \[thm:mainconvex\] For almost all $\alpha>0$ and every $\mathbf{x} \in I^2$, every convex set $S \subset I^2$ whose boundary $\partial S$ is a twice continuously differentiable curve with positive curvature at every point is a bounded remainder set for the continuous irrational rotation with slope $\alpha$ and starting point $\mathbf{x}$. We will see from the proofs that Theorems \[thm:mainpoly\] and \[thm:mainconvex\] hold for all $\alpha$ whose continued fraction expansion $\alpha=[a_0; a_1, a_2,\cdots]$ satisfies $$\sum_{l=0}^s \frac{a_{l+1}}{q_{l}^{1/2}} \sum_{k=1}^{l+1}a_k < C ,$$ where $C$ is a constant independent of $s$. Here, $(q_l)_{l {\geqslant}0}$ is the sequence of best approximation denominators for $\alpha$. The following results are immediate consequences of Theorems \[thm:mainpoly\] and \[thm:mainconvex\]. \[cor:mainpoly\] Let $S$ be a polygon in $I^2$. Then $S$ is a bounded remainder set with respect to continuous irrational rotation for almost every $\alpha>0$ and every starting point $\mathbf{x} \in I^2$. \[cor:mainconvex\] Let $S$ be a convex set in $I^2$ whose boundary $\partial S$ is a twice continuously differentiable curve with positive curvature at every point. Then $S$ is a bounded remainder set with respect to continuous irrational rotation for almost every $\alpha>0$ and every starting point $\mathbf{x} \in I^2$. In light of Corollaries \[cor:mainpoly\] and \[cor:mainconvex\], it is tempting to raise the question of whether every convex set $S \subset I^2$ is a bounded remainder set with respect to continuous irrational rotation for almost every slope $\alpha>0$ and every starting point $\mathbf{x} \in I^2$. We leave this question open. Theorems \[thm:mainpoly\] and \[thm:mainconvex\] above are, in a certain sense, optimal. First of all, the slope condition in Theorem \[thm:mainpoly\] on the edges of the polygon $S$ cannot be omitted. To see this, fix some $\alpha>0$, and let $S$ be the parallelogram shown in Figure \[fig:fullpgram\] with $p \notin {\mathbb{Z}}\alpha ({\operatorname{mod}}1)$ and $\lambda (S) = p$. (-.15,0) – (1.15, 0); (0,-.1) – (0, 1.1); (0,1) – (1,1) – (1,0); (0,1) node\[left\][$1$]{}; (1,0) node\[below\][$1$]{}; (0,0) – (1, [1/(2\sqrt(2))]{}) – (1, [1/(2\sqrt(2))+.4]{}) – (0,.4) – (0,0); (.5, [.2+1/(4\sqrt(2))]{}) node[$S$]{}; (-.1,0) – (-.1,.4); (-.1, .2) node\[left\][$p$]{}; (1.1,0) – (1.1, [1/(2\sqrt(2))]{}); (1.1,[1/(4\sqrt(2))]{}) node\[right\][$\alpha$]{}; It is not difficult to show that for such a set $S$, with two edges of slope $\alpha$, we have $$\left\| \int_0^T \chi_S \left( \{t\}, \{\alpha t\} \right) \, dt - \sum_{n=1}^{{\lfloor T \rfloor}} \chi_{[0,p)}(\{n\alpha\}) \right\| {\leqslant}1.$$ We recall from the discrete setting that if $p \notin {\mathbb{Z}}\alpha ({\operatorname{mod}}1)$, then the difference $$\left\| \sum_{n=1}^{{\lfloor T \rfloor}} \chi_{[0,p)}(\{n\alpha\}) - p {\lfloor T \rfloor} \right\|$$ is unbounded as $T \rightarrow \infty$ [@kesten], and accordingly so is $$\left\| \Delta_T(S, \alpha, 0) \right\| = \left\| \int_0^T \chi_S \left( \{t\}, \{ \alpha t\}\right) - pT\right\| .$$ Thus, the set $S$ in Figure \[fig:fullpgram\] is not of bounded remainder for the continuous irrational rotation with slope $\alpha$ starting at the origin. By an equivalent argument, all sets $S'$ similar to the examples shown in Figure \[fig:modpgram\] with $p \notin {\mathbb{Z}}\alpha ({\operatorname{mod}}1)$ are not* bounded remainder sets. (-.15,0) – (1.15, 0); (0,-.1) – (0, 1.1); (0,1) – (1,1) – (1,0); (0,1) node\[left\][$1$]{}; (1,0) node\[below\][$1$]{}; (0,0) – (1, [1/(2\sqrt(2))]{}); (1, [1/(2\sqrt(2))+.4]{}) – (0,.4); (.1,[.1/(2\sqrt(2))]{}) – (.4, [.4/(2\sqrt(2))]{}) – (.4, [.4+.4/(2\sqrt(2))]{}) – (.1, [.4+.1/(2\sqrt(2))]{}) – (.1, [.1/(2\sqrt(2))]{}); (.25, [.2+.25/(2\sqrt(2))]{}) node[$S'$]{}; (.6, [.6/(2\sqrt(2))]{}) – (.9, [.9/(2\sqrt(2))]{}) – (.8, [.35+.3/(2\sqrt(2))]{}) – (.95, [.5+.3/(2\sqrt(2))]{}) – (.9, [.4+.9/(2\sqrt(2))]{}) – (.6, [.4+.6/(2\sqrt(2))]{}) –(.65,.5) – (.5,.35)– (.6, [.6/(2\sqrt(2))]{}); (.72, .45) node[$S'$]{}; (-.1,0) – (-.1,.4); (-.1, .2) node\[left\][$p$]{}; (1.1,0) – (1.1, [1/(2\sqrt(2))]{}); (1.1,[1/(4\sqrt(2))]{}) node\[right\][$\alpha$]{}; Secondly, in neither Theorem \[thm:mainpoly\] nor \[thm:mainconvex\] can we replace “for almost all $\alpha$” by “for all irrational $\alpha$”. This is clarified by the following: \[thm:mainneg\] \[it:negpoly\] For uncountably many $\alpha>0$ there exist triangles in $I^2$ with no edge of slope $\alpha$ which are not bounded remainder sets for the continuous irrational rotation with slope $\alpha$ and arbitrary starting point. \[it:negconvex\] For uncountably many $\alpha>0$ there exist discs in $I^2$ which are not bounded remainder sets for the continuous irrational rotation with slope $\alpha$ and arbitrary starting point. \[it:specset\] The triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ is a bounded remainder set for every slope $\alpha >0$ and every starting point $\mathbf{x} \in I^2$. Theorem \[thm:mainneg\] illustrates that for very special polygons $S$, Theorem \[thm:mainpoly\] does actually hold for all irrational $\alpha$. Other trivial examples of such special sets are rectangles of the form $[0,\gamma) \times [0,1)$ (or $[0,1)\times[0,\gamma)$), where $0 < \gamma {\leqslant}1$. Finally, let us point out that Theorems \[thm:mainpoly\] and \[thm:mainconvex\], and their proofs, give information on the behavior of discrepancies of the continuous irrational rotation on the unit square. Let $\mathcal{B}$ denote a certain class of measurable subsets of $I^2$. Then by the discrepancy $D_T^{(\mathcal{B})}$ of a continuous irrational rotation with slope $\alpha>0$ and starting point $\mathbf{x} \in I^2$ with respect to $\mathcal{B}$ we mean $$D_T^{(\mathcal{B})} := \sup_{S \in \mathcal{B}} \Delta_T (S, \alpha, \mathbf{x}),$$ with $\Delta_T (S, \alpha, \mathbf{x})$ defined in . The most extensively studied case in the classical theory of irregular distribution is that when $\mathcal{B}$ is the class of axis-parallel rectangles. Theorem \[thm:mainpoly\] tells us that in this case, we have $$\Delta_T (S, \alpha, \mathbf{x}) = O(1)$$ for all $\mathbf{x}$, almost all $\alpha$ and all $S \in \mathcal{B}$. Moreover, by a careful consideration of the constants involved in the proof of Theorem \[thm:mainpoly\], one can verify that the $O$-constant will depend only on $\alpha$, and not on the choice of rectangle $S$. As a consequence, we obtain the following result, previously shown by Drmota [@drmota] (see also [@tichy]). \[cor:discr1\] The discrepancy $D_T^{(\mathcal{B})}$ of the continuous irrational rotation with slope $\alpha$ and starting point $\mathbf{x}$ with respect to the class $\mathcal{B}$ of axis-parallel rectangles in $I^2$ is $$D_T^{(\mathcal{B})} = O(1)$$ for all $\mathbf{x} \in I^2$ and almost all $\alpha >0$. As clarified by the example in Figure \[fig:fullpgram\], an analogous result does not hold if $\mathcal{B}$ is the class of all* rectangles. It follows that the isotropic discrepancy, i.e. the discrepancy with respect to the class of all convex sets, cannot be bounded. However, if we let $\mathcal{B}$ be the class $\mathcal{D}$ of all discs in $I^2$, then we can attain a result analogous to Corollary \[cor:discr1\]. Theorem \[thm:mainconvex\] tells us that for all $S \in \mathcal{D}$, we have $$\Delta_T(S, \alpha, \mathbf{x}) = O(1)$$ for all $\mathbf{x}$ and almost all $\alpha$, and from the proof of Theorem \[thm:mainconvex\] it is not difficult to see that the $O$-constant can be made independent of the size and position of the disc $S$. We thus get: \[cor:discr2\] The discrepancy $D_T^{(\mathcal{D})}$ of the continuous irrational rotation with slope $\alpha$ and starting point $\mathbf{x}$ with respect to the class $\mathcal{D}$ of discs in $I^2$ is $$D_T^{(\mathcal{D})}=O(1)$$ for all $\mathbf{x} \in I^2$ and almost all $\alpha>0$. The rest of the paper is organized as follows. In Section \[sec:proofs\] we present necessary preliminary material, and give the proofs of Theorems \[thm:mainpoly\] and \[thm:mainconvex\]. Section \[sec:neg\] is devoted to the proof of Theorem \[thm:mainneg\]. Preliminaries and proofs of Theorems \[thm:mainpoly\] and \[thm:mainconvex\] {#sec:proofs} ============================================================================ Continued fractions ------------------- We begin by briefly reviewing some well-known facts about continued fractions. For an irrational $\alpha \in (0,1)$, let $$[0; a_1, a_2, a_3, \ldots ]$$ be its continued fraction expansion, and denote by $p_n / q_n$ its $n$th convergent. The numerators $p_n$ and denominators $q_n$ are given recursively by $$\begin{aligned} q_0 = 1, \quad &q_1 = a_1 , \quad &q_{n+1} = a_{n+1} q_n + q_{n-1} , \\ p_0 = 0, \quad &p_1 = 1 , \quad &p_{n+1} = a_{n+1} p_n + p_{n-1} . \\ \end{aligned}$$ It follows readily from these recurrences that $$\label{eq:cfminusone} p_nq_{n+1} - p_{n+1} q_n = (-1)^{n+1}.$$ The $n$th convergent $p_n/q_n$ is greater than $\alpha$ for every odd value of $n$, and smaller than $\alpha$ for every even value of $n$. It is easy to see that $\lim_{n \rightarrow \infty} p_n / q_n = \alpha$, and moreover we have the error bounds $$\label{eq:cferror} \frac{1}{(a_{n+1}+2)q_n^2} {\leqslant}\left\| \alpha - \frac{p_n}{q_n} \right\| {\leqslant}\frac{1}{a_{n+1}q_n^2} .$$ Every non-negative integer $N$ has a unique expansion $$N = \sum_{i=0}^s b_i q_i, \text{ with } b_s>0 ; \quad 0 {\leqslant}b_i {\leqslant}a_{i+1} , \quad 0 {\leqslant}i {\leqslant}s .$$ We will refer to this as the Ostrowski expansion of $N$ to base $\alpha$. Finally, we will need the following result, which follows from well-known facts in metric theory for continued fractions (see e.g. [@khinchin]). \[lem:cfsum\] For almost every irrational $\alpha \in (0,1)$ and every $m>0$, the sum $$\sum_{l=0}^s \frac{a_{l+1}}{q_l^{1/m}} \sum_{k=1}^{l+1} a_k$$ is uniformly bounded in $s$. Functions of bounded remainder ------------------------------ It is not difficult to show that the question of whether $S \subset I^2$ is a bounded remainder set for the continuous two-dimensional irrational rotation is essentially a one-dimensional problem. By making an appropriate projection, the question can be restated as that of whether a certain associated function is of bounded remainder. \[def:brf\] Let $f : {\mathbb{R}}\mapsto {\mathbb{C}}$ be a $1$-periodic function which is integrable over $[0,1]$. We say that $f$ is a bounded remainder function with respect to $\alpha \in {\mathbb{R}}\setminus {\mathbb{Q}}$ if there is a constant $C=C(f,\alpha)$ such that $$\left\| \sum_{k=0}^{N-1} f(k\alpha) - N \int_0^1 f(x) \, dx \right\| {\leqslant}C$$ for all integers $N>0$. Bounded remainder functions have been studied by several authors, see e.g. [@hellekalek], or [@schoissen] and the references therein. We will consider two special classes of functions: hat functions and dome functions. \[def:hatfnc\] We say that $T \,: \, {\mathbb{R}}\mapsto [0, \infty)$ is a hat function if $T$ is supported on the interval $[0,b]$, with $b >0$, and $$\label{eq:hatfnc} T(x) = \begin{cases} \frac{H}{a} x \text{ ,} & 0 {\leqslant}x {\leqslant}a \text{ ;}\\ -\frac{H}{b-a} (x-b) \text{ ,} & a < x {\leqslant}b \text{ ,} \end{cases}$$ for some $0<a<b$ and $H>0$. \[def:domefnc\] We say that a continuous function $T \,: \, {\mathbb{R}}\mapsto [0, \infty)$ supported on $[0,B]$, with $B>0$, is a dome function if it satisfies the following two conditions: 1. \[it:domediff\] $T$ is concave and twice differentiable on the open interval $(0,B)$. 2. \[it:domegrowth\] There exist $\varepsilon >0$, $m>0$ and $c>0$ such that $$\label{eq:growthcond} \begin{aligned} &\left\| T(x) \right\| {\leqslant}c \cdot x^{1/m} \quad &\text{ for all } \, 0 {\leqslant}x < \varepsilon ; \\ &\left\| T(B-x)\right\| {\leqslant}c \cdot x^{1/m} \quad &\text{ for all } \, 0 {\leqslant}x < \varepsilon .\\ \end{aligned}$$ We will establish and prove the following two results, which will be crucial for the proofs of Theorems \[thm:mainpoly\] and \[thm:mainconvex\] later on. \[prop:hatfncbrf\] Let $\tau (x) = \sum_{m\in {\mathbb{Z}}} T(x+m)$, where $T$ is a hat function. Then $\tau$ is a bounded remainder function with respect to almost every $\alpha \in {\mathbb{R}}\setminus {\mathbb{Q}}$. \[prop:domefncbrf\] Let $\tau (x) = \sum_{m\in {\mathbb{Z}}} T(x+m)$, where $T$ is a dome function. Then $\tau$ is a bounded remainder function with respect to almost every $\alpha \in {\mathbb{R}}\setminus {\mathbb{Q}}$. \[rem:shifts\] For sufficiently regular functions, including periodizations of hat and dome functions, the bounded remainder property is not affected by shifting the function (see [@schoissen p. 128–129]). It thus follows from Propositions \[prop:hatfncbrf\] and \[prop:domefncbrf\] that for almost every $\alpha \in {\mathbb{R}}\setminus {\mathbb{Q}}$, we have $$\left\| \sum_{k=0}^{N-1} \tau (k\alpha +x_0) - N \int_0^1 \tau (x) \right\| {\leqslant}C$$ for all $N>0$ and every $x_0 \in {\mathbb{R}}$ whenever $\tau$ is the periodization of a hat or dome function. The constant $C$ may depend on $\tau$ and $\alpha$, but not on $N$ or $x_0$. Later on we explain how Theorems \[thm:mainpoly\] and \[thm:mainconvex\] follow from the results above. For the proof of Proposition \[prop:hatfncbrf\], we will need the following lemma. \[lem:altsum\] Let $f \, : \, {\mathbb{R}}\mapsto {\mathbb{R}}$ be a $1$-periodic function, $\alpha$ be irrational and $N$ be a nonnegative integer with Ostrowski expansion $$N= b_s q_s + \ldots + b_0 q_0$$ to base $\alpha$. We then have $$\label{eq:altsum} \sum_{k=0}^{N-1} f(k\alpha) = \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} f \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right),$$ for some $\rho_{k,l}$ satisfying $-1<\rho_{k,l}<2$. Let $n(0)=0$ and $n(l)=b_{l-1} q_{l-1} + \ldots +b_0q_0$ for $1{\leqslant}l{\leqslant}s$. It is straightforward to show that $$\label{eq:ostrowskisum} \sum_{k=0}^{N-1} f(k\alpha) = \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} f(k\alpha +(n(l)+bq_l)\alpha).$$ We define $\theta_l$ from the equation $$\frac{\theta_l}{a_{l+1}q_l^2} = \alpha -\frac{p_l}{q_l} ,$$ and observe that by we have $1/3{\leqslant}\|\theta_l\|{\leqslant}1$. Moreover, we find $x_l\in [0,1)$ and $m_l\in \{ 0, \ldots , q_l-1 \}$, $m_l = m_l(b,x,\alpha)$, such that $$\left\{(n(l)+bq_l)\alpha \right\} = \frac{m_l}{q_l} + \frac{x_l}{q_l}.$$ We can then rewrite the summand on the right hand side in as $$\label{eq:rewrittensummand} f(k\alpha +(n(l)+bq_l)\alpha) = f\left(\frac{k p_l+m_l}{q_l} + \frac{k\theta_l}{a_{l+1}q_l^2}+ \frac{x_l}{q_l} \right) .$$ Using the substitution $kp_l + m_l = t \, ({\operatorname{mod}}q_l)$, which by gives $$k=(t-m_l)q_{l-1}(-1)^{l-1} \, ({\operatorname{mod}}q_l),$$ we get $$\label{eq:rewrittenfill} \left\{ \frac{kp_l+m_l}{q_l} + \frac{k\theta_l}{a_{l+1}q_l^2}+\frac{x_l}{q_l} \right\} = \left\{ \frac{t}{q_l} + \frac{\rho_{t,l}}{q_l} \right\},$$ where $$\label{eq:rho} \rho_{t,l} := \left\{ (t-m_l)(-1)^{l-1}\frac{q_{l-1}}{q_l} \right\} \frac{\theta_l}{a_{l+1}} + x_l .$$ With this definition we have $$- \frac{1}{a_{l+1}} < \rho_{t,l} < \frac{1}{a_{l+1}}+1 ,$$ and hence $-1 < \rho_{t,l} < 2$. Combining , and , we thus arrive at . It will be sufficient to prove Proposition \[prop:hatfncbrf\] for the case when $b {\leqslant}1$ in Definition \[def:hatfnc\]. To see this, observe that any general hat function $T$ can be written as a sum of shifted hat functions $T_i$ with support $[0,b]$, $b {\leqslant}1$. Since any finite sum of bounded remainder functions is again a bounded remainder function, the general case follows from the special case $\tau(x) = \sum_{m \in {\mathbb{Z}}} T_i(x+m)$. Our goal is to show that for almost every $\alpha \in {\mathbb{R}}\setminus {\mathbb{Q}}$, we can find a constant $C=C(\alpha, \tau)$ such that $$\label{eq:brhatgoal} \left\| \sum_{k=0}^{N-1} \tau (k\alpha) - N \int_0^1 \tau (x) \, dx \right\| {\leqslant}C$$ for every integer $N>0$. It will be enough to verify this for $\alpha \in (0,1)$, as the sum in depends only on the fractional part of $\alpha$. By Lemma \[lem:altsum\] we may rewrite this sum as $$\sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right),$$ where $N=b_sq_s + \cdots + b_0q_0$ is the Ostrowski expansion of $N$ to base $\alpha$ and $-1 <\rho_{k,l} < 2$. We verify in two steps: First we show that $$\label{eq:hatstep1} \left\| \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right) - N \int_0^1 \tau (x) \, dx \right\| {\leqslant}C , \quad N=1,2, \ldots ,$$ for almost every irrational $\alpha \in (0,1)$. We then show that $$\label{eq:hatstep2} \left\| \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \left( \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right)- \tau \left( \frac{k}{q_l}\right) \right) \right\| {\leqslant}C , \quad s=1,2, \ldots ,$$ for almost every irrational $\alpha \in (0,1)$. Combining and , we immediately obtain . Let us first verify that holds. On the interval $I$, the function $\tau$ is of the form with $b {\leqslant}1$, so we can find $u_l, v_l \in \{ 0,1, \ldots , q_l-1 \}$ and $\xi_l , \eta_l \in (0,1]$ such that $$\label{eq:aandb} a=\frac{u_l + \xi_l}{q_l} \quad \text{ and } \quad b=\frac{v_l+\eta_l}{q_l} .$$ For sufficiently large $l>l_0$ (where $l_0=l_0(\tau)$ depends only on $\tau$), we have $u_l < v_l$, and a straightforward calculation gives $$\sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right) = \frac{Hb}{2} q_l + \frac{Ha\eta_l(1-\eta_l)-Hb\xi_l(1-\xi_l)}{2a(b-a)q_l}.$$ Thus we have $$\left\| \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right) - q_l \int_0^1 \tau(x) \, dx \right\| {\leqslant}C \frac{1}{q_l},$$ where $C=C(\tau)$ (this is trivially true also when $l {\leqslant}l_0$), and it follows that $$\left\| \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right) - N \int_0^1 \tau (x) \, dx \right\| {\leqslant}C \sum_{l=0}^s \frac{b_l}{q_l} .$$ Since $b_l < a_{l+1}$, it follows from Lemma \[lem:cfsum\] that the right hand side above is uniformly bounded in $s$ for almost every $\alpha \in (0,1)$. This confirms . We go on to verify . We will assume below that $b < 1$ in ; the proof when $b=1$ is slightly simpler, but essentially the same. Let $Q_l = \{ 0,1, \ldots , q_l-1\}$, and define $u_l, v_l \in Q_l$ as in . Denote by $E$ a set of “exceptional” indices $$E = \{ 0,u_l-1, u_l, u_l+1, v_l-1, v_l, v_l+1, q_l-1\}$$ (for sufficiently large $l {\geqslant}l_0$, these are all distinct). We have $$\label{eq:indexsplit} \sum_{k=0}^{q_l -1} \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right) = \sum_{k \in Q_l \setminus E} \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right) + \sum_{k \in E} \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right),$$ and since $\tau$ is everywhere linear (with bounded slope) it is clear that $$\label{eq:indexsplit1} \sum_{k \in E} \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right) = \sum_{k \in E} \tau \left( \frac{k}{q_l} \right) + O \left( \frac{1}{q_l} \right).$$ The second sum on the right hand side in can be rewritten using the specific form of $\tau$ on $I$. We get $$\label{eq:indexsplit2} \sum_{k \in Q_l \setminus E} \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right) = \sum_{k \in Q_l \setminus E} \tau \left( \frac{k}{q_l} \right) + \Sigma_1 ,$$ where $$\Sigma_1 := \frac{1}{q_l} \left( \frac{H}{a} \sum_{k=1}^{u_l-2} \rho_{k,l} - \frac{H}{b-a} \sum_{k=u_l+2}^{v_l-2} \rho_{k,l} \right) ,$$ and $\rho_{k,l}$ is defined in . To verify , we will need to find an appropriate bound on $\Sigma_1$. We now show that $\Sigma_1 = O( \sum_{i=1}^l a_i /q_l)$. By defining $\alpha_l$ and $\gamma_l$ as $$\label{eq:alphgam} \begin{aligned} \alpha_l &:= (-1)^{l-1} \frac{q_{l-1}}{q_l} , \\ \gamma_l &:= -m_l(-1)^{l-1}\frac{q_{l-1}}{q_l} , \end{aligned}$$ we can rewrite $\rho_{k,l}$ in as $$\rho_{k,l} = \omega_{k,l} \cdot \frac{\theta_l}{a_{l+1}} + x_l ,$$ where $\omega_{k,l} := \{ k\alpha_l + \gamma_l \}$. Using and the fact that $x_l \in [0,1)$, it is an easy task to show that $$\frac{H}{a} \sum_{k=1}^{u_l-2} x_l - \frac{H}{b-a} \sum_{k=u_l+2}^{v_l-2} x_l = O(1) .$$ We thus have $$\label{eq:S1calc1} \begin{aligned} \Sigma_1 &= \frac{\theta_l}{q_l a_{l+1}} \left( \frac{H}{a} \sum_{k=1}^{u_l-2} \omega_{k,l} - \frac{H}{b-a} \sum_{k=u_l+2}^{v_l-2} \omega_{k,l} \right) + O \left( \frac{1}{q_l} \right) \\ &= \frac{H\theta_l}{q_l a_{l+1}} \left( \frac{1}{a} \sum_{k=0}^{u_l-1} \omega_{k,l} - \frac{1}{b-a} \sum_{k=u_l}^{v_l-1} \omega_{k,l} \right) + O \left( \frac{1}{q_l} \right) ,\\ \end{aligned}$$ where the last equality follows from boundedness of the terms $\omega_{k,l}$. To further approximate $\Sigma_1$, we employ Koksma’s inequality for the sequence $\{\omega_{k,l} \}_{k=0}^{q_l-1}$ and the linear function $f(x) = \{ x\}$ (see [@kuipers Theorem 5.1]). For $1 {\leqslant}N {\leqslant}q_l$, we have $$\left\| \sum_{k=0}^{N-1} \omega_{k,l} - N \int_0^1 x \, dx \right\| = \left\| \sum_{k=0}^{N-1} \omega_{k,l} - \frac{N}{2} \right\| {\leqslant}ND_N^(\omega_{k,l}) V_I(f),$$ where $V_I(f) =1$ is the total variation of $f$ over $I$, and $D_N^(\omega_{k,l})$ denotes the star-discrepancy of the point set $\{\omega_{k,l} \}_{k=0}^{N-1}$. The extreme discrepancy $D_N$ of $\{ \omega_{k,l} \}_{k=0}^{N-1}$ equals that of $\{ k\alpha_l\}_{k=0}^{N-1}$. Note that $\|\alpha_l\| = q_{l-1}/q_l$ has continued fraction expansion $$\|\alpha_l \| = \left[ 0; a_l, a_{l-1}, \ldots a_1 \right].$$ It thus follows that $$\label{eq:boundomega2} ND_N^* (\omega_{k,l}) {\leqslant}ND_N(\omega_{k,l}) = ND_N(k\alpha_l) {\leqslant}1 + 2 \sum_{i=1}^l a_i$$ for $1 {\leqslant}N {\leqslant}q_l$ (see [@kuipers p. 126] for the last inequality). Hence, we have $$\left\| \sum_{k=0}^{N-1} \omega_{k,l} - \frac{N}{2} \right\| {\leqslant}1 + 2 \sum_{i=1}^l a_i ,$$ and from this and it follows that $$\begin{aligned} \Sigma_1 &= \frac{H \theta_l}{q_l a_{l+1}} \left( \left( \frac{1}{a} + \frac{1}{b-a} \right) \sum_{k=0}^{u_l-1} \omega_{k,l} - \frac{1}{b-a} \sum_{k=0}^{v_l-1} \omega_{k,l} \right) + O \left( \frac{1}{q_l} \right) \\ &= \frac{H\theta_l}{q_l a_{l+1}} \left( \frac{b}{a(b-a)} \cdot \frac{u_l}{2} - \frac{1}{b-a} \cdot \frac{v_l}{2} \right) + O\left( \frac{\sum_{i=1}^l a_i}{q_l} \right) \\ &= \frac{H\theta_l}{2q_l a_{l+1}} \left( \frac{b}{a(b-a)} \left( q_l a-\xi_l\right) - \frac{1}{b-a} \left( q_lb-\eta_l\right) \right) + O\left( \frac{\sum_{i=1}^l a_i}{q_l} \right) \\ &= O\left( \frac{\sum_{i=1}^l a_i}{q_l} \right) . \end{aligned}$$ Let us finally see that this bound on $\Sigma_1$ implies . Inserting and in , we get $$\left\| \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right) - \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right)\right\| {\leqslant}\frac{C}{q_l} \sum_{i=1}^l a_i ,$$ for $l {\geqslant}l_0=l_0(\tau)$ and some constant $C$ which depends only on $\tau$ and $\alpha$ (this bound holds trivially also when $l < l_0$). We thus have $$\begin{aligned} \left\| \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \left( \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right)- \tau \left( \frac{k}{q_l}\right) \right) \right\| &{\leqslant}C' a_1 + C \sum_{l=1}^s \frac{b_l}{q_l} \sum_{i=1}^l a_i \\ &{\leqslant}C \sum_{l=0}^s \frac{a_{l+1}}{q_l} \sum_{i=1}^{l+1} a_i. \end{aligned}$$ By Lemma \[lem:cfsum\], the sum on the right hand side above is bounded uniformly in $s$ for almost every irrational $\alpha \in (0,1)$. This verifies , and completes the proof of Proposition \[prop:hatfncbrf\]. Before we embark on the proof of Proposition \[prop:domefncbrf\], we establish the following preliminary result. \[lem:boundedvar\] Suppose $f : {\mathbb{R}}\mapsto {\mathbb{R}}$ is a dome function as given in Definition \[def:domefnc\], and let $q>2/B$. Denote by $f_q'$ the function $$\label{eq:derivq} f_q'(x) = \begin{cases} f'(x) & 1/q {\leqslant}x {\leqslant}B-1/q , \\ 0 & \text{ otherwise. } \end{cases}$$ Then for $q>1/\varepsilon$, with $\varepsilon$ as in , the total variation $V_I(f_q')$ of $f_q'$ over $I$ satisfies $$V_I(f_q') {\leqslant}Cq^{1-1/m} ,$$ where $C=C(c)$ with $c$ as in . The function $f$ is concave and twice differentiable on $(0,B)$, from which it follows that $f'$ is monotonically nonincreasing and $$V_I(f_q') = 2 \left( f'\left( \frac{1}{q} \right) - f'\left( B-\frac{1}{q} \right) \right).$$ Moreover, we have that $$f' \left( \frac{1}{q} \right) {\leqslant}\frac{f\left( 1/q\right)-f(0)}{1/q} = qf\left(\frac{1}{q}\right) ,$$ and likewise $$f'\left(B-\frac{1}{q} \right) {\geqslant}-qf\left(B-\frac{1}{q} \right).$$ By the conditions on $f$ it thus follows that $$V_I(f_q') {\leqslant}4cq^{1-1/m} \quad \text{ for all } q> \frac{1}{\varepsilon} .$$ It will be sufficient to prove Proposition \[prop:domefncbrf\] for the case when $B {\leqslant}1$ in Definition \[def:domefnc\]. To see this, observe that any general dome function $T$ can be written as a sum of shifted hat functions, and shifted dome functions with support in $I$. This is illustrated for the case $1 < B {\leqslant}2$ in Figure \[fig:domedecomp\]; we may write the function $T$ as $$T= T_1 + T_2 + T_3 ,$$ where $T_1$ is the hat function in with $a=1, b=B$ and $H=T(1)$, and $T_2$ and $T_3$ are the dome functions $T_2 = \chi_{[0,1]}\cdot (T-T_1)$ and $T_3=\chi_{[1,B]}\cdot (T-T_1)$. As the sum of finitely many bounded remainder functions is again a bounded remainder function, the general case follows from the special case $T=T_3$ and Proposition \[prop:hatfncbrf\]. In other words, it is sufficient to consider the case when, restricted to the unit interval, $\tau$ is simply a dome function with support $[0,B]$, $B {\leqslant}1$. (-.15,0) – (2.3, 0); (0,-.1) – (0, 1.2); (1,-.03) – (1, .03); (1,-.03) node\[below\][$1$]{}; (2,-.03) – (2, .03); (2,-.03) node\[below\][$2$]{}; (0,0) to \[out=90, in=180\] (1,1) to \[out=0, in=90\] (1.6,0); (.25, .8) node\[black\][$T$]{}; (0,0) – (1,1) – (1.6,0); (.6,.75) node\[[black!70!white]{}\][$T_1$]{}; plot (, [sqrt(1-(-1)\^2)-]{}); (.8,.3) node\[[black!50!white]{}\][$T_2$]{}; plot (, [sqrt(1-((-1)/.6)\^2)+(5/3)\(-1.6)]{}); (1.1,.3) node\[[black!50!white]{}\][$T_3$]{}; Let $\tau$ be such a function. We want to show that for almost every $\alpha \in {\mathbb{R}}\setminus {\mathbb{Q}}$, we can find a constant $C=C(c,m,\alpha)$ such that $$\label{eq:brfbowler} \left\| \sum_{k=0}^{N-1} \tau(k \alpha) - N \int_0^1 \tau(x) \, dx \right\| {\leqslant}C$$ for every integer $N>0$. Again it will be enough to verify this for $\alpha \in (0,1)$, as the sum in depends only on the fractional part of $\alpha$. By Lemma \[lem:altsum\], we may rewrite this sum as $$\sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right),$$ where $N=b_sq_s + \cdots + b_0q_0$ is the Ostrowski expansion of $N$ to base $\alpha$ and $-1<\rho_{k,l}<2$. We verify in two steps: First we show that $$\label{eq:bowler1} \left\| \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right) - N \int_0^1 \tau (x) \, dx \right\| {\leqslant}C , \quad N=1,2, \ldots ,$$ for almost every irrational $\alpha \in (0,1)$. We then show that $$\label{eq:bowler2} \left\| \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \left( \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right)- \tau \left( \frac{k}{q_l}\right) \right) \right\| {\leqslant}C , \quad s=1,2, \ldots ,$$ for almost every irrational $\alpha \in (0,1)$. Combining and , we immediately obtain . Let us first see that holds. On the interval $I$, the function $\tau$ is supported on $[0,B]$ with $0<B{\leqslant}1$, so we can find $u_l \in \{ 0, 1, \ldots , q_l-1\}$ and $\xi_l \in (0,1]$ such that $$\label{eq:ulxil} B = \frac{u_l + \xi_l}{q_l}.$$ We begin by considering the inner sum $$\sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right) = \sum_{k=1}^{u_l-1} \tau \left( \frac{k}{q_l} \right) + \tau \left( \frac{u_l}{q_l} \right) .$$ It is not difficult to show, for instance using integration by parts, that $$\begin{aligned} \sum_{k=1}^{u_l-1} \tau \left( \frac{k}{q_l} \right) &= q_l \int_{1/q_l}^{(u_l-1)/q_l} \tau(x) \, dx + \frac{1}{2} \left( \tau \left( \frac{1}{q_l}\right)+ \tau \left( \frac{u_l-1}{q_l}\right) \right)\\ &+ \int_{1/q_l}^{(u_l-1)/q_l} \left( \{ q_lx \} - \frac{1}{2} \right) \tau'(x) \, dx , \end{aligned}$$ and hence $$\begin{aligned} \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right) - q_l\int_0^1 \tau(x) \, dx &= \tau \left( \frac{u_l}{q_l} \right) + \frac{1}{2} \left( \tau \left( \frac{1}{q_l}\right) + \tau \left( \frac{u_l-1}{q_l} \right)\right) \\ &- q_l \left( \int_0^{1/q_l} \tau(x) \, dx + \int_{(u_l-1)/q_l}^B \tau(x) \, dx \right)\\ &+ \int_{1/q_l}^{(u_l-1)/q_l} \left( \{ q_lx\} - \frac{1}{2} \right) \tau'(x) \, dx . \end{aligned}$$ Now let $l > l_0=l_0(\tau)$ be sufficiently large for $q_l >2/\varepsilon$. It is then clear from the conditions on $\tau$ that all but the last term on the right hand side above are bounded by $C q_l^{-1/m}$ in absolute value (where $C=C(c,m)$). In fact, the same bound holds also for the last term, as $$\begin{aligned} &\left\| \int_{1/q_l}^{(u_l-1)/q_l} \left( \{q_l x\} - \frac{1}{2} \right) \tau'(x) \, dx \right\| \\ &{\leqslant}\sum_{i=1}^{u_l-2} \left\| \int_{i/q_l}^{(i+1)/q_l} \left( \{q_l x\} - \frac{1}{2} \right) \tau'(x) \, dx \right\| \\ &{\leqslant}\sum_{i=1}^{u_l-2} \Big\| \max_{x \in [\frac{i}{q_l}, \frac{i+1}{q_l}]} \tau'(x) - \min_{x \in [\frac{i}{q_l}, \frac{i+1}{q_l}]} \tau'(x) \Big\| \int_{(2i+1)/2q_l}^{(i+1)/q_l} \left( \{ q_lx\} - \frac{1}{2} \right) \, dx \\ &{\leqslant}\frac{1}{8q_l} V_I(\tau'_{q_l}) , \end{aligned}$$ with $\tau'_{q_l}$ defined as in . Since $q_l> 2/\varepsilon$, it follows from Lemma \[lem:boundedvar\] that $$\frac{1}{8q_l} V_I(\tau'_{q_l}) {\leqslant}C q_l^{-1/m},$$ where $C=C(c)$, and hence we get $$\left\| \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right) - q_l\int_0^1 \tau(x) \, dx \right\| {\leqslant}C q_l^{-1/m},$$ for some constant $C(c, m)$ and $l>l_0$ (and this bound holds trivially also when $l {\leqslant}l_0$). It follows that $$\left\| \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \tau \left( \frac{k}{q_l} \right) - N \int_0^1 \tau (x) \, dx \right\| {\leqslant}C \sum_{l=0}^s \frac{b_l}{q_l^{1/m}},$$ and by Lemma \[lem:cfsum\] the latter sum is uniformly bounded in $s$ for almost every irrational $\alpha \in (0,1)$. This confirms . We now show that holds. We assume below that $B < 1$; the proof when $B=1$ is slightly simpler, but essentially the same. Again we begin by treating the inner sum $$\label{eq:b2inner} \sum_{k=0}^{q_l-1} \left( \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right) - \tau \left( \frac{k}{q_l} \right) \right),$$ which we will show is bounded in absolute value by $$\label{eq:wantedbound} \sum_{i=1}^l a_i \left( C_1q_l^{-1} + C_2q_l^{-1/m} \right),$$ for constants $C_1=C_1(m,c,\alpha)$ and $C_2=C_2(m,c,\alpha)$. Let $u_l$ be defined as in , and denote by $E$ a set of “exceptional” indices $$E= \left\{ 0,1,u_l-2, u_l-1, u_l, u_l+1, q_l-1 \right\}$$ (for sufficiently large $l$, these are all distinct). We split the sum into two parts $$\Sigma_1 := \sum_{k \in E} \left( \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right) - \tau \left( \frac{k}{q_l} \right) \right)$$ and $$\Sigma_2 := \sum_{k=2}^{u_l-3} \left( \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right) - \tau \left( \frac{k}{q_l} \right) \right) .$$ Now let $l>l_1$ be sufficiently large for $q_l > 4/\varepsilon$. Since $-1 < \rho_{k,l} < 2$, it follows from the conditions on $\tau$ that $$\label{eq:S1bound} \left\| \Sigma_1 \right\| {\leqslant}C q_l^{-1/m} ,$$ where $C=C(c)$. To find a bound on $\Sigma_2$, we first rewrite the sum using the mean value theorem. We have that $$\Sigma_2 = \sum_{k=2}^{u_l-3} \tau'(r_k) \frac{\rho_{k,l}}{q_l} ,$$ where $r_k \in (k/q_l, (k+\rho_{k,l})/q_l)$ if $\rho_{k,l}>0$ and $r_k \in ((k+\rho_{k,l})/q_l, k/q_l)$ if $\rho_{k,l}<0$. It follows that $$\label{eq:S2ineq} \begin{aligned} \left\| \Sigma_2 - \sum_{k=2}^{u_l-3} \tau' \left(\frac{k}{q_l} \right) \frac{\rho_{k,l}}{q_l} \right\| &= \left\| \sum_{k=2}^{u_l-3} \left( \tau'(r_k) - \tau' \left(\frac{k}{q_l} \right)\right) \frac{\rho_{k,l}}{q_l} \right\| \\ &{\leqslant}\frac{2}{q_l} \sum_{k=2}^{u_l-3} \max_{x,y \in \left[ \frac{k-1}{q_l}, \frac{k+2}{q_l} \right]} \left\| \tau'(x) - \tau'(y) \right\| \\ &{\leqslant}\frac{6}{q_l} V_I(\tau'_{q_l}) {\leqslant}Cq_l^{-1/m}, \end{aligned}$$ where $C=C(c)$, and for the last inequality we have used Lemma \[lem:boundedvar\]. Finally, we need to find a bound on $$\sum_{k=2}^{u_l-3} \tau' \left(\frac{k}{q_l} \right) \frac{\rho_{k,l}}{q_l} .$$ Recall from the proof of Proposition \[prop:hatfncbrf\] that we may write $\rho_{k,l}$ as $$\rho_{k,l} = \omega_{k,l} \frac{\theta_l}{a_{l+1}}+x_l ,$$ where $\omega_{k,l} = \{k\alpha_l+\gamma_l\}$, and $\alpha_l$ and $\gamma_l$ are given in . Let us define the two-dimensional sequence $\omega := (\omega_1(k), \omega_2(k))_{k=0}^{q_l-1}$, where $$\omega_1(k) = \frac{k}{q_l} , \quad \omega_2(k) = \omega_{k,l} .$$ Moreover, let $G \, : \, I^2 \mapsto {\mathbb{R}}$ be the function given by $$G(x,y) := \chi_{[2/q_l, (u_l-3)/q_l]}(x) \tau'(x) \cdot h(y),$$ where $h \, : \, I \mapsto {\mathbb{R}}$ is the linear function $$h(y) := \frac{\theta_l}{a_{l+1}}y + x_l .$$ We then have $$\label{eq:Gsum} \sum_{k=2}^{u_l-3} \tau' \left(\frac{k}{q_l} \right) \frac{\rho_{k,l}}{q_l} = \frac{1}{q_l} \sum_{k=0}^{q_l-1} G \left( \omega_1 (k), \omega_2(k) \right).$$ From the two-dimensional Koksma-Hlawka inequality [@kuipers p. 151, p. 100] we get $$\label{eq:KH2dim} \begin{aligned} &\left\| \frac{1}{q_l} \sum_{k=0}^{q_l-1} G(w_1(k), w_2(k)) - \int_0^1 \int_0^1 G(x,y) \, dx dy \right\| \\ &{\leqslant}D_{q_l}^(\omega_1) V_I(\chi_{[2/q_l, (u_l-3)/q_l]} \tau') + D_{q_l}^(\omega_2)V_I(h)+D_{q_l}^(\omega)V_{I^2}(G) \\ &{\leqslant}D_{q_l}^(\omega) \left( V_I(\chi_{[2/q_l, (u_l-3)/q_l]} \tau') + V_I(h) + V_{I^2}(G) \right) . \end{aligned}$$ We now use this inequality to find a bound on the sum . It is not difficult (see e.g. [@kuipers p. 106]) to show that $$\label{eq:qDq} q_l D_{q_l}^(\omega) {\leqslant}2 q_l D_{q_l}^(\omega_2) {\leqslant}2 \left( 1+2\sum_{i=1}^l a_i \right),$$ where for the second inequality we have used . Moreover, we have $$\label{eq:Vh} V_I(h) = \frac{\|\theta_l\|}{a_{l+1}} {\leqslant}1,$$ and using monotonicity of $\tau'$ and Lemma \[lem:boundedvar\] we get $$\label{eq:Vtau} V_I(\chi_{[2/q_l, (u_l-3)/q_l]} \tau') {\leqslant}V_I(\tau_{q_l}') {\leqslant}C q_l^{1-1/m},$$ where $C=C(c)$ with $c$ as in . It follows that $$\label{eq:VG} V_{I^2}(G) {\leqslant}V_I(\chi_{[2/q_l, (u_l-3)/q_l]} \tau') \cdot V_I(h) {\leqslant}C q_l^{1-1/m}.$$ Lastly, we have that $$\left\| \int_0^1 \int_0^1 G(x,y) \, dx dy \right\| = \left\| \left( \tau \left( \frac{u_l-3}{q_l} \right) - \tau \left( \frac{2}{q_l}\right)\right) \cdot \left( \frac{\theta_l}{2a_{l+1}}+x_l \right)\right\|,$$ which by is bounded by $Cq_l^{-1/m}$, $C=C(c,m)$, when $l>l_1$. Inserting – and this integral estimate in , we get $$\begin{aligned} \left\| \frac{1}{q_l} \sum_{k=0}^{q_l-1} G \left( \omega_1 (k), \omega_2(k) \right) \right\| &{\leqslant}Cq_l^{-1/m} + \frac{2}{q_l}\Big( 1+2 \sum_{i=1}^l a_i \Big)\left(1+2Cq_l^{1-1/m} \right) \\ &{\leqslant}\sum_{i=1}^l a_i \left( C_1q_l^{-1} + C_2 q_l^{-1/m} \right), \end{aligned}$$ where the constants $C_1$ and $C_2$ depend only on $c$ and $m$ in . It thus follows from and that $\|\Sigma_2\|$ satisfies the bound for $l>l_1$. The same is true for $\|\Sigma_1\|$ by , and hence $\Sigma_1 +\Sigma_2$ in obeys the bound as well. We get $$\begin{aligned} &\left\| \sum_{l=0}^{s} \sum_{b=0}^{b_l-1} \sum_{k=0}^{q_l-1} \left( \tau \left( \frac{k}{q_l} + \frac{\rho_{k,l}}{q_l} \right)- \tau \left( \frac{k}{q_l}\right) \right) \right\| \\ &{\leqslant}C' a_1 + C_1 \sum_{l=1}^s \frac{b_l}{q_l} \sum_{i=1}^l a_i + C_2 \sum_{l=1}^s \frac{b_l}{q_l^{1/m}} \sum_{i=1}^l a_i \\ &{\leqslant}C_1 \sum_{l=0}^s \frac{a_{l+1}}{q_l} \sum_{i=1}^{l+1} a_i + C_2 \sum_{l=0}^s \frac{a_{l+1}}{q_l^{1/m}} \sum_{i=1}^{l+1} a_i , \end{aligned}$$ and from Lemma \[lem:cfsum\] it follows that the latter expression is bounded uniformly in $s$ for almost every irrational $\alpha \in (0,1)$. This verifies , and completes the proof of Proposition \[prop:domefncbrf\]. Proof of Theorems \[thm:mainpoly\] and \[thm:mainconvex\] --------------------------------------------------------- We now turn to the proofs of Theorems \[thm:mainpoly\] and \[thm:mainconvex\]. We will begin by proving a lemma showing that the question of whether $S \subset I^2$ is a bounded remainder set can be restated as a question of whether an associated function is of bounded remainder. Let $S \subset I^2$ be either a polygon or a set satisfying the conditions in Theorem \[thm:mainconvex\]. We can then associate to $S$ a function $\tau_S : [0,1) \mapsto [0,\infty)$ defined as $$\label{eq:tau} \tau_S(x) := \int_0^1 \chi_S(t, \{t\alpha+x\}) \, dt .$$ A geometric interpretation of $\tau_S$ is illustrated in Figure \[fig:tauS\]. It is easy to show that $$\int_0^1 \tau_S(x) \, dx = \lambda (S) .$$ Moreover, we have the following: \[lem:equivs\] The set $S\subset I^2$ is a bounded remainder set for the irrational rotation with slope $\alpha>0$ and starting point ${\mathbf{x}}= (x_1, x_2) \in I^2$ if and only if $\tau_S$ is a bounded remainder function with respect to $\alpha$. (-.1,0) – (1.15, 0); (0,-.15) – (0, 1.1); (0,1) – (1,1) – (1,0); (0,1) node\[left\][$1$]{}; (1,0) node\[below\][$1$]{}; (.25,.45) – (.5,.2) – (.75,.45) – (.6,.7) – (.4,.7) – (.25,.45); (.5,.5) node\[above left\][$S$]{}; (-.02,.25) – (.02,.25); (-.01, .25) node\[left\][$y$]{}; (0,.25) – (1, 0.6); (.33, [.25+.35\.33]{}) – (.72, [.25+.35\.72]{}); (.33, [.25+.35\.33]{}) – (.33,0); (.72, [.25+.35\.72]{}) – (.72,0); (0,.25) – (1,.25); (.33,0) – (.72, 0); (.33,.015) – (.33,-.015); (.72, .015) – (.72, -.015); (.33,-.1) – (.72,-.1); ([(.33+.72)/2]{}, -.1) node\[below\][$\tau_S(y)$]{}; (1.1,.25) – (1.1, .6); (1.1,[(.25+.6)/2]{}) node\[right\][$\alpha$]{}; By Remark \[rem:shifts\], it will be sufficient to show that $S \subset I^2$ is a bounded remainder set if and only if $\tau_S(x+x_0)$ is a bounded remainder function for some shift $x_0 \in I$. We will verify this for $x_0=\{x_2-x_1 \alpha \}$. Recall from Definition \[def:cbrs\] that $S$ is a bounded remainder set if the difference $$\Delta_T(S, \alpha, {\mathbf{x}}) = \int_0^T \chi_S \left( \{x_1+t\}, \{x_2+t \alpha \} \right) \, dt - T \lambda (S)$$ is uniformly bounded in $T$. For a given $T>0$ we let $N = {\lfloor T \rfloor}$, and denote by $S_N(\alpha, x_0)$ the difference $$S_N(\alpha, x_0) = \sum_{k=0}^{N-1} \tau_S \left( \{ k \alpha + x_0 \}\right) - N \lambda (S) .$$ By Definition \[def:brf\], the function $\tau_S(x+x_0)$ is of bounded remainder if $S_N(\alpha, x_0)$ is bounded uniformly in $N$. Thus, to prove Lemma \[lem:equivs\] it is sufficient to show that $$\label{eq:boundeddiff} \left\| S_N(\alpha, x_0) - \Delta_T(S, \alpha, {\mathbf{x}}) \right\| {\leqslant}C ,$$ where $C$ is a constant independent of $T$ (or equivalently, of $N$). To verify , we observe that $$\begin{aligned} S_N(\alpha, x_0) &= \sum_{k=0}^{N-1} \int_0^1 \chi_S \left( t, \{(t+k)\alpha + x_0 \}\right) \, dt - N \lambda (S) \\ &= \int_0^N \chi_S \left( \{t\}, \{t\alpha + x_0 \} \right) \, dt - N \lambda (S) \\ &= \int_{-x_1}^{{\lfloor T \rfloor}-x_1} \chi_S \left( \{x_1+t\}, \{x_2+t\alpha\}\right) \, dt - {\lfloor T \rfloor} \lambda (S) . \end{aligned}$$ It is now easy to see that the difference in must be bounded by $$\begin{aligned} \left\| \int_{-x_1}^0 \chi_S \left( \{x_1+t\}, \{x_2+t\alpha\}\right) \, dt \right\| &+ \\ \left\| \int_{{\lfloor T \rfloor}-x_1}^T \chi_S \left( \{x_1+t\}, \{x_2+t\alpha\}\right) \, dt \right\| &+ \{T\} \lambda (S) {\leqslant}4 , \end{aligned}$$ thus verifying and completing the proof of Lemma \[lem:equivs\]. With Lemma \[lem:equivs\] established, we are equipped to prove Theorem \[thm:mainpoly\]. It will be sufficient to consider the special case when $S$ is a triangle. This is easy to see when $S$ is a convex polygon; $S$ can then be partitioned into finitely many triangles which are disjoint (up to boundaries), and which all have the property that no edge has slope $\alpha$. Finally, since any union of finitely many disjoint bounded remainder sets is again a bounded remainder set for the irrational rotation with slope $\alpha$, the result follows. A similar, but slightly more involved argument can be given to show that also the case when $S$ is non-convex follows from the triangle case. We thus aim to prove that for almost all $\alpha>0$ and every $\mathbf{x} \in I^2$, every triangle $S$ with no edge of slope $\alpha$ is a bounded remainder set for the continuous irrational rotation with slope $\alpha$ and starting point $\mathbf{x}$. Fix some $\alpha$, and let $S$ be a triangle with no edge of slope $\alpha$. Denote by $l(y)$ the intersection in the plane of $S$ and the straight line with slope $\alpha$ through the point $(0,y)$, and let $T_S \, : \, {\mathbb{R}}\mapsto [0,\infty)$ be the function $$T_S(y) = \frac{\|l(y)\|}{\sqrt{1+\alpha^2}} .$$ Then $T_S$ is a (possibly shifted) hat function as defined in and $\tau_S$ in is given by $$\tau_S(x) = \sum_{m \in {\mathbb{Z}}} T_S(x+m) .$$ Let ${\mathbf{x}}\in I^2$ be any given starting point for the irrational rotation. By Lemma \[lem:equivs\], the triangle $S$ is a bounded remainder set if and only if $\tau_S$ is a bounded remainder function with respect to $\alpha$. By Proposition \[prop:hatfncbrf\], this is indeed the case for every irrational $\alpha>0$ whose continued fraction expansion satisfies $$\label{eq:alphcond} \sum_{l=0}^s \frac{a_{l+1}}{q_{l}^{1/2}} \sum_{k=1}^{l+1}a_k {\leqslant}C$$ for some constant $C$ independent of $s$, i.e. a set of full measure. This completes the proof of Theorem \[thm:mainpoly\]. We complete this section with the proof of Theorem \[thm:mainconvex\]. Recall that this result says that for every $\mathbf{x} \in I^2$ and almost all $\alpha>0$, every convex set $S$ whose boundary is a twice differentiable curve with positive curvature at every point is a bounded remainder set for the continuous irrational rotation with slope $\alpha$ and starting point $\mathbf{x}$. We have seen in Lemma \[lem:equivs\] that the set $S$ is of bounded remainder for the irrational rotation with slope $\alpha$ and starting point ${\mathbf{x}}\in I^2$ if and only if the associated function $\tau_S$ in is of bounded remainder with respect to $\alpha$. Suppose that $\tau_S$ is of the form $$\label{eq:csformoftau} \tau_S(x) = \sum_{m \in {\mathbb{Z}}} T_S(x+m),$$ where $T_S$ is the shift of a dome function as given in Definition \[def:domefnc\]. Then this would be an immediate consequence of Proposition \[prop:domefncbrf\] and Remark \[rem:shifts\] for every $\mathbf{x} \in I^2$ and every irrational $\alpha >0$ satisfying . Our proof is thus complete if we can show that $\tau_S$ is of the form for some shifted dome function $T_S$. As in the proof of Theorem \[thm:mainpoly\], we let $l(y)$ be the intersection in the plane of the set $S$ and the straight line with slope $\alpha$ through the point $(0,y)$, and we let $T_S \, : \, {\mathbb{R}}\mapsto [0, \infty)$ be the function $$T_S(y) = \frac{\|l(y)\|}{\sqrt{1+\alpha^2}} .$$ Then $\tau_S$ is given in . It is clear that $T_S$ is a continuous function supported on some interval $[B_1, B_2]$, and that an appropriate shift of $T_S$ would satisfy condition in Definition \[def:domefnc\]. We will show that also condition is satisfied for this shift of $T_S$; that is, we can find $c>0$, $m>0$ and $\varepsilon>0$ such that $$T_S(B_1+x) {\leqslant}cx^{1/m},$$ and $$T_S(B_2-x) {\leqslant}cx^{1/m},$$ whenever $0 {\leqslant}x < \varepsilon$. We verify only the latter inequality (the argument for the former is equivalent). Let $C=(C_1(s), C_2(s))$ denote the boundary of $S$ parametrized by arc length, and denote by $L$ its total length. We then have $\|C'(s)\|=1$ and $C'(s) \perp C''(s)$ for all $s \in [0,L]$. The curvature $\kappa (s)$ at the point $C(s)$ is given by $\kappa (s) = \|C''(s)\|$, and assumed positive for all $s \in [0,L]$. We let $$\label{eq:defkappa} k := \min_{s \in [0,L]} \kappa (s) .$$ The line with slope $\alpha$ through the point $(0,B_2)$ in the plane will intersect the curve $C$ at a single point $p$. We let this line be the $x$-axis in a new coordinate system $(x,y)$ where $p$ is the origin (see Figure \[fig:defH\]), and view $C$ as a curve in these coordinates with $C(0) = (C_1(0), C_2(0)) = (0,0)$. (-.5,0) – (5,0); (0,-.5) – (0,5); (2, 2.5) ellipse (1cm and 1.5cm); (-.5, [(10+3\sqrt(3))/4-sqrt(3)]{}) – (3.5, [(10+3\sqrt(3))/4+sqrt(3)]{}); (.5, [(10+3\sqrt(3))/4+2/sqrt(3)]{}) – (3.5, [(10+3\sqrt(3))/4-4/sqrt(3)]{}); (1.5, [(10+3\sqrt(3))/4]{}) circle (.05); (1.5, [(10+3\sqrt(3))/4+.07]{}) node\[above\][$p$]{}; (0, [10/4]{}) circle (.05); (0, [10/4+.1]{}) node\[left\][$B_2$]{}; (-.5, [(10+3\sqrt(3))/4-sqrt(3)]{}) node\[below left\][$x$]{}; (3.5, [(10+3\sqrt(3))/4-4/sqrt(3)]{}) node\[below right\][$y$]{}; (2.9,3.9) node[$C$]{}; We may then think of a section of $C$ around $p$ as the graph of the function $H \, : \, (-\delta, \delta) \mapsto [0,\infty)$ given by $$H(x) = C_2\left( C_1^{-1}(x)\right) .$$ We have $C_2'(0)=0$ and $C_1'(0)=1$, and since $C_1$ and $C_2$ are both twice continuously differentiable it follows that $$H'(x) = \frac{C_2'(C_1^{-1}(x))}{C_1'(C_1^{-1}(x))}$$ and $$H''(x) = \frac{C_2''(s)C_1'(s)-C_2'(s)C_1''(s)}{\left( C_1'(s) \right)^3}, \quad s=C_1^{-1}(x),$$ are both well-defined and continuous on some interval $(-\delta, \delta)$. By choosing $\delta$ sufficiently small we can ensure that $$\left\| C_1'(C_1^{-1}(x)) \right\| {\geqslant}\frac{1}{2}, \quad x \in (-\delta, \delta),$$ which (for $s=C_1^{-1}(x)$ and recalling that $C'(s) \perp C''(s)$) in turn implies $$\label{eq:boundHprime2} \|H''(x)\| = \frac{\|C''(s)\| \cdot \|C'(s)\|}{\|C_1'(s)\|^3} {\geqslant}\frac{k}{8} , \quad x \in (-\delta, \delta),$$ with $k$ given in . We now use this lower bound on $\|H''(x)\|$ to find an upper bound on $T_S(B_2-z)$ for sufficiently small $z>0$. We have $$\label{eq:TSl} T_S(B_2-z)=\frac{\|l\|}{\sqrt{1+\alpha^2}} ,$$ where $l$ is the intersection of $S$ with the line of slope $\alpha$ through the point $(0,B_2-z)$, illustrated in Figure \[fig:defell\]. (0,1.5) – (0,5); (0, [10/4]{}) – (3, [(10+3\sqrt(3))/4+3/4\sqrt(3)]{}); (.5, [(10+3\sqrt(3))/4+2/sqrt(3)]{}) – (2.5, [(10+3\sqrt(3))/4-2/sqrt(3)]{}); (2.5, [(10+3\sqrt(3))/4]{}) arc (60:180:1cm and 1.5cm); (0, [10/4-.6]{}) – (1.02, [10/4-.6+1.02\sqrt(3)/2]{}); (1.02, [10/4-.6+1.02\sqrt(3)/2]{}) – (2.33, [10/4-.6+2.33\sqrt(3)/2]{}); (2.33, [10/4-.6+2.33\sqrt(3)/2]{}) – (3, [(10+3\sqrt(3))/4-.6+3/4\sqrt(3)]{}); (1.5, [(10+3\sqrt(3))/4]{}) circle (.05); (1.5, [(10+3\sqrt(3))/4+.07]{}) node\[above\][$p$]{}; (0, [10/4]{}) circle (.05); (0, [10/4+.1]{}) node\[left\][$B_2$]{}; (0, [10/4-.6]{}) circle (.05); (0, [10/4-.6]{}) node\[left\][$B_2-z$]{}; (1.8, [(10+3\sqrt(3))/4-.3\2/sqrt(3)-.15]{}) node\[below left\][$l$]{}; (2.5, [(10+3\sqrt(3))/4-2/sqrt(3)]{}) node\[below right\][$y$]{}; The line segment $l$ is at height $y=z/\sqrt{1+\alpha^2}$ above the point $p$ (see Figure \[fig:defell\]). If $y < \min \{ H(\delta), H(-\delta) \}$, then we may denote by $x_1, x_2$ the two values of $x \in (-\delta, \delta)$ satisfying $H(x)=y$, and $$\label{eq:absl} \|l\| {\leqslant}2 \max \left\{ \|x_1\|, \|x_2\| \right\} .$$ By Taylor’s theorem we have $$y = H(x_i) =H(0) + \frac{H'(0)}{1!}x_i + \frac{H''(r_i)}{2!}x_i^2 = \frac{H''(r_i)}{2!}x_i^2 ,$$ for $i=1,2$ and some $r_i \in (-\delta, \delta)$, and from it thus follows that $$\|x_i\| = \left( \frac{2y}{H''(r_i)} \right)^{1/2} {\leqslant}\frac{4}{\sqrt{k}} \cdot y^{1/2}, \quad i=1,2 .$$ Hence, from we get $$\label{eq:boundonell} \|l\| {\leqslant}\frac{8}{\sqrt{k}} \cdot y^{1/2} {\leqslant}\frac{8}{\sqrt{k}} \cdot z^{1/2},$$ and by and we have $$T_S(B_2-z) {\leqslant}\frac{8}{\sqrt{k(1+\alpha^2)}} \cdot z^{1/2} .$$ This verifies that a shift of the function $T_S$ satisfies the growth condition in Definition \[def:domefnc\] with $c=8/\sqrt{k(1+\alpha^2)}$, $m=2$ and some $\varepsilon >0$ (for instance, $\varepsilon = \min \{ H(\delta), H(-\delta) \}$ will suffice). The function $\tau_S$ is thus of the form , where $T_S$ is the shift of a dome function, and this completes the proof of Theorem \[thm:mainconvex\]. Proof of Theorem \[thm:mainneg\] {#sec:neg} ================================ In this section we present the proof of Theorem \[thm:mainneg\]. For the proof of part we simply give an outline, as this proof largely follows the proof given above for Proposition \[prop:hatfncbrf\]. Part , on the other hand, is proven in full detail. Lastly, we present the proof of part . Fix an irrational $\alpha>0$ with continued fraction expansion $\alpha = [0; a_1, a_2, a_3, \cdots]$ satisfying $a_1=1$ and $a_{l+1}{\geqslant}q_l^7$. One can show that there are uncountably many such irrationals. Let $S$ be the triangle with vertices $(0,0)$, $(0,1)$ and $(K,1)$ for some $0<K<1$ to be determined. We will assume that $1-K\alpha >0$. Denote by $\tau_S$ the function in associated to $S$; this is a hat function as defined in , with $a=1-K\alpha$ and $b=1$. By Lemma \[lem:equivs\], the triangle $S$ is a bounded remainder set for the continuous irrational rotation with slope $\alpha$ and some arbitrary starting point $\mathbf{x} \in I^2$ if and only if $\tau_S$ is a bounded remainder function with respect to $\alpha$. In what follows we show that the latter is not* the case, and accordingly $S$ is not a bounded remainder set. For $N=\sum_{l=0}^s b_l q_l$, one can show by calculations analogous to those in the proof of Proposition \[prop:hatfncbrf\] that $$\label{eq:negpoly1} \left\| \sum_{k=0}^{N-1} \tau_S(\{k \alpha\})- \frac{NK}{2} \right\| = C \sum_{l=0}^s \xi_l (1-\xi_l) \frac{b_l}{q_l} + O(1) ,$$ where $C$ depends only on $K$ and $\alpha$, and $\xi_l =\{q_l a\} = \{q_l(1-K\alpha)\}$. For $x \in {\mathbb{R}}$, let ${\left\lVertx\right\rVert}$ denote the minimal distance from $x$ to an integer, and note that $$\xi_l (1-\xi_l) {\geqslant}\frac{1}{2} {\left\lVertq_la\right\rVert}.$$ It is a well-known fact (see e.g. [@khinchin p. 69]) that for almost all $a \in (0,1)$ one can find a positive constant $c$ such that $${\left\lVertn\cdot a\right\rVert} {\geqslant}\frac{c}{n^2}$$ for all $n{\geqslant}2$. Thus, one can indeed find $K\in (0,1)$ such that $a= 1-K\alpha >0$, and moreover $$C \sum_{l=0}^s \xi_l (1-\xi_l) \frac{b_l}{q_l} {\geqslant}C \sum_{l=0}^s {\left\lVertq_la\right\rVert} \frac{b_l}{q_l} > C \sum_{l=0}^s \frac{b_l}{q_l^3}.$$ Now let $b_l:=q_l^4$. Then the sum on the right hand side in is bounded from below by $C \sum_{l=0}^s q_l$, which tends to infinity as $s \rightarrow \infty$. For the sequence of integers $N_s = \sum_{l=0}^s q_l^5$, we thus have $$\left\| \sum_{k=0}^{N_s-1} \tau_S(\{k \alpha\})- N_s \lambda(S) \right\| \rightarrow \infty$$ as $s \rightarrow \infty$. This shows that $\tau_S$ is not a bounded remainder function with respect to $\alpha$, and completes the proof of Theorem \[thm:mainneg\] \[it:negpoly\]. Fix an irrational $\alpha \in (1/4, 1/2)$ with continued fraction expansion $\alpha=[0;a_1, a_2, \ldots ]$ satisfying $a_{l+1}>q_l^{100}$ and $p_l$ even for an infinite number of odd indices $l$, say for the sequence $l_1 < l_2 < l_3 \ldots$. One can show that there are uncountably many such irrationals. Let $S$ be the disc with diameter $d:=\alpha / \sqrt{1+\alpha^2}$ illustrated in Figure \[fig:negdisc\]. By Lemma \[lem:equivs\], the set $S$ is of bounded remainder for the continuous irrational rotation with slope $\alpha$ and arbitrary starting point $\mathbf{x} \in I^2$ if and only if the associated function $\tau_S$ in is a bounded remainder function with respect to $\alpha$. In what follows, we will show that there exists an $x \in I$ and a sequence of integers $N_1 < N_2 < N_3 \ldots$ such that $$\left\| \sum_{k=0}^{N_i-1} \tau_S \left( \{ k\alpha +x \} \right) - N_i \lambda (S) \right\| \rightarrow \infty$$ as $i \rightarrow \infty$. By Remark \[rem:shifts\], this proves $\tau_S$ is not a bounded remainder function, and accordingly $S$ is not a bounded remainder set. (-.15,0) – (1.15, 0); (0,-.1) – (0, 1.1); (0,1) – (1,1) – (1,0); (0,1) node\[left\][$1$]{}; (1,0) node\[below\][$1$]{}; ([2/3]{}, [1/(3\sqrt(2))+1/(4\sqrt(2))]{}) circle\[radius=[sqrt(3)/(4\sqrt(7)]{}\]; ([2/3]{}, [1/(3\sqrt(2))+1/(4\sqrt(2))]{}) node[$S$]{}; (0,0) – (1,[1/(2\sqrt(2))]{}); (0,[1/(2\sqrt(2))]{}) – (1, [2/(2\sqrt(2))]{}); (0,[1/(2\sqrt(2))]{}) – (1,[1/(2\sqrt(2))]{}); (-.1, 0) – (-.1, [1/(2\sqrt(2))]{}); (-.1, [1/(4\sqrt(2))]{}) node\[left\][$\alpha$]{}; (1.1,0) – (1.1, [1/(2\sqrt(2))]{}); (1.1,[1/(4\sqrt(2))]{}) node\[right\][$\alpha$]{}; The function $\tau_S$ associated to $S$ is given explicitly by $$\tau_S (y) = \begin{cases} \frac{\alpha}{1+\alpha^2} \sqrt{1-(1-2y/\alpha)^2} \text{ ,} &0{\leqslant}y{\leqslant}\alpha \text{ ;}\\ 0 \text{ ,} &\alpha<y{\leqslant}1 \text{ ,} \end{cases}$$ and we note that $$\label{eq:negdiscmes} \lambda(S) = \int_0^1 \tau_S (y) \, dy = \frac{\pi}{4} \cdot \frac{\alpha^2}{1+\alpha^2} .$$ We introduce the notation $S_N(x)$ for the sum $$S_N(x) := \sum_{k=0}^{N-1} \tau_S \left( \{ k\alpha +x\}\right) .$$ Let us now fix some $i$ (and thereby an odd index $l_i$), put $$p:=p_{l_i}=2m \quad (m \in {\mathbb{N}}), \quad q:= q_{l_i},$$ and evaluate the sum $S_N(x)$ for $N:=q^{11}$ and some $x \in [0,1/q]$. We then have $$\label{eq:SNdouble} S_N(x) = \sum_{j=0}^{q^{10}-1} \sum_{k=0}^{q-1} \tau_S \left( \{ (jq+k)\alpha + x \}\right) .$$ Recall from that $$\left\| \alpha - \frac{p}{q} \right\| {\leqslant}\frac{1}{q^2 a_{l_i+1}} {\leqslant}\frac{1}{q^{102}}.$$ Using this fact, we get $${\left\lVertjq\alpha\right\rVert} < j {\left\lVertq\alpha\right\rVert} {\leqslant}\frac{j}{q^{101}} < \frac{1}{q^{91}},$$ where ${\left\lVertx\right\rVert}$ denotes the minimal distance from $x \in {\mathbb{R}}$ to an integer. It follows that $$\begin{aligned} {\left\lVert\left\{ (jq+k)\alpha + x \right\} - \left\{ k \cdot \frac{p}{q} + x\right\}\right\rVert} &{\leqslant}{\left\lVertjq\alpha\right\rVert}+k{\left\lVert\alpha-\frac{p}{q}\right\rVert} \\ &< \frac{1}{q^{91}}+\frac{q}{q^{102}} < \frac{1}{q^{90}} , \end{aligned}$$ and hence $$\left\| \tau_S \left( \{(jq+k)\alpha+x\} \right) - \tau_S \left( \left\{ k\cdot \frac{p}{q}+x \right\} \right) \right\| {\leqslant}\left\| \tau_S \left( \frac{1}{q^{90}} \right) \right\| < \frac{1}{q^{44}}.$$ Combining this bound with , we get $$\label{eq:SNbound1} \left\| S_N (x) - q^{10}\sum_{k=0}^{q-1} \tau_S \left( \left\{ k \cdot \frac{p}{q}+x\right\} \right)\right\| < q^{11} \cdot \frac{1}{q^{44}} = \frac{1}{q^{33}} .$$ In light of , we introduce the function $$\sigma (y) := \begin{cases} \frac{\alpha}{1+\alpha^2} \sqrt{1-(1-2qy/p)^2} \text{ ,} &0{\leqslant}y{\leqslant}p/q \text{ ;}\\ 0 \text{ ,} &p/q<y{\leqslant}1 \text{ .} \end{cases}$$ Since the index $l_i$ is odd, we have $\alpha < p/q$ and $\sigma (y)=\tau_S (\alpha qy/p)$ for all $y \in [0,1)$. From $$\left\| \frac{\alpha q}{p} -1 \right\| = \frac{q}{p} \left\| \alpha - \frac{p}{q} \right\| < \frac{1}{\alpha} \cdot \frac{1}{q^{102}} < \frac{1}{q^{101}}$$ it thus follows that $$\left\| \sigma (y) - \tau_S(y) \right\|= \left\| \tau_S \left( \frac{\alpha p}{q} y \right) - \tau_S(y) \right\| < \left\| \tau_S \left( \frac{1}{q^{101}} \right) \right\| < \frac{1}{q^{50}} .$$ Combining this bound with , we get $$\label{eq:SNbound2} \left\| S_N (x) - q^{10}\sum_{k=0}^{q-1} \sigma \left( \left\{ k \cdot \frac{p}{q}+x\right\} \right)\right\| < \frac{1}{q^{33}}+ \frac{q^{11}}{q^{50}} < \frac{1}{q^{32}} .$$ Note that some of the above estimates hold only for $q$ greater than some lower threshold $q>q_0$. Let us now have a closer look at the sum over $\sigma$ in . We have $$\label{eq:sigmabound} \begin{aligned} \sum_{k=0}^{q-1} \sigma \left( \left\{ k \cdot \frac{p}{q}+x\right\} \right) &= \sum_{k=0}^{p-1} \sigma \left( \frac{k}{q} +x\right) \\ &=\frac{\alpha}{1+\alpha^2} \sum_{k=0}^{p-1} \sqrt{1- \left( 1-\frac{2k}{p}-\frac{2q}{p} x \right)^2} \\ &=\frac{\alpha}{1+\alpha^2} \sum_{k=0}^{2m-1} \sqrt{1- \left( 1-\frac{k}{m}-\frac{q}{m} x \right)^2} \\ &=\frac{\alpha}{1+\alpha^2} \cdot 2mG_m\left( \frac{q}{m} x \right), \end{aligned}$$ where $$G_m(x) := \frac{1}{2m} \sum_{k=0}^{2m-1} \sqrt{1-\left( 1-\frac{k}{m} - x\right)^2} \text{ ,} \quad x \in \left[0, \frac{1}{m}\right) .$$ (-.15,0) – (2.3, 0); (2.3,0) node\[below\][$x$]{}; (0,-.1) – (0, 1.4); (0,1.2) node\[left\][$G_m(x)$]{}; (1,-.03) – (1, .03); (1,-.03) node\[below\][$1/2m$]{}; (2,-.03) – (2, .03); (2,-.03) node\[below\][$1/m$]{}; (2,0.3) arc\[radius=1, start angle=0, end angle=180\]; The function $G_m$ is illustrated in Figure \[fig:Gm\]. It is clear that $G_m(x) = G_m(1/m-x)$, and by elementary analysis one can show that $G_m$ increases on $[0,1/(2m))$ in such a way that $$G_m \left( \frac{1}{3m} \right) > G_m \left( \frac{1}{6m} \right) + \frac{2c}{m^{3/2}}$$ for some $c>0$. From this inequality one can deduce that there exists a subinterval $\Lambda \subset [0,1/(2m)]$ of length at least $1/(6m)$ such that either $$\label{eq:Gcond1} G_m(x) > \frac{1}{2} \int_0^2 \sqrt{1-(1-y)^2} \, dy + \frac{c}{m^{3/2}} = \frac{\pi}{4} + \frac{c}{m^{3/2}}$$ or $$\label{eq:Gcond2} G_m(x) < \frac{1}{2} \int_0^2 \sqrt{1-(1-y)^2} \, dy - \frac{c}{m^{3/2}} = \frac{\pi}{4} - \frac{c}{m^{3/2}}$$ for all $x \in \Lambda$. We assume in what follows that holds for all $x \in \Lambda$ (the case when holds is treated similarly). Then for $\tilde{x} \in \tilde{\Lambda}$, where $$\tilde{\Lambda}:= (m/q)\Lambda \subset [0,1/(2q)) ,$$ we have $q\tilde{x}/m \in \Lambda$, and from and it follows that $$\label{eq:sigmabound2} \sum_{k=0}^{q-1} \sigma \left( \left\{ k \cdot \frac{p}{q}+\tilde{x} \right\} \right) > \frac{\alpha}{1+\alpha^2} \cdot 2m \left( \frac{\pi}{4}+\frac{c}{m^{3/2}} \right).$$ In the following we let $c_1, c_2, \ldots$ denote positive absolute constants. From and we get $$\begin{aligned} S_N(\tilde{x}) &> q^{10} \sum_{k=0}^{q-1} \sigma \left( \left\{ k \cdot \frac{p}{q}+\tilde{x} \right\} \right) - \frac{1}{q^{32}} \\ &> q^{10} \cdot 2m \cdot \frac{\alpha}{1+\alpha^2} \left( \frac{\pi}{4}+\frac{c}{m^{3/2}}\right) - \frac{1}{q^{32}} \\ &> N \cdot \frac{p}{q}\cdot \frac{\pi\alpha}{4(1+\alpha^2)} + c_1q^9 \\ &> N \cdot \frac{\pi \alpha^2}{4(1+\alpha^2)} + c_1q^9 = N \lambda(S) + c_1 q^9, \end{aligned}$$ where we recall from that $\lambda (S)$ is the integral over $\tau_S$ and the measure of the disc $S$ in Figure \[fig:negdisc\]. Thus, we have shown that $$\label{eq:SNtilde} S_N(\tilde{x}) - N \lambda (S) > c_1q^9 \text{ ,} \quad \tilde{x} \in \tilde{\Lambda} .$$ Finally, we define the set $\bar{\Lambda} \subset I$ by $$\tilde{\Lambda}^{(j)} := \tilde{\Lambda} + \frac{j}{q} \text{ ,} \quad \bar{\Lambda}:= \bigcup_{j=0}^{q-1} \tilde{\Lambda}^{(j)} .$$ Since $\lambda(\tilde{\Lambda}) {\geqslant}1/(6q)$, we have $\lambda (\bar{\Lambda}) {\geqslant}1/6$. Choose some $x \in \bar{\Lambda}$, and find $j \in \{0,1, \ldots, q-1\}$ such that $$x=\tilde{x}+\frac{j}{q} \text{ ,} \quad \tilde{x} \in \tilde{\Lambda} .$$ Furthermore, choose $k_j \in \{0,1, \ldots , q-1\}$ such that $k_jp \equiv q-j ({\operatorname{mod}}q)$, and note that $${\left\lVertk_j\alpha+\frac{j}{q}\right\rVert} = {\left\lVertk_j\alpha - \frac{k_jp}{q}\right\rVert} {\leqslant}k_j {\left\lVert\alpha - \frac{p}{q}\right\rVert} < \frac{1}{q^{101}} .$$ From this and the fact that $\|\tau_S\|{\leqslant}1$, it follows that $$\begin{aligned} S_N(x) &> \sum_{k=0}^{k_j} \tau_S \left( \{ k\alpha +x\} \right) + \sum_{k=0}^{N-1} \tau_S \left(\{k\alpha + k_j \alpha +x\} \right) -q \\ &> \sum_{k=0}^{N-1} \tau_S \left( \left\{ k\alpha + x -\frac{j}{q}\right\} \right) - c_2q \\ &= \sum_{k=0}^{N-1} \tau_S \left( \{k\alpha + \tilde{x} \}\right) - c_2q = S_N(\tilde{x})-c_2q , \end{aligned}$$ and from we thus get $$\label{eq:SNfinal} S_N(x) - N \lambda (S) > c_3 q^9$$ for all $x \in \bar{\Lambda}$. The above analysis can be carried out for each $l_i$ (given that $q_{l_i}$ is above the threshold $q_{l_i}>q_0$). That is, for each $i$, we find $\bar{\Lambda}_i \subset I$ of measure $\lambda(\bar{\Lambda}_i){\geqslant}1/6$ such that holds for all $x \in \bar{\Lambda}_i$ with $q=q_{l_i}$ and $N=q^{11}$. Now fix $x \in I$ such that $x \in \bar{\Lambda}_i$ for infinitely many $i$, and for each such $i$ let $q_i = q_{l_i}$ and $N_i = q_i^{11}$. Then for these $N_i$, we have $$\left\| S_{N_i}(x)-N_i \lambda (S) \right\| = \left\| \sum_{k=0}^{N_i-1} \tau_S \left( \{k \alpha+x\}\right)-N_i \lambda (S)\right\| \rightarrow \infty$$ as $i \rightarrow \infty$. This verifies that $\tau_S$ is not a bounded remainder function with respect to $\alpha$, and completes the proof of Theorem \[thm:mainneg\] \[it:negconvex\]. Let $S$ be the triangle with vertices $(0,0)$, $(0,1)$ and $(1,0)$. Fix some slope $\alpha>0$ and starting point $\mathbf{x} \in I^2$. For simplicity we assume that $\alpha<1$ (the proof is similar when $\alpha{\geqslant}1$). By Lemma \[lem:equivs\], the set $S$ is of bounded remainder for the continuous irrational rotation with slope $\alpha$ and starting point $\mathbf{x}$ if and only if the associated function $\tau_S$ in is of bounded remainder with respect to $\alpha$. For the specific triangle $S$, we have $$\label{eq:specsettau} \tau_S(x) = \begin{cases} \frac{1-x}{1+\alpha} \text{ ,} & 0 {\leqslant}x {\leqslant}1-\alpha \text{ ;}\\ \frac{1-x}{1+\alpha} + \frac{2-x}{1+\alpha}-\frac{1-x}{\alpha} \text{ ,} & 1-\alpha <x{\leqslant}1 \text{ .} \end{cases}$$ It is a well-known fact that a $1$-periodic function $f$ which is integrable over the unit interval $I$ is a bounded remainder function with respect to $\alpha$ if and only if there exists a bounded and measurable $1$-periodic function $g$ satisfying the equation $$f(x) - \int_0^1 f(t) \, dt = g(x) - g(x+\alpha)$$ for almost every $x$. This is known as the cohomological equation for $f$. By a classical result of Gottschalk and Hedlund [@gottschalk Theorem 14.11], the function $g$ can be chosen to be continuous whenever $f$ is continuous. Thus, our proof is complete if we can find a continuous $1$-periodic function $g$ such that $$\label{eq:specsetg} \tau_S(x) - \int_0^1 \tau_S(t) \, dt = g(x) - g(x+\alpha),$$ where $\tau_S$ is given in . Let $g$ be the continuous $1$-periodic function defined on $I$ by $$g(x) = \frac{x(x-1)}{2\alpha(1+\alpha)} .$$ It is straightforward to check that this function satisfies . This confirms that $\tau_S$ is a bounded remainder function with respect to $\alpha$, and completes the proof of Theorem \[thm:mainneg\] \[it:specset\]. [10]{} J. Beck, Super-uniformity of the Typical Billiard Path, Bolyai Society Mathematical Studies 21, An Irregular Mind ([Szemerédi]{} is 70), 39–129, Springer, 2010. J. Beck, Strong uniformity, Radon Series on Computational and Applied Mathematics 15, Uniform Distribution and Quasi-Monte Carlo Methods, 17–44, De Gruyter, Berlin, 2014. J. Beck, From Khinchin’s conjecture on strong uniformity to superuniform motions., Mathematika 61 (2015), 591–707. M. Drmota, Irregularities of continuous distributions, Ann. Inst. Fourier 39 (1989), 501–527. M. Drmota and R. Tichy, Sequences, Discrepancies and Applications, Springer Lecture Notes in Mathematics 1651, Springer, 1997. W. H. Gottschalk and G. A. Hedlund, Topological dynamics, (Colloquium Publications of the American Mathematical Society (AMS). Vol. 36) Providence, R.I.: American Mathematical Society (AMS). VIII, 1955. S. Grepstad and N. Lev, Sets of bounded discrepancy for multi-dimensional irrational rotation, Geom. Funct. Anal. 25 (2015), 87–133. E. [Hecke]{}, [Über analytische Funktionen und die Verteilung von Zahlen mod. eins]{}, [Abh. Math. Semin. Univ. Hamb.]{} 1 (1921), 54–76. (German). P. Hellekalek and G. Larcher, On functions with bounded remainder, Ann. Inst. Fourier 39 (1989), 17–26. H. [Kesten]{}, [On a conjecture of Erdős and Szüsz related to uniform distribution mod 1]{}, [Acta Arith.]{} 12 (1966), 193–212. A. Khinchin, Continued fractions, The University of Chicago Press, 1964. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974. A. Ostrowski, [Mathematische Miszellen IX: Notiz zur Theorie der Diophantischen Approximationen]{}, Jahresber. Dtsch. Math.-Ver. 36 (1927), 178–180 (German). , [Mathematische Miszellen. XVI: Zur Theorie der linearen diophantischen Approximationen]{}, Jahresber. Dtsch. Math.-Ver. 39 (1930), 34–46 (German). J. Schoissengeier, [Regularity of distribution of $(n\alpha)$-sequences]{}, Acta Arith. 133 (2008), no. 2, 127–157.
--- author: - '[Mauro Garavello]{} [^1]' date: \| SISSA-ISAS\ Via Beirut, 2-4\ 34014 Trieste, Italy\ April 2002 title: 'Verification Theorems for Hamilton-Jacobi-Bellman equations' --- Introduction. ============= int.tex [Acknowledgments]{} The author wishes to thank Prof. B. Piccoli, for having proposed him the study of this problem and for his useful advice, and the referees, for their improving suggestions. Preliminaries. {#se2} ============== We consider a control system: $$\label{cs} \dot x(t)=f(t,x(t),u(t)),\quad (t,x)\in\Omega,\quad u(t)\in U$$ where 1. $\Omega$ is an open and connected subset of ${\mathbb R}\times{\mathbb R}^n$.\[a1\] 2. $U$ is a non-empty subset of ${\mathbb R}^q$, for some $q{\geqslant}1$, $q\in{\mathbb N}$.\[a2\] 3. ${\mathcal U}=L^p({\mathbb R};U)$ with $1{\leqslant}p<+\infty$ is the set of admissible controls.\[a3\] 4. \[a4\] $f:\Omega\times U\to{\mathbb R}^n$ is measurable in $t$, continuous in $(x,u)$, differentiable in $x$ and, for each $u\in U$, $D_xf(\cdot,\cdot,u)$ is bounded on compact sets. Moreover there exists ${\varphi}_1:{\mathbb R}\to{\mathbb R}^+$ integrable and for every $K$, compact subset of $\Omega$, there exist a modulus of continuity $\omega_K$ and a constant $L_K>0$ such that, if $(t,x)\in K$ and $(t,y)\in K$, then for all $u$ $$\label{fc} \begin{cases} {\left\vert f(t,x,u)-f(t,y,u) \right\vert}{\leqslant}\omega_K({\left\vert x-y \right\vert})\\ \left(f(t,x,u)-f(t,y,u)\right)\cdot (x-y){\leqslant}L_K{\left\vert x-y \right\vert}^2\\ {\left\vert f(t,x,u) \right\vert}{\leqslant}L_K({\varphi}_1(t)+{\left\vert u \right\vert}^p). \end{cases}$$ We consider a function $L:\Omega\times U\to{\mathbb R}$ and assume: 1. $L$ is measurable in $t$ and continuous in $(x,u)$. Moreover, there exist ${\varphi}_2 :{\mathbb R}\to{\mathbb R}^+$ integrable and, for every $R{\geqslant}0$, $C_R{\geqslant}0$ such that $$\label{lc} {\left\vert L(t,x,u) \right\vert}{\leqslant}C_R({\varphi}_2(t)+{\left\vert u \right\vert}^p),\quad {\left\vert (t,x) \right\vert}{\leqslant}R.$$ In this paper we indicate with $x(\,\cdot\,;u,t_0,x_0)$ the solution to (\[cs\]) such that $x(t_0;u,t_0,x_0)=x_0$. Define the value function: $$\label{vf} V(t_0,x_0):=\!\!\!\inf_{\substack{u\in{\mathcal U}\vspace{3pt}\\ (T,x(T;u,t_0,x_0))\in S} }\!\!\!\left\{\int_{t_0}^T\!\!\!\!\!\!L(s,x(s;u,t_0,x_0),u(s))ds\!+\!\psi (T,x(T;u,t_0,x_0))\right\}$$ where $S$ - the target - is a closed subset of ${\mathbb R}\times{\mathbb R}^n$ contained in $\Omega$, $\psi :S\to{\mathbb R}$ is the final cost. We recall the following definition:\ A subset $A$ of $\Omega$ is a countably $\mathcal H^n$-rectifiable set if there exist $A_1$ and $A_2$ such that $A=A_1\cup A_2$, $A_1$ is a finite or countable union of connected $\mathcal C^1$ submanifolds of positive codimension, and ${\mathcal H}^n(A_2)=0$, where ${\mathcal H}^k$ is the $k$-dimensional Hausdorff measure. Examples of syntheses. {#se6} ====================== In next sections we give sufficient conditions for a candidate value function $W$ to coincide with $V$. Beside some regularity conditions, we ask a HJB inequality outside a countably $\mathcal H^n$-rectifiable set. This regularity is shared by every function $W$ obtained from a regular synthesis, thus it can be used to prove the optimality of the synthesis itself. In this section we give various examples to which Theorem \[teo1\] is applicable. First of all, we need some definitions. A synthesis $\Gamma$ is a collection $\{(x_{(\bar t,\bar y)}(\cdot), u_{(\bar t,\bar y)})\}_{(\bar t, \bar y)\in \Omega}$ such that $x_{(\bar t,\bar y)}(\cdot)=x(\cdot;u_{(\bar t,\bar y)},\bar t,\bar y):[\bar t,\tau(\bar t,\bar y)]\to{\mathbb R}^n$, $u_{(\bar t,\bar y)}\in{\mathcal U}$ for every $(\bar t,\bar y)\in\Omega$, $x_{(\bar t,\bar y)}(\tau(\bar t,\bar y))\in S$ and for every $t\in[\bar t,\tau(\bar t,\bar y)]$ $$u_{(t,x_{(\bar t,\bar y)}(t))}(s)=u_{(\bar t,\bar y)}(s+t) \quad\textrm{a.e.}$$ and $$x_{(t,x_{(\bar t,\bar y)}(t))}(\cdot) =x_{(\bar t,\bar y)}(\cdot+t)$$ A synthesis $\Gamma$ is optimal if every $u_{(\bar t,\bar y)}$ is an optimal control. There is a standard method in geometric control theory to construct an optimal synthesis, see [@BP-98]. This consists of four steps: 1) using Pontryagin Maximum Principle and other geometric tools to study the properties of optimal trajectories, 2) derive a sufficient family of extremal trajectories (i.e. trajectories satisfying PMP), 3) construct a synthesis formed by extremal trajectories and 4) prove its optimality. In many cases, for autonomous systems, it happens that the extremal synthesis is associated to a feedback $u:{\mathbb R}^n\to U$ that is smooth on each stratum of a stratification, see [@ps] for details. Roughly speaking a stratification is a locally finite collection of disjoint regular submanifolds, of various dimensions, that is a partition and such that the boundary of each manifold is union of manifolds of higher codimensions. In this case the synthesis is called regular in the sense of Boltyanskii-Brunovský, see [@Bolt; @Brun; @ps]. Step 4) of the geometric control approach can thus be obtained in essentially two ways: either using the regularity of the synthesis, see [@ps], or proving that the candidate value function $W$ associated to the synthesis coincides with $V$. The latter is exploited in [@F-R] for a continuous $W$, defined on a subset of ${\mathbb R}^n$, that is differentiable and satisfies the HJB equation outside a locally finite union of smooth submanifolds of positive codimension. Then the optimality is granted for initial points for which all admissible trajectories remains in the domain of $W$. A mild generalization is obtained in [@Bressan], where trajectories can exit the domain of $W$, but the boundary of the domain of $W$ is a level set of $W$ itself. Another approach is the one of nonsmooth analysis, using which various verification theorems can be proved, see for example [@Vinter]. Our main results, see Theorems \[teo1\] and \[teo2\], generalize previous results in the following way: 1. As in [@Bressan] we assume that $W$ can be defined on a subset and the boundary of its domain is a level curve of $W$. 2. We ask $W$ to be differentiable and satisfy HJB only outside a countably ${\mathcal H}^n$-rectifiable set. 3. $W$ is only lower semicontinuous (satisfying other weak continuity assumptions). A direct comparison with results of nonsmooth analysis is difficult. However, we point out that the value function fails in general to be locally Lipschitz continuous, see Example 3.1, for regular synthesis. In case of locally Lipschitz regularity, our result is consequence of those obtained by nonsmooth analysis methods, see for example [@Clarke; @Vinter]. We give now some examples to illustrate the applicability of our results. A whole class of examples can be find in [@BressanPiccoli; @bpic]. The first example shows a typical regular synthesis with a non locally Lipschitz continuous value function. In the second, the value function is not continuous and it is differentiable only outside a countably ${\mathcal H}^n$-rectifiable set. Last example shows the well known Fuller phenomenon. In this case optimal trajectories have an infinite number of switchings and the methods of Boltyanskii-Brunovský do not work (while it does the result of [@ps]). Let $x\in{\mathbb R}$ and $u\in[-1,1]$. Consider the control system $$\ddot x+x=u$$ and the problem of reaching the origin in minimum time. If we define $x_1=x$ and $x_2=\dot x$ we obtain the following first-order system: $$\label{sin1} \left\{ \begin{array}{l} \dot x_1=x_2\\ \dot x_2=-x_1+u. \end{array} \right.$$ Every optimal trajectory is a bang-bang trajectory, i.e. formed by arcs corresponding to control $+1$ or $-1$. The synthesis is illustrated in Figure \[fig2\]. There are some “switching curves”: - all semi-circles of radius $1$ contained in $\{(x_1,x_2):x_2{\leqslant}0\}$ and centered at $(2n+1,0)$, with $n\in{\mathbb N}\setminus\{0\}$; - all semi-circles of radius $1$ contained in $\{(x_1,x_2):x_2{\geqslant}0\}$ and centered at $(-2n-1,0)$, with $n\in{\mathbb N}\setminus\{0\}$. Optimal trajectories switch along these curves, i.e. change control from $+1$ to $-1$ or viceversa. Let $\gamma^{\pm}$ be the trajectory that switches at points $(\pm 2,0)$ (defined say on $[-\infty,0]$). Then the value function is not locally Lipschitz continuous at any point of supp $(\gamma^\pm)$, but however it satisfies all the hypotheses of Theorem \[teo1\]. ![Synthesis of system (\[sin1\]).[]{data-label="fig2"}](sintesi1.eps){height="6cm"} Let $\Omega={\mathbb R}^2$, $f\equiv0$, $L\equiv1$. Consider the target: $$S=\left\{(t,x):x\ne0, t=\sin(1/x)\right\}\cup\left\{x=0,-1{\leqslant}t{\leqslant}1\right\}\cup\left\{t{\geqslant}1\right\}$$ and the final cost $\psi$ constantly equal to $0$. The value function for this problem is given by: $$V(t,x)=\left\{ \begin{array}{lrl} \sin(1/x)-t & \textrm{if} & x\ne0,t{\leqslant}\sin(1/x)\\ 1-t & \textrm{if} & x\ne0,\sin(1/x)<t<1\\ 0 & \textrm{if} & t{\geqslant}1\\ -1-t & \textrm{if} &x=0,t{\leqslant}-1\\ 0 & \textrm{if} &x=0,-1<t<1. \end{array} \right.$$ This function satisfies all the hypotheses of Theorem \[teo1\] and clearly it is not continuous. Moreover it is differentiable outside a countably $\mathcal H^n$-rectifiable set $A$, which is not a locally finite union of regular manifolds. (Fuller phenomenon). Let us consider the system $$\left\{ \begin{array}{l} \dot x_1=x_2\\ \dot x_2=u \end{array} \right.$$ with ![Synthesis of Fuller phenomenon.[]{data-label="Ful.Ph."}](fuller.eps){height="6cm"} ${\left\vert u \right\vert}{\leqslant}1$, $\Omega={\mathbb R}\times{\mathbb R}^2$, $S={\mathbb R}\times\{0\}$, $\psi\equiv0$, $L(t,x_1,x_2,u)=x_1^2$. This problem is well-known in the literature, see for example [@Z-B]. Every optimal trajectory is composed by an infinite number of bang-bang arcs, while the time for reaching the origin of ${\mathbb R}^2$ is finite. There are two switching curves $\zeta^+$ and $\zeta^-$ which separate ${\mathbb R}^2$ into two regions $Z^+$ and $Z^-$ where the optimal trajectory uses respectively the control $u=+1$ and $u=-1$, see Figure \[Ful.Ph.\]. The value function of this problem satisfies all the hypotheses of Theorem \[teo1\]. Some useful results. {#se3} ==================== We start by recalling without proofs some classical results about ODEs. (Local existence and uniqueness of the trajectory). Assume (A-1)-(A-4). Fixed $u\in{\mathcal U}$ and $(t_0,x_0)\in\Omega$, there exist $\delta>0$ and a unique absolutely continuous function $x(\cdot;u,t_0,x_0):[t_0,t_0+\delta]\to{\mathbb R}^n$ solution to (\[cs\]). (Continuous dependence by data). Assume (A-1)-(A-4). Let $(t_0,x_0)\in\Omega$, $(t_0,x_n)\in\Omega$ for every $n\in{\mathbb N}$ and $u\in{\mathcal U}$, $u_n\in{\mathcal U}$ for every $n\in{\mathbb N}$. Let us suppose that there exists a time $T>t_0$ such that $x(\cdot;u,t_0,x_0)$ and $x(\cdot;u_n,t_0,x_n)$ are defined in $[t_0,T]$. If $x_n\to x_0$ and $u_n\to u$ in the strong topology of $L^p([t_0,T];U)$ as $n\to+\infty$, then $x(\cdot;u_n,t_0,x_n)\to x(\cdot;u,t_0,x_0)$ uniformly in $[t_0,T]$ as $n\to+\infty$. Now, we present two technical lemmas used to prove the theorems of the next sections. \[fin\] Fix an element $\omega\in U$, $t'<t''$ and $x\in{\mathbb R}^n$ with $(t'',x)\in\Omega$. Assume that there exists ${\mathcal W}$, an open neighborhood of $x$ in ${\mathbb R}^n$, such that $\zeta^y(\cdot)$, the solution to $\dot\zeta^y(t)=f(t,\zeta^y(t),\omega )$ with $\zeta^y(t'')=y$, is defined on $[t', t'']$ for any $y\in{\mathcal W}$ and $(t,\zeta^y(t))\in\Omega\,\,\,\,\forall t\in[t',t'']$. Let $A$ be a countable $\mathcal H^n$-rectifiable set.\ Then for a.e. $y\in{\mathcal W}$ the set $B^y:=\left\{t\in [t',t'']:(t,\zeta^y(t))\in A\right\}$ is finite or countable. This lemma is a slight generalization of a result proved in Theorem 2.14 of [@ps], since here we consider the trajectory coupled with time. We can write $A=A_1\cup A_2$, where $A_1=\cup_jM_j$ and $\{M_j\}_{j\in J}$ is a finite or countable family of connected submanifolds of ${\mathbb R}^{n+1}$ of codimension $d_j>0$, and ${\mathcal H}^n (A_2)=0$. After replacing each $M_j$ by a finite or countable family of open submanifolds of $M_j$, we may assume that the $M_j$ are embedded. Define $\widetilde {\mathcal W}:=]t',t''[\times{\mathcal W}$ and let $\Phi$ be the map $\widetilde {\mathcal W}\ni (t,y)\mapsto (t,\zeta^y(t))\in \Omega$. The Jacobian of $\Phi$ is $$\label{jphi} {\bf J}\Phi\, =\,\left(\! \begin{array}{l\|c} 1&0\,\,\cdots\cdots\,\, 0\\ \hline {\bf b} & \begin{matrix} \, & & \,\\[.1pt] & {\bf V} ^\zeta (t;t',{\bf Id})&\\ & & \end{matrix} \end{array}\right)\!$$ where ${\bf b}$ is the column vector $f(t,\zeta^y(t),\omega)$ and ${\bf V}^\zeta (t;t', {\bf Id})$ is the fundamental matrix solution to the linear system $$\label{lin} \dot v(t)=-D_xf(t,\zeta^y (-t+t'+t''),w)\cdot v(t)$$ such that ${\bf V}^\zeta (t';t',{\bf Id})={\bf Id}$. So the determinant of ${\bf J}\Phi$ is equal to the determinant of ${\bf V}^\zeta (t;t',{\bf Id})$, which is equal to $exp\int_{t'}^ttr(-D_xf(s,\zeta^y(-s+t'+t''),\omega ))ds$, by Liouville’s theorem (see [@hart]). In particular $det ({\bf J}\Phi )$ is strictly positive for any $t\in [t',t'']$. Moreover, by (A-\[a4\]) $tr(-D_xf)$ is bounded on compact sets and then there exist $c>0$, $C>0$ such that $0<c{\leqslant}det ({\bf J}\Phi){\leqslant}C$.\ So $\Phi$ is a Lipschitz diffeomorphism. In particular we have ${\mathcal H}^n(\Phi^{-1}(A_2))=0$. Now for each $j$ consider $\widetilde M_j :=\Phi^{-1}(M_j)$. It is an embedded submanifold of codimension $d_j>0$. Let $\Pi :\widetilde{\mathcal W}\to{\mathcal W}$ be the canonical projection. Consider the set $S_j$ consisting of the points $s\in\widetilde M_j$ such that $\Pi{{\setbox0 =\hbox{$\mathsurround=0pt\vert$}\hbox {\lower\dp0 \copy0 }}}_{\widetilde M_j}$ is not regular. Thus, by Sard’s theorem, $\mathcal L^n(\Pi (S_j))=0$. Moreover ${\mathcal H}^n(\Pi(\Phi^{-1}(A_2)))=0$. So the set $\mathcal B:=\Pi(\Phi^{-1}(A_2))\cup (\bigcup_j\Pi (S_j))$ has Lebesgue measure $0$ in ${\mathbb R}^n$.\ Let $y\in{\mathcal W}\setminus\mathcal B$. Then $(t,\zeta^y(t))\not\in A_2$ if $t'<t<t''$. To obtain the thesis, it is sufficient to show that, for each $j$, the set $E_j=\left\{ t\in ]t',t''[: (t,\zeta^y(t))\in M_j\right\}$ is at most countable. Fix $j$ and suppose $t\in E_j$. $\widetilde M_j$ has codimension $d_j>0$, so the dimension $\nu_j$ of $\widetilde M_j$ is less or equal to $n$. Since $y\not\in\mathcal B$, the map $d\Pi (t,y):T_{(t,y)}\widetilde M_j\to{\mathbb R}^n$ is onto, thus $\nu_j =n$ and $d\Pi (t,y)$ is injective. Obviously $d\Pi (t,y)(\frac{\partial}{\partial t})=0$, so $\frac{\partial}{\partial t}\not\in T_{(t,y)}\widetilde M_j$ and, consequently, $(\tilde t,y)\not\in\widetilde M_j$ if $0<{\left\vert \tilde t-t \right\vert}{\leqslant}{\varepsilon}$ for ${\varepsilon}>0$ sufficiently small. Therefore $t$ is an isolated point of $E_j$ and so the lemma is proved. \[lerv\] Let $g$ be a real-valued function on a compact interval $[a,b]$. Assume that there exists a finite or countable subset $E$ of $[a,b]$ with the following properties: 1. $\liminf_{h\downarrow 0}\frac{g(x+h)-g(x)}{h}{\geqslant}0$ for all $x\in[a,b[\setminus E$,\[gc1\] 2. $\liminf_{h\downarrow 0}g(x+h){\geqslant}g(x)$ for all $x\in[a,b[$,\[gc2\] 3. $\liminf_{h\downarrow 0}g(x-h){\leqslant}g(x)$ for all $x\in]a,b]$.\[gc3\] Then $g(b){\geqslant}g(a)$. For a proof of this lemma see [@ps Lemma B.1]. Problem with finite time. {#se4} ========================= We indicate with $\partial Q$ the topological boundary of an arbitrary $Q{\subseteq}{\mathbb R}\times{\mathbb R}^n$. Before stating the theorem we need the following definition Suppose that we have a time-varying Lipschitz-continuous vector field $X$ on ${\mathbb R}^n$ and $W:\Omega\to{\mathbb R}\cup\{\pm\infty\}$. We say that $W$ has the no downward jumps property (NDJ) along $X$ if for any $[a,b]\ni t\mapsto\gamma(t)$, solution to $\dot\gamma(t)=X(t,\gamma(t))$ such that $(t,\gamma(t))\in\Omega$ $\forall t\in [a,b]$, we have $\liminf_{h\downarrow 0} W(t-h,\gamma(t-h)){\leqslant}W(t,\gamma(t))$, whenever $t\in ]a,b]$. \[teo1\] Suppose (A-\[a1\])-(A-5) hold. Let $Q{\subseteq}\Omega$ be an open subset containing $S$. Let $W:\overline Q\to{\mathbb R}$ be a lower semicontinuous function such that: 1. \[NU1\] $W$ has the NDJ property along every time-varying vector field of the type $f(t,x,u)$ with $u\in U$ fixed and for each $t$ $${\textrm{ess-liminf}}_{y\to x}W(t,y){\leqslant}W(t,x).$$ 2. $W{\leqslant}\psi$ on $S$.\[i2\] 3. \[i3\] At every point $(t,x)\in\partial Q$ one has $$W(t,x)=\sup_{(s,y)\in Q}W(s,y).$$ 4. \[i4\] There exists a countably $\mathcal H^n$-rectifiable set $A{\subseteq}\Omega$ such that $W$ is differentiable on $Q\setminus A$ and satisfies $$W_s(s,y)+\inf_{\omega\in U}\left\{ W_y(s,y)\cdot f(s,y,\omega)+L(s,y,\omega)\right\}{\geqslant}0\quad \textrm{on }Q\setminus A.$$ 5. $L{\geqslant}0$.\[i5\] Then $W{\leqslant}V$ on $Q$. If $Q=\Omega$ we can drop hypotheses \[i3\]) and \[i5\]). Suppose by contradiction that there exists $(t_0,x_0)\in Q$ such that $W(t_0,x_0)>V(t_0,x_0)$. In particular $V(t_0,x_0)<+\infty$. First of all, let us consider the case $V(t_0,x_0)>-\infty$. So we can find ${\varepsilon}>0$, $\delta>0$ such that $$\label{1} V(t_0,x_0){\leqslant}W(t_0,x_0)-2{\varepsilon}$$ and, by the lower semicontinuity of $W$, $$\label{2} {\left\vert x-x_0 \right\vert}<\delta \quad{\Rightarrow}\quad W(t_0,x)>V(t_0,x_0)+{\varepsilon}.$$ We can find $u^\ast\in{\mathcal U}$ such that $x^\ast (\cdot ):=x(\cdot; u^\ast,t_0,x_0)$ satisfies $(T,x^\ast (T))\in S$ and $$\label{3} \int_{t_0}^TL(s,x^\ast (s), u^\ast (s))ds+\psi (T,x^\ast (T)){\leqslant}V(t_0,x_0)+\frac{{\varepsilon}}{2}\,\,.$$ Moreover, for every $l\in{\mathbb N}$ there exists $u_l\in{\mathcal U}$ such that $\\| u_l-u^\ast\\|_{L^p([t_0,T])}{\leqslant}\frac{1}{l}$, $u_l$ piecewise constant and left continuous. By [@brez Théorèm IV.9], there exists a subsequence of $(u_l)_l$, denoted again by $(u_l)_l$, and a function $h\in L^p([t_0,T])$ such that $\vert u_l\vert{\leqslant}h$ a.e. and $u_l$ converges to $u^\ast$ a.e. as $l\to+\infty$. Hence, if we denote by $x_l(\cdot)$ the trajectory $x(\cdot;u_l,T,x^\ast(T))$, for $l$ sufficiently big, we have (see Proposition 4.2), $$\label{5} {\left\vert x_l (t)-x^\ast (t) \right\vert}<\frac{\delta}{2}\quad\quad\forall t\in [t_0,T]$$ and $$\label{4} {\left\vert \int_{t_0}^T[L(s,x_l(s),u_l(s))-L(s,x^\ast (s),u^\ast (s))]ds \right\vert}{\leqslant}\frac{{\varepsilon}}{2}.$$ Fix $l$ such that (\[5\]) and (\[4\]) hold and an interval $]t',t'']$ such that $u_l (t)\equiv\omega$ on $]t',t'']$. Suppose that $(t,x_l (t))\in Q\,\,\,\forall t\in [t',t'']$. Let $\zeta^y(t)$ be the trajectory associated to the constant control $\omega$ such that $\zeta ^y(t'')=y$. By the fact that $d(\partial Q,\{ (t,x_l (t)):t\in [t',t'']\} )>0$, we can find an open neighborhood ${\mathcal W}$ of $x_l(t'')$ in ${\mathbb R}^n$ such that $(t'',y)\in Q\,\,\,\forall y\in{\mathcal W}$ and $\left\{ (t,\zeta^y(t)):t\in[t',t'']\right\}{\subseteq}Q\,\,\,\forall y\in{\mathcal W}$. By Lemma \[fin\], we have that for a.e. $y\in{\mathcal W}$ the set $B^y:=\left\{t\in [t',t'']:(t,\zeta^y(t))\in A\right\}$ is at most countable.\ Therefore,since for every fixed $t$ $\textrm{ess-liminf}_{y\to x}W(t,y){\leqslant}W(t,x)$, then for every $\delta_j\to 0$, $\delta_j>0$ there exists a sequence $(y_j^l)_j\in{\mathbb N}$ such that $y_j^l\to x_l(t'')$, $W(t'',y_j^l){\leqslant}W(t'',x_l(t''))+\delta_j$ and $B^{y_j^l}$ is at most countable. Consider the following function defined on $[t',t'']$: $${\varphi}^l_j(t):=W(t,\zeta^{y_j^l}(t))+\int_{t'}^tL(s,\zeta^{y_j^l}(s),\omega )ds.$$ By the choice of $y_j^l$ and the hypotheses $\ref{i4})$, ${\varphi}_j^l$ is differentiable a.e. with a nonnegative derivative. By the lower semicontinuity of $W$ and the NDJ condition, it follows that ${\varphi}^l_j$ verifies the hypotheses of Lemma \[lerv\] and so ${\varphi}^l_j(t'){\leqslant}{\varphi}^l_j(t'')$. Thus $$\label{7} W(t',\zeta^{y_j^l}(t')){\leqslant}W(t'',\zeta^{y_j^l}(t''))+\int_{t'}^{t''}L(s,\zeta^{y_j^l}(s),\omega )ds.$$ Now, using the fact that $\zeta^{y_j^l}(t'')=y_j^l$ we obtain $$\begin{aligned} W(t',\zeta^{y_j^l}(t')) & {\leqslant}& W(t'',y_j^l)+\int_{t'}^{t''}L(s,\zeta^{y_j^l}(s),\omega )ds\nonumber\\ & {\leqslant}& W(t'',x_l (t''))+\delta_j+\int_{t'}^{t''}L(s,\zeta^{y_j^l}(s),\omega )ds.\label{7.1} \end{aligned}$$ By Proposition 4.2, $\zeta^{y_j^l}(\cdot )\to x_l(\cdot)$ as $j\to+\infty$ and so by the Lebesgue theorem and the lower semicontinuity of $W$, passing to the limit as $j\to +\infty$ we obtain: $$\label{8} W(t',x_l (t')){\leqslant}W(t'',x_l (t''))+\int_{t'}^{t''}L(s,x_l (s),\omega )ds.$$ First consider the case $\left\{ (t,x_l (t)):t\in[t_0,T]\right\}{\subseteq}Q$. Summing (\[8\]) over each interval on which $u_l$ is constant we have $$\label{9} W(t_0,x_l (t_0)){\leqslant}W(T,x_l (T))+\int_{t_0}^TL(s,x_l (s),u_l (s))ds.$$ Now, $x_l (T)=x^\ast (T)$ by definition and so, using (\[2\]-\[4\]) and \[i2\]) $$\begin{aligned} W(t_0,x_l (t_0)) & {\leqslant}& W(T,x^\ast (T))+\int_{t_0}^TL(s,x_l (s),u_l (s))ds\\ & {\leqslant}& \psi (T,x^\ast (T))+\int_{t_0}^TL(s,x_l (s),u_l (s))ds\nonumber\\ & {\leqslant}& V(t_0,x_0)+\frac{{\varepsilon}}{2}-\int_{t_0}^TL(s,x^\ast (s),u^\ast (s))ds\nonumber\\ & & +\int_{t_0}^TL(s,x_l (s),u_l (s))ds\nonumber\\ & {\leqslant}& V(t_0,x_0)+{\varepsilon}< W(t_0,x_l (t_0))\nonumber.\end{aligned}$$ This is a contradiction.\ Suppose now $\left\{ (t,x_l (t)):t\in[t_0,T]\right\}\not{\subseteq}Q$. Define $$\label{tau} \hat\tau :=\inf\left\{ t{\leqslant}T: (s,x_l (s))\in Q\quad\forall s\in [t,T]\right\}.$$ In particular $(\hat \tau,x_l (\hat\tau ))\in\partial Q$. Using the same argument to pass from (\[8\]) to (\[9\]), we obtain that for every $\tau >\hat\tau$ $$\label{bo} W(\tau ,x_l (\tau)){\leqslant}W(T,x^\ast (T))+\int_\tau^TL(s,x_l (s),u_l (s))ds$$ and so, using \[i2\]) and (\[3\]) $$\begin{aligned} W(\tau ,x_l (\tau)) & {\leqslant}& \psi(T,x^\ast (T))+\int_\tau^TL(s,x_l (s),u_l (s))ds\nonumber\\ & {\leqslant}& V(t_0,x_0)+\frac{{\varepsilon}}{2}-\int_{t_0}^TL(s,x^\ast (s),u^\ast (s))ds\nonumber\\ & & +\int_\tau^TL(s,x_l (s),u_l (s))ds.\label{lpos}\end{aligned}$$ Using (\[1\]), (\[4\]) and \[i5\]), we obtain for all $\tau >\hat\tau$ $$W(\tau ,x_l (\tau)){\leqslant}V(t_0,x_0)+{\varepsilon}{\leqslant}W(t_0,x_0)-{\varepsilon}.\label{Ref1}$$ Passing to the liminf as $\tau\to\hat\tau$ and using the lower semicontinuity of $W$, we conclude $$\label{Ref2} W(\hat\tau ,x_l (\hat\tau)) {\leqslant}W(t_0,x_0)-{\varepsilon}$$ and so by $\ref{i3})$ $$\label{Ref3} W(t_0,x_0){\leqslant}\sup_{(t,x)\in Q} W(t,x){\leqslant}W(t_0,x_0)-{\varepsilon}$$ which is a contradiction. Now, we have to treat the case $V(t_0,x_0)=-\infty$. Since $W(t_0,x_0)>-\infty$ and $W$ is lower semicontinuous, we may find two constants $M>1$ and $\delta>0$ such that: $$W(t_0,x)>-M$$ for every $x$ so that ${\left\vert x-x_0 \right\vert}<\delta$. Moreover we can find $u^\ast\in{\mathcal U}$ such that $x^\ast(\cdot):=x(\cdot;u^\ast,t_0,x_0)$ satisfies $(T,x^\ast(T))\in S$ and $$\int_{t_0}^TL(s,x^\ast(s),u^\ast(s))ds+\psi(T,x^\ast(T)){\leqslant}-2M.$$ With the same arguments of the first part of the proof we may find a control $u_l\in{\mathcal U}$ piecewise constant and left continuous such that, if $x_l(\cdot)$ is the trajectory $x(\cdot;u_l,T,x^\ast(T))$, $${\left\vert x_l(t)-x^\ast(t) \right\vert}<\frac{\delta}{2}\quad\forall t\in [t_0,T]$$ and $${\left\vert \int_{t_0}^T[L(s,x_l(s),u_l(s))-L(s,x^\ast(s),u^\ast(s))]ds \right\vert}{\leqslant}1.$$ Repeating the same calculations as before, we obtain that $$\begin{aligned} -M & {\leqslant}& W(T,x^\ast(T))+\int_{t_0}^TL(s,x_l(s),u_l(s))ds\\ & {\leqslant}& \psi(T,x^\ast(T))+\int_{t_0}^TL(s,x^\ast(s),u^\ast(s))ds+1\\ & {\leqslant}& -2M+1\end{aligned}$$ which gives $M{\leqslant}1$, a contradiction. This concludes the proof of the theorem. Let us suppose that $W$ satisfies all the hypotheses of the previous theorem. If moreover $W{\geqslant}V$ then $W=V$. If $W$ is produced by a synthesis procedure, the inequality $W{\geqslant}V$ always holds and so if $W$ satisfies all the hypotheses of Theorem \[teo1\] then $W$ coincides with the value function. Notice that the value function of an optimal control problem has the NDJ property along every possible direction as a consequence of the Dynamic Programming Principle. Indeed, for every $(t,y)\in\Omega\setminus S$ and for every admissible control $u\in{\mathcal U}$ (in particular for every control $\omega\chi_I$, where $\omega\in U$ and $I$ bounded interval), the function $$h\mapsto\int_t^{t+h}L(s,x(s;u,t,y),u(s))ds+V(t+h,x(t+h;u,t,y))$$ is non decreasing for $h\in[0,\delta]$ and $\delta$ small enough. Instead, the hypothesis $$\textrm{ess-liminf}_{y\to x}W(t,y){\leqslant}W(t,x)$$ for each $t$ fixed, says that, for every ${\varepsilon}>0$ there exists a subset $V{\subseteq}\{y\in{\mathbb R}^n:{\left\vert y-x \right\vert}{\leqslant}{\varepsilon}\}$ of strictly positive Lebesgue measure such that $$\inf_{y\in V}W(t,y){\leqslant}W(t,x).$$ So, if we consider a set $V_1{\subseteq}{\mathbb R}^n$ of zero Lebesgue measure with $x$ as a cluster point, the set $V\setminus V_1$ has a strictly positive Lebesgue measure. In the proof of Theorem \[teo1\] this fact is used to avoid the points $y$ for which $B^y$ is not countable. Moreover this hypothesis, coupled with the lower semicontinuity of $W$, gives the following: - for each $t$, $$W(t,x)=\liminf_{y\to x}W(t,y)=\textrm{ess-liminf}_{y\to x}W(t,y).$$ \[rip3\] Hypothesis \[i3\]) of Theorem \[teo1\] says that, in the case $Q\ne \Omega$, the boundary of $Q$ must be a level set of the function $W$. We can relax the same hypothesis in the following way: - At every point $(t,x)\in\partial Q$ one has $$\liminf_{\substack{\tau\to t,y\to x\\(\tau,y)\in Q}}W(\tau, y){\geqslant}\sup_{(s,y)\in Q}W(s,y)$$ and the conclusion of the theorem remains valid. Moreover if we define with $R(t,x)$ the set of point reachable with an admissible control from $(t,x)$, the previous condition can be replaced by $$\inf_{(s,y)\in R(t,x)\cap\partial Q}W(s,y){\geqslant}W(t,x)$$ and the conclusion still holds. \ The hypotheses of the positiveness of $L$ is almost optimal as the next example shows. However, the Lagrangian $L$ may be negative on some region if trajectories can not stay for too long in such a region and one can relax the assumption \[i5\]) as shown in Remark \[RemL\]4.\ \[ExaLpos\] ![$L(x,0)$ of Example 5.1.[]{data-label="fig"}](funct1.eps){height="6cm"} Consider the system $\dot x=u$, $U=[-1,1]\,$ and $\,{\mathcal U}=L^1({\mathbb R};U)$, $\Omega={\mathbb R}^2$, $S={\mathbb R}\times\{ 0\}$, $Q={\mathbb R}\times ]-1,1[$ with the Lagrangian $L(t,x,u)=u^2+x^4-6x^3+7x^2$ (see Figure \[fig\]) and $\psi\equiv 0$ on $S$. Since the Lagrangian is negative in a region where the system can stay for an arbitrary interval of times, clearly the value function for this problem is equal to $-\infty$. If $W\equiv C$ on $\overline Q$ with $C$ negative constant, then $W$ verifies all the hypotheses of the Theorem \[teo1\], but \[i5\]). In fact \[NU1\]), \[i2\]), \[i3\]) are obvious, while \[i4\]) holds because $L$ is positive on $Q$ and $W$ is differentiable on $Q$. So there exist infinitely many functions $W$ defined on $\overline Q$ verifying the hypotheses of Theorem \[teo1\], but \[i5\]), which are not lower or equal to the value function $V$. \ \[RemL\] If one wants to eliminate hypothesis \[i5\]) from the previous theorem, one may assume one of the following conditions: 1. Fix ${\varepsilon}>0$ and $(\bar t,\bar x)\in Q$. We call $x_{\varepsilon}:[\bar t,T]\to{\mathbb R}^n$ an ${\varepsilon}$-quasi optimal trajectory (${\varepsilon}$-q.o.t.) for $(\bar t,\bar x)$ if: 1. $\exists\,u_{\varepsilon}\in{\mathcal U}$ such that $\dot x_{\varepsilon}(s)=f(s,x_{\varepsilon}(s),u_{\varepsilon}(s))$ for a.e. $s\in[\bar t,T]$, 2. $x_{\varepsilon}(\bar t)=\bar x$, 3. $(T,x_{\varepsilon}(T))\in S$, 4. $V(\bar t,\bar x)+{\varepsilon}{\geqslant}\int_{\bar t}^TL(s,x_{\varepsilon}(s),u_{\varepsilon}(s))ds+\psi(T,x_{\varepsilon}(T))$. Now define $Q_1$ as the set of point $(\bar t,\bar x)\in Q$ such that, for every ${\varepsilon}>0$, there exists $x_{\varepsilon}$, an ${\varepsilon}$-q.o.t. for $(\bar t,\bar x)$, satisfying $(s,x_{\varepsilon}(s))\in Q$ for any $s\in[\bar t,T]$. What we need is that $L{\geqslant}0$ in $\Omega\setminus Q_1$. In fact, under this assumption, we may suppose that $(s,x (s))\in\Omega\setminus Q_1$ for every $s\in[t_0,\hat\tau [$, where $x$ is the trajectory defined in the proof of Theorem \[teo1\] and the time $\hat\tau$ is defined in (\[tau\]). So the integral $\int_{t_0}^{\hat\tau}L(s,x (s),u (s))ds$ is positive. Otherwise we can assume $Q_1=Q$. 2. We can also use an hypothesis similar to one given in [@ma]. For any $(\bar t,\bar x)\in \Omega$ and $u\in{\mathcal U}$, let $x_{\bar t,\bar x}(\cdot;u):=x(\cdot;u,\bar t,\bar x)$ be the solution to (\[cs\]) associated to the control $u$. Consider the set $P$ consisting of those points $(\bar t,\bar x)$ of $Q$ such that $$\int_{\bar t}^TL(s,x_{\bar t,\bar x}(s;u),u(s))ds{\geqslant}0\qquad\forall T>t\quad\forall u\in{\mathcal U}.$$ We have to suppose that, if $(\bar t,\bar x)\in Q\setminus[P\cup S]$, there exist a bounded and open set $B$, $(\bar t,\bar x)\in B{\subseteq}Q$, $B\cap S=\emptyset$, so that $\partial B{\subseteq}Q$, and a positive number $M$ strictly less than $$\inf_{u\in{\mathcal U}}\left\{ T>0:d((\bar t+T,x_{\bar t,\bar x}(\bar t+T;u)),\partial B){\leqslant}d((\bar t,\bar x),\partial B)/2\right\}$$ such that, for all $u\in{\mathcal U}$, $(M+\bar t,x_{\bar t,\bar x}(M+\bar t;u))\in Q\cap P$ and $$\int_{\bar t}^{\bar t+M}L(s,x_{\bar t,\bar x}(s),u(s))ds{\geqslant}0,$$ and this allow to conclude the proof of Theorem \[teo1\] without using $L{\geqslant}0$ on the whole space. Consider the system $\dot x=u$, $U=[-1,1]$, ${\mathcal U}=L^1({\mathbb R};U)$, $\Omega={\mathbb R}^+\times{\mathbb R}$, $S={\mathbb R}^+\times\{0\}$, $Q={\mathbb R}^+\times ]-1,1[$, $\psi=0$ on $S$ and the Lagrangian defined by $$L(t,x,u):= \left\{ \begin{array}{lrl} u^2+x^2 & \textrm{if} & x{\leqslant}1\\ (u^2+1)(2-x)+(x-1)(u^2+Ct) & \textrm{if} & 1<x<2\\ u^2+x^2-6x+8+Ct & \textrm{if} & x{\geqslant}2 \end{array} \right.$$ It is clear that this Lagrangian, for $C$ sufficiently big, satisfies the conditions a) and b) of the previous remark, even if it is not positive outside $Q$. We can relax hypotheses \[i3\]) and \[i5\]) with the following: iii’) : the boundary $\partial Q$ is a level set of $W$; v’) : $L{\geqslant}0$ on $\Omega\setminus Q$. With these hypotheses, we can obtain an inequality of type (\[8\]) for each interval where the couple time-trajectory is in $Q$ and then, using $iii')$, $v')$, the lower semicontinuity of $W$ and the NDJ property we can obtain (\[9\]). Problem with infinite time. {#se5} =========================== In this section we consider the control system (\[cs\]) and assume that (A-\[a1\])-(A-5) hold with $0\leq C_R\leq C$ for some $C>0$ and every $R>0$. Moreover we suppose that the target $S$ is a closed subset of ${\mathbb R}\times{\mathbb R}^n$ which satisfies the structural property: $(\ast)$ : For any $T>0$, there exists $(t,x)\in S$ with $t{\geqslant}T$. Let $S_1$ be an open neighborhood of $S$ contained in $\Omega$. Assume that the final cost $\psi$ is defined on $S_1$ and, if $d((t,x(t;u,t_0,x_0)),S)\to 0$ as $t\to+\infty$, then the trajectory $x(\cdot;u,t_0,x_0)$ is definitively in $S_1$, that is: $(\ast\ast)$ : $\!\!\!\exists\, T>t$ such that $(s,x(s;u,t_0,x_0))\in S_1$ for all $s{\geqslant}T$. Define the value function: $$\label{vfit} V(t_0,x_0):=\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\inf_{\substack{u\in{\mathcal U}\\[5pt]d((t,x(t;u,t_0,x_0)),S)\to 0\\ \text{as }t\to+\infty}}\!\!\!\!\!\!\!\!\!\!\!\!\!\left\{\!\int_{t_0}^{+\infty}\!\!\!\!\!\!\!\!\!\!\!L(s,x(s;u,t_0,x_0),u(s))ds\!+\!\limsup_{t\to+\infty}\psi(t,x(t;u,t_0,x_0))\right\}$$ In other words, we consider only the trajectories that approach the target $S$ in infinite time. Notice that this condition does not imply that $(T,x(T))\not\in S$ for any $T{\geqslant}t_0$.\ The introduction of an open neighborhood of the target $S$ is due to a technical reason and precisely to the fact that it is necessary to compare the candidate value function to the final cost near the target. Notice that in the following theorem the set $Q$ must contain $S_1$. For example we consider $\Omega={\mathbb R}^+\times{\mathbb R}$, $S={\mathbb R}^+\times\{0\}$, $Q=\{(t,x):t>0,x<1/t\}$ and $S_1=\{(t,x):t>0,x<3/t\}$. If $(t,2/t)$, with $t>0$, is a trajectory, then it is definitely in $S_1$, but it is never in $Q$. \[teo2\] Let $Q{\subseteq}\Omega$ be an open subset containing $S_1$. Let $W:\overline Q\to{\mathbb R}$ be a lower semicontinuous function such that 1. $W$ has the NDJ property along every time-varying vector field of the type $f(t,x,u)$ with $u\in U$ fixed and for each $t$, $$\textrm{ess-liminf}_{y\to x}W(t,y){\leqslant}W(t,x).$$ 2. $W{\leqslant}\psi$ on $S_1$.\[ii2\] 3. \[ii3\] At every point $(t,x)\in\partial Q$ one has $$W(t,x)=\sup_{(s,y)\in Q}W(s,y).$$ 4. \[ii4\] There exists a countable $\mathcal H^n$-rectifiable set $A{\subseteq}\Omega$ such that $W$ is differentiable in $Q\setminus A$ and satisfies $$W_s(s,y)+\inf_{\omega\in U}\left\{ W_y(s,y)\cdot f(s,y,\omega)+L(s,y,\omega)\right\}{\geqslant}0\quad \textrm{in }Q\setminus A.$$ 5. $L{\geqslant}0$.\[ii5\] Then $W{\leqslant}V$ on $Q$. If $Q=\Omega$ we can drop hypotheses \[ii3\]) and \[ii5\]). Suppose by contradiction that there exists $(t_0,x_0)\in Q$ such that $W(t_0,x_0)>V(t_0,x_0)$. In particular $V(t_0,x_0)<+\infty$. First of all, let us consider the case $V(t_0,x_0)>-\infty$. As in the first part of the proof of Theorem \[teo1\], we can find ${\varepsilon}>0$ and $\delta>0$ such that the following holds: $$\begin{gathered} V(t_0,x_0){\leqslant}W(t_0,x_0)-2{\varepsilon}\\ {\left\vert x-x_0 \right\vert}<\delta\quad{\Rightarrow}\quad W(t_0,x)>V(t_0,x_0)+\frac{3{\varepsilon}}{2}\,\,.\label{21}\end{gathered}$$ We can choose $u^\ast\in{\mathcal U}$, with the property that the trajectory $(t,x^\ast(t))$ approaches the target when $t\to+\infty$, and such that $$\label{eq2} \int_{t_0}^{+\infty}L(s,x^\ast(s), u^\ast(s))ds+\limsup_{t\to+\infty}\psi(t,x^\ast (t)){\leqslant}V(t_0,x_0)+\frac{{\varepsilon}}{2}\,\,,$$ where $x^\ast(\cdot)$ is the trajectory corresponding to the control $u^\ast$ such that $x^\ast(t_0)=x_0$.\ Consider, now, a strictly increasing sequence of times $T_j>t_0$ converging to $+\infty$. We may suppose that $(t,x^\ast(t))\in Q$ for every $t{\geqslant}T_1$. Fix $j\in{\mathbb N}$. For every $l\in{\mathbb N}$, there exists $u_j^l\in{\mathcal U}$ piecewise constant and left continuous such that $\\| u_j^l-u^\ast\\|_{L^p([t_0,T_j])}{\leqslant}\frac{1}{l}$. So, by [@brez Théorèm IV.9], we can extract a subsequence of $(u_j^l)_l$, denoted again with $(u_j^l)_l$, and we can find a function $h_j\in L^p([t_0,T_j])$ such that $\vert u_j^l\vert{\leqslant}h_j$ a.e. for every $l\in{\mathbb N}$ and $u_j^l\to u^\ast$ for a.e. $t\in [t_0,T_j]$ as $l\to+\infty$. Thus denoting with $x_j^l(\cdot)$ the trajectory $x(\cdot;u_j^l,T_j,x^\ast(T_j))$, for $l$ sufficiently big we have (see Proposition 3.2) $$\label{lim1} {\left\vert x_j^l(t)-x^\ast(t) \right\vert}{\leqslant}\frac{\delta}{2}\qquad\forall t\in [t_0,T_j]$$ and then $$\label{eq1} {\left\vert \int_{t_0}^{T_j}\left[ L(s,x_j^l(s),u_j^l(s))-L(s,x^\ast(s),u^\ast (s))\right]ds \right\vert}{\leqslant}\frac{{\varepsilon}}{2}\,\,.$$ Now, fix $l\in{\mathbb N}$ such that (\[lim1\]) and (\[eq1\]) hold. First, let us suppose that $\{ (t,x_j^l(t)):t\in[t_0,T_j]\}{\subseteq}Q$. So, using Lemma \[fin\], Lemma \[lerv\], the same arguments as in the proof of Theorem \[teo1\] and (\[eq1\]) we conclude $$\begin{aligned} W(t_0,x_j^l(t_0)) & {\leqslant}& W(T_j,x_j^l(T_j))+\int_{t_0}^{T_j}L(s,x_j^l(s),u_j^l(s))ds\\ & {\leqslant}& W(T_j,x^\ast(T_j))+\int_{t_0}^{T_j}L(s,x^\ast(s),u^\ast(s))ds+\frac{{\varepsilon}}{2}\,\,.\end{aligned}$$ Using (\[21\]) and (\[lim1\]) we have $$V(t_0,x_0)+\frac{3{\varepsilon}}{2}<W(T_j,x^\ast(T_j))+\int_{t_0}^{T_j}L(s,x^\ast(s),u^\ast(s))ds+\frac{{\varepsilon}}{2}\,\,.$$ Now consider the other case and precisely $\{ (t,x_j^l(t)):t\in[t_0,T_j]\}\not{\subseteq}Q$. Define $$\tau_j^l:=\inf\left\{ t{\geqslant}t_0:(s,x_j^l(s))\in Q\,\,\,\,\forall s\in[t,T_j]\right\}.$$ Given $\tau_j^l<t<T_j$ $$W(t,x_j^l(t)){\leqslant}W(T_j,x_j^l(T_j))+\int_t^{T_j}L(s,x_j^l(s),u_j^l(s))ds.$$ Considering the fact that $(t,x_j^l(t))\to (\tau_j^l,x_j^l(\tau_j^l))$ as $t\to\tau_j^l$, $(\tau_j^l,x_j^l(\tau_j^l))\in\partial Q$ and (\[ii3\]) we obtain $$W(t_0,x_0){\leqslant}W(T_j,x^\ast(T_j))+\int_{\tau_j^l}^{T_j}L(s,x_j^l(s),u_j^l(s))ds.$$ We can now use the hypothesis $\ref{ii5})$, (\[21\]) and (\[eq1\]) in order to have $$\begin{gathered} V(t_0,x_0)+\frac{3{\varepsilon}}{2} < W(t_0,x_0){\leqslant}\nonumber\\ {\leqslant}W(T_j,x^\ast(T_j))+\int_{t_0}^{T_j}L(s,x^\ast (s),u^\ast(s))ds+\frac{{\varepsilon}}{2}\,\,.\label{29}\end{gathered}$$ In all cases we have that, for every $j\in{\mathbb N}$, $$V(t_0,x_0)+\frac{3{\varepsilon}}{2} < W(T_j,x^\ast(T_j))+\int_{t_0}^{T_j}L(s,x^\ast (s),u^\ast(s))ds+\frac{{\varepsilon}}{2}\,\,.$$ So, applying the limsup as $j\to+\infty$ we get $$\begin{aligned} V(t_0,x_0)+\frac{3{\varepsilon}}{2} & {\leqslant}& \limsup_{j\to+\infty}W(T_j,x^\ast(T_j))+\int_{t_0}^{+\infty}\!\!\!L(s,x^\ast(s),u^\ast(s))ds+\frac{{\varepsilon}}{2}\nonumber\\ & {\leqslant}& \limsup_{t\to+\infty}W(t,x^\ast(t))+\int_{t_0}^{+\infty}\!\!\!L(s,x^\ast(s),u^\ast(s))ds+\frac{{\varepsilon}}{2}\,\,.\nonumber\end{aligned}$$ For $t$ sufficiently big, $(t,x^\ast(t))\in S_1$ and so, using (\[ii2\]) and (\[eq2\]), $$\begin{aligned} V(t_0,x_0)+\frac{3{\varepsilon}}{2} & {\leqslant}& \limsup_{t\to+\infty}\psi(t,x^\ast(t))+\int_{t_0}^{+\infty}L(s,x^\ast(s),u^\ast(s))ds+\frac{{\varepsilon}}{2}\\ & {\leqslant}& V(t_0,x_0)+{\varepsilon}\end{aligned}$$ which implies $$V(t_0,x_0){\leqslant}V(t_0,x_0)-\frac{{\varepsilon}}{2}$$ which is a contradiction. It remains the case $V(t_0,x_0)=-\infty$. Since $W(t_0,x_0)>-\infty$ and $W$ is lower semicontinuous, we may find two constants $M>1$ and $\delta>0$ such that $$W(t_0,x)>-M$$ for every $x$ so that ${\left\vert x-x_0 \right\vert}<\delta$. Moreover we can find $u^\ast\in{\mathcal U}$ such that $x^\ast(\cdot):=x(\cdot;u^\ast,t_0,x_0)$ approaches the target when $t\to+\infty$ and $$\int_{t_0}^{+\infty}L(s,x^\ast(s),u^\ast(s))ds+\limsup_{t\to+\infty}\psi(t,x^\ast(t)){\leqslant}-2M.$$ Consider a strictly increasing sequence of times $T_j>t_0$ converging to $+\infty$ and repeat the previous arguments in order to find a control $u_j^l\in{\mathcal U}$ piecewise constant, left continuous and such that, if $x_j^l(\cdot):=x(\cdot;u_j^l,T_j,x^\ast(T_j))$, $${\left\vert x_j^l(t)-x^\ast(t) \right\vert}{\leqslant}\frac{\delta}{2}\quad\forall t\in[t_0,T_j]$$ and $${\left\vert \int_{t_0}^{T_j}[L(s,x_j^l(s),u_j^l(s))-L(s,x^\ast(s),u^\ast(s))]ds \right\vert}{\leqslant}1.$$ Proceeding as before we obtain that $$\begin{aligned} -M & {\leqslant}& W(T_j,x^\ast(T_j))+\int_{t_0}^{T_j}L(s,x_j^l(s),u_j^l(s))ds\\ & {\leqslant}& W(T_j,x^\ast(T_j))+\int_{t_0}^{T_j}L(s,x^\ast(s),u^\ast(s))ds+1\end{aligned}$$ for every $j\in{\mathbb N}$. Passing to the limit we have: $$\begin{aligned} -M & {\leqslant}& \limsup_{j\to+\infty}W(T_j,x^\ast(T_j))+\int_{t_0}^{+\infty}L(s,x^\ast(s),u^\ast(s))ds+1\\ & {\leqslant}& \limsup_{t\to+\infty}W(t,x^\ast(t))+\int_{t_0}^{+\infty}L(s,x^\ast(s),u^\ast(s))ds+1\\ & {\leqslant}& \limsup_{t\to+\infty}\psi(t,x^\ast(t))+\int_{t_0}^{+\infty}L(s,x^\ast(s),u^\ast(s))ds+1\\ & {\leqslant}& -2M+1\end{aligned}$$ which gives $M{\leqslant}1$, a contradiction. So the theorem is proved. Let $W$ satisfies all the hypotheses of the previous theorem and moreover $W{\geqslant}V$ where $V$ is defined in (\[vfit\]). Then $W$ coincides with the value function. In theorem \[teo2\] the condition \[ii2\]) can be relaxed in the following way: $$\limsup_{t\to+\infty}W(t,x(t)){\leqslant}\limsup_{t\to+\infty}\psi(t,x(t))$$ for every $x(\cdot)$ solution to (\[cs\]) such that $d((t,x(t)),S)\to0$ as $t\to+\infty$. So, if one wants to minimize a Lagrangian cost without final cost, the condition becomes $$\limsup_{t\to+\infty}W(t,x(t)){\leqslant}0$$ for every $x(\cdot)$ with the above property. If we assume that there exists $\eta>0$ such that $S+B(0,\eta){\subseteq}S_1$, where $B(0,\eta)$ is the ball in ${\mathbb R}^{n+1}$ centered in $0$ with radius $\eta$, then hypothesis $(\ast\ast)$ obviously holds. In fact suppose $d((t,x(t;u,t_0,x_0)),S)\to 0$ as $t\to+\infty$. Then there exists $T>0$ such that $d((s,x(s;u,t_0,x_0)),S)<\frac{\eta}{2}$ for all $s{\geqslant}T$. So we can choose an element $(t(s),y(s))\in S$ in order to have $d((s,x(s;u,t_0,x_0)),(t(s),y(s)))<\frac{\eta}{2}$ for all $s{\geqslant}T$. So the points $(s,x(s;u,t_0,x_0))\in S+B(0,\eta){\subseteq}S_1$ for every $s{\geqslant}T$. We obtain a generalization of Theorems \[teo1\] and \[teo2\] considering the same problem (\[cs\]) with assumptions (A-\[a1\])-(A-\[a4\]), but we accept at the same time all the trajectories that hit the target in finite time or that tend to the target in infinite time. Obviously an analogous theorem as \[teo1\] and \[teo2\] holds. Also in this case we can substitute hypothesis $\ref{ii3})$ of Theorem \[teo2\] in an analogous way as in Remark 5.3. Moreover we can eliminate hypothesis \[ii5\]) of Theorem \[teo2\] in the same way as in Remark \[RemL\]4. Viscosity solutions and value functions. ======================================== This appendix is intended to recall the notion of viscosity sub- and super-solution and to state some known properties of the value function. Proof are analogous to those of [@bardi]. Let $\Omega_1$ be an open subset of ${\mathbb R}\times{\mathbb R}^n$. We need the following definitions: Let $f:A\to\overline{\mathbb R}$ be a function where $A$ is an open subset of ${\mathbb R}^l$, for some $l\in{\mathbb N}\setminus\{ 0\}$. The lower semicontinuous envelope $f_\ast$ and the upper semicontinuous envelope $f^\ast$ of $f$ are defined by: $$\begin{gathered} f_\ast(x):=\lim_{r\to0^+}\inf\left\{ f(y):y\in A,{\left\vert y-x \right\vert}{\leqslant}r\right\},\\ f^\ast(x):=\lim_{r\to0^+}\sup\left\{ f(y):y\in A,{\left\vert y-x \right\vert}{\leqslant}r\right\}.\end{gathered}$$ The lower semicontinuous (resp. upper semicontinuous) envelope of a function $f$ is a lower semicontinuous (resp. upper semicontinuous) function. More precisely, it is the greatest (resp. least) lower semicontinuous (resp. upper semicontinuous) function less or equal (resp. greater or equal) to $f$. Moreover $f$ is continuous if and only if $f_\ast=f^\ast$. We say that a lower semicontinuous function $V:\Omega_1\to\overline{\mathbb R}$ is a viscosity super-solution to $F(t,x,D_tV,D_xV)=0$ in $\Omega_1$ if, for any ${\varphi}\in\mathcal C^1(\Omega_1)$ and for any $(t_0,x_0)\in\Omega_1$ point of local minimum for $V-{\varphi}$, one has $F^\ast(t_0,x_0,D_t{\varphi}(t_0,x_0),D_x{\varphi}(t_0,x_0)){\geqslant}0$. We say that an upper semicontinuous function $V:\Omega_1\to\overline{\mathbb R}$ is a viscosity sub-solution to $F(t,x,D_tV,D_xV)=0$ in $\Omega_1$ if, for any ${\varphi}\in\mathcal C^1(\Omega_1)$ and for any $(t_0,x_0)\in\Omega_1$ point of local maximum for $V-{\varphi}$, one has $F_\ast (t_0,x_0,D_t{\varphi}(t_0,x_0),D_x{\varphi}(t_0,x_0)){\leqslant}0$. We say that a function $V:\Omega_1\to\overline{\mathbb R}$ is a viscosity solution to $F(t,x,D_tV,D_xV)=0$ in $\Omega_1$ if $V_\ast$ is a viscosity super-solution and $V^\ast$ is a viscosity sub-solution to the equation. Note that the notion of viscosity solution is not bilateral, in the sense that the set of viscosity solution to $F=0$ and $-F=0$ in general are different. Let us consider the following hypotheses: (H-1) : The functions $f$ and $L$ are continuous in all the variables. (H-2) : $U$ is a bounded set. We have the following: \[pp1\] Let us assume (A-1)-(A-5) and (H-1)-(H-2). Then the value function $V$ defined in (\[vf\]) satisfies the dynamic programming principle, that is $$V(t_0,x_0)=\!\!\!\!\!\!\!\inf_{\substack{u\in{\mathcal U}\vspace{3pt}\\(T,x(T;u,t_0,x_0))\in S}}\!\!\!\!\!\left\{\int_{t_0}^{T_1}\!\!\!\!\!\!L(s,x(s;u,t_0,x_0),u(s))ds\!+\!V(T_1,x(T_1;u,t_0,x_0))\right\}$$ for every $(t_0,x_0)\in\Omega\setminus S$ and for every $T_1$ less than the minimum time to reach the target. An analogous proposition holds for the value function $V$ defined in (\[vfit\]). Let us now state without proof the result that ensure that the value function is a viscosity solution to a Hamilton-Jacobi-Bellman equation. \[teoA\] Let us assume (A-1)-(A-5) and (H-1)-(H-2). Then the value functions (\[vf\]) and (\[vfit\]) are viscosity solutions of $$-V_s(t,x)-\inf_{\omega\in U}\left\{ f(t,x,\omega)\cdot V_y(t,x)+L(t,x,\omega)\right\}=0\quad \text{in }\Omega\setminus S.$$ [99]{} M. Bardi, I. Capuzzo-Dolcetta, , Birkhäuser, 1997. V. G. Boltyanskii, Sufficient conditions for optimality and the justification of the dynamic programming principle, SIAM J. Control Optim. 4 (1966), pp. 326-361. U. Boscain, B. Piccoli, Geometric control approach to synthesis theory. Control theory and its applications, Rend. Sem. Mat. Univ. Politec. Torino 56 (1998), pp. 53-68. A. Bressan, , S.I.S.S.A., Trieste, 1994. A. Bressan, B. Piccoli, A generic classification of time-optimal planar stabilizing feedbacks, SIAM J. Control Optim. 36 (1998), pp. 12-32. H. Brezis, , Masson, 1987. P. Brunovský, Existence of regular syntheses for general problems, J. Differential Equations 38 (1980), pp. 317-343. P. Cannarsa, A. Mennucci, C. Sinestrari, Regularity results for Solutions of a Class of Hamilton-Jacobi Equations, Arch. Rational Mech. Anal., 140 (1997), pp. 197-223. F. H. Clarke, , Canadian Mathematical Society series of monographs and advanced texts \[Wiley\], 1983. L. C. Evans, R. F. Gariepy, , Studies in Advanced Mathematics, CRC press. W. H. Fleming, R. W. Rishel, , Springer-Verlag, 1975. G. B. Folland, , J. Wiley and sons, 1984. P. Hartman, , S. H. Hartman, Baltimore, 1973. M. Malisoff, On the Bellman equation for control problems with exit times and unbounded cost functionals, Proceedings of the $38^{th}$ Conference on Decision & Control, Phoenix, Arizona USA, December 1999. M. Malisoff, H. J. Sussmann, Further Results on the Bellman Equation for Optimal Control Problems with Exit Times and Nonnegative Instantaneous Costs, to appear. B. Piccoli, Classification of Generic Singularities for the Planar Time-Optimal Synthesis, SIAM J. Control Optim., 34 (1996), pp. 1914-1946. B. Piccoli, Infinite time regular synthesis, ESAIM, Control Optim. Calc. Var. 3, (1998), pp. 381-405. B. Piccoli, H. J. Sussmann, Regular Synthesis and Sufficient Conditions for Optimality, SIAM J. Control Optim., 39 (2000), pp. 359-410. R. Vinter, , Birkhäuser, Boston, 2000. M. I. Zelikin, V. F. Borisov, , Birkhäuser, Boston, 1994. [^1]: E-mail: `mgarav@sissa.it`.
--- abstract: 'It is known that the normal three-dimensional (3D) Ising model on a cubic lattice is dual to the Wegner’s 3D $Z_2$ lattice gauge theory. Here we find an unusual $Z_2$ lattice gauge theory which is dual to the 3D Ising model with not only nearest-neighbor (nn) coupling, but also next-nearest-neighbor (nnn) coupling. Our gauge theory has on each edge four $Z_2$ variables that have product $+1$, each located on a vector perpendicular to the edge. The nn coupling in the Ising model maps to the plaquette term in the gauge theory where the four variables multiplied have their vectors pointing inward, while the nnn coupling maps to the coupling between the $Z_2$ variables on nearby vectors on each edge in the gauge theory. A Wilson loop observable in the gauge theory depends on a framing of a loop, and maps to a surface of flipped-sign nn and nnn couplings in the Ising model. Further numerical simulations could be made to explore the universality at the phase transition.' author: - Changnan bibliography: - 'library.bib' title: 'Dual $Z_2$ Lattice Gauge Theory of the 3D Ising Model with both Nearest- and Next-Nearest-Neighbor Couplings' --- Introduction {#Introduction} ============ In high energy physics, symmetries play an essential role. The language used to describe the gauge symmetry is the gauge theory. Gauge theories accurately describe three of the four fundamental forces of nature - the electromagnetic force, the weak force, and the strong force [@Gaillard1999]. However, some gauge theories such as quantum chromodynamics (QCD) are so complicated that it is very hard to solve them analytically. In order to solve these gauge theories, people discretize the spacetime into a lattice and use numerical methods to solve them. The lattice gauge theories give people more insights at the level of elementary particles. For example, the confinement of quarks can be displayed by the scaling property of Wilson loops in the lattice QCD [@Wilson1974]. As a simpler lattice gauge theory, the $Z_2$ lattice gauge theory, which describes the symmetry when the field $\phi$ is changed to its negation $-\phi$, also has interesting properties with Wilson loops [@Wegner1971; @Wegner2014]. In 1971, F. J. Wegner discovered the duality of the $Z_2$ lattice gauge theory and the Ising model with nearest-neighbor coupling in a three dimensional cubic lattice [@Wegner1971; @Kramers1941; @Wannier1945]. Also, a Wilson loop in the $Z_2$ lattice gauge theory is dual to a surface of frustrated links in the Ising model [@Wegner1971; @Bill2013]. Ising models are statistical models in which spins with value $\{\pm 1\}$ are placed on the lattice nodes and there are interactions between the spins. People are interested in Ising models because they display phase transitions and they are easy to simulate numerically [@Onsager1944]. They are also good models to investigate the critical phenomena at the phase transition point. The critical phenomena are the universal behaviors of the models at the phase transition point. For example, at the phase transition point of the Ising models, the correlation function of the square of the spins (or the plaquette for $Z_2$ lattice gauge theory) decays with the distance between the spins in a power law, and the exponent of this power law should be independent to the microscopic structure of the model [@Pelissetto2002]. We would like to test this universality for different Ising models and their dual gauge theories. The 3D Ising model with both nearest- and next-nearest-neighbor couplings will be a good model to work with. By changing the ratio between the couplings, we can get a continuous spectrum of models. However, the dual gauge theory of such Ising model was not reported. In this paper, we show an unusual gauge theory that is dual to the 3D Ising model with both nearest- and next-nearest-neighbor couplings. In our gauge theory, each edge is associated with four $Z_2$ variables, each located on a perpendicular vector. The nearest- and next-nearest-neighbor couplings are dual to the four-vector plaquette coupling and the two-vector on-edge coupling, respectively. The duality between the defect bonds in the Ising model and the Wilson loops in the gauge theory is also discussed. Model {#Model} ===== The action of the 3D Ising model with both nearest-neighbor (nn) and next-nearest-neighbor (nnn) couplings is: $$S_\Ising = -\beta_1\sum_{\vv\in nn}s_v^z s_{v'}^z - \beta_2\sum_{\vvv\in nnn}s_v^z s_{v''}^z, \label{S-Ising}$$ where $v$, $v'$, and $v''$ are vertices in the 3D cubic lattice, $nn$ and $nnn$ respectively stand for the nearest- and next-nearest-neighbor vertices pairs, $s_v^z=\pm 1$ is the Ising variable placed on the vertex $v$, and $\beta_1$ and $\beta_2$ are the reduced coupling coefficients. In this paper, we only consider the cases when $\beta_1>0$ and $\beta_2\ge0$. Before we talk about the action of the dual $Z_2$ lattice gauge theory, we first need to introduce the setup of the variables in this unusual lattice gauge theory. The dual $Z_2$ lattice gauge theory has Ising variables with values $\pm 1$ placed on the edges of the dual cubic lattice. Each edge is associated with four variables, which are placed at the four sides of the edge. We show these variables as four perpendicular vectors to the edge, as shown in Fig. 1 and Fig. 2. (The vectors are only for illustrating the relative position of the variables. The values of the variables are not vectors; they are $\pm1$.) There is a constraint on these variables: the product of four variables on each edge should equal to $+1$; i.e. there are even numbers of $+1$’s and $-1$’s among the four variables on each edge. ![The variables in the dual model. The grey cubes denote the lattice in the Ising model. The crosses at both ends of the black line denote the centers of the grey cubes. (The crosses are the vertices of the dual lattice.) The black line is the edge of the dual lattice. Four variables with values $\pm 1$ are placed on the black line, denoted as the four vectors in the middle of the black line.](figures/1.png){width="7cm"} ![More variables in the dual model. The black lines are the edges of the dual cubic lattice. The $a_i$’s, $b_i$’s, etc. are the variables in the dual model. Their values can be either $1$ or $-1$, and they need to satisfy the constraint - the product of four variables on each edge should be $1$, e.g. $a_1 a_2 a_3 a_4 = b_1 b_2 b_3 b_4 = \cdots = 1$.](figures/2.png){width="7cm"} The action of the dual gauge model is: $$\label{S-gauge} S_\gauge = - \alpha_1\sum_{\textrm{plaquettes}}[a_2 b_4 c_4 d_2] - \alpha_2\sum_{\textrm{edges}}([a_1 a_2] + [a_2 a_3]),$$ where “$[\cdot]$” means an example of the general idea, the “$[a_2 b_4 c_4 d_2]$” in the first term represents the product of the four variables on the edges of a plaquette, each pointing inward. (See Fig. 2 for the meaning of the letters.) The “$[a_1 a_2]$” or “$[a_2 a_3]$” in the second term represent the product of two nearby (but not opposite) variables on one edge. Because of the constraint on the four variables on an edge, i.e. $a_1 a_2 a_3 a_4 = 1$, there are $a_1 a_2 = a_3 a_4$ and $a_2 a_3 = a_4 a_1$. Therefore, the second term is actually symmetric: $a_1 a_2 + a_2 a_3 = (a_1 a_2 + a_2 a_3 + a_3 a_4 + a_4 a_1)/2$. ![The first term in Eq. \[S-gauge\] is shown by the plaquette on the left. The four variables on the edges of the plaquette should all point inward. The order of the subscripts in the second term in Eq. \[S-gauge\] is shown on the right.](figures/3-1.png "fig:"){height="2cm"} ![The first term in Eq. \[S-gauge\] is shown by the plaquette on the left. The four variables on the edges of the plaquette should all point inward. The order of the subscripts in the second term in Eq. \[S-gauge\] is shown on the right.](figures/3-2.png "fig:"){height="2cm"} The first and second terms in the dual action (Eq. \[S-gauge\]) are better illustrated in Fig. 3. Some examples of the second term are shown in Fig. 4. The coupling coefficients satisfy the dual relations: $$\begin{aligned} \sinh(2 \beta_1)\sinh(2 \alpha_1) &=& 1, \\ \sinh(2 \beta_2)\sinh(2 \alpha_2) &=& 1.\end{aligned}$$ Note that when $\beta_2 = 0$, i.e. there is no nnn coupling, we will get $\alpha_2 = \infty$, i.e. the second term in the dual action dominates and makes the system always at its “most probable state” (the state with the minimal action). From the examples of the second term in Fig. 4 we can see that the “most probable states” are when the four variables on each edge have the same value, which means we can assign a single value to each edge. This goes back to Wegner’s $Z_2$ lattice gauge theory [@Wegner1971]. ![The possible configurations of the four variables on an edge, and their corresponding contributions in Eq. \[S-gauge\]. Note that the “most probable states” are when $a_1 a_2 + a_2 a_3$ reaches the maximum value, i.e. the four variables being the same.](figures/3.png){width="5cm"} In our unusual gauge theory, the local gauge transformation is to flip the values of the variables on the edges that all connect to one vertex in the dual lattice. As in Fig. 2, it means that we change the signs of the $a_i$’s, $d_i$’s, $e_i$’s, $f_i$’s, $g_i$’s, and $h_i$’s. It is not hard to check that Eq. \[S-gauge\] is invariant under this local gauge transformation. The first and second terms in Eq. \[S-gauge\] are basic gauge invariants in our gauge theory. We can multiply the basic gauge invariants to get more complicated gauge invariants. Since the square of an Ising variable is always $1$, we can “glue” the plaquette terms with the edge terms in Eq. \[S-gauge\], and erase the vectors that appear twice. Repeating this process, we can see that any loop of edges with one vector on each edge is gauge invariant. Some examples are shown in Fig. 5. ![Examples of gauge invariants. We can decompose a single-vector-framing loop into basic gauge invariant terms by tilling the loop with plaquette terms and then changing the direction of the vectors on the framing by multiplying edge terms.](figures/4.png){width="7cm"} Proof of Duality {#Proof} ================ To prove the duality, we consider the high temperature expansion (HTE) of one model and the low temperature expansion (LTE) of the other. The HTE partition function of the Ising model is: $$\begin{aligned} \label{Z-Ising} Z_\Ising &=& \sum_\config e^{-S_\Ising} \nonumber \\ &=& \sum_\config \exp\left(\beta_1\sum_{nn}s_v^z s_{v'}^z + \beta_2\sum_{nnn}s_v^z s_{v''}^z\right) \nonumber \\ &\propto& \sum_\config \prod_{nn}\left(1+s_v^z s_{v'}^z\tanh\beta_1\right) \nonumber \\ && \qquad \times\prod_{nnn}\left(1+s_v^z s_{v''}^z\tanh\beta_2\right).\end{aligned}$$ In the last step, we have use the trick that $\exp(\beta s)=\cosh\beta\ (1 + s\tanh\beta)$ when the value of $s$ can only be $\pm1$. Because there is a sum over all configurations of the $s_v^z$’s in Eq. \[Z-Ising\], after the expansion of the product, any term that does not cancel out all $s_v^z$’s will be eliminated by the sum. The terms that cancel out all $s_v^z$’s are either $1$, or the loops constructed by the nn and nnn bonds. The basic element of such loops is the triangle with two nn bonds and one nnn bond. Such a basic loop contributes a term of $\tanh(\beta_1)^2 \tanh(\beta_2)$ in the HTE of the Ising partition function. All larger HTE loops can be tilled by the basic triangle loops. Therefore, the HTE Ising partition function becomes: $$Z_\Ising \propto 1+\sum_{\textrm{basic triangles}}\tanh(\beta_1)^2 \tanh(\beta_2)+\cdots.$$ The LTE partition function of the gauge theory is: $$\begin{aligned} \label{Z-gauge} Z_\gauge &=& \sum_\config e^{-S_\gauge} \nonumber \\ &=& \left.e^{-S_\gauge}\right\|_{\max} + \sum \left.e^{-S_\gauge}\right\|_{\textrm{excitations}} \nonumber \\ &\propto& 1+\sum \exp(S_\gauge\|_{\min}-S_\gauge\|_{\textrm{excitations}}).\end{aligned}$$ The simplest excitation of our dual gauge theory is to flip the signs of a pair of nearby variables on an edge. (Not to flip just one variable, because there is a constraint.) This flipping of signs affects two plaquettes and one edge, and thus contributes a term of $\exp(-4\alpha_1-2\alpha_2)$ in the LTE the partition function. Therefore, the LTE gauge partition function becomes: $$Z_\gauge \propto 1+\sum_{\textrm{simplest excitations}}\exp(-4\alpha_1-2\alpha_2)+\cdots.$$ There is a one-to-one correspondence between the basic triangle loops in the Ising model and the simplest excitations in the gauge theory. As shown in Fig. 6, the pair of flipped variables are on the edge that penetrates the triangle. The two variables are the ones that point towards the nn bonds. It is not hard to see that there are also correspondences between larger loops in the Ising model and higher excitations in the dual gauge theory. Therefore, we can see that the two models are dual if we choose $\tanh(\beta_1)=e^{-2\alpha_1}$ and $\tanh(\beta_2)=e^{-2\alpha_2}$. These conditions are equivalent to the dual relations we have shown in the previous section (Eq. 3 and 4). ![The duality of high temperature expansion (HTE) of the Ising model and the low temperature expansion (LTE) of the gauge theory.](figures/5.png){width="7cm"} Similarly, in the HTE of the gauge theory, we consider the “loops” (or “wraps”) constructed by the terms in the dual action, i.e. the plaquettes and the two-vector edges. The unit of such a wrap is a cube in the dual lattice with six plaquettes and twelve two-vector edges. Such a basic wrap contributes a term of $\tanh(\alpha_1)^6 \tanh(\alpha_2)^{12}$ in the HTE gauge partition function. As shown in Fig. 7, The corresponding LTE of the Ising model is to flip the sign of the spin that is at the center of the cube (or more generally, inside the wrap). This flipping of sign affects six nn bonds and twelve nnn bonds, and thus adds a term of $\exp(-12\beta_1-24\beta_2)$ in the LTE of the Ising partition function. There are also correspondences between larger wraps in the gauge theory and higher excitations in the Ising model. We can see that the two models are dual if we choose $\tanh(\alpha_1)=e^{-2\beta_1}$ and $\tanh(\alpha_2)=e^{-2\beta_2}$. These conditions are also equivalent to the dual relations Eq. 3 and 4. ![The duality of low temperature expansion (LTE) of the Ising model and the high temperature expansion (HTE) of the gauge theory. For clearness, not all plaquettes and edges are shown in the HTE of the gauge theory.](figures/6.png){width="7cm"} Wilson Loop =========== Like the Wilson loop in Wegner’s $Z_2$ lattice gauge theory [@Wegner1971; @Wilson1974], the gauge invariants in our gauge theory are also loops, but with vectors on the framing. We start with the two basic gauge invariants in our gauge theory, the all-inward-vector plaquettes and the two-vector edges. First, for a plaquette $a_2 b_4 c_4 d_2$ (see Fig. 2), the average value is: $$\begin{aligned} &&\frac{1}{Z_\gauge}\sum_\config a_2 b_4 c_4 d_2 \nonumber \\ &&\quad\times\exp\left(\alpha_1\sum_{\textrm{p}}[a_2 b_4 c_4 d_2] + \alpha_2\sum_{\textrm{e}}([a_1 a_2] + [a_2 a_3])\right) \nonumber\\ &=& \sum_\config a_2 b_4 c_4 d_2 \prod_{\textrm{p}}(1+[a_2 b_4 c_4 d_2]\tanh\alpha_1) \nonumber \\ &&\quad\times\prod_{\textrm{e}}(1+[a_1 a_2]\tanh\alpha_2)(1+[a_2 a_3]\tanh\alpha_2) \nonumber \\ &&/\left\{\sum_\config \prod_{\textrm{p}}(1+[a_2 b_4 c_4 d_2]\tanh\alpha_1)\right. \nonumber \\ &&\quad\left.\times\prod_{\textrm{e}}(1+[a_1 a_2]\tanh\alpha_2)(1+[a_2 a_3]\tanh\alpha_2)\right\}.\end{aligned}$$ The expansion of the numerator can be split into two terms: $$\begin{aligned} &&\sum_\config a_2 b_4 c_4 d_2 \prod_{\textrm{p}}(1+[a_2 b_4 c_4 d_2]\tanh\alpha_1) \nonumber \\ &&\quad\times\prod_{\textrm{e}}(1+[a_1 a_2]\tanh\alpha_2)(1+[a_2 a_3]\tanh\alpha_2) \nonumber \\ &=&\sum_\config a_2 b_4 c_4 d_2 \ (1+a_2 b_4 c_4 d_2\tanh\alpha_1) \nonumber \\ &&\quad\times\prod_{\textrm{p except } abcd}(1+[a_2 b_4 c_4 d_2]\tanh\alpha_1) \nonumber \\ &&\quad\times\prod_{\textrm{e}}(1+[a_1 a_2]\tanh\alpha_2)(1+[a_2 a_3]\tanh\alpha_2) \nonumber \\ &=&\sum_\config a_2 b_4 c_4 d_2 \prod_{\textrm{p except } abcd}(1+[a_2 b_4 c_4 d_2]\tanh\alpha_1) \nonumber \\ &&\quad\times\prod_{\textrm{e}}(1+[a_1 a_2]\tanh\alpha_2)(1+[a_2 a_3]\tanh\alpha_2) \nonumber \\ &&+\sum_\config \tanh\alpha_1 \prod_{\textrm{p except } abcd}(1+[a_2 b_4 c_4 d_2]\tanh\alpha_1) \nonumber \\ &&\quad\times\prod_{\textrm{e}}(1+[a_1 a_2]\tanh\alpha_2)(1+[a_2 a_3]\tanh\alpha_2) \nonumber \\ &=&\frac{1}{\tanh\alpha_1}\sum_\config a_2 b_4 c_4 d_2 \tanh\alpha_1 \nonumber \\ &&\quad\times\prod_{\textrm{p except } abcd}(1+[a_2 b_4 c_4 d_2]\tanh\alpha_1) \nonumber \\ &&\quad\times\prod_{\textrm{e}}(1+[a_1 a_2]\tanh\alpha_2)(1+[a_2 a_3]\tanh\alpha_2) \nonumber \\ &&+\tanh\alpha_1\sum_\config \prod_{\textrm{p except } abcd}(1+[a_2 b_4 c_4 d_2]\tanh\alpha_1) \nonumber \\ &&\quad\times\prod_{\textrm{e}}(1+[a_1 a_2]\tanh\alpha_2)(1+[a_2 a_3]\tanh\alpha_2) \nonumber \\ &=&\frac{1}{\tanh\alpha_1}\sum_{\textrm{wraps with }abcd}(\tanh\alpha_1)^{\cdots}(\tanh\alpha_2)^{\cdots} \nonumber \\ &&+\tanh\alpha_1\sum_{\textrm{wraps without }abcd}(\tanh\alpha_1)^{\cdots}(\tanh\alpha_2)^{\cdots}. \nonumber \\\end{aligned}$$ There are two types of terms in Eq. 10: one corresponding to a wrap that contains the plaquette $a_2 b_4 c_4 d_2$, and the other does not. Note that a wrap in the gauge theory maps to the flipping of the Ising variables inside the wrap. We denote the two Ising variables which are closest to the plaquette $a_2 b_4 c_4 d_2$ as $v$ and $v'$, and note that they are on the two sides of the plaquette and $\vv\in nn$. In the first type of terms, where the corresponding wrap contains the plaquette, the two Ising variables on the two sides of this plaquette must be one inside the wrap and one outside the wrap, and thus they have different signs. On the other hand, in the second type of terms, where the wrap does not contain the plaquette, these two Ising variables must have the same sign. Using the dual relation Eq. 3, we can rewrite Eq. 10 in the Ising model side as $$\begin{aligned} &&e^{2\beta_1}\sum_{v,v'\textrm{diff. signs}}\exp(S_\Ising\|_{\min}-S_\Ising) \nonumber \\ &&+e^{-2\beta_1}\sum_{v,v'\textrm{same sign}}\exp(S_\Ising\|_{\min}-S_\Ising) \nonumber \\ &=&\sum_\config e^{-2\beta_1 s_v^z s_{v'}^z}\exp(S_\Ising\|_{\min}-S_\Ising).\end{aligned}$$ Note that the denominator in Eq. 9 can be rewrite as $$\sum_\config \exp(S_\Ising\|_{\min}-S_\Ising).$$ Therefore, Eq. 9, the average value of a plaquette, becomes $$\frac{1}{Z_\Ising}\sum_\config e^{-2\beta_1 s_v^z s_{v'}^z}\exp(-S_\Ising),$$ which is the average value of $e^{-2\beta_1 s_v^z s_{v'}^z}$. $v$ and $v'$ are the two vertices on the two sides of the plaquette. The values on the Ising model side are easier to compute in a numerical simulation. ![The duality between the defect bonds in the Ising model and the gauge invariant loops in the gauge theory. Note that a two-vector edge can also be seen as a loop of two single-vector edges, going back and forth.](figures/8.png){width="7cm"} We can also combine the exponentials in Eq. 13, and make Eq. 9 into $$\frac{1}{Z_\Ising}\sum_\config \exp(-S_\Ising^)=\frac{Z_\Ising^}{Z_\Ising},$$ where $S_\Ising^=S_\Ising+2\beta_1 s_v^z s_{v'}^z$. Considering the definition of $S_\Ising$ in Eq. 1, we can interpret $S_\Ising^$ as that the coupling coefficient between $v$ and $v'$ becomes $-\beta_1$, i.e. that there is a defect nn bond. With similar methods, we can show that the average value of a two-vector edges in our gauge theory equals to the average value of $e^{-2\beta_2 s_v^z s_{v''}^z}$ where $v$ and $v''$ are nnn pair. It also equals to the partition function when there is a defect nnn bond. In Fig. 8, we summarize the relation between defect bonds in the Ising model and the average value of Wilson loops in our gauge theory. Any more complicated Wilson loop is a combination of these two basic cases. Discussion ========== In the previous sections, we have only talked about the 3D Ising model on an isotropic cubic lattice. Actually the duality even works for the very anisotropic case, in which there are three different nn coupling coefficients and six different nnn coupling coefficients. The dual gauge theory will have three different coefficients for the plaquettes in different orientations, and will have different coefficients for $[a_1 a_2]$ and $[a_2 a_3]$ in Eq. \[S-gauge\]. The dual relations keep simple as $\sinh(2\beta_{\cdots})\sinh(2\alpha_{\cdots})=1$. An interesting anisotropic case is when nnn coupling only occurs in the x-y plane. The third dimension can be seen as the time direction: it becomes an Ising model with 2D Hamiltonian. Because the nnn coupling in the x-z and y-z planes is zero, the edge term coefficients in the y and x edges in the gauge theory become infinity, which force the four variables on those edges become the same. Therefore, the dual gauge theory only has vector variables on the z-direction edges. With this simplification, we can write down a Hamiltonian form for this gauge theory. We do not know whether there is a Hamiltonian form for the gauge theory we discussed in the previous sections. The proof of duality in Sec. \[Proof\] can also be made with the method of path integral, however not as intuitive as the high temperature expansion and the low temperature expansion of the two models. For the future work, we will use numerical simulations to explore how the phase transition point changes with the ratio between the nnn and nn couplings. We will investigate the scaling laws of the Wilson loops around the critical region, and try to find out whether there is universality in our model. Theoretical approach will also be helpful to understand the features of this model. The author would like to express sincere gratitude to the mentor Prof. Anton Kapustin for the guidance. This work was supported by the Caltech SURF program. The author acknowledges the generous support of Dr. Neher and his family. Thanks to Yu-An Chen for many fruitful discussion. Some computations in this project were conducted on the Caltech High Performance Computing Cluster.
--- abstract: 'Let $E^n$ denote the (real) $n$-dimensional Euclidean space. It is not known whether an equilateral set in the $\l_1$ sum of $E^a$ and $E^b$, denoted here as $E^a \oplus_1 E^b$, has maximum size at least $\dim(E^a \oplus_1 E^b) + 1 = a + b + 1$ for all pairs of $a$ and $b$. We show, via some explicit constructions of equilateral sets, that this holds for all $a \le 27$, as well as some other instances.' author: - 'Aaron Lin[^1]' title: 'Equilateral sets in the $\l_1$ sum of Euclidean spaces' --- The Problem {#sec:problem} =========== An equilateral set in a normed space $(X, \lVert \cdot \rVert)$ is a subset $S \subset X$ such that given a fixed $\lambda > 0$, we have $\lVert x -y \rVert = \lambda$ for all distinct $x,y \in S$. Since norms respect scalar multiplication, the maximum size of an equilateral set in a normed space $X$ is well-defined, and we denote it by $e(X)$. When $\dim(X) = n$, we have the tight upper bound $e(X) \le 2^n$, proved in [@Petty] by Petty over 40 years ago. However, the following conjecture concerning a lower bound on $e(X)$, formulated also by Petty (amongst others), remains open for $n \ge 5$. (The $n=2$ case is easy; see [@Petty; @Vaisala] and [@Makeev] for the $n = 3$ and $4$ cases respectively.) \[conj:lower\_bound\] Let $X$ be an $n$-dimensional normed space. Then $e(X) \ge n + 1$. We wish to verify this conjecture for the Cartesian product $\RR^a \times \RR^b$, equipped with the norm $\lVert \cdot \rVert$ given by $$\lVert (x, y) \rVert = \lVert x \rVert_2 + \lVert y \rVert_2,$$ where $x \in \RR^a$, $y \in \RR^b$, and $\lVert \cdot \rVert_2$ denotes the Euclidean norm. We denote this space by $E^a \oplus_1 E^b$, and refer to it as the $\l_1$ sum of the Euclidean spaces $E^a$ and $E^b$. This was considered originally by Roman Karasev of the Moscow Institute of Physics and Technology, as a possible counterexample to Conjecture \[conj:lower\_bound\]. See [@Swanepoel Section 3] for more background on equilateral sets. The Results {#sec:result} =========== Observe that we need only construct $a + b + 1$ points in $E^a \oplus_1 E^b$ which form an equilateral set to show that $e(E^a \oplus_1 E^b) \ge \dim(E^a \oplus_1 E^b) + 1 = a + b + 1$. We will work with these points in the form $(x_i, y_i) \in \RR^a \times \RR^b$, since we can then examine the $x_i$’s and $y_i$’s separately when necessary. By abuse of notation, we will denote the origin of any Euclidean space by $o$. Let $d_n$ denote the circumradius of a regular $n$-simplex ($n \ge 1$) with unit side length. Note that $$d_n = \left( \sqrt{2+\frac{2}{n}} \right)^{-1}$$ is a strictly increasing function of $n$, and we have $1/2 \le d_n < 1/\sqrt{2}$. The $a = 1$ case is easy. \[prop:a=1\] $e(E^1 \oplus_1 E^b) \ge b + 2$. Let $y_1, \dotsc, y_{b+1}$ be the vertices of a regular $b$-simplex with unit side length centred on the origin. Then the points $(o, y_1), \dotsc, (o, y_{b+1}), (1-d_b, o)$ are pairwise equidistant. We next deal with the case where $b = a$. \[prop:a=b\] $e(E^a \oplus_1 E^a) \ge 2a + 1$. We first describe an equilateral set of size $2a$ in $E^a \oplus_1 E^a$: consider the set of points $\{ (v_i, \frac{1}{2}e_i) : i = 1, \dotsc, a\} \cup \{ (v_i, -\frac{1}{2}e_i) : i = 1, \dotsc, a\}$, where $v_1, \dotsc, v_a$ are the vertices of a regular simplex of codimension one, centred on the origin with side length $1 - 1/\sqrt{2}$, and $e_1, \dotsc, e_a$ are the standard basis vectors. Note that the $2a$ vectors $\pm \frac{1}{2} e_i$ for $i = 1, \dotsc, a$ form a cross-polytope in $E^a$, centred on the origin. We now want to add a point of the form $(x, o)$ to the above set, a unit distance away from every other point. Note that we must have $\lVert x - v_i \rVert_2 = 1/2$ for $i = 1, \dotsc, a$, and $x$ must lie on the one-dimensional subspace orthogonal to the $(a-1)$-dimensional subspace spanned by the $v_i$’s. This is realisable if $\lVert x - v_i \rVert_2 \ge (1 - 1/\sqrt{2})d_{a-1}$ (note that the $(a-1)$-simplex formed by the $v_i$’s has side length $1 - 1/\sqrt{2}$), in which case we have an equilateral set of size $2a + 1$ in $E^a \oplus_1 E^a$. But we have $$\frac{1}{2} > \frac{1}{\sqrt{2}} \left( 1 - \frac{1}{\sqrt{2}} \right) > \left( 1 - \frac{1}{\sqrt{2}} \right)d_{a-1}$$ for all $a \ge 2$. In the remaining case and our main result, we have $b > a \ge 2$, and we find sufficient conditions for an equilateral set of size $a + b + 1$ to exist in $E^a \oplus_1 E^b$. \[thm\] Let $b > a \ge 2$. Let $c = \lfloor 1 + b/(a + 1) \rfloor$, $\beta = b \pmod{a+1} \in \{0, \dotsc, a\}$, and $\alpha = a + 1 - \beta$. If $\beta = 0$, $1$, or $a$, or $$\label{eqn:hard_cond_ugly} \frac{\alpha - 1}{2\alpha} \left( 1 - \sqrt{\frac{c-1}{c}} \right)^2 + \frac{\beta - 1}{2\beta} \left( 1 - \sqrt{\frac{c}{c+1}} \right)^2 \le \left( 1 - \sqrt{\frac{1}{2} \left( \frac{c-1}{c} + \frac{c}{c+1} \right)} \right)^2$$ holds, then $e(E^a \oplus_1 E^b) \ge a + b + 1$. Note that if inequality (\[eqn:hard\_cond\_ugly\]) is satisfied by all pairs of $a$ and $b$ with $b > a \ge 2$ and $b \ne 0$, $1$, or $a \pmod{a+1}$, then Proposition \[prop:a=1\], Proposition \[prop:a=b\], and Theorem \[thm\] cover all possible cases, as $E^a \oplus_1 E^b$ is isometrically isomorphic to $E^b \oplus_1 E^a$. Unfortunately, this is not true, and we explore its limitations after the proof of Theorem \[thm\]. We are going to describe an equilateral set of size $a + b + 1$ with unit distances between points. Noting that $\alpha \cdot (c-1) + \beta \cdot c = b$, consider the following decomposition of $E^b$ into pairwise orthogonal subspaces: $$E^b = U_1 \oplus \dotsb U_\alpha \oplus V_1 \oplus \dotsb \oplus V_\beta,$$ where $\dim U_i = c - 1$ for $i = 1, \dotsc, \alpha$ and $\dim V_j = c$ for $j = 1, \dotsc, \beta$. Let $u_1^{(i)}, \dotsc, u_c^{(i)}$ be the vertices of a regular $(c-1)$-simplex with unit side length centred on the origin in $U_i$, and let $v_1^{(j)}, \dotsc, v_{c+1}^{(j)}$ be the vertices of a regular $c$-simplex with unit side length centred on the origin in $V_j$. The $a + b + 1$ points of our equilateral set will be $$\left \{ \left(w_i, u_k^{(i)} \right) : 1 \le i \le \alpha, 1 \le k \le c \right \} \cup \left \{ \left(z_j, v_\ell^{(j)} \right) : 1 \le j \le \beta, 1 \le \ell \le c + 1 \right \}.$$ Note here that $\alpha \cdot c + \beta \cdot (c+1) = a + b + 1$, and we have $\lVert u_k^{(i)} - u_{k'}^{(i)} \rVert_2 = \lVert v_\ell^{(j)} - u_{\ell'}^{(j)} \rVert_2 = 1$ for $k \ne k'$ and $\ell \ne \ell'$. All that remains is then to calculate how far apart the $w_i$’s and $z_j$’s should be in $E^a$, and see if such a configuration is realisable. We only have three non-trivial distances to calculate: - the distance between $\left(z_j, v_\ell^{(j)} \right)$ and $\left(z_{j'}, v_{\ell'}^{(j')} \right)$ for $j \ne j'$ should be one, and so $$\lVert z_j - z_{j'} \rVert_2 = 1 - \sqrt{d_{c}^2 + d_{c}^2} = 1 - \sqrt{\frac{c}{c+1}} =: f(c),$$ - the distance between $\left(w_i, u_k^{(i)} \right)$ and $\left(w_{i'}, u_{k'}^{(i')} \right)$ for $i \ne i'$ should be one, and so $$\lVert w_i - w_{i'} \rVert_2 = 1 - \sqrt{d_{c-1}^2 + d_{c-1}^2} = 1 - \sqrt{\frac{c-1}{c}} = f(c-1),$$ - finally, the distance between $\left(w_i, u_k^{(i)} \right)$ and $\left(z_{j}, v_{\ell}^{(j)} \right)$ should also be one, and so $$\lVert w_i - z_{j} \rVert_2 = 1 - \sqrt{d_{c-1}^2 + d_c^2} = 1 - \sqrt{\frac{1}{2} \left( \frac{c-1}{c} + \frac{c}{c+1} \right)} =: g(c).$$ What we need in $E^a$ is thus a regular $(\alpha - 1)$-simplex with side length $f(c-1)$ and a regular $(\beta - 1)$-simplex with side length $f(c)$, with the distance between any point from one simplex and any point from the other being $g(c)$. Note that here we consider the $(-1)$-simplex to be empty. We now show that this configuration is realisable (in $E^a$) if the conditions in the statement of the theorem are satisfied. We first consider the special cases $\beta = 0$ and $\beta = 1$ or $a$, and then the main case $2 \le \beta \le a-1$. It is trivial if $\beta = 0$: then $\alpha = a + 1$ and we only need to find a regular $a$-simplex with side length $f(c-1)$ in $E^a$. If $\beta = 1$, in which case $\alpha = a$, consider the decomposition $E^a = E^{a-1} \oplus E^1$. Consider the points $(p_1, o), \dotsc, (p_a, o)$, where $p_1, \dotsc, p_a$ are the vertices of a regular $(a-1)$-simplex with side length $f(c-1)$, centred on the origin in $E^{a-1}$. We want to add a point $(o, \zeta)$ for some $\zeta \in E^1$ such that, for any $i = 1, \dotsc, a$, we have $$\lVert (p_i, o) - (o, \zeta) \rVert_2 = g(c),$$ or equivalently, $$d_{a-1}^2 f(c-1)^2 + \zeta^2 = g(c)^2.$$ Noting that $d_{a-1} < 1/\sqrt{2}$, it suffices to show, for all $c \ge 2$, that $$f(c-1)^2 < 2 g(c)^2.$$ But this is easily verifiable to be true, and so the desired $a$-simplex exists in $E^a$. By symmetry and the fact that $f(c)^2 < f(c-1)^2$, the desired $a$-simplex also exists if $\beta = a$. Now suppose $2 \le \beta \le a - 1$ so that $\alpha, \beta \ge 2$. Consider this time, the decomposition $E^a = E^{\alpha - 1} \oplus E^{\beta - 1} \oplus E^1$, noting that $\alpha + \beta = a + 1$. Suppose $p_1, \dotsc, p_\alpha$ are the vertices of a regular $(\alpha-1)$-simplex with side length $f(c-1)$, centred on the origin in $E^{\alpha-1}$, and $q_1, \dotsc, q_\beta$ are the vertices of a regular $(\beta-1)$-simplex with side length $f(c)$, centred on the origin in $E^{\beta-1}$. Consider then the set of points $\{ (p_i, o, o) : i = 1, \dotsc, \alpha \} \cup \{ (o, q_j, \zeta) : j = 1, \dotsc, \beta \}$, where $\zeta \in E^1$ is to be determined. As before, we want a $\zeta$ such that for all $i$ and $j$, we have $$\lVert (p_i, o, o) - (o, q_j, \zeta) \rVert_2 = g(c),$$ or equivalently $$\label{eqn:hard_cond_nice} \left( d_{\alpha-1} f(c-1) \right)^2 + \left( d_{\beta-1} f(c) \right)^2 \le g(c)^2.$$ But this is exactly inequality (\[eqn:hard\_cond\_ugly\]). As mentioned above, inequality (\[eqn:hard\_cond\_ugly\]), and thus inequality (\[eqn:hard\_cond\_nice\]), does not hold for all pairs of $a$ and $b$. However, we have the following result. \[lem\] If $b \ge a ^2 + a$, then inequality (\[eqn:hard\_cond\_nice\]) holds. Since $f(n)$ is a decreasing function of $n$, inequality (\[eqn:hard\_cond\_nice\]) holds if $a$ and $b$ satisfy $$\left( d_{\alpha-1}^2 + d_{\beta-1}^2 \right) f(c-1)^2 < g(c)^2.$$ Using the fact that $\alpha = a + 1 - \beta$ implies $d_{\alpha-1}^2 + d_{\beta-1}^2 \le (a-1)/(a+1)$, we therefore just need $a$ and $b$ to satisfy $$\frac{a-1}{a+1} < \left( \frac{g(c)}{f(c-1)} \right)^2.$$ But the latter expression is an increasing function of $c$, and so if $c \ge a$, or equivalently, when $b \ge a^2 + a$, we need only consider the inequality $$\frac{a-1}{a+1} < \left( \frac{g(a)}{f(a-1)} \right)^2,$$ which is then easily verifiable to be true. It can be checked (by computer) that inequality (\[eqn:hard\_cond\_nice\]) holds for all $a \le 27$, but does not hold for $a = 28$ and $b = 40$, $a = 29$ and $39 \le b \le 44$, and $a = 30$ and $40 \le b \le 47$. The spaces of smallest dimension where we could not find an equilateral set of size $a + b + 1$ are $E^{28} \oplus_1 E^{40}$ and $E^{29} \oplus_1 E^{39}$. Acknowledgements {#acknowledgements .unnumbered} ---------------- The author would like to thank Konrad Swanepoel for introducing him to this problem, and for the numerous helpful suggestions in writing this up. [^1]: Department of Mathematics, London School of Economics and Political Science, London, United Kingdom.
--- abstract: 'This paper presents a novel autonomous surface vessel (ASV), called Roboat II for urban transportation. Roboat II is capable of accurate simultaneous localization and mapping (SLAM), receding horizon tracking control and estimation, and path planning. Roboat II is designed to maximize the internal space for transport, and can carry payloads several times of its own weight. Moreover, it is capable of holonomic motions to facilitate transporting, docking, and inter-connectivity between boats. The proposed SLAM system receives sensor data from a 3D LiDAR, an IMU, and a GPS, and utilizes a factor graph to tackle the multi-sensor fusion problem. To cope with the complex dynamics in the water, Roboat II employs an online nonlinear model predictive controller (NMPC), where we experimentally estimated the dynamical model of the vessel in order to achieve superior performance for tracking control. The states of Roboat II are simultaneously estimated using a nonlinear moving horizon estimation (NMHE) algorithm. Experiments demonstrate that Roboat II is able to successfully perform online mapping and localization, plan its path and robustly track the planned trajectory in the confined river, implying that this autonomous vessel holds the promise on potential applications in transporting humans and goods in many of the waterways nowadays.' author: - ' Wei Wang, Tixiao Shan, Pietro Leoni, David Fernández-Gutiérrez, Drew Meyers, Carlo Ratti and Daniela Rus [^1] [^2] [^3]' bibliography: - 'HalfScaleRoboatAutonomyIROS2020.bib' title: 'Roboat II: A Novel Autonomous Surface Vessel for Urban Environments ' --- Introduction ============ The increasing needs for water-based navigation in areas such as oceanic monitoring, marine resource exploiting, and hydrology surveying have all led to a strong demand from commercial, scientific, and military communities for the development of innovative autonomous vessels (ASVs) [@Corke2007a; @Doniec1648; @Leonard1639838; @paull2014auv; @Dhariwal4399056; @LIU201671]. ASVs also have a promising role in the future of transportation for many coastal and riverside cities such as Amsterdam and Venice, where some of the existing infrastructures like roads and bridges is always overburdened. A fleet of eco-friendly self-driving vessels could shift the transport behaviors from the roads to waterways, possibly reducing street traffic congestion in these water-related cities. Much progress has been made on ASV autonomy in the last several decades [@LIU201671], such as localization [@Corke2007a; @Dhariwal4399056; @paull2014auv], object detection [@Sukhatme5980509], path planning [@Dhariwal4399056; @Sukhatme21767; @shan2020receding] and tracking control [@klinger2017control; @GuerreiroSilvestre1868]. However, current ASVs are usually developed for open waters[@GuerreiroSilvestre1868; @Manley5289429; @LIU201671] and thus cannot satisfactorily meet the autonomy requirements for applications in narrow and crowded urban water environments such as Amsterdam canals. Developing an autonomy system for vessels in urban waterways is more challenging than for traditional ASVs in open water environments. This paper focuses on the desig of localization and control problems for urban ASVs. First, to safely navigate in urban waterways, an ASV should localize itself with centimeter-level or decimeter-level accuracy. Current ASVs usually use GPS and IMU (fused by an extended Kalman filter (EKF) or unscented Kalman filter (UKF)) which typically results in a meter-level precision [@LIU201671]. These GPS-IMU-based approaches can be unstable in urban waterways, where GPS signals are often severely attenuated. A reliable multi-sensor navigation system which includes GPS, compass, speed log, and a depth sensor to account for sensor failure was proposed, but it cannot guarantee high accuracy [@naeem2012integrated]. To date, there is no feasible solution for accurate urban ASV localization. Second, a number of tracking control methods such as sliding mode method [@ashrafiuon2008sliding], integrator back-stepping method [@khalil1996noninear; @klinger2017control] and adaptive control [@skjetne2004nonlinear] have been proposed for ASVs. However, most of current controllers are either verified by simulation or partly verified in open waters which do not care too much of the tracking accuracy. Moreover, many controllers use kinematic model instead of a dynamical one for the vessels. The control performance will always decline a lot due to the highly non-linearity of the water and the persistent disturbances in real environments. Our recently launched Roboat project aims at developing a fleet of autonomous vessels for transportation, and constructing dynamic floating infrastructure [@WeiICRA2018; @WeiIROS2019; @Mateos8793525; @WangMRS2020] (e.g., bridges and stages) in the city of Amsterdam. In our previous work [@WeiICRA2018; @WeiIROS2019], we have designed a quarter-scale Roboat that was able to localize itself using LiDAR and track on the reference trajectory using NMPC. By contrast, this paper develops a new large-scale vessel, Roboat II, which is potentially capable of carrying passengers. Moreover, we develop the new SLAM algorithm, NMHE state estimation algorithm and adapt the NMPC for large-scale Roboat II. The main contributions of our work can be summarized as follows:\ $\tiny {\bullet}$ Designing and building of a new vessel, Roboat II;\ $\tiny {\bullet}$ NMHE state estimation for autonomous vessels;\ $\tiny {\bullet}$ Simultaneous localization and mapping (SLAM) algorithm for urban vessels;\ $\tiny {\bullet}$ NMPC-NMHE control strategy with full state integration for accurate tracking;\ $\tiny {\bullet}$ Extensive experiments to validate the developed autonomy system in rivers. This paper is structured as follows. Section II overviews the Roboat prototype. Section III describes the framework of the developed autonomy system for urban ASVs. NMPC and NMHE are described in Section IV. SLAM algorithm is presented in Section V. Experiments are presented in Section VI. Section VII concludes this paper. Roboat Design ============= Hull Design ----------- Roboat II is designed starting from two main principles: simple construction and inter-connectivity between vessels. The former principle leads to define the hull shape using single-curvature surfaces. Regarding the latter, the ability to connect multiple units in the water to build larger structures, side by side or perpendicularly, dictated the 1:2 ratio. In this way, two vessels are able to dock with their short sides on the long, lateral side of a third one if needed. Also, this means an even distribution of the location of the connectors, leaving as empty bays spaced along the sides for easy accommodation of latching modules, as shown in Fig. \[Roboathull\]. ![Mechanical design of Roboat II. []{data-label="Roboathull"}](Figures/roboathull2){width="1.0\linewidth"} Bolted to the structure, these modules are fast to replace, iteration after iteration, without further interventions on the hull itself. Being those the points where the vessel are connected, they are also particularly subjected to mechanical stress. Thus, a structural rib is placed on each of their sides to distribute the forces. With a connector every 500 mm, it is their position that dictate the internal ribs distribution. Other than this, the need for a unit capable of maximizing the internal space and not required to move fast (cruise speed is 7.5 km/h) resulted in a bulky shape, perfectly symmetrical on the two main axes, and with the control system able to adjust the thrust to move in any desired direction. Marine plywood has been preferred because its lower cost and easier workability. Similar as if we were using aluminum foil, the easier option for the assembly was to design the hull as a combination of CNC cut sheets, 4 to 6 mm thick and bent if needed along a single direction, rather than pressed to achieve double curvature. A fiberglass coating was still applied on the exterior, with the interior space being easily accessible through 8 waterproof hatches on the top deck. Furthermore, the whole top deck was not glued permanently to the hull but bolted instead, with a neoprene gasket in the middle. This allows accessing the internal volume in situations where the hatches are not large enough, such as for extended intervention on the inside structure or for permanent placement of large components. Hardware -------- We use a small-form-factor barebone computer, Intel^^ NUC (NUC7i7DNH) as the main processor of our vessel which runs a robotics middleware, Robot Operating System (ROS). An auxiliary STM32 processor is used for converting the calculated forces from the controller to actuator signals. Roboat also has several onboard sensors, as shown in Fig. \[HalfscaleRoboat\]. More specifications of Roboat are listed in Table \[technicalspecifications\]. ![Hardware design of Roboat II.[]{data-label="HalfscaleRoboat"}](Figures/sensor-setup.pdf){width="0.7\linewidth"} Items Characteristics --------------------------------- ------------------------------------------- Dimension (L$\times$W$\times$H) 2.0 m$ \times $ 1.00 m $ \times $ 0.475 m Total mass $\sim$ 80.0 kg Center of gravity height $\sim$ 175 mm Drive mode Four T200 thrusters Onboard sensors 3D LIDAR, IMU, Camera, GPS Power supply 14.8 V, 22A$\cdot$h Li-Po battery Operation time $\sim$ 2.0 hours Control mode Autonomous/Wireless mode Maximum speed 1.0 Body Length/s : Technical specifications of the prototype[]{data-label="technicalspecifications"} Hydrodynamic Analysis with CFD Simulation ----------------------------------------- The mechanical model described in the previous sections allows us to determine the hydrostatic coefficients of the prototype. Figure \[fig:vol\_WS\] shows the mass displacement and wetted surface as functions of the vessel draught assuming it floats in brackish water (density $\sim1010$ kg/m$^3$), which have a direct impact on its dynamics. ![Displaced mass and wetted surface values as function of draught.[]{data-label="fig:vol_WS"}](Figures/hydrostatics_mass_WS){width="0.7\linewidth"} Based on the total mass and center of gravity height of the tested prototype from Table \[technicalspecifications\], the expected draught of the prototype from Fig. \[fig:vol\_WS\] is 107 mm. Given that these Roboat units will operate in environments with wave heights below 10 cm, which yield roll and pitch moments $\sim$5 and $\sim$10 kg$\cdot$m, respectively, we expect roll and pitch angles below 5$^{\circ}$. These outcomes validate the assumption of planar motion followed in the dynamic model described in Sec. \[ss:dynamic\_model\]. Moreover, \[fig:vol\_WS\] also shows the capability to carry a much larger payload than the quarter-scale model [@WeiICRA2018]. Roboat II is able to carry two people on-board (see supplementary material). Additionally, we used the SolidWorks^^ Flow Simulation package to evaluate numerically the hydrodynamic response of the Roboat under the design operating conditions. Figure \[fig:CFD\_sim\] shows a side view with the pressure distribution over the hull as the Roboat moves forward at 0.8 m/s, as well as the velocity streamlines around it. The results show how the smooth hull efficiently deflects the incident flow, minimally perturbing the free surface and avoiding high dynamic pressure concentrations. ![CFD results from SolidWorks^^ Flow Simulation with the Roboat moving forward at 0.8 m/s.[]{data-label="fig:CFD_sim"}](Figures/F08_rough_crop){width="1.0\linewidth"} Autonomy Framework of Urban Vessels =================================== In this section, we describe the autonomy framework of our Roboat II which can move in urban waterways, as is shown in Fig. \[AutonomyFramework\]. ![Current autonomy framework of Roboat II. It mainly contains a planner,a SLAM module, an NMPC tracking module, an NMHE state estimator, and a simple object detector. []{data-label="AutonomyFramework"}](Figures/AutonomyFramework.pdf){width="0.9\linewidth"} The autonomous Roboat conducts a transportation task in the canals as follow. When a task such as passenger delivery is required from a user at a specific position, the system coordinator will assign this delivery task to an unoccupied Roboat that is closest to the passenger. When Roboat picks up the passenger, it will first plan its path to the destination and then generate a feasible path based on the current traffic condition. Then, Roboat starts to localizes itself by running the developed SLAM algorithm which utilizes LiDAR, IMU and GPS sensors. The NMPC controller will accurately and robustly track the reference trajectories from the planner during the whole task in an urban waterway. The planner will avoid obstacles if it receives obstacle information from a obstacle detector. We use a simple A$^{\ast}$ planner for the path generation and focus on SLAM, dynamics modeling and identification, NMPC tracking and NMHE state estimation in this study. Receding Horizon control and Estimation ======================================= Nonlinear model predictive control (NMPC) is a dynamic optimization-based strategy for feedback control which determines the current control action by optimizing the system behavior over a finite window, often referred to as prediction horizon. The NMPC controller is responsible for tracking the calculated optimal trajectory from the path planner. The states of the vessel are simultaneously estimated using an nonlinear moving horizon estimation (NMHE) algorithm. These estimated states are further fed to the NMPC, as shown in Fig. \[AutonomyFramework\]. The performance of the NMPC largely relies on the selected system model. Therefore, we build a dynamical model for Roboat II, and experimentally identify the unknown parameters in the model to achieve superior performance for NMPC control. Dynamical Model {#ss:dynamic_model} --------------- The dynamics of our vessel is described by the following nonlinear differential equation [@WeiICRA2018] $$\begin{aligned} \label{MFCPredynamics} &&\dot{\mathbf{x}}=\mathbf{T}(\mathbf{x})\mathbf{v} \label{MFCPredynamicsA}\\ &&\dot{\mathbf{v}}=\mathbf{M}^{-1}(\mathbf{\tau}+\mathbf{\tau}_{\text{env}})-\mathbf{M}^{-1}(\mathbf{C}(\mathbf{v})+\mathbf{D}(\mathbf{v}))\mathbf{v} \label{MFCPredynamicsB}\end{aligned}$$ where $\mathbf{x}=[x~y~\psi]^{T} \in\mathbb{R}^{3\times1}$ is the position and heading angle of the vessel in the inertial frame; $\mathbf{v}=[u~v~r]^{T}\in \mathbb{R}^{3\times1}$ denotes the vessel velocity, which contains the surge velocity ($u$), sway velocity ($v$), and yaw rate ($r$) in the body fixed frame; $\mathbf{T}(\mathbf{x})\in \mathbb{R}^{3\times3}$ is the transformation matrix converting a state vector from body frame to inertial frame; $\mathbf{M} \in \mathbb{R}^{3\times3}$ is the positive-definite symmetric added mass and inertia matrix; $\mathbf{C}(\mathbf{v})\in\mathbb{R}^{3\times3}$ is the skew-symmetric vehicle matrix of Coriolis and centripetal terms; $\mathbf{\tau}_{\text{env}} \in\mathbb{R}^{3\times1}$ is the environmental disturbances from the wind, currents and waves; $\mathbf{D}(\mathbf{v})\in\mathbb{R}^{3\times3}$ is the positive-semi-definite drag matrix-valued function; $\mathbf{\tau}=[\tau_u~ \tau_v~ \tau_r]^{T} \in\mathbb{R}^{3\times1}$ is the force and torque applied to the vessel in all three DOFs, which is defined as follow $$\begin{aligned} \label{AppliedForceMaxtrix} \mathbf{\tau} =\mathbf{B}\mathbf{u} = \left[ \begin{array}{cccc} 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 1\\ \dfrac{a}{2}&-\dfrac{a}{2} & \dfrac{b}{2} &-\dfrac{b}{2} \end{array} \right] \left( \begin{array}{c} f_1\\ f_2\\ f_3\\ f_4 \end{array} \right)\end{aligned}$$ where $\mathbf{B}\in\mathbb{R}^{4\times3}$ is the control matrix describing the thruster configuration and $\mathbf{u}=[f_1~f_2~f_3~f_4]^{T}\in\mathbb{R}^{4\times1}$ is the control vector where $f_1$, $f_2$, $f_3$ and $f_4$ represent the left, right, anterior, and rear thrusters, respectively; $a$ is the distance between the transverse propellers and $b$ is the distance between the longitudinal propellers. $\mathbf{M}$, $\mathbf{C}(\mathbf{v})$ and $\mathbf{D}(\mathbf{v})$ are mathematically described as follows: $$\begin{aligned} \mathbf{M}=\text{diag}\{m_{11}, m_{22}, m_{33}\} \label{InertiaMaxtrix}\end{aligned}$$ $$\begin{aligned} \mathbf{C}(\mathbf{v})= \left[ \begin{array}{ccc} 0 & 0 & -m_{22}v\\ 0 &0 & m_{11}u\\ m_{22}v &-m_{11}u &0 \end{array} \right]\end{aligned}$$ $$\begin{aligned} \mathbf{D}(\mathbf{v})=\text{diag} \{X_u, Y_v, N_r\}\label{LinearDrag}\end{aligned}$$ Further, by combining (\[MFCPredynamicsA\]) and (\[MFCPredynamicsB\]), the complete dynamic model of the vessel is reformulated as follow $$\begin{aligned} \label{MPCdynamics} \dot{\mathbf{q}}(t)=f(\mathbf{q}(t),\mathbf{u}(t))\end{aligned}$$ where $\mathbf{q}=[x~y ~\psi~ u~ v~ r]^{T}\in \mathbb{R}^{6\times1}$ is the state vector of the vessel, and $f(\cdot, \cdot, \cdot): \mathbb{R}^{n_q}\times \mathbb{R}^{n_u} \times \mathbb{R}^{n_p}\longrightarrow \mathbb{R}^{n_q}$ denote the continuously differentiable state update function. The system model describes how the full state $\mathbf{q}$ changes in response to applied control input $\mathbf{u} \in\mathbb{R}^{4\times1}$. Similarly, a nonlinear measurement model denoted $\mathbf{h}(t)$ can be described with the following equation: $$\begin{aligned} \label{measurementequation} \mathbf{z}(t)=h(\mathbf{q}(t),\mathbf{u}(t))\end{aligned}$$ where $\mathbf{z}=[ x~y ~\psi~ r~f_1~f_2~f_3~f_4] \in \mathbb{R}^{8\times1}$ denotes the measurement vector, and $h(\cdot, \cdot): \mathbb{R}^{n_q}\times \mathbb{R}^{n_u} \longrightarrow \mathbb{R}^{n_z}$ denote measurement function. Next, the unknown hydrodynamic parameter vector, $\xi=[m_{11}~m_{22}~m_{33}~X_u~Y_v~N_r]^{T}$, in the dynamical model is required to be identified before applying it to the controller. The estimation of $\xi$ using the experimental data set ${\mathbf{v}^{\text{s}},\mathbf{u}^{\text{s}}}$ is a grey-box identification problem. The identification can be treated as an optimization problem described below \[modelidentificationformulation\] $$\begin{aligned} &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\underset{\xi}{\text{min}}~{\sum}_{t=0}^{T_s} \varepsilon(t)^{T}w\varepsilon(t),\\ \text{s.t.}~~ &&\xi_{l} \leq \xi \leq \xi_{u}, \\ &&\mathbf{q}(t)=f(\mathbf{q}(t),\mathbf{u}(t),\xi), t\in[0 ~T_s],\end{aligned}$$ where $\varepsilon(t)\in\mathbb{R}^{3\times1}$ is the deviation between the experimental velocity $\mathbf{v}^s(t)$ and the simulated velocity $\mathbf{v}^m(t)$ at time $t$. $\xi_l$ and $\xi_u$ are the lower and upper bounds of $\xi$, respectively. $w\in\mathbb{R}^{3\times3}$ represents the weight matrix for the optimization. Different from our previous work [@WeiICRA2018], we adopt Sequential Quadratic Programming (SQP) method to numerically solve (\[modelidentificationformulation\]) in this study because SQP satisfies bounds at all iterations and (\[modelidentificationformulation\]) is not a large-scale optimization problem. Nonlinear Model Predictive Control {#NMPCsection} ---------------------------------- For our trajectory tracking problem involving online dynamics learning, we formulate the optimal control problem for NMPC in the form of a least square function to penalize the deviations of predicted state ($\mathbf{q}_k$) and control ($\mathbf{u}_k$) trajectories from their specified references, over the given prediction horizon window $N_c$ ($t_j\leq t \leq t_{j+N_c}$): \[OCP\] $$\begin{aligned} \underset{\mathbf{q}_k,\mathbf{u}_k}{\text{min}}\dfrac{1}{2}\Big\{\sum_{k=j}^{j+N_c-1}(\\|\mathbf{q}_k-\mathbf{q}_k^{\text{ref}}\\|^2_{W_q}+\\|\mathbf{u}_k-\mathbf{u}_k^{\text{ref}}\\|^2_{W_u})+ \label{OCPequation}\\ \\|\mathbf{q}_{N_c}-\mathbf{q}_{N_c}^{\text{ref}}\\|^2_{W_{N_c}}\Big\} \nonumber\\ \text{s.t.}~~\mathbf{q}_j= \hat{\mathbf{q}}_j,\\ \mathbf{q}_{k+1}=f(\mathbf{q}_k,\mathbf{u}_k), k=j,\cdot \cdot \cdot, j+N_c-1,\\ \mathbf{q}_{k,\text{min}}\leq\mathbf{q}_k\leq \mathbf{q}_{k,\text{max}}, k=j,\cdot \cdot \cdot, j+N_c,\\ \mathbf{u}_{k,\text{min}}\leq\mathbf{u}_k\leq \mathbf{u}_{k,\text{max}}, k=j,\cdot \cdot \cdot, j+N_c-1,\end{aligned}$$ where $\mathbf{q}_k \in \mathbb{R}^{n_q}$ denotes the vessel state, $\mathbf{u}_k \in \mathbb{R}^{n_u}$ denotes the control input, $ \hat{\mathbf{q}}_j \in \mathbb{R}^{n_q}$ denotes the current state estimate, $\mathbf{q}_k^{\text{ref}}$ and $\mathbf{u}_k^{\text{ref}}$ denote the time-varying state and control references, respectively; $\mathbf{q}_{N_c}^{\text{ref}}$ denotes the terminal state reference; $W_q\in \mathbb{R}^{n_q\times n_q}$, $W_u\in \mathbb{R}^{n_u\times n_u}$, and $W_{N_c}\in \mathbb{R}^{n_q\times n_q}$ are the positive definite weight matrices that penalize deviations from the desired values. These weight matrices are assumed constant for a certain scale vessel in this study. Moreover, $\mathbf{p}$ is the parameter vector in the model which is referred as the payload of the vessel in this study. Furthermore, $\mathbf{q}_{k,\text{min}}$ and $\mathbf{q}_{k,\text{max}}$ denote the lower and upper bounds of the states, respectively; $\mathbf{u}_{k,\text{min}}$ and $\mathbf{u}_{k,\text{max}}$ denote the lower and upper bounds of the control input, respectively. The weighting matrices $W_q$, $W_u$ and $W_{N_c}$ for the NMPC used in the experiments are selected as $$\begin{aligned} &&W_q=\text{diag}\{200,200,100,10,10,10\}\\ &&W_{N_c}=\text{diag}\{1000,1000,500,50,50,150\} \\ && W_u=\text{diag}\{1,1,1,1\} \end{aligned}$$ The prediction horizon $N_c = 4$ s, and the constraints on the control input $\mathbf{u}$ used in the experiments are chosen as follow $$\begin{aligned} \label{forcelimits} \mathbf{-50}_{4\times 1}~\text{N} \leq\mathbf{u}_{\text{quart}}\leq \mathbf{50}_{4\times 1} ~\text{N}\end{aligned}$$ Nonlinear Moving Horizon Estimation ----------------------------------- Online state estimation is employed to further generate more accurate states considering the dynamics and state constraints of the vessel. Nonlinear Moving Horizon Estimation (NMHE) is an online optimization-based state estimation approach that can handle nonlinear systems and satisfy inequality constraints on the estimated states and parameters [@Rao1178905]. In a similar manner, we utilize a least square function to penalize the deviation of estimated outputs from the measurements, formulating the NMHE problem as follows: At current time $t_j$ there shall be $N_c$ measurements $\mathbf{z}_{j-M+1}, ..., \mathbf{z}_j \in \mathbb{R}^{n_z}$ available, associated to the time instants $t_{j-N_c+1}< ...< t_j$ in the past. $T_{\text{E}}=t_j-t_{j-N_c+1}$ is the length of the horizon. Finally, the discrete time dynamic optimization problem to estimate the constrained states ($\hat{\mathbf{q}}$) at time $t_j$ using the available measurements within the horizon, is solved by \[ObjectiveFunctionMHE\] $$\begin{aligned} \underset{\hat{\mathbf{q}}_k}{\text{min}}\\|\hat{\mathbf{q}}_{j-N_c+1}-\bar{\mathbf{q}}_{j-N_c+1})\\|^{2}_{P_L}\!\!+\!\!\!\!\!\!\sum^{j}_{k=j-N_c+1}\!\!\!\!\\|\mathbf{z}_k-\mathbf{h}(\hat{\mathbf{q}}_k,\mathbf{u}_k)\\|^{2}_{R_T}\\ \text{s.t.}~ \hat{\mathbf{q}}_{k+1}=f(\hat{\mathbf{q}}_{k+1},\mathbf{u}_k)+\mathbf{w}_k, k=j-N_c+1,..., j-1, \\ \hat{\mathbf{q}}_{k,\text{min}}\leq\hat{\mathbf{q}}_k\leq \hat{\mathbf{q}}_{k,\text{max}}, k=j-N_c+1,..., j,\end{aligned}$$ where $\hat{\mathbf{q}}_{k,\text{min}}$ and $\hat{\mathbf{q}}_{k,\text{max}}$ denote the lower and upper bounds on the estimated states of the vessel, respectively. The weighting matrices $P_{N_c}$ and $R_k$ are usually interpreted as the inverses of the measurement and process noise covariance matrices, respectively. $P_{N_c}$ and $R_k$ should be selected adequately to achieve good state estimates based on the knowledge or prediction of the error distributions. Moreover, $\bar{\mathbf{q}}_{j-N_c+1}$ represents the estimated state at the start of estimation horizon $t_{j-N_c+1}$. The weighting matrices $P_L$ and $R_T$ are the inverses of the process and measurement noise covariance matrices, respectively. Considering the noise characteristics of sensors listed in Section \[hardware\], the weighting matrix $R_T$ used in the experiments is chosen as follow $$\begin{aligned} \!\! \!\!R_T\!\!\!\!\!\!&\!\!=\!\!&\!\!\!\!\!\!\text{diag}\{\sigma_{x}^{2},\sigma_{y}^{2},\sigma_{\psi}^{2},\sigma_{r}^{2},\sigma_{f_1}^{2},\sigma_{f_2}^{2},\sigma_{f_3}^{2},\sigma_{f_4}^{2}\}^{-1} \nonumber \\ \!\!\!\!\!\!&=&\!\!\!\!\!\!\text{diag}\{0.0005,0.0005,0.0005,0.0001,1, 1, 1, 1\}^{-1} \end{aligned}$$ Moreover, the weighting matrix $P_L$ used in the experiments are selected as follow $$\begin{aligned} \!\! \!\!P_L\!\!\!\!\!\!&\!\!=\!\!&\!\!\!\!\!\!\text{diag}\{\sigma_{x}^{2},\sigma_{y}^{2},\sigma_{\psi}^{2},\sigma_{u}^{2},\sigma_{v}^{2},\sigma_{r}^{2}\}^{-1} \nonumber \\ \!\!\!\!\!\!&=&\!\!\!\!\!\!\text{diag}\{1, 1, 1, 0.1, 0.1, 1\}^{-1} \end{aligned}$$ Simultaneous Localization and Mapping {#sec::localization} ===================================== ![image](Figures/factor.pdf){width=".95\textwidth"} An overview of the proposed SLAM system, which is adpated from [@shan2020lio], is shown in Fig. \[fig::factor-graph\]. The system receives sensor data from a 3D lidar, an IMU, and a GPS. SLAM algorithm aims to estimate the state $\mathbf{x}=[x~y~\psi]\in\mathbb{R}^{3\times1}$ of the vessel given the sensor measurements. This state estimation problem can be formulated as a posteriori (MAP) problem. In order to seamlessly incorporate measurements from various sensors, we utilize a factor graph to model this problem. Then solving the MAP problem is equivalent to solving a nonlinear least-squares problem [@dellaert2017factor]. We introduce three types of $factors$ along with one $variable$ type for factor graph construction. This variable, representing the vessel’s state, is referred to as $nodes$ of the graph. The three types of factors are: (a) lidar odometry factors, (b) GPS factors, and (c) loop closure factors. To limit the memory usage and improve the efficiency of the localization system, we add new node ${\bold{x}}\in\mathbb{R}^{3\times1}$ to the graph using a simple but effective heuristic approach. A new node ${\bold{x}}$ is only added when the position or rotation change of the vessel exceeds a user-defined threshold. We use incremental smoothing and mapping with the Bayes tree (iSAM2) [@kaess2012isam2] to optimize the factor graph upon the insertion of a new node. The process for generating the aforementioned factors is described in the following sections. Lidar Odometry Factor {#sec::lidar-odometry} --------------------- We perform feature extraction using the raw point cloud from the 3D lidar. Similar to the process introduced in [@LEGO-LOAM], we first project the raw point cloud onto a range image as the arriving point cloud may not be organized. Each point in the point cloud is associated with a pixel in the range image. Then we calculate the roughness value of a pixel in the range image using its neighboring range values. Points with a large roughness value are classified as edge features. Similarly, a planar feature is determined by a small roughness value. We denote the extracted edge and planar features from a lidar scan at time $t$ as ${\text{F}}^{e}_{t}$ and ${\text{F}}^{p}_{t}$ respectively. All the features extracted at time $t$ compose a lidar frame $\mathbb{F}_{t}$, where $\mathbb{F}_{t} = \{{\text{F}}^{e}_{t}, {\text{F}}^{p}_{t}\}$. Note that $\mathbb{F}_{t}$ is represented in the local sensor frame. A more detailed description of the feature extraction process can be found in [@LEGO-LOAM]. It is computationally intractable to add factors for every lidar frame. Thus we adopt the concept of keyframe selection, which is widely used in the visual SLAM field. We select a lidar frame $\mathbb{F}_{t+1}$ as a keyframe when the vessel’s pose change exceeds a user-defined threshold. In this paper, we select a new lidar keyframe when the vessel’s position change exceeds $1m$ or the rotation change exceeds $10^\circ$. The lidar frames between two keyframes are discarded. When a lidar keyframe is selected, we perform scan-matching to calculate the relative transformation between the new keyframe with the previous sub-keyframes. The sub-keyframes are obtained by transforming the previous $n$ keyframes into the global world frame. These transformed keyframes are then merged together into a voxel map ${\bold{M}}_{t}$. Note that ${\bold{M}}_{t}$ is composed of two sub-voxel maps because we extract two types of features in the previous step. These two sub-voxel maps are denoted as ${\bold{M}}^{e}_{t}$, the edge feature voxel map, and ${\bold{M}}^{p}_{t}$, the planar feature voxel map. The relationships between the feature sets and the voxel maps can be represented as: $$\begin{aligned} \nonumber &{\bold{M}}_{t} = \{{\bold{M}}^{e}_{t}, {\bold{M}}^{p}_{t}\} \\ \nonumber &where: {\bold{M}}^{e}_{t} =\; {'{\text{F}}}^{e}_{t} \cup {'{\text{F}}}^{e}_{t-1} \cup...\cup{'{\text{F}}}^{e}_{t-n} \\ \nonumber &\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\bold{M}}^{p}_{t} =\; {'{\text{F}}}^{p}_{t} \cup {'{\text{F}}}^{p}_{t-1} \cup ... \cup {'{\text{F}}}^{p}_{t-n},\end{aligned}$$ where ${'{\text{F}}}^{e}_{t}$ and ${'{\text{F}}}^{p}_{t}$ are the transformed edge and planar features in the global frame. $n$ is chosen to be 25 for all the tests. The new lidar keyframe $\mathbb{F}_{t+1}$ is transformed into the global world frame using the initial guess from IMU. The transformed new keyframe in the world frame is denoted as $'\mathbb{F}_{t+1}$. With $'\mathbb{F}_{t+1}$ and the voxel map ${\bold{M}}_{t}$, we perform scan-matching using the method proposed in [@LOAM] and obtain the relative transformation $\Delta{\bold{T}}_{t,\;t+1}$ between them. At last, $\Delta{\bold{T}}_{t,\;t+1}$ is added as the lidar odometry factor into the factor graph. GPS Factor {#sec::gps} ---------- Though we can achieve low-drift state estimation solely by using lidar odometry factors, the localization system still suffers from drift during long-duration navigation tasks. We thus utilize the absolute measurements from GPS and incorporate them as GPS factors into the factor graph. When the GPS measurements are available, we first transform them into Cartesian coordinate frame using the method proposed in [@moore2016generalized]. Upon the insertion of a new node to the factor graph, we then add a new GPS factor and associate it with the new node. Loop Closure Factor {#sec::loop-closure} ------------------- Due to the utilization of a factor graph, loop closures can also be seamlessly incorporated into the proposed system. A successful detection of loop closure is introduced as a loop closure factor in the factor graph. For the purpose of illustration, we introduce a naive but effective loop closure detection approach that is based on Euclidean distance. When a new state ${\bold{x}}_{i+1}$ is added into the factor graph, as is shown in Fig. \[fig::factor-graph\], we search the graph and find the prior states that are within a certain distance to ${\bold{x}}_{i+1}$. For example, ${\bold{x}}_3$ is the returned candidate state. We then extract sub-keyframes $\{\mathbb{F}_{3-m}, ..., \mathbb{F}_{3}, ..., \mathbb{F}_{3+m}\}$ and merge them into a local voxel map, which is similar to the voxel map introduced in Sec. \[sec::lidar-odometry\]. If a successful match can be found between $\mathbb{F}_{i+1}$ and the voxel map, we obtain the relative transformation $\Delta{\bold{T}}_{3,\;i+1}$ and add it as a loop closure factor to the graph. Throughout the map, we choose $m$ to be 12, and the search distance for loop closures is set to be $15$ m. In practice, we find adding loop closure factors is especially useful for eliminating the drift when conducting navigation tasks in GPS-denied regions. Experiments and Results ======================= We performed several experiments in canals and rivers to verify the algorithms as well as demonstrate the effectiveness of the developed autonomous system. All the algorithms including SLAM, NMPC, NMHE, path planner are executed on an onboard computer (described in Section \[hardware\]). These algorithms are updated at a rate of 10 Hz. SLAM Results ------------ In order to test the performance of the proposed SLAM system, we mount the sensor suite, which includes a Lidar, an IMU, and a GPS, on a manned boat and cruised along the canals of Amsterdam for 3 hours. We start and finish the data-gathering process at the same location. Altogether, we collected 107,656 lidar scans with an estimated trajectory length of 19,065 m. Performing SLAM in the canal environment is challenging due to several reasons. Many bridges over the canals pose degenerate scenarios, as there are few useful features when the boat is under them, similar to moving through a long, featureless corridor. Bridges and buildings obstruct the reception of GPS data and result in intermittent GPS availability throughout the dataset. We also observe numerous false directions from the lidar when direct sunlight is in the sensor field-of-view. We test the performance of the system by disabling the insertion of GPS factor, loop closure factor, or both. When we solely use lidar odometry factors with two other factors being disabled, the system fails to produce meaningful results and suffers from great drift in featureless environment. When we use both lidar odometry factors and GPS factors, the system is able to provide accurate pose estimation in the horizontal plane while suffers from drift in the vertical direction. This is because the elevation measurement from GPS is very inaccurate - giving rise to altitude errors approximating 100 m in our tests, which further motivates our usage of loop closure factors. Upon finishing processing the dataset, the proposed SLAM system achieves a relative translation error of 0.17 m when returning to the start location. We also note that the average runtime for registering a new lidar keyframe and optimizing the graph is only 79.3ms, which is suitable for deploying the proposed system on a low-powered embedded hardware. Results of NMPC and NMHE ------------------------ First, we experimentally estimated the dynamics model of the vessel in order to achieve superior performance of NMPC for tracking control. The data was gathered when the vessel was remotely controlled to perform sinusoidal movements in Charles River (Fig. \[fig::charles-river\]), which couples the surge, sway, and rotation motions. The input force and the vessel velocity were recorded at a rate of 10 Hz in the experiments. The trials were repeated five times. The duration of each trial was around 150 s. We utilized the optimization algorithm described in Section \[ss:dynamic\_model\] and identified the hydrodynamic parameters $\xi$ for Roboat II as follows: $m_{11} =172$ kg, $m_{22} =188$ kg, $m_{33} =24~ \text{kg}\cdot\text{m}^{2}$, $X_{u}=38$ kg/s, $Y_{v}=168$ kg/s and $N_{r}=16$ kg$\cdot$m$^{2}$/s. Next, we tested the performance of the NMPC tracking and the NMHE state estimation on Roboat II in Charles River. We use an $A^{\ast}$ planner to generate the reference paths for NMPC. The experiment was conducted as follows. First, we assigned a goal point for the planner using a ROS graphical interface, RVIZ. The planner will generate a feasible path starting from the current position and ending with the goal position of the vessel. Then, Roboat II starts to track the reference trajectories to approach the destination. After Roboat II reached the destination, we selected another goal for Roboat II which initiates a new tracking section. We repeated the same assignments several times in the experiments. NMHE was running at the same time to estimate the states of Roboat II during the experiments. Trajectory and heading angle tracking performances of Roboat II are shown in Fig. \[MPCTracking\]. ![Performance of the NMPC tracking in the river test. (a) trajectory; (b)heading angle; (c) forces. []{data-label="MPCTracking"}](Figures/MPCTracking){width="0.96\linewidth"} This demonstrates that the MPC can successfully track on the references even with environmental noises in the river. The position tracking RMSE (Root Mean Square Error) value for NMPC on Roboat II is 0.096 m while the heading angle tracking RMSE value for NMPC is 0.149 rad. Note that we penalize (the weight coefficient) less for the heading angle than that of the positions in (\[OCPequation\]). The reason is that we focus more on tracking a desired path accurately. Moreover, less penalization on the heading angle avoid oscillations around the desired path. The simple planner generates the whole reference path to the goal position at the rate of 10 Hz. In Fig. \[MPCTracking\](a) and (b), we only show 10 reference points starting from the current position of Roboat II because the generated reference path is always slightly oscillates. A more careful inspection from Fig. \[MPCTracking\](b) indicates that the noisy reference heading angle at each time instant during the experiments. The control forces for Roboat II are shown in Fig.\[MPCTracking\](a). These generated control signals by the NMPC is restrained within the lower and upper bounds specified in (\[forcelimits\]) in Section \[NMPCsection\]. It is clear that the left and right thrusters contribute significantly as the system in on-track. If the system was not on-track, all the four thrusters would contribute significantly (at around 42 s and 84 s) to help the system rapidly reach the reference positions and orientations. The indirectly measured and estimated linear and angular velocities for Roboat II are shown in Fig. \[NMHEvelocity\]. ![Performance of the NMHE state estimation in the river test. (a) linear speed $u$; (b)linear speed $v$; (c)angular speed $r$. []{data-label="NMHEvelocity"}](Figures/NMHEvelocity){width="1.0\linewidth"} This demonstrates that the MHE can successfully deal with noises on the measurements. The reference for linear longitudinal velocity $u$ and lateral velocity $v$ are respectively set to 0.6 m/s and 0 m/s throughout the reference path generation. The estimates always track around these reference values as the system is on-track, which also indicates the effectiveness of the NMPC tracking. The peak and valley areas in the velocities suggest the process that Roboat II approaches the current goal position and heads to the next goal by fast turning itself. Conclusion and Future Work ========================== In this paper, we have developed a novel autonomous surface vessel (ASV), called Roboat II, for urban transportation. Roboat II is capable of performing accurate simultaneous localization and mapping (SLAM), receding horizon tracking control and estimation, and path planning in the urban waterways. Our work will be extended in the following directions in the near future. First, we will explore more efficient planning algorithms to enable the vessel to handle complicated scenarios in the waterways. Second, we will apply active object detection and identification to improve Roboat’s understanding of its environment for robust navigation. Third, we will estimate disturbances such as currents and waves to further improve the tracking performance in more noisy waters. Fourth, we will develop algorithms for multi-robot formation control and self-assembly on the water, enabling the construction of on-demand large-scale infrastructure. [^1]: This work was supported by grant from the Amsterdam Institute for Advanced Metropolitan Solutions (AMS) in Netherlands. [^2]: W. Wang, T. Shan, P. Leoni, D. Gutierrez, D. Meyers and C. Ratti are with the SENSEable City Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. [{wweiwang, shant, leoni, davidfg, drewm, ratti}@mit.edu]{} [^3]: W. Wang, T. Shan and D. Rus are with the Computer Science and Artificial Intelligence Lab (CSAIL), Massachusetts Institute of Technology, Cambridge, MA 02139 USA. [{wweiwang, shant, rus}@mit.edu]{}
--- abstract: \| In 2018, Pott, at al. have studied in \[IEEE Transactions on Information Theory. Volume: 64, Issue: 1, 2018\] the maximum number of bent components of vectorial function. They have presented serval nice results and suggested several open problems in this context. This paper is in the continuation of their study in which we solve two open problems raised by Pott et al. and partially solve an open problem raised by the same authors. Firstly, we prove that for a vectorial function, the property of having the maximum number of bent components is invariant under the so-called CCZ equivalence. Secondly, we prove the non-existence of APN plateaued having the maximum number of bent components. In particular, quadratic APN functions cannot have the maximum number of bent components. Finally, we present some sufficient conditions that the vectorial function defined from $\mathbb{F}_{2^{2k}}$ to $\mathbb{F}_{2^{2k}}$ by its univariate representation: $$\alpha x^{2^i}\left(x+x^{2^k}+\sum\limits_{j=1}^{\rho}\gamma^{(j)}x^{2^{t_j}} +\sum\limits_{j=1}^{\rho}\gamma^{(j)}x^{2^{t_j+k}}\right)$$ has the maximum number of [ components bent functions, where $\rho\leq k$]{}. Further, we show that the differential spectrum of the function $ x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}})$ (where $i,t_1,t_2$ satisfy some conditions) is different from the binomial function $F^i(x)= x^{2^i}(x+x^{2^k})$ presented in the article of Pott et al. Finally, we provide sufficient and necessary conditions so that the functions $$Tr_1^{2k}\left(\alpha x^{2^i}\left(Tr^{2k}_{e}(x)+\sum\limits_{j=1}^{\rho}\gamma^{(j)}(Tr^{2k}_{e}(x))^{2^j} \right)\right)$$ are bent. author: - 'Sihem Mesnager$^{1}$, Fengrong Zhang$^2$, Chunming Tang$^3$, Yong Zhou$^2$' title: Further study on the maximum number of bent components of vectorial functions --- [Keywords:]{} Vectorial functions, Boolean functions, Bent functions, Nonlinearity, APN functions, Plateaued functions, CCZ equivalence. Introduction ============ Vectorial (multi-output) Boolean functions, that is, functions from the vector space $\mathbb{F}^{n}_{2}$ (of all binary vectors of length $n$) to the vector space $\mathbb{F}^{m}_{2}$, for given positive integers $n$ and $m$. These functions are called $(n,m)$-functions and include the (single-output) Boolean functions (which correspond to the case $m =1$). In symmetric cryptography, multi-output functions are called S-boxes. They are fundamental parts of block ciphers. Being the only source of nonlinearity in these ciphers, S-boxes play a central role in their robustness, by providing confusion (a requirement already mentioned by C. Shannon), which is necessary to withstand known (and hopefully future) attacks. When they are used as S-boxes in block ciphers, their number $m$ of output bits equals or approximately equals the number $n$ of input bits. They can also be used in stream ciphers, with $m$ significantly smaller than $n$, in the place of Boolean functions to speed up the ciphers. We shall identify $\mathbb{F}_{2}^n$ with the Galois field $\mathbb{F}_{2^n}$ of order $2^n$ but we shall always use $\mathbb{F}_{2}^n$ when the field structure will not really be used. The [component functions]{} of $F$ are the Boolean functions $v\cdot F$, that is, $x\in\mathbb{F}_{2^n}\mapsto {Tr_{1}^{m}}(v F(x))$, where “$\cdot$“ stands for an inner product in $\mathbb{F}_{2^m}$ (for instance: $u\cdot v:={Tr_{1}^{m}}(uv), \forall u\in \mathbb{F}_{2^m}, v\in \mathbb{F}_{2^m}$ where ”${Tr_{1}^{m}}$" denotes the absolute trace over $ \mathbb{F}_{2^m}$). In order to classify vectorial Boolean functions that satisfy desirable nonlinearity conditions, or to determine whether, once found, they are essentially new (that is, inequivalent in some sense to any of the functions already found) we use some concepts of equivalence. For vectorial Boolean functions, there exist essentially two kinds concepts of equivalence: the extended affine EA-equivalence and the CCZ-equivalence (Carlet-Charpin-Zinoviev equivalence). Two $(n,r)$-functions $F$ and $F^{\prime}$ are said to be EA-equivalent if there exist affine automorphisms $L$ from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$ and $L^{\prime}$ from $\mathbb{F}_{2^r}$ to $\mathbb{F}_{2^r}$ and an affine function $L''$ from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^r}$ such that $F'=L^{\prime} \circ F \circ L + L''$. EA-equivalence is a particular case of CCZ-equivalence [@CCZ98]. Two $(n,r)$-functions $F$ and $F^{\prime}$ are said to be CCZ-equivalent if their graphs $G_F:=\{(x,F(x)),~ x\in \mathbb{F}_{2^n}$} and $G_F^{\prime}:=\{(x,F^{\prime}(x)),~ x\in \mathbb{F}_{2^n}$} are affine equivalent, that is, if there exists an affine permutation $\mathcal{L}$ of $\mathbb{F}_{2^n} \times \mathbb{F}_{2^m}$ such that $\mathcal{L}(G_F)=G_F^{\prime}$. A standard notion of nonlinearity of an $(n,m)$-function $F$ is defined as $$\label{N_1} \mathcal{N}(F)=\underset{v\in \F_{2^m}^\star}{\min}nl(v\cdot F),$$ where $v\cdot F$ denotes the usual inner product on $\F_{2^m}$ and $nl(\cdot)$ denotes the nonlinearity of Boolean functions (see definition in Section \[Preliminaries\]). From the covering radius bound, it is known that $\mathcal{N}(F)\leqslant 2^{n-1}-2^{n/2-1}$. The functions achieving this bound are called $(n,m)$-bent functions. Equivalently, a vectorial Boolean function $F: \mathbb{F}_{2^n}\rightarrow \mathbb{F}_{2^m}$ is said to be a vectorial bent function if all nonzero component functions of $F$ are bent (Boolean) functions. Bent Boolean functions have maximum Hamming distance to the set of affine Boolean functions. The notion of bent function was introduced by Rothaus [@RO76] and attracted a lot of research of more than four decades. Such functions are extremal combinatorial objects with several areas of application, such as coding theory, maximum length sequences, cryptography. A survey on bent function can be found in [@CarletMesnagerDCC2016] as well as the book [@MesnagerBook]. In [@CC-Nyberg], it is shown that $(n,m)$-bent functions exist only if $n$ is even and $m\leqslant n/2$. The notion of nonlinearity in (\[N\_1\]) (denoted by $\mathcal{N}$), was first introduced by Nyberg in [@CC-Nyberg], which is closely related to Matsui’s linear attack [@Matsui] on block ciphers. It has been further studied by Chabaud and Vaudenay [@ChabaudVaudenay95]. The nonlinearity is invariant under CCZ equivalence (and hence under extended affine equivalence). Budaghyan and Carlet have proved in [@BC09] that for bent vectorial Boolean functions, CCZ-equivalence coincides with EA-equivalence. The problem of construction vectorial bent functions has been considered in the literature. Nyberg [@CC-Nyberg] investigated the constructions of vectorial bent functions; she presented two constructions based on Maiorana-McFarland bent functions and $\mathcal{PS}$ bent functions, respectively. In [@SatohIwataKurosawa99], Satoh, Iwata, and Kurosawa have improved the first method of construction given in [@CC-Nyberg] so that the resulting functions achieve the largest degree. Further, serval constructions of bent vectorial functions have been investigated in some papers [@CarletMesnagerBentVectorial; @Pasaliczh2012; @Feng2011; @MesnagerDCC2015; @Wu2005]. A complete state of the art can be found in [@MesnagerBook] (Chapter 12). Very recently, Pott et al.[@Pott2017] considered functions $ \mathbb{F}_{2^n}\rightarrow \mathbb{F}_{2^n}$ of the form $F^i(x) = x^{2^i}(x + x^{2^{k}})$, where $n=2k, i=0,1,\cdots,n-1$. They showed that the upper bound of number of bent component functions of a vectorial function $F: \mathbb{F}_{2^n}\rightarrow \mathbb{F}_{2^n}$ is $2^n-2^{n/2}$ ($n$ even). In addition, they showed that the binomials $F^i(x)= x^{2^i}(x + x^{2^k})$ have such a large number of bent components, and these binomials are inequivalent to the monomials $x^{2^{k}+1}$ if $0 < i < k$. Further, the properties (such as differential properties and complete Walsh spectrum) of the functions $F^i$ were investigated. In this paper, we will consider three open problems raised by Pott et al [@Pott2017]. In the first part, we prove that CCZ equivalence is preserved for vectorial functions having the maximum number of bent components. Next, we consider APN plateaued functions and investigate if they can have the maximum number of bent components. We shall give a negative answer to this question. Finally, we consider the bentness property of functions $\mathbb{F}_{2^{2k}}\rightarrow \mathbb{F}_{2^{2k}}$ of the form $$\label{equa main} G(x)=\alpha x^{2^i}\left(x+x^{2^k}+\sum\limits_{j=1}^{\rho}\gamma^{(j)}x^{2^{t_j}} +\sum\limits_{j=1}^{\rho}\gamma^{(j)}x^{2^{t_j+k}}\right),$$ where $m\leq k$, $\gamma^{(j)}\in {\Bbb F}_{2^{k}}$ and $0\leq t_j\leq k$ be a nonnegative integer. In particular, [ we show the functions $ x^{2^{t_2}}\left(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}}\right)$ are inequivalent to $x^{2^{t_2}}(x + x^{2^k}) $, where $t_1=1$ and $\gcd(t_2,k) \neq 1$ ]{}. Here we use the concept of CCZ-equivalence when we speak about the equivalence of functions. The rest of the paper is organized as follows. Some preliminaries are given in Section \[Preliminaries\]. In Section \[stability\], we prove our result on the stability under CCZ equivalence of a function having the maximum number of bent components which solve Problem 4 in [@Pott2017]. Next, in Section \[APNplateaued\], we prove that APN plateaued functions cannot have the maximum number of bent components, which partially solves Problem 8 in [@Pott2017]. Finally, in Section \[newbent\] we investigate Problem 2 in [@Pott2017]. To this end, we provide several functions defined as $G(x) = x L(x)$ on ${\Bbb F}_{2^{2k}}$ (where $L(x)$ is a linear function on ${\Bbb F}_{2^{2k}}$) such that the number of bent components $Tr^{2k}_1(\alpha F(x))$ is maximal. Preliminaries and notation {#Preliminaries} ========================== Throughout this article, $\\| E\\|$ denotes the cardinality of a finite set $E$, the binary field is denoted by $\mathbb{F}_2$ and the finite field of order $2^n$ is denoted by ${\Bbb F}_{2^n}$. The multiplicative group ${\Bbb F}^_{2^n}$ is a cyclic group consisting of $2^n-1$ elements. The set of all Boolean functions mapping from ${\Bbb F}_{2^n}$ (or $ {\Bbb F}_2^n$) to ${\Bbb F}_{2}$ is denoted by $B_n$. Recall that for any positive integers $k$, and $r$ dividing $k$, the trace function from ${{\mathbb F}_{2^{k}}}$ to ${{\mathbb F}_{2^{r}}}$, denoted by $Tr_{r}^{k}$, is the mapping defined as: $$Tr_{r}^{k}(x):=\sum_{i=0}^{\frac kr-1} x^{2^{ir}}=x+x^{2^r}+x^{2^{2r}}+\cdots+x^{2^{k-r}}.$$ In particular, the [absolute trace]{} over $\mathbb{F}_2$ of an element $x \in \mathbb{F}_{2^n}$ equals $Tr_1^{n}(x)=\sum_{i=0}^{n-1} x^{2^i}$. There exist several kinds of possible [univariate representations]{} (also called trace, or polynomial, representations) of Boolean functions which are not all unique and use the identification between the vector-space ${\Bbb F}_2^n$ and the field ${{\mathbb F}_{2^{n}}}$. Any Boolean function over ${\Bbb F}_2^n$ can be represented in a unique way as a polynomial in one variable $x\in \mathbb{F}_{2^n}$ of the form $f(x) =\sum_{j=0}^{2^n-1} a_j x^j$, where $a_0,a_{2^n-1}\in{\Bbb F}_2$, $a_j$’s are elements of ${{\mathbb F}_{2^{n}}}$ for $1\leq j <2^n-1$ such that ${a_j}^2=a_{2i\mod (2^n-1)}$. The binary expansion of $j$ is $j=j_0+j_12+\cdots j_{n-1}2^{n-1}$ and we denote $\bar{j}=(j_0,j_1, \cdots,j_{n-1})$. The algebraic degree of $f$ equals $max \{wt(\bar i)\mid a_j\not=0, 0\leq j < 2^n\}$ where $wt(\bar i)=j_0+j_1+\cdots+j_{n-1}$. Affine functions (whose set is denoted by $A_n$) are those of algebraic degree at most $1$. The Walsh transform of $f\in{ B}_{n}$ at $ \ll \in \mathbb{F}_{2^n}$ is defined as $$\begin{array}{l}W_f(\ll) = \sum\limits_{x \in \mathbb{F}_{2^n}}(-1)^{f(x) + Tr^n_1(\ll x)}.\end{array}$$ A Boolean function $f(x_1, \ldots, x_n)$ defined on ${\Bbb F}_2^n$ is commonly represented as a multivariate polynomial over $\F_2$ called the [algebraic normal form]{} (ANF) of $f$. More precisely, $f(x_1, \ldots, x_n)$ can be written as $$\label{eq:anf} f(x_1,\dots,x_n)=\sum_{u\in \F_2^n}\lambda_u\left(\prod_{i=1}^n x_i^{u_i}\right)~,$$ for $\lambda_u\in \F_2~, u=(u_1,\ldots,u_n)$. The algebraic degree of $f$, denoted by $\deg(f)$, is equal to the maximum Hamming weight of $u \in \F_2^n$ for which $\lambda_u \neq 0$. An $n$-variable Boolean function of the form $a \cdot x=a_{1}x_{1}\oplus\cdots\oplus a_{n}x_{n}$, where $a_i \in {\Bbb F}_{2}$ and whose degree is at most one, is called linear function. The function $f(x)=a \cdot x \oplus b$, where $b \in {\Bbb F}_{2}$ is a fixed constant, is called affine and the set of all $n$-variable affine functions is denoted by $A_{n}$. The nonlinearity* of $f\in B_{n}$ is defined as the minimum Hamming distance to the set of all $n$-variable affine functions, i.e., $$\begin{array}{c} nl(f)=\min_{g\in A_{n}}d(f,g). \end{array}$$ where $d(f,g)$ is the Hamming distance between $f$ and $g$. Following is the relationship between nonlinearity and Walsh spectrum of $f \in {B}_n$ $$nl(f) = 2^{n-1} - \frac{1}{2}\max_{\ll \in \mathbb{F}_{2^n}}\|W_f(\ll)\|.$$ By Parseval’s identity $\sum_{\ll \in \mathbb{F}_{2^n}} W_f(\ll)^2 = 2^{2n}$, it can be shown that $ \max\{ \|W_f(\ll) \| : \ll \in \mathbb{F}_{2^n} \} \ge 2^{\frac{n}{2}}$ which implies that $nl(f) \le 2^{n-1} - 2^{\frac{n}{2} - 1}$. If $n$ is an even integer a function $f \in \mathcal{B}_n$ is said to be bent if $W_f(\ll) \in \{2^{\frac{n}{2}}, -2^{\frac{n}{2}}\}$, for all $\ll \in \mathbb{F}_{2^n}$. Moreover, a function $f \in \mathcal{B}_n$ is said to be $t$-plateaued if $W_f(\ll) \in\{ 0, \pm 2^\frac{n+t}{2}\}$, for all $\ll \in \mathbb{F}_{2^n}$. The integer $t$ $(0\leq t\leq n$) is called the amplitude of $f$. Note that a bent function is a 0-plateaued function. In the following, $``<,>"$ denotes the standard inner (dot) product of two vectors, that is, $<\ll , x>=\lambda_1x_1+ \ldots + \lambda_nx_n$, where $\ll,x\in F_2^n$. If we identify the vector space $F_2^n$ with the finite field $F_{2^n}$, we use the trace bilinear form $Tr^n_1(\ll x)$ instead of the dot product, that is, $<\ll , x>=Tr^n_1(\ll x)$, where $\ll,x\in F_{2^n}$. For vectorial functions $F: \mathbb{F}_2^n\rightarrow \mathbb{F}_2^m$, the extended Walsh-Hadamard transform defined as, $$\begin{array}{l} W_F(u,v) = \sum\limits_{x \in \mathbb{F}_2^n}(-1)^{<v,F(x)> + <u, x>},\end{array}$$ where $F(x)=(f_1(x),f_2(x),\cdots,f_m(x)),u\in \mathbb{F}_2^n, v\in \mathbb{F}_2^m$. Let $F$ be a vectorial function from $\mathbb{F}_{2^n}$ into $\mathbb{F}_{2^m}$. The linear combinaison of the coordinates of $F$ are the Boolean functions $f_\lambda: x\mapsto Tr_1^{m} (\lambda F(x))$, $\lambda \in \mathbb{F}_{2^m}$, where $f_{0}$ is the null function. The functions $f_\lambda$ ($\lambda\not=0$) are called the components of $F$. A vectorial function is said to be bent (resp. $t$-plateaued) if all its components are bent (resp. $t$-plateaued). A vectorial function $F$ is called vectorial plateaued if all its components are plateaued with possibly different amplitudes. Let $F: \mathbb{F}_2^n\rightarrow \mathbb{F}_2^n $ be an $(n,n)$-function. For any $a\in {\Bbb F}_2^n, b\in {\Bbb F}_2^n$, we denote $$\begin{array}{c} \Delta_F(a,b)=\{x\|x\in {\Bbb F}_2^n,F(x\oplus a)\oplus F(x)=b\},\\ \delta_F(a,b)=\\|\Delta_F(a,b)\\|, \end{array}$$ Then, we have $\delta(F):=max_{a\not=0, b\in {\Bbb F}_2^n } \delta_F(a,b)\geq 2$ and the functions for which equality holds are said to be almost perfect nonlinear(APN). A nice survey on Boolean and vectorial Boolean functions for cryptography can be found in [@Cbook1] and [@Cbook], respectively. The stability of a function having the maximum number of bent components under CCZ equivalence {#stability} ============================================================================================== In [@Pott2017], Pott et al. have shown that the maximum number of bent components of a vectorial $(n,n)$-function $F$ is $2^n -2^k$ where $k:=\frac{n}2$ ($n$ even). They left open the problem whether the property of a function having the maximum number of bent components is invariant under CCZ equivalence or not. In this section we solve this problem by giving a positive answer in the following theorem. \[theo:CCZ\] Let $n=2k$ and $F, F'': \mathbb F_2^n \rightarrow \mathbb F_2^n$ be CCZ-equivalent functions. Then $F$ has $2^n -2^k$ bent components if and only if $F''$ has $2^n -2^k$ bent components. Let $F$ be a function with $2^n -2^k$ bent components. Define $$S=\{v\in \mathbb F_2^n: x \rightarrow <v, F(x)> \text{ is not bent} \}.$$ By Theorem 3 of [@Pott2017], $S$ is a linear subspace of dimension $k$. Then, let $U$ be any $k$-dimensional subspace of $\mathbb F_2^n$ such that $U \cap S =\{0\}$. Let $v_1, \cdots, v_k$ be a basis of $S$ and $u_1, \cdots, u_k$ be a basis of $U$. Define a new function $F': \mathbb F_2^n \rightarrow \mathbb F_2^n$ as $$F'(x)=(H(x), I(x))$$ where $H(x)=(<v_1, F(x)>, \cdots, <v_k, F(x)>)$ and $I(x)=(<u_1, F(x)>, \cdots, <u_k, F(x)>)$. Then, $F'$ is EA-equivalent to $F$. Recall that the property of a function having the maximum number of bent components is invariant under EA equivalence. Thus, $F'$ has $2^n -2^k$ bent components. Since $F$ and $F''$ are CCZ-equivalent functions, $F''$ is CCZ-equivalent to $F'$, which has $2^n -2^k$ bent components. Let $\mathcal L(x, y,z) = (L_1(x, y,z), L_2(x, y,z), L_3(x, y,z))$, (with $L_1 :\mathbb F^n_2 \times F^k_2 \times F^k_2 \rightarrow \mathbb F^n_2$, $L_2 :\mathbb F^n_2 \times F^k_2 \times F^k_2 \rightarrow \mathbb F^k_2$ and $L_3 :\mathbb F^n_2 \times F^k_2 \times F^k_2 \rightarrow \mathbb F^k_2$) be an affine permutation of $\mathbb F^n_2 \times F^k_2 \times F^k_2$ which maps the graph of $F'$ to the graph of $F''$. Then, the graph $\mathcal G_{F''}=\{\mathcal L (x, H(x), I(x)): x \in \mathbb F_2^n \}$. Thus $L_1(x,H(x), I(x))$ is a permutation and for some affine function $L_1': \mathbb F^n_2 \times F^k_2 \rightarrow \mathbb F^n_2$ and linear function $L_1'': \mathbb F^k_2 \rightarrow \mathbb F^n_2$ we can write $L_1(x, y,z)=L_1'(x, y)+L_1''(z)$. For any element $v$ of $\mathbb F^n_2$ we have $$\begin{aligned} \label{eq:v-comp} <v , L_1(x, H (x), I(x))> =& <v , L_1'(x,H(x))> + <v , L_1''(I (x))> \nonumber\\ =& <v , L_1'(x,H(x))> + <L_1''^(v) , I (x)> \nonumber\\ =& <L_1''^(v) , I (x)>+<v',H(x)>+<v'',x>+a, \end{aligned}$$ where $a\in \mathbb F_2$, $v'\in \mathbb F_2^k$, $v''\in \mathbb F_2^n$ and $L_1''^$ is the adjoint operator of $L_1''$, in fact, $L_1''^$ is the linear permutation whose matrix is transposed of that of $L_1''$. Since $L_1(x,H(x), I(x))$ is a permutation, then any function $<v , L_1(x, H (x), I(x))>$ is balanced (recall that this property is a necessary and sufficient condition) and, hence, cannot be bent. From the construction of $F'$, $<L_1''^(v) , I (x)>+<v',H(x)>+<v'',x>+a$ is not bent if and only if $L_1''^(v)=0$. Therefore, $L_1''^(v)=0$ for any $v\in \mathbb F_2^n$. This means that $L_1''$ is null, that is, $L_1(x, H (x), I(x))=L_1'(x, H (x)) $. We can also write $L_i(x, y,z)=L_i'(x, y)+L_i''(z)$ for $i\in \{2, 3\}$ where $L_i': \mathbb F^n_2 \times F^k_2 \rightarrow \mathbb F^k_2$ are affine functions and $L_i'': F^k_2 \rightarrow \mathbb F^k_2$ are linear functions. Set $F_1''(x)=L_1(x, H (x), I(x))=L_1'(x, H (x))$ and $F_2''(x)=(L_2'(x, H(x))+L_2''(I(x)),L_3'(x, H(x))+L_3''(I(x)))$. Then, $F''(x)=F_2''\circ F_1''^{-1}(x)$. For any $v\in \mathbb F_2^n$ and $u=(u',u'')\in \mathbb F_2^k \times \mathbb F_2^k $, $$\begin{aligned} W_{F''}(v,u)=& \sum_{x \in \mathbb F_2^n} (-1)^{<u, F''(x)>+<v, x>}\\ =& \sum_{x \in \mathbb F_2^n} (-1)^{<u, F''\circ F_1''(x)>+<v, F_1''(x)>}\\ =&\sum_{x \in \mathbb F_2^n} (-1)^{<u, F''_2(x)>+<v, F_1''(x)>}\\ =& \sum_{x \in \mathbb F_2^n} (-1)^{<u, (L_2''(I(x)),L_3''(I(x)))>+ <u, (L_2'(x, H(x)),L_3'(x, H(x)))> +<v, L_3'(x, H(x))>}\\ =& \sum_{x \in \mathbb F_2^n} (-1)^{< L_2''^(u')+L_3''^(u''), I(x)>+ <v',H(x)>+<v'',x>+a}. \end{aligned}$$ By the construction of $I(x)$, if $L_2''^(u')+L_3''^(u'')\neq 0$, $< L_2''^(u')+L_3''^(u''), I(x)>+ <v',H(x)>$ is bent. Thus, $<u, F''(x)>$ is bent when $L_2''^(u')+L_3''^(u'')\neq 0$, where $u=(u',u'')\in \mathbb F_2^k \times \mathbb F_2^k $. For $i=2, 3$, let $A_i$ be the matrices of size $k \times k$ defined as $$\begin{aligned} L_i''(z)=z A_i,\end{aligned}$$ where $z=(z_1, \cdots, z_k) \in \mathbb F_2^k$. Then, $$\begin{aligned} L_2''^(u')+L_3''^(u'')=&u'A_2^T+u'' A_3^T \nonumber \\ =& (u',u'') \left [\begin{matrix} A_2^T \\ A_3^T \end{matrix} \right ].\end{aligned}$$ Recall that $\mathcal L$ is a affine permutation. Hence, the rank of the linear function $(L_1''(z), L_2''(z), L_3''(z))$ $=(0,L_2''(z), L_3''(z))$ from $\mathbb F_2^k$ to $\mathbb F^n_2 \times F^k_2 \times F^k_2$ is $k$. By $(L_1''(z), L_2''(z), L_3''(z))=z \left [ \begin{matrix} 0\| A_2 \|A_3 \end{matrix} \right]$, the rank of the matrix $\left [ \begin{matrix} A_2 \|A_3 \end{matrix} \right]$ is $k$. Thus, the rank of the matrix $\left [\begin{matrix} A_2^T \\ A_3^T \end{matrix} \right ]=\left [ \begin{matrix} A_2 \|A_3 \end{matrix} \right]^T$ is also $k$. Set $$\begin{aligned} S'=& \{(u',u'')\in \mathbb F_2^k \times \mathbb F_2^k: L_2''^(u')+L_3''^(u'')=0\}\\ =&\{(u',u'')\in \mathbb F_2^k \times \mathbb F_2^k: (u',u'') \left [\begin{matrix} A_2^T \\ A_3^T \end{matrix} \right ]=0\}.\end{aligned}$$ Then, $S'$ is a linear subspace of dimension $k$. By the previous discussion, if $u=(u',u'')\in \mathbb F_2^n \setminus S'$, the component function $<u, F''(x)>$ is bent. Thus, $F''(x)$ has at least $2^n -2^k$ bent components. From Theorem 3 in [@Pott2017], $F''(x)$ has exactly $2^n -2^k$ bent components, which completes the proof. The non-existence of APN plateaued functions having the maximum number of bent components {#APNplateaued} ========================================================================================= In [@Pott2017], the authors asked if APN functions could have the maximum number of bent components or not. In this section we investigate the case of all APN plateaued functions. The result is given the following theorem. Let $F$ be a plateaued APN function defined on $\mathbb{F}_2^n$ (where $n\geq 4$ is an even positive integer). Then $F$ cannot have the maximum number of bent components. Let $F$ be a plateaued APN function on $\mathbb{F}_2^n$. Denote $$N_t=\{v\in \mathbb{F}_2^n: W_F(u,v) =\pm 2^{\frac{n+t}{2}}\},$$ where $W_F(u,v)=\sum_{x\in \mathbb{F}_2^n} (-1)^{v\cdot F(x)+u\cdot x}$ and $t$ is a positive integer ($0\leq t \leq n$). We have $$\begin{aligned} \label{equ1} \sum_{u,v\in \mathbb{F}_2^n} W_F^4(u,v) =&\sum_{v\in F_2^n}(2^{\frac{n+t_v}{2}})^2 \sum_{u\in \mathbb{F}_2^n}W_F^2(u,v)\nonumber\\ =&2^n\sum_{v\in \mathbb{F}_2^n}2^{t_v} \sum_{u\in \mathbb{F}_2^n}W_F^2(u,v)\nonumber\\ =& 2^{3n}\sum_{v\in \mathbb{F}_2^n}2^{t_v}\nonumber\\ =&2^{3n}(N_0+N_22^2+\cdots +N_n2^n).\end{aligned}$$ Since $F$ is APN, we have $$\label{equ2} \sum_{u,v\in \mathbb{F}_2^n} W_F^4(u,v) =2^{3n}(3\cdot 2^n-2).$$ From Equations (\[equ1\]) and (\[equ2\]), we have $$N_0+N_22^2+\cdots+N_n 2^n=3\cdot 2^n-2.$$ Therefore, we have $N_0\equiv 2\mod 4$. Since $n\geq 4$, $2^n-2^{\frac{n}{2}} \equiv 0 \mod 4$. Hence, $$N_0\neq 2^n-2^{\frac{n}{2}}.$$ Thus, $F$ does not have the maximum number of bent components. In particular, quadratic APN functions cannot have the maximum number of bent components. New constructions of bent component functions of vectorial functions {#newbent} ===================================================================== In this section we provide several functions defined as $G(x) = x L(x)$ on ${\Bbb F}_{2^{2k}}$ such that the number of bent components $Tr^{2k}_1(\alpha F(x))$ equals $2^{2k}-2^k$, where $L(x)$ is a linear function on ${\Bbb F}_{2^{2k}}$. We first recall two lemmas which will be useful in our context. [@Pott2017]\[lemma adjoint\] Let $V = {\Bbb F}_{2^{2k}}$ and let $ <,>$ be a nondegenerate symmetric bilinear form on $V$. If $\mathcal{L}:V\rightarrow V$ is linear, we denote the adjoint operator by $\mathcal{L}^$, i.e., $<x,\mathcal{L}(y)>=<\mathcal{L}^(x),y>$ for all $x,y\in V$. The function $f : V \rightarrow {\Bbb F}_2$, defined by $x\mapsto <x,\mathcal{L}(x)>$, is bent if and only if $\mathcal{L}+\mathcal{L}^$ is invertible. \[lemma L\][@Pott2017] Let $V = {\Bbb F}_{2^{2k}}$ and $<x , y>=Tr^n_1(xy)$ be the trace bilinear form. If $\mathcal{L}:V\rightarrow V$ is defined by $\mathcal{L}(x)=\alpha x^{2^i} $, $\alpha\in V$ and for any $i=0,1,\cdots,n-1$, then $\mathcal{L}^(x)= \alpha^{2^{n-i}} x^{2^{n-i}}$. In [@Pott2017], the authors presented a construction of bent functions through adjoint operators. We start by providing a simplified proof of [@Pott2017 Theorem 4] (which is the main result of their article). \[theo in POtt\][@Pott2017 Theorem 4] Let $V = {\Bbb F}_{2^{2k}}$ and $i$ be a nonnegative integer. Then, the mapping $\mathrm{F}_\alpha^{i} $ defined by $$\mathrm{F}_\alpha^{i}(x)=Tr_{1}^{2k}\left(\alpha x^{2^i}(x+x^{2^k})\right)$$ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}} $. From the proof of [@Pott2017 Theorem 4], we know $$\mathcal{L}(x)+\mathcal{L}^(x)=Tr^{2k}_k(\alpha x^{2^i})+\alpha^{2^{2k-i}}\left(Tr^{2k}_k( x)\right)^{2^{k-i}}.$$ From Lemma \[lemma adjoint\], we need to show that $\mathcal{L}(x)+\mathcal{L}^(x)=0$ if and only if $x=0$. Let $\nabla_{a}=\{x\|Tr^{2k}_k(x)=a, a\in {\Bbb F}_{2^{k}} \} $. We all know $ Tr^{2k}_k( x) $ is a surjection from ${\Bbb F}_{2^{2k}} $ to ${\Bbb F}_{2^{k}}$ and $\\|\nabla_{a}\\|=2^{2k-k} $ for any $a\in {\Bbb F}_{2^{k}}$. We also know $\nabla_{0}= {\Bbb F}_{2^{k}}$. If $\alpha\notin {\Bbb F}_{2^{k}} $, then $\mathcal{L}(x)+\mathcal{L}^(x)=0$ if and only if $$\label{equ 1bent2} \left\{\begin{array}{c} Tr^{2k}_k(\alpha x^{2^i})=0,\\ Tr^{2k}_k( x)=0, \end{array} \right.$$ i.e., $x=0$. If for any $ x\neq 0$, we always have $\mathcal{L}(x)+\mathcal{L}^(x)\neq 0$, then $\alpha\notin {\Bbb F}_{2^{k}} $. In fact, if $\alpha\in {\Bbb F}_{2^{k}} $, then $$\mathcal{L}(x)+\mathcal{L}^(x)=\alpha Tr^{2k}_k( x^{2^i})+\left(\alpha Tr^{2k}_k( x)\right)^{2^{k-i}}=\alpha Tr^{2k}_k( x)+\left(\alpha Tr^{2k}_k( x)\right)^{2^{k-i}}.$$ Further, $\mathcal{L}(x)+\mathcal{L}^(x)= 0$ for any $x\in {\Bbb F}_{2^{k}}$. Thus, we have $\mathrm{F}_\alpha^{i}(x)$ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}}$. Now, we are going to present a first new family of bent functions through adjoint operators. \[theo-new\_bent\] Let $V = {\Bbb F}_{2^{2k}}$ and $i$ be a nonnegative integer. Let $t_1,t_2$ be two positive integers such that $0\leq t_1,t_2\leq k$ and both $ z^{2^{k-t_1}-1}+z^{2^{k-t_2}-1}+1=0$ and $ z^{2^{t_1}-1}+ z^{2^{t_2}-1}+1=0$ have no solutions on ${\Bbb F}_{2^{k}}$. Then, the function $\mathrm{F}_\alpha^{i} $ defined on $V$ by $$\label{equa bent0} \mathrm{F}_\alpha^{i}(x)=Tr_1^{2k}\left(\alpha x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}})\right)$$ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}} $. We have $$\label{equa bent1} \begin{array}{rl} \mathrm{F}_\alpha^{i}(x)=&Tr_{1}^{2k}\left(\alpha x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}})\right)\\ =&Tr_{1}^{2k}(x\alpha x^{2^i})+Tr_{1}^{2k}(x\alpha^{2^k} x^{2^{i+k}})+Tr_{1}^{2k}(x\alpha^{2^{2k-t_1}} x^{2^{i+2k-t_1}})\\ &+Tr_{1}^{2k}(x\alpha^{2^{k-t_1}} x^{2^{i+k-t_1}})+Tr_{1}^{2k}(x\alpha^{2^{2k-t_2}} x^{2^{i+2k-t_2}})+Tr_{1}^{2k}(x\alpha^{2^{k-t_2}} x^{2^{i+k-t_2}})\\ =&Tr_{1}^{2k}(x\mathcal{L}(x)), \end{array}$$ where $$\begin{array}{rl} \mathcal{L}(x)=&\alpha x^{2^i}+\alpha^{2^k} x^{2^{i+k}}+\alpha^{2^{2k-t_1}} x^{2^{i+2k-t_1}}+\alpha^{2^{k-t_1}} x^{2^{i+k-t_1}}\\ &+\alpha^{2^{2k-t_2}} x^{2^{i+2k-t_2}}+\alpha^{2^{k-t_2}} x^{2^{i+k-t_2}}\\ =&\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}+\alpha^{2^{k-t_1}} x^{2^{i+k-t_1}}+(\alpha^{2^{k-t_1}} x^{2^{i+k-t_1}})^{2^k}\\ &+\alpha^{2^{k-t_2}} x^{2^{i+k-t_2}}+(\alpha^{2^{k-t_2}} x^{2^{i+k-t_2}})^{2^k}\\ =&\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)+\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)^{2^{k-t_1}}+\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)^{2^{k-t_2}} \end{array}$$ According to Lemma \[lemma L\], the adjoint operator $ \mathcal{L}^(x)$ is $$\label{equa adjoint1} \begin{array}{rl} \mathcal{L}^(x)=&\alpha^{2^{2k-i}} x^{2^{2k-i}}+\alpha^{2^{2k-i}} x^{2^{k-i}}+\alpha^{2^{2k-i}} x^{2^{t_1-i}}\\ &+\alpha^{2^{2k-i}} x^{2^{k+t_1-i}}+\alpha^{2^{2k-i}} x^{2^{t_2-i}}+\alpha^{2^{2k-i}} x^{2^{k+t_2-i}}\\ = &\alpha^{2^{2k-i}} \left(x^{2^{2k-i}}+ x^{2^{k-i}}+ x^{2^{t_1-i}} + x^{2^{k+t_1-i}}+ x^{2^{t_2-i}}+ x^{2^{k+t_2-i}}\right)\\ =&\alpha^{2^{2k-i}} \left( x^{2^{k-i}}+(x^{2^{k-i}})^{2^k}+ x^{2^{t_1-i}} + (x^{2^{t_1-i}})^{2^k}+ x^{2^{t_2-i}}+ (x^{2^{t_2-i}})^{2^k}\right)\\ =&\alpha^{2^{2k-i}} \left( (x+x^{2^k})^{2^{k-i}}+ (x+x^{2^k})^{2^{t_1-i}}+(x+x^{2^k})^{2^{t_2-i}} \right)\\ =&\alpha^{2^{2k-i}} \left( (x+x^{2^k})^{2^{k-i}}+ (x+x^{2^k})^{2^{t_1+k-i}}+(x+x^{2^k})^{2^{t_2+k-i}} \right) \end{array}$$ Thus, we have $$\begin{array}{rl} \mathcal{L}(x)+\mathcal{L}^(x) =&\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)+\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)^{2^{k-t_1}}+\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)^{2^{k-t_2}} \\ &+\alpha^{2^{2k-i}} \left( (x+x^{2^k})^{2^{k-i}}+ (x+x^{2^k})^{2^{t_1+k-i}}+(x+x^{2^k})^{2^{t_2+k-i}} \right). \end{array}$$ Note that we have $\mathcal{L}(x)\in {\Bbb F}_{2^{k}} $ and $ (x+x^{2^k})^{2^{k-i}}+ (x+x^{2^k})^{2^{t_1+k-i}}+(x+x^{2^k})^{2^{t_2+k-i}}\in {\Bbb F}_{2^{k}}$ for any $x\in {\Bbb F}_{2^{2k}}$. From Lemma \[lemma adjoint\], it is sufficient to show that $\mathcal{L}(x)+\mathcal{L}^(x)$ is invertible. That is, we need to show that $\mathcal{L}(x)+\mathcal{L}^(x)=0$ if and only if $x=0$. Since both $ z^{2^{k-t_1}-1}+z^{2^{k-t_2}-1}+1=0$ and $ z^{2^{t_1}-1}+ z^{2^{t_2}-1}+1=0$ have no solution in ${\Bbb F}_{2^{k}}$, we have both $\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)+\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)^{2^{k-t_1}}+\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)^{2^{k-t_2}}=0$ and $ (x+x^{2^k})^{2^{k-i}}+ (x+x^{2^k})^{2^{t_1+k-i}}+(x+x^{2^k})^{2^{t_2+k-i}}=0$ if and only if $$\label{equ bent2} \left\{\begin{array}{c} \alpha x^{2^i}+(\alpha x^{2^i})^{2^k}=0,\\ x+x^{2^k}=0. \end{array} \right.$$ From the proof of Theorem \[theo in POtt\], when both $ z^{2^{k-t_1}-1}+z^{2^{k-t_2}-1}+1=0$ and $ z^{2^{t_1}-1}+ z^{2^{t_2}-1}+1=0$ have no solution in ${\Bbb F}_{2^{k}}$, we have $\mathrm{F}_\alpha^{i}(x) $ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}} $. We immediately have the following statement by setting $t_2= k-t_1$ in the previous theorem. \[cor thr\] Let $V = {\Bbb F}_{2^{2k}}$ and $i$ be a nonnegative integer. Let $t_1,t_2$ be two positive integers such that $t_1+t_2= k$ and $ z^{2^{t_1}-1}+z^{2^{t_2}-1}+1=0$ has no solution in ${\Bbb F}_{2^{k}}$. Then, the mapping $\mathrm{F}_\alpha^{i} $ defined on $V$ by $$\mathrm{F}_\alpha^{i}(x)=Tr_{1}^{2k}\left(\alpha G(x)\right)$$ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}} $, where $G(x)=x^{2^i}\left(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}}\right)$. The previous construction given by Theorem \[theo-new\_bent\] can be generalized as follows. \[theorem 3\] Let $V = {\Bbb F}_{2^{2k}}$ and $i$ be a nonnegative integer. Let $t_1,t_2$ be two positive integers such that $0\leq t_1,t_2\leq k$ and both $ (\gamma^{(1)})^{2^{k-t_1}} z^{2^{k-t_1}-1}+(\gamma^{(2)})^{2^{k-t_2}}z^{2^{k-t_2}-1}+1=0$ and $ (\gamma^{(1)})^{2^{k-i}} z^{2^{t_1}-1}+(\gamma^{(2)})^{2^{k-i}} z^{2^{t_2}-1}+1=0$ have no solution in ${\Bbb F}_{2^{k}}$, where $\gamma^{(1)}, \gamma^{(2)}\in {\Bbb F}_{2^{k}}$. Then, the mapping $\mathrm{F}_\alpha^{i} $ defined by $$\label{equa bent 30} \mathrm{F}_\alpha^{i}(x)=Tr_{1}^{2k}\left(\alpha x^{2^i}\left(x+x^{2^k}+\gamma^{(1)}(x^{2^{t_1}}+x^{2^{t_1+k}}) +\gamma^{(2)}(x^{2^{t_2}}+x^{2^{t_2+k}})\right)\right)$$ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}} $. We have $$\label{equa bent31} \begin{array}{rl} \mathrm{F}_\alpha^{i}(x) =&Tr_{1}^{2k}(x\mathcal{L}(x)), \end{array}$$ where $$\begin{array}{rl} \mathcal{L}(x)=&\alpha x^{2^i}+\alpha^{2^k} x^{2^{i+k}}+(\gamma^{(1)})^{2^{k-t_1}} \left(\alpha^{2^{2k-t_1}} x^{2^{i+2k-t_1}}+\alpha^{2^{k-t_1}} x^{2^{i+k-t_1}}\right)\\ +&(\gamma^{(2)})^{2^{k-t_2}} \left(\alpha^{2^{2k-t_2}} x^{2^{i+2k-t_2}}+\alpha^{2^{k-t_2}} x^{2^{i+k-t_2}}\right)\\ =&\left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)+(\gamma^{(1)})^{2^{k-t_1}} \left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)^{2^{k-t_1}}\\ +&(\gamma^{(2)})^{2^{k-t_2}} \left(\alpha x^{2^i}+(\alpha x^{2^i})^{2^k}\right)^{2^{k-t_2}} \end{array}$$ The adjoint operator $ \mathcal{L}^(x)$ is $$\label{equa adjoint31} \begin{array}{rl} \mathcal{L}^(x)=&\alpha^{2^{2k-i}} x^{2^{2k-i}}+\alpha^{2^{2k-i}} x^{2^{k-i}}+(\gamma^{(1)})^{2^{k-i}}\left(\alpha^{2^{2k-i}} x^{2^{t_1-i}}\right.\\ &\left.+\alpha^{2^{2k-i}} x^{2^{k+t_1-i}}\right)+(\gamma^{(2)})^{2^{k-i}}\left(\alpha^{2^{2k-i}} x^{2^{t_2-i}}+\alpha^{2^{2k-i}} x^{2^{k+t_2-i}}\right)\\ =&\alpha^{2^{2k-i}} \left( (x+x^{2^k})^{2^{k-i}}+ (\gamma^{(1)})^{2^{k-i}}(x+x^{2^k})^{2^{t_1-i}}+(\gamma^{(2)})^{2^{k-i}}(x+x^{2^k})^{2^{t_2-i}} \right)\\ =&\alpha^{2^{2k-i}} \left( (x+x^{2^k})^{2^{k-i}}+ (\gamma^{(1)})^{2^{k-i}}(x+x^{2^k})^{2^{t_1+k-i}} +(\gamma^{(2)})^{2^{k-i}}(x+x^{2^k})^{2^{t_2+k-i}} \right) \end{array}$$ Note that we have $\mathcal{L}(x)\in {\Bbb F}_{2^{k}} $ and $ (x+x^{2^k})^{2^{k-i}}+ (\gamma^{(1)})^{2^{k-i}}(x+x^{2^k})^{2^{t_1+k-i}} +(\gamma^{(2)})^{2^{k-i}}(x+x^{2^k})^{2^{t_2+k-i}}\in {\Bbb F}_{2^{k}}$ for any $x\in {\Bbb F}_{2^{2k}}$. In order to show that $\mathcal{L}(x)+\mathcal{L}^(x)$ is invertible, we need to show that $\mathcal{L}(x)+\mathcal{L}^(x)=0$ if and only if $x=0$. Since both $ (\gamma^{(1)})^{2^{k-t_1}} z^{2^{k-t_1}-1}+(\gamma^{(2)})^{2^{k-t_2}}z^{2^{k-t_2}-1}+1=0$ and $ (\gamma^{(1)})^{2^{k-i}} z^{2^{t_1}-1}+(\gamma^{(2)})^{2^{k-i}} z^{2^{t_2}-1}+1=0$ have no solutions on ${\Bbb F}_{2^{k}}$, we have both $\mathcal{L}(x)=0$ and $ (x+x^{2^k})^{2^{k-i}}+ (\gamma^{(1)})^{2^{k-i}}(x+x^{2^k})^{2^{t_1+k-i}} +(\gamma^{(2)})^{2^{k-i}}(x+x^{2^k})^{2^{t_2+k-i}}=0$ if and only if $$\label{equ bent32} \left\{\begin{array}{c} \alpha x^{2^i}+(\alpha x^{2^i})^{2^k}=0,\\ x+x^{2^k}=0. \end{array} \right.$$ By using the proof of Theorem \[theo in POtt\], we have $\mathrm{F}_\alpha^{i}(x) $ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}} $. By the same process used to prove Theorem \[theorem 3\], one can get the following result. \[theo bentkla\] Let $V = {\Bbb F}_{2^{2k}}$ and $i,\rho$ be two nonnegative integers such that $\rho\leq k$. Let $\gamma^{(j)}\in {\Bbb F}_{2^{k}}$ and $0\leq t_j\leq k$ be a nonnegative integer, where $j=1,2,\cdots,\rho$. Assume that both equations $ \sum\limits_{j=1}^{\rho}(\gamma^{(j)})^{2^{k-t_j}} z^{2^{k-t_j}-1}+1=0$ and $ \sum\limits_{j=1}^{\rho}(\gamma^{(j)})^{2^{k-i}} z^{2^{t_j}-1}+1=0$ have no solution in ${\Bbb F}_{2^{k}}$. Then, the mapping $\mathrm{F}_\alpha^{i} $ defined on $V$ by $$\label{equa bent 41} \mathrm{F}_\alpha^{i}(x)=Tr_{1}^{2k}\left(\alpha G(x)\right) $$ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}} $, where $G(x)= x^{2^i}\left(Tr^{2k}_k(x)+\sum\limits_{j=1}^{\rho}\gamma^{(j)}(Tr^{2k}_k(x))^{2^{t_j}}\right)$. [@Pott2017]\[lemma vectorial bent\] Let $F_\alpha(x) = Tr^{2k}_1(\alpha G(x))$, be a Boolean bent function for any $\alpha \in {\Bbb F}_2^{2k}\setminus {\Bbb F}_2^{k}$, where $G : {\Bbb F}_2^{2k}\rightarrow {\Bbb F}_2^{2k}$. Then, $F : {\Bbb F}_2^{2k}\rightarrow {\Bbb F}_2^{k} $, defined as $F(x)=Tr^{2k}_k(\alpha G(x))$ is a vectorial bent function for any $\alpha \in {\Bbb F}_2^{2k}\setminus {\Bbb F}_2^{k}$. According to Theorem \[theo bentkla\] and Lemma \[lemma vectorial bent\], we immediately get the following theorem. \[theo bentvector\] Let $G(x)$ be defined as in Theorem \[theo bentkla\]. Then, the mapping $\mathrm{F}_\alpha$ defined by $$\label{equa bent 41} \mathrm{F}_\alpha(x)=Tr^{2k}_k\left(\alpha G(x)\right)$$ is a vectorial bent function for any $\alpha \in {\Bbb F}_{2^{2k}}\setminus {\Bbb F}_{2^k}$. In [@Pott2017], the authors presented the differential spectrum of the functions $ G: {\Bbb F}_{2^{2k}}\rightarrow {\Bbb F}_{2^{2k}}$ defined by $G(x)=x^{2^i}(x+x^{2^k})$. Their result is given below. [@Pott2017]\[lemma diff\] Let $i$ be a nonnegative integer such that $i<k$. The differential spectrum of the functions $G(x)=x^{2^i}(x+x^{2^k}), G: {\Bbb F}_{2^{2k}}\rightarrow {\Bbb F}_{2^{2k}} $, is given by, $$\label{equ lem pott} \delta_{G}(a,b)\in \left\{ \begin{array}{cl} \{0,2^k\} & ~~\text{if}~~a\in {\Bbb F}^_{2^k},\\ \{0,2^{\gcd(i,k)}\}& ~~\text{if}~~a\in {\Bbb F}_{2^{2k}}\setminus {\Bbb F}_{2^k}. \end{array}\right.$$ In particular, $\delta_{G}(a,b)=2^k$ only for $a\in {\Bbb F}^_{2^k}$ and $b\in {\Bbb F}_{2^k}$. Now we are going to show that the differential spectrum of the functions $x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}}) $ is different from the one of the functions $x\mapsto x^{2^i}(x+x^{2^k}) $. Let $ \mathrm{F}_\alpha^{i}(x)=Tr^{2k}_1\left(\alpha G(x)\right)$ be defined as Theorem \[theo-new\_bent\], where $G(x)= x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}})$. If there exists $t_1=1$ and $gcd(t_2,k)\neq 1$ such that both $ z^{2^{k-t_1}-1}+z^{2^{k-t_2}-1}+1=0$ and $ z^{2^{t_1}-1}+ z^{2^{t_2}-1}+1=0$ have no solutions on ${\Bbb F}_{2^{k}}$, then for $i=t_2$, there exist elements $a\in {\Bbb F}_{2^{2k}}\setminus {\Bbb F}_{2^k}$ such that the number $\delta_G(a,b)$ is equal to $2$ for any $b\in {\Bbb F}_{2^{2k}}$, which is neither $2^{\gcd(i,k)}$ nor $ 2^k$. Let $a\in {\Bbb F}_{2^{2k}}\setminus {\Bbb F}_{2^k}$ such that $\tau^{2^{t_1}}=\tau$, where $\tau=a+a^{2^k}\neq 0$ (since $t_1\|k$). We have $$\label{equ bent51} \begin{array}{rl} G(x)+G(x+a)=&a^{2^i}\left(x+x^{2^k}+(x+x^{2^k})^{2^{t_1}}+(x+x^{2^k})^{2^{t_2}}\right)\\ &+(x+a)^{2^i}\left(a+a^{2^k}+(a+a^{2^k})^{2^{t_1}}+(a+a^{2^k})^{2^{t_2}}\right). \end{array}$$ Thus, for any $x'\in {\Bbb F}_{2^{2k}}$, there must be one element $b\in {\Bbb F}_{2^{2k}}$ such that $ G(x')+G(x'+a)=b$. Let $x',x''$ are the solutions of $G(x)+G(x+a)=b $. Hence $$\label{equ bent52} \begin{array}{rl} &G(x')+G(x'+a)+G(x'')+G(x''+a)\\ =&a^{2^i}\left(x'+x'^{2^k}+(x'+x'^{2^k})^{2^{t_1}}+(x'+x'^{2^k})^{2^{t_2}} +x''+x''^{2^k}\right.\\ &\left.+(x''+x''^{2^k})^{2^{t_1}}+(x''+x''^{2^k})^{2^{t_2}}\right)\\ &+(x'+x'')^{2^i}\left(a+a^{2^k}+(a+a^{2^k})^{2^{t_1}}+(a+a^{2^k})^{2^{t_2}}\right)=0. \end{array}$$ Since $x+x^{2^k}\in {\Bbb F}_{2^k}$ for any $x\in {\Bbb F}_{2^{2k}}$, (\[equ bent52\]) implies that $(x'+x'')^{2^i}\left(a+a^{2^k}+(a+a^{2^k})^{2^{t_1}}\right.$ $\left.+(a+a^{2^k})^{2^{t_2}}\right)$ belongs to the multiplicative coset $a^{2^i}{\Bbb F}^_{2^k}$. Thus, we necessarily have $x'+x''=a\nu$, where $\nu\in {\Bbb F}^_{2^k}$. Further, $x'+x''+(x'+x'')^{2^k}=a\nu+a^{2^k}\nu$. Since $t_1\|t_2$, from (\[equ bent52\]), we have $$\label{equ bent53} \begin{array}{rl} &\left(\tau+\tau^{2^{t_1}}+\tau^{2^{t_2}}\right) \nu^{2^i} +\tau \nu+\tau^{2^{t_1}}\nu^{2^{t_1}}+\tau^{2^{t_2}}\nu^{2^{t_2}}\\ =&\tau \left(\nu^{2^{i}} + \nu+ \nu^{2^{t_1}}+ \nu^{2^{t_2}}\right)=0. \end{array}$$ If we set $i=t_2$, then from (\[equ bent53\]) we have $\delta_G(a,b)=2\neq 2^{\gcd(i,k)}$ since $\gcd(i,k)=gcd(t_2,k)\neq 1$. For one element $b\in {\Bbb F}_{2^{2k}}$, if for any $x'\in {\Bbb F}_{2^{2k}}$, we always have $ G(x')+G(x'+a)\neq b$, then $\delta_G(a,b)=0$. Let $i,\rho$ be two nonnegative integers such that $\rho\leq k$. Let $0\leq t_j\leq k$ be a nonnegative integer, where $j=1,2,\cdots,\rho$. Assume that both $ \sum\limits_{j=1}^{\rho}z^{2^{k-t_j}-1}+1=0$ and $ \sum\limits_{j=1}^{\rho} z^{2^{t_j}-1}+1=0$ have no solution in ${\Bbb F}_{2^{k}}$. Then, the mapping $\mathrm{F}_\alpha^{i} $ defined by $$\mathrm{F}_\alpha^{i}(x)=Tr_{1}^{2k}\left(\alpha G(x)\right) $$ where $G(x)= x^{2^i}\left(Tr^{2k}_k(x)+\sum\limits_{j=1}^{\rho}(Tr^{2k}_k(x))^{2^{t_j}}\right)$ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}} $. Further, if the number of the solutions of $ \sum\limits_{j=1}^{\rho} z^{2^{t_j}}+z+z^{2^i}=0$ on ${\Bbb F}_{2^{k}}$ is not equal to $2^{\gcd(i,k)}$, then there exist elements $a\in {\Bbb F}_{2^{2k}}\setminus {\Bbb F}_{2^k}$ such that the number $\delta_G(a,b)$ does not equal $2^{\gcd(i,k)}$ for any $b\in {\Bbb F}_{2^{2k}}$. From Theorem \[theo bentkla\], we know $ \mathrm{F}_\alpha^{i}(x) $ is bent if and only if $\alpha\notin {\Bbb F}_{2^{k}} $. We have $$\label{equ bent61} \begin{array}{rl} G(x)+G(x+a)=&a^{2^i}\left(Tr^{2k}_k(x)+\sum\limits_{j=1}^{\rho}(Tr^{2k}_k(x))^{2^{t_j}}\right)\\ &+(x+a)^{2^i}\left(Tr^{2k}_k(a)+\sum\limits_{j=1}^{\rho}(Tr^{2k}_k(a))^{2^{t_j}}\right)=b. \end{array}$$ Let $a\in {\Bbb F}_{2^{2k}}\setminus {\Bbb F}_{2^k}$ such that $a+a^{2^k}=1 $. We need to show the number of solutions of $ G(x)+G(x+a)=b$ is not equal to $ 2^{\gcd(i,k)}$ for any $b\in {\Bbb F}_{2^{2k}}$. Let $\rho=\gcd(i,k)$. We suppose $\delta_G(a,b)=2^{\rho}$ and let $x',x''$ are the solutions of (\[equ bent61\]) for some $b$. Hence $$\label{equ bent62} \begin{array}{rl} &G(x')+G(x'+a)+G(x'')+G(x''+a)\\ =&a^{2^i}\left(Tr^{2k}_k(x'+x'')+\sum\limits_{j=1}^{\rho}(Tr^{2k}_k(x'+x''))^{2^{t_j}}\right)+(x'+x'')^{2^i}=0 \end{array}$$ since $ \sum\limits_{j=1}^{\rho} z^{2^{t_j}-1}+1=0$ have no solution in ${\Bbb F}_{2^{k}} $, that is, $ \left(Tr^{2k}_k(a)+\sum\limits_{j=1}^{\rho}(Tr^{2k}_k(a))^{2^{t_j}}\right)=1$. For any $x\in {\Bbb F}_{2^{2k}}$, (\[equ bent62\]) implies that $(x'+x'')^{2^i}$ belongs to the multiplicative coset $a^{2^i}{\Bbb F}^_{2^k}$. Thus, we necessarily have $x'+x''=a\nu$, where $\nu\in {\Bbb F}^_{2^k}$. Further, $Tr^{2k}_k(x'+x'')=x'+x''+(x'+x'')^{2^k}=a\nu+a^{2^k}\nu$. From (\[equ bent62\]), we have $$\label{equ bent63} \nu^{2^i} +\nu+\sum\limits_{j=1}^{\rho}\nu^{2^{t_j}}=0.$$ We also know that the number of the solutions of $ \sum\limits_{j=1}^{\rho} z^{2^{t_j}}+z+z^{2^i}=0$ on ${\Bbb F}_{2^{k}}$ is not equal to $2^{\gcd(i,k)}$, thus, if $a\in \{ x\|Tr^{2k}_k(x)=1,x\in {\Bbb F}_{2^{2k}}\}$, the number $\delta_G(a,b)$ is not equals $2^{\gcd(i,k)}$ for any $b\in {\Bbb F}_{2^{2k}}$ \[theo genealized\] Let $n=2k,e$ be two positive integers. Let $V = {\Bbb F}_{2^{2k}}$ and $i$ be a nonnegative integer. Let $E=\{x\|x\in {\Bbb F}_{2^{2k}}, Tr_k^{2k}(x)\in {\Bbb F}_{2^{e}} \} $ and $O=\{x\in {\Bbb F}_{2^{2k}}, Tr_k^{2k}(x)\in M \} $, where $M=\{y+Tr^k_e(y)\| y\in{\Bbb F}_{2^{k}}\}$. Let $\mathrm{F}_\alpha^{i} $ be the function defined on $V$ by $$\label{equa bent70} \mathrm{F}_\alpha^{i}(x)=Tr_{1}^{2k}\left(\alpha x^{2^i}Tr^{2k}_e(x)\right).$$ If $\frac{k}{e}$ is even, then $\mathrm{F}_\alpha^{i} $ is bent if and only if $\alpha\notin E$. If $\frac{k}{e}$ is odd, then $\mathrm{F}_\alpha^{i} $ is bent if and only if $\alpha\notin O$. Further, if $k$ is odd and $e=2$, then $\mathrm{F}_\alpha^{i} $ is bent if and only if $\alpha\notin O$. We have $$\label{equa bent71} \begin{array}{rl} \mathrm{F}_\alpha^{i}(x)=&Tr_{1}^{2k}\left(\alpha x^{2^i}(x+x^{2^e}+x^{2^{2e}}+\cdots+x^{2^{2k-e}})\right)\\ =&Tr_{1}^{2k}(x\alpha x^{2^i})+Tr_{1}^{2k}(x\alpha^{2^{2k-e}} x^{2^{i+2k-e}})+Tr_{1}^{2k}(x\alpha^{2^{2k-2e}} x^{2^{i+2k-2e}})\\ &+\cdots +Tr_{1}^{2k}(x\alpha^{2^{e}} x^{2^{i+e}})\\ =&Tr_{1}^{2k}(x\mathcal{L}(x)), \end{array}$$ where $$\begin{array}{rl} \mathcal{L}(x)=&\alpha x^{2^i}+\alpha^{2^{2k-e}} x^{2^{i+2k-e}}+\alpha^{2^{2k-2e}} x^{2^{i+2k-2e}} +\cdots +\alpha^{2^{e}} x^{2^{i+e}}\\ =&\alpha x^{2^i}+(\alpha x^{2^i})^{2^{2k-e}}+(\alpha x^{2^i})^{2^{2k-2e}}+\cdots+(\alpha x^{2^i})^{2^{e}}\\ =&Tr^{2k}_e(\alpha x^{2^i}). \end{array}$$ According to Lemma \[lemma L\], the adjoint operator $ \mathcal{L}^(x)$ is $$\label{equa adjoint71} \begin{array}{rl} \mathcal{L}^(x)=&\alpha^{2^{2k-i}} x^{2^{2k-i}}+\alpha^{2^{2k-i}} x^{2^{e-i}}+\alpha^{2^{2k-i}} x^{2^{2e-i}} +\cdots+ \alpha^{2^{2k-i}} x^{2^{2k-e-i}}\\ = &\alpha^{2^{2k-i}} \left(x^{2^{2k-i}}+ x^{2^{e-i}}+ x^{2^{2e-i}} +\cdots+ x^{2^{2k-e-i}}\right)\\ = &\alpha^{2^{2k-i}} \left(x+ x^{2^{e}}+ x^{2^{2e}} +\cdots+ x^{2^{2k-e}}\right)^{2^{2k-i}}\\ \\ =&\alpha^{2^{2k-i}}\left(Tr^{2k}_e( x)\right)^{2^{2k-i}}. \end{array}$$ Thus, we have $$\begin{array}{rl} \mathcal{L}(x)+\mathcal{L}^(x) =&Tr^{2k}_e(\alpha x^{2^i})+\alpha^{2^{2k-i}}\left(Tr^{2k}_e( x)\right)^{2^{2k-i}}. \end{array}$$ From Lemma \[lemma adjoint\], it is sufficient to show that $\mathcal{L}(x)+\mathcal{L}^(x)$ is invertible. That is, we need to show that $\mathcal{L}(x)+\mathcal{L}^(x)=0$ if and only if $x=0$. For $\frac{k}{e}$ being even, we have $ Tr^{2k}_e( x)=0$ if and only if $x\in E$. If $\alpha\notin E$, then $\mathcal{L}(x)+\mathcal{L}^(x)=0$ is only if $$\label{equ bent72} \left\{\begin{array}{c} Tr^{2k}_e(\alpha x^{2^i})=Tr^{k}_e\left(Tr^{2k}_k(\alpha x^{2^i})\right)=0,\\ Tr^{2k}_e( x)=0, \end{array} \right.$$ i.e., $x=0$. If for any $x\neq 0$, we have $\mathcal{L}(x)+\mathcal{L}^(x)\neq0 $, then $\alpha\notin E$. In fact, if suppose $\alpha\in E $, then $$\begin{array}{rl} \mathcal{L}(x)+\mathcal{L}^(x) =&Tr^{2k}_e(\alpha x^{2^i})+\alpha^{2^{2k-i}}\left(Tr^{2k}_e( x)\right)^{2^{2k-i}}\\ =& Tr^{2k}_k(\alpha )Tr^{k}_e\left(Tr^{2k}_k( x^{2^i})\right)+\alpha^{2^{2k-i}}\left(Tr^{2k}_e( x)\right)^{2^{2k-i}}\\ =&0 \end{array}$$ for any $x\in E$. Hence, $ \mathrm{F}_\alpha^{i}(x)$ is bent if and only if $\alpha\notin E$. Similarly, for $\frac{k}{e}$ odd, we have $ Tr^{2k}_e( x)=0$ if and only if $x\in O$. We can prove $ \mathrm{F}_\alpha^{i}(x)$ is bent if and only if $\alpha\notin O$. Similarly, for $k$ odd and $e=2$, we have $ Tr^{2k}_2( x)=0$ if and only if $x\in O$. We can prove $ \mathrm{F}_\alpha^{i}(x)$ is bent if and only if $\alpha\notin O$. Note that Theorem \[theo in POtt\] is special case of Theorem \[theo genealized\]. It corresponds to the case where $e=k$. Similary to Theorem \[theo bentkla\], we have the following statement. \[theo bentkla generalized\] Let $i,\rho$ be two nonnegative integers such that $\rho\leq k$. Let $\gamma^{(j)}\in {\Bbb F}_{2^{k}}$ and $0\leq t_j\leq k$ be a nonnegative integer, where $j=1,2,\cdots,\rho$. Let both $ \sum\limits_{j=1}^{\rho}(\gamma^{(j)})^{2^{k-t_j}} z^{2^{k-t_j}-1}+1=0$ and $ \sum\limits_{j=1}^{\rho}(\gamma^{(j)})^{2^{k-i}} z^{2^{t_j}-1}+1=0$ have no solution in ${\Bbb F}_{2^{k}}$. Let $E$ and $O$ be defined as Theorem \[theo genealized\]. Let the function $\mathrm{F}_\alpha^{i} $ be defined by $$\label{equa bent 80} \mathrm{F}_\alpha^{i}(x)=Tr_{1}^{2k}\left(\alpha x^{2^i}\left(Tr^{2k}_e(x) +\sum\limits_{j=1}^{\rho}\gamma^{(j)}(Tr^{2k}_e(x))^{2^{t_j}}\right)\right).$$ If $\frac{k}{e}$ is even, then $\mathrm{F}_\alpha^{i} $ is bent if and only if $\alpha\notin E$. If $\frac{k}{e}$ is odd, then $\mathrm{F}_\alpha^{i} $ is bent if and only if $\alpha\notin O$. Further, if $k$ is odd and $e=2$, then $\mathrm{F}_\alpha^{i} $ is bent if and only if $\alpha\notin O$. Conclusions =========== This paper is in the line of a very recent paper published in the IEEE-transactions Information Theory by Pott et al [@Pott2017] in which several open problems have been raised. In the present paper, we have established that the property of a function having the maximal number of bent components is invariant under CCZ-equivalence which gives an answer to an open problem in [@Pott2017]. 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Vaudenay.: Links between differential and linear cryptanalysis. Proceedings of [EUROCRYPT’94, Lecture Notes in Computer Science]{} 950, pp. 356-365 (1995). A. Pott, E. Pasalic, A. Muratović-Ribić and S. Bajrić.: On the maximum number of bent components of vectorial functions, IEEE Transactions on Information Theory. Volume: 64, Issue: 1, pp. 403-411, 2018. O. S. Rothaus.: On bent functions, J. Combin. Theory, Ser. A, vol. 20, pp. 300–305, May 1976. Pasalic, E. and Zhang, W. G.: On Multiple Output Bent Functions, Inf. Process. Lett. vol. [112]{}, no. [21]{}, pp. [811–815]{}, nov, [2012]{}. K. Feng and J. Yang.: Vectorial boolean functions with good cryptographic properties. Int. J. Found. Comput. Sci., vol. 22(6):1271-1282, 2011 C. Tang, Y. Qi, M. Xu.: New Quadratic Bent Functions in Polynomial Forms with Coefficients in Extension Fields, https://eprint.iacr.org/2013/405.pdf M. 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--- abstract: 'The angle of rotation of any target about the radar line of sight (LOS) is known as the polarization orientation angle. The orientation angle is found to be non-zero for undulating terrains and man-made targets oriented away from the radar LOS. This effect is more pronounced at lower frequencies (eg. L- and P- bands). The orientation angle shift is not only induced by azimuthal slope but also by range slope. This shift increases the cross-polarization (HV) intensity and subsequently the covariance or the coherency matrix becomes reflection asymmetric. Compensating this orientation angle prior to any model-based decomposition technique for geophysical parameter estimation or classification is crucial. In this paper a new method is proposed to estimate the orientation angle based on the maximization of the degree of polarization. The proposed method is then used to infer the change in the degree of polarization with the associated orientation angle.' author: - \| Avik Bhattacharya\ Centre of Studies in Resources Engineering (CSRE),\ Indian Institute of Technology Bombay, Powai-400076, India. bibliography: - 'mybibfile.bib' title: 'Maximization of the degree of polarization to estimate the polarization orientation angle from PolSAR data: An insight into the real and complex rotation of the coherency matrix' --- Introduction ============ There are two main effects of surface slope and oriented urban structures on synthetic aperture radar (SAR) signals. One effect is related to the change in the radiometric property and the other is related to the change in the polarization states of the backscattered signal. The angle of rotation of any target about the radar line of sight (LOS) is known as the polarization orientation angle, $\theta$. In general, the basic principle of orientation compensation is to rotate the data about the LOS by $-\theta$. This process is analogous to rotating the antenna basis about the LOS by an angle $\theta$ such that the cross-polarized return (HV) is minimum. It has been observed that the orientation is non zero for undulating terrains and man-made targets tilted away from the radar LOS. These shifts of the orientation angle from zero are more pronounced in the low frequency (eg. L- and P- band) polarimetric synthetic aperture radar (PolSAR) data. It has been shown in [@Kimura08] that the orientation shift is not only induced by terrain slope but also by artificial structures in urban area tilted away from radar LOS. Compensating this orientation prior to any model-based decomposition [@freeman98] [@YAMAGUCHI2011] [@singh13] technique for geophysical parameter estimation or classification is very crucial. The decomposition methods proposed by Yamaguchi [@Yamaguchi05], An [@Wentao10] and Lee and Ainsworth [@Lee2011] reduced the number of independent parameters of the coherency matrix $\mathbf{[T]}$ from nine to eight by real rotating the coherency matrix. The decomposition yielded better results than before by using six out of eight parameters. The real and imaginary parts of $(T_{13})$ are unaccounted for in all these decomposition. The general four-component decomposition proposed by Singh [@singh13] uses a complex unitary transformation to the already real unitary rotated coherency matrix. This transformation completely eliminate the $(T_{23})$ element and accounts for seven out of seven independent parameters of the coherency matrix. In a similar way as before, the complex orientation angle is estimated by minimizing the cross-polarized term $(T_{33}(\theta))$. On the other hand, the eigenvector-eigenvalue decomposition parameters like the average scattering mechanism ($\alpha$), the entropy ($H$), the anisotropy ($A$) of the Cloude-Pottier decomposition [@CLOUDE97] and the scattering mechanism ($\alpha_{s}$), the scattering mechanism phase ($\phi_{\alpha_{s}}$) and the helicity ($\tau_{m}$) of the Touzi decomposition [@TOUZI2007] are all roll invariant. The orientation angle estimation methods can be broadly categorized into two groups: (1) orientation angle derived from Digital Elevation Model (DEM) and (2) orientation angle derived from PolSAR data. The DEM obtained from SAR interferometry can also be used to compensate PolSAR data. The slope and the azimuth estimated from the DEM can then be used to obtain the orientation angle. Apart from the DEM derived orientation angle there are few other methods available in the literature which directly uses PolSAR data to compute the orientation angle. The co-polarized peak shift in polarization signature proposed in [@Schuler96] is used to estimate the orientation angle in the range $[-\frac{\pi}{4},\frac{\pi}{4}]$. The phase difference between the RR-LL (Right-Right and Left-Left) circular polarizations has been used in [@Lee2000] to estimate the orientation angle. The angle is in the range of $[-\frac{\pi}{4},\frac{\pi}{4}]$ and the method is computationally simpler than the co-polarized peak shift method. The minimization of the cross-pol power to estimate the orientation angle has been proposed in [@Xu05]. The estimated orientation angle is dependent on $\mbox{Re}(T_{23})$, $T_{22}$ and $T_{33}$ elements of the 3$\times$3 coherency matrix, where $\mbox{Re}(X)$ denotes the real part of $X$. It has been shown that the cross-pol power minimization produces the same results as the circular polarization method. The orientation angle can also be estimated from the Cloude-Pottier eigenvalue/eigenvector decomposition [@Cloude96]. The above mentioned methods have been widely used to compensate the polarization orientation effects prior to any geophysical parameter extraction or classification. In the following sections we will look into the estimation of the real (coherency matrix rotated by a real unitary matrix) and complex (real rotated coherency matrix rotated by a complex unitary matrix) orientation angles by maximizing the degree of polarization with an insight into the change of the degree of polarization with the associated orientation angles. Methodology =========== Real orientation angle ---------------------- To estimate the real orientation angle, the multi-looked Hermitian positive semi-definite 3$\times$3 coherency matrix $\mathbf{[T]}$ which is obtained from the averaged outer product of the target vector with its conjugate is rotated by a real unitary matrix $\mathbf{[U_{3R}]}$ given as, $$\begin{split} \mathbf{[T(\theta)]} = \mathbf{[U_{3R}]}&\mathbf{[T]}{\mathbf{[U_{3R}]^{-1}}} \\ \\ \mathbf{[T]} = \left[ \begin{array}{ccc} T_{11} &\quad T_{12} &\quad T_{13} \\ T_{12}^{} &\quad T_{22} &\quad T_{23} \\ T_{13}^{} &\quad T_{23}^{} &\quad T_{33} \end{array}\right] ; \qquad & \mathbf{[U_{3R}]} = \left[ \begin{array}{ccc} 1 &\quad 0 &\quad 0 \\ 0 &\quad \cos(2\theta) &\quad \sin(2\theta) \\ 0 &\quad -\sin(2\theta) &\quad \cos(2\theta) \end{array}\right] \end{split} \label{eq:orienting_T_matrix}$$ where $\theta \in [-\frac{\pi}{4},\frac{\pi}{4}]$. In [@Lee2011], the orientation angle, $\theta$ is estimated by minimizing the cross-polarization $(\mbox{HV})$ response given in equation . The effect of $\theta$ on the three diagonal elements of the coherency matrix shows that: (1) $T_{11}=\left\|\mbox{HH} + \mbox{VV}\right\|^{2}/2$ is roll invariant for any $\theta$, (2) $T_{22}=\left\|\mbox{HH} - \mbox{VV}\right\|^{2}/2$ always increases or remains the same after the orientation angle compensation, (3) $T_{33}=2\left\|\mbox{HV}\right\|^{2}$ always decreases or remains the same after orientation angle compensation. $$\theta = \frac{1}{4}\tan^{-1}\left(\frac{-2\mbox{Re}(T_{23})}{T_{33}-T_{22}}\right) \label{eq:lee_angle}$$ Besides, multi-look PolSAR data in general can be represented by a 4$\times$4 Mueller matrix $\mathbf{[M]}$, which can be directly deduced from a 4$\times$4 coherency matrix $\mathbf{[T]}$. Since our interest centers on the special case of backscatter (BSA convention), the fourth row and column becomes zero due to reciprocity ($S_{\mbox{HV}}=S_{\mbox{VH}})$. In this case, $\mathbf{[T]}$ reduces to a 3$\times$3 coherency matrix, although $\mathbf{[M]}$ is a real 4$\times$4 matrix. The Mueller matrix is a linear mapping between the input and the output Stokes vectors. The received Stokes vector $\mathbf{G_{H}^{r}(\theta)} = [g_{H1}^{r} \quad g_{H2}^{r} \quad g_{H3}^{r} \quad g_{H4}^{r}]^{T}$ for a linear horizontally (H) polarized EM wave on transmit and the received Stokes vector $\mathbf{G_{V}^{r}(\theta)} = [g_{V1}^{r} \quad g_{V2}^{r} \quad g_{V3}^{r} \quad g_{V4}^{r}]^{T}$ for a linear vertically (V) polarized EM wave on transmit are related by the Mueller matrix as shown in equation , $$\begin{split} \mathbf{G_{H}^{r}(\theta)} &= \mathbf{[M(\theta)]}\mathbf{G_{H}^{t}} \\ \mathbf{G_{V}^{r}(\theta)} &= \mathbf{[M(\theta)]}\mathbf{G_{V}^{t}} \end{split} \label{eq:stokes_transmit_receive}$$ where $\mathbf{G_{H}^{t}} = [1 \quad 1 \quad 0 \quad 0]^{T}$ and $\mathbf{G_{V}^{t}} = [1 \quad {-1} \quad 0 \quad 0]^{T}$ are the transmitted linear horizontal and vertical polarized Stokes vectors respectively and $\mathbf{[M(\theta)]}$ is the rotated Mueller matrix derived from $\mathbf{[T(\theta)]}$. Here, the superscript $T$ denotes the vector transpose. The state of polarization of an EM wave is characterized in terms of the degree of polarization $(0\le p\le 1)$. The degree of polarization is defined as the ratio of the (average) intensity of the polarized portion of the wave to that of the (average) total intensity of the wave. For a completely polarized EM wave, $p=1$ and for a completely unpolarized EM wave, $p=0$. In between these two extreme cases, the EM wave is said to be partially polarized, $0\le p \le 1$. The degree of polarization of a received EM wave for a horizontally and a vertically transmitted wave is defined as $p_{H}(\theta)$ and $p_{V}(\theta)$ respectively as given in equation . $$p_{H}(\theta) = \frac{\sqrt{(g_{H2}^{r})^{2}+(g_{H3}^{r})^{2}+(g_{H4}^{r})^{2}}}{g_{H1}^{r}} \quad;\qquad p_{V}(\theta) = \frac{\sqrt{(g_{V2}^{r})^{2}+(g_{V3}^{r})^{2}+(g_{V4}^{r})^{2}}}{g_{V1}^{r}}, \label{eq:dop_h_v}$$ where the effective degree of polarization $p_{E}(\theta)$ is defined as, $$p_{E}(\theta) = \sqrt{\frac{p_{H}^{2}(\theta)+p_{V}^{2}(\theta)}{2}}. \label{eq:effective_dop}$$ The orientation angle $\theta$ is estimated by maximizing this effective degree of polarization $p_{E}(\theta)$ in the range of $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ as, $$\theta = \operatorname{\arg\!\max}_{-\pi/4\le\theta\le \pi/4}\left\{ p_{E}(\theta)\right\}. \label{eq:real_oa}$$ Finally this estimated orientation angle is restricted in the range $\left[-\frac{\pi}{8}, \frac{\pi}{8}\right]$ by the procedure given in equation so as to compare it with the orientation angle estimated by the method given in [@Lee2011] which lies in the range $\left[-\frac{\pi}{8}, \frac{\pi}{8}\right]$, $$\theta_{0}= \begin{cases} \theta+\pi/4, & \text{if}\ \theta<-\pi/8 \\ \theta-\pi/4, & \text{if}\ \theta>\pi/8 \\ \theta, & \text{otherwise}. \end{cases} \label{eq:final_OA}$$ To demonstrate the proposed methodology we have considered a coherency matrix $\mathbf{[T]}$ from a rotated urban area as an example, $$\mathbf{[T]} = \left[ \begin{array}{ccc} 23.66 &\qquad 2.46 + 0.61i &\qquad -0.01 - 2.03i \\ 2.46 - 0.61i &\qquad 20.58 &\qquad 6.74 - 0.06i \\ -0.01 + 2.03i &\qquad 6.74 + 0.06i &\qquad 15.15 \end{array}\right]. \label{eq:T_matrix_example}$$ The real orientation angle estimated from the proposed method is, $\theta_{0}=17^\circ$, which is similar to the one estimated by the method proposed in [@Lee2011]. The variation of $p_{H}$, $p_{V}$ and $p_{E}$ with $\theta$ is shown in Figure \[fig:theory\_results\_R\](a) and the vertical green and red lines in the zoomed Figure \[fig:theory\_results\_R\](b) shows the estimated angles from the proposed and the method in [@Lee2011] respectively. Complex orientation angle ------------------------- The complex orientation angle has been primarily used in the generalized four component decomposed proposed by [@singh13] to completely eliminate the $T_{23}$ component which is responsible for helical scattering. By eliminating the $T_{23}$ component of the coherency matrix, the number of independent information becomes seven for which the proposed decomposition performs better than the three component Freeman and Durden [@freeman98] decomposition and the four component scattering power decomposition with real rotation of the coherency matrix by Yamaguchi et al. [@YAMAGUCHI2011]. The idea of complex orientation is to rotate an already real rotated $\mathbf{[T(\theta)]}$ matrix by a complex unitary matrix $\mathbf{[U_{3C}]}$ given as, $$\begin{aligned} \mathbf{[T(\phi)]} & = & \mathbf{[U_{3C}]}\mathbf{[T(\theta_{0})]}{\mathbf{[U_{3C}]^{-1}}}\\ \mathbf{[T(\theta_{0})]} & = & \left[ \begin{array}{ccc} T_{11}(\theta_{0}) &\quad T_{12}(\theta_{0}) &\quad T_{13}(\theta_{0}) \\ T_{12}^{}(\theta_{0}) &\quad T_{22}(\theta_{0}) &\quad T_{23}(\theta_{0}) \\ T_{13}^{}(\theta_{0}) &\quad T_{23}^{}(\theta_{0}) &\quad T_{33}(\theta_{0}) \end{array}\right]\\ \mathbf{[U_{3C}]} & = & \left[ \begin{array}{ccc} 1 &\quad 0 &\quad 0 \\ 0 &\quad \cos(2\phi) &\quad j\sin(2\phi) \\ 0 &\quad j\sin(2\phi) &\quad \cos(2\phi) \end{array}\right] \label{eq:orienting_T_matrix_complex}\end{aligned}$$ where $\phi \in [-\frac{\pi}{4},\frac{\pi}{4}]$. Unlike before, where the real part of the $T_{23}$ element ($\mbox{Re}(T_{23})$) was used to compute the real orientation angle, the complex orientation angle is estimated by using the imaginary part of the $T_{23}(\theta_{0})$ element $(\mbox{Im}(T_{23}(\theta_{0})))$ instead as shown below, $$\phi = \frac{1}{4}\tan^{-1}\left(\frac{-2\mbox{Im}(T_{23}(\theta_{0}))}{T_{33}(\theta_{0})-T_{22}(\theta_{0})}\right). \label{eq:lee_angle_complex}$$ Here, the effective degree of polarization is a function of complex orientation angle and as before, it is obtained from the horizontally and vertically polarized transmitted EM wave for which the degree of polarization are $p_{H}(\phi)$ and $p_{V}(\phi)$ respectively using the similar expression as in . Further, following the similar procedure used for the estimation of real orientation angle, the complex orientation angle is estimated by maximizing the effective degree of polarization $p_{E}(\phi)$ in the range $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. Now, to compare the estimated angle $\phi$ with the one obtained in [@singh13], it is restricted in the range $\left[-\frac{\pi}{8}, \frac{\pi}{8}\right]$ using and the complex orientation angle lying in this range is denoted by $\phi_{0}$. We have again considered the same $\mathbf{[T]}$ matrix given in and rotated it with the estimated real orientation angle $(\theta_{0}=17^\circ)$. The real rotated coherency matrix, $\mathbf{[T(\theta_{0})]}$ is then subsequently unitary rotated by $\mathbf{[U_{3C}]}$. The complex orientation angle estimated from the proposed method is, $\phi_{0}=-0.11^\circ$, which is similar to the one estimated by the method proposed in [@singh13]. The variation of $p_{H}$, $p_{V}$ and $p_{E}$ with $\phi$ is shown in Figure \[fig:theory\_results\_C\](a) and the vertical green and red lines in the zoomed Figure \[fig:theory\_results\_C\](b) shows the estimated angles from the proposed and the method in [@singh13] respectively. Study area and dataset ====================== For this study, a subset of an UAVSAR image acquired over Hayward, California, USA has been chosen. The sensor was operated in L-Band, fully polarimetric mode with a bandwidth of 80 MHz capable of delivering a pixel spacing of 0.6m x 1.6m in Single Look Complex (SLC) mode. The multilooked product which is used in this study has 3 looks in range and 12 looks in azimuth applied to it to produce an image of resolution 5m x 7.2m. The extracted scene has a center latitude of $37^\circ46'01.65"$N and longitude of $122^\circ12'39.78"$W. This particular subset from the UAVSAR image over Hayward is chosen because it consists of clustered areas with varying orientation angles. The radar LOS and flight direction is marked with an arrow in Figure \[fig:dataset\](a) and the subset of the scene is shown with a black border. As seen from the street map, there are variations in the orientation angle of the urban structures with respect to the radar LOS. The top part of the scene consists of an urban patch which is approximately $15^\circ$ to $20^\circ$ oriented away from the radar LOS. The central portion of the scene consists of non-oriented urban areas, while in the lower half of the scene, the urban areas are approximately $-5^\circ$ to $-10^\circ$ oriented away from the radar LOS. The right portion of the scene consists of a forested area over an undulating topography and the left portion consists of a water area. The Pauli RGB image in Figure \[fig:dataset\](b) clearly shows the dominance of $\mbox{HV}$ component (in green) in the urban area rotated along the LOS. Results ======= The real orientation angle obtained by the proposed method and the one given in [@Lee2011] for the study area (Figure \[fig:dataset\](a)) is shown in Figure \[fig:results\_ROA\](a) and Figure \[fig:results\_ROA\](b) respectively. A 3$\times$3 boxcar filter has been applied to the PolSAR data before the orientation angle estimation. The orientation angle estimated by the proposed method shows complete similarity with that given in Lee et al.*. This is illustrated in Figure \[fig:results\_ROA\](c) for a transect “A” (shown in Figure \[fig:dataset\](b)) over a rotated urban area. The histogram of the difference in orientation angle computed for the entire scene by the two methods shows a mean of 0.06$^\circ$ with a standard deviation of 4.2$^\circ$ as shown in Figure \[fig:results\_ROA\](d). The orientation angle obtained from complex rotation by the method in [@singh13] and the proposed method are shown in Figure \[fig:results\_ROC\](a) and Figure \[fig:results\_ROC\](b) respectively. The complex orientation angle roughly varies from 0$^\circ-$ 2$^\circ$ for most areas of the image, with the exception of the water surface which shows an orientation angle of about 7$^\circ$. The histogram of the difference in orientation angle computed for the entire scene by the two methods shows a mean of $-$0.04$^\circ$ with a standard deviation of 4.3$^\circ$ as shown in Figure \[fig:results\_ROC\](d). The Dop after real orientation compensation will either increase or remain the same as the unrotated case, and the Dop after complex orientation compensation will subsequently either increase or remain the same as the real rotated case. Figure \[fig:dop\_change\_R\_C\](a) shows the change in Dop between the real rotated and the unrotated image, while Figure \[fig:dop\_change\_R\_C\](b) depicts the Dop change between the complex rotated and unrotated image. It can be seen that for a urban patch in the top-left portion of the image, the Dop increases by $\sim8\%$ for the real rotation case, whereas the same area shows an increase of $\sim12\%$ in case of the complex rotation. This is illustrated by the transect drawn over the urban patch, shown in Figure \[fig:dop\_change\_R\_C\](c). A similar rise of approximately $\sim2\%$ is also seen in the forested areas growing over undulating terrain in the right portion of the image. There is negligible or no rise in Dop observed for the water area that dominates the left portion of the image. This is shown in Figure \[fig:dop\_change\_R\_C\](c), where the portion of the transect to the left of the dotted line represents the water surface. This interesting insight into the variation of Dop with real and complex orientation compensation will be useful in characterizing a target. Conclusion ========== An orientation compensation method for PolSAR data has been developed and analyzed with a L-band UAVSAR data. We have found that the angle extracted from the proposed method is similar to the one proposed by an earlier method. An insight into the effect of orientation compensation is revealed by the variation of Dop with real and complex orientation. For a completely polarized incident wave, the output Dop is dependent on the Mueller matrix which describes the scattering process. In this regard, further research is needed to estimate the maximum achievable Dop for a target under consideration.
--- abstract: 'We prove that the group of Hamiltonian automorphisms of a symplectic $4$-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if $M$ is a simply connected symplectic $4$-manifold with $b_{2}\geq 3$, and if $\widetilde{M}_{\delta}$ denotes a blow-up of $M$ of small enough capacity $\delta$, then the rational cohomology algebra of the Hamiltonian group $\operatorname{Ham}(\widetilde{M}_{\delta})$ is not finitely generated. Both results are based on the fact that in a symplectic $4$-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable. Some applications and examples are given.' address: 'Department of Mathematics, University of Toronto, Toronto, Canada, M5S 3G3.' author: - Martin Pinsonnault title: 'Maximal compact tori in the Hamiltonian groups of $4$-dimensional symplectic manifolds' --- [^1] Introduction ============ Let $(M,{\omega})$ be a closed symplectic manifold of dimension $2n$. Given a time-dependent function $H:M\times{\mathbb{R}}\to {\mathbb{R}}$, define the associate Hamiltonian vector field $X_{t}^{H}$ by the formula $dH_{t}={\omega}(X_{t}^{H},-)$. A Hamiltonian automorphism $\phi$ of $(M,{\omega})$ is a symplectomorphism of $M$ which can be represented as the time-$1$ map of the flow of some Hamiltonian vector field $X_{t}^{H}$. Denote by $\operatorname{Ham}(M,{\omega})$ the group of all Hamiltonian automorphisms equipped with the standard $C^{\infty}$-topology. This is an infinite dimensional Fréchet Lie group whose Lie algebra consists of the space of smooth functions on $M$ with zero mean. The Hamiltonian group shares two remarkable properties with compact Lie groups: its Lie algebra comes with a natural bi-invariant nondegenerate inner product given by $$(f,g) = \int_{M} fg{\omega}^{n}$$ while the group itself admits a bi-invariant Finsler metric, called the Hofer metric, induced by the $L_{\infty}$ norm on its Lie algebra. For this reason, it is generally believed that the Hamiltonian group $\operatorname{Ham}(M,{\omega})$ is tamer than the diffeomorphism group $\operatorname{Diff}(M)$ and constitutes an intermediate object between compact Lie groups and more general diffeomorphism groups, see [@Bl], [@MD:LecturesSymp], and [@Re]. To develop a better understanding of this principle, it is natural to look at effective Hamiltonian group actions on symplectic manifolds. Since any compact Lie group $G$ contains a unique conjugacy class of maximal tori, and because the algebraic and homotopical properties of $G$ are, to a large extent, determined by the action of the Weyl group $W_{T}=N_{T}/T$ on a fixed maximal torus $T$, the first step in this direction is to understand maximal Hamiltonian actions by tori or, in other words, to classify conjugacy classes of maximal compact tori in $\operatorname{Ham}(M,{\omega})$. Because the symplectomorphism group $\operatorname{Symp}(M,{\omega})$ acts by conjugation on $\operatorname{Ham}(M,{\omega})$, this problem naturally breaks into two parts: one can first try to classify conjugacy classes of maximal tori in $\operatorname{Ham}(M,{\omega})$ with respect to the action of $\operatorname{Symp}(M,{\omega})$ and, afterward, with respect to the action of $\operatorname{Ham}(M,{\omega})$ itself. The case of $2$-dimensional manifolds is classical: the only effective Hamiltonian torus action on a surface is, up to symplectic conjugation, the standard orthogonal $S^1$-action on $S^2\subset{\mathbb{R}}^{3}$. By Smale’s theorem, $\operatorname{Symp}(S^{2})$ retracts onto $\operatorname{SO}(3)$, showing that $\operatorname{Symp}(S^{2})=\operatorname{Ham}(S^{2})$ and that the standard circle action behaves homotopically as $S^{1}\subset \operatorname{SO}(3)$. The case of $4$-manifolds is more interesting since the Hamiltonian group of a $4$-manifold can have distinct conjugacy classes of maximal tori. For example, the product $S^{2}\times S^{2}$ equipped with a product symplectic form $\mu\sigma\oplus\sigma$, $\mu\geq 1$, admits $\lceil \mu\rceil$ inequivalent Hamiltonian $T^{2}$-actions, where $\lceil \mu\rceil$ is the smallest integer larger than or equal to $\mu$, see [@Ka-Hirzebruch]. Moreover, by performing a sequence of symplectic blow-ups of appropriate capacities on $S^{2}\times S^{2}$, we can also create Hamiltonian $S^{1}$-actions that cannot be extended to an action of $T^{2}$, that is, maximal circles subgroups in $\operatorname{Ham}(M,{\omega})$. Indeed, by blowing up rational manifolds, we can construct $4$-manifolds with any given number of inequivalent maximal $S^{1}$- or $T^{2}$-actions. However, our first result shows that we can create, up to symplectic equivalence, at most finitely many such actions, namely, \[Finiteness\] The Hamiltonian group of a symplectic $4$-manifold contains at most finitely many conjugacy classes of maximal tori with respect to the action of the full symplectomorphism group. In contrast, the groups of volume preserving diffeomorphisms of several rational symplectic $4$-manifolds contain infinitely many conjugacy classes of maximal tori. So, from this point of view, Hamiltonian groups of $4$-manifolds do form intermediate objects between Lie groups and diffeomorphisms groups. The proof of Theorem \[Finiteness\] relies on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on $4$-manifolds, and on the analysis of exceptional curves in symplectic $4$-manifolds. More precisely, recall that a homology class $E\in H_{2}(M;{\mathbb{Z}})$ is a symplectic exceptional class if it can be represented by a symplectic sphere $S$ with $c_{1}(S)=1$ and $S\cdot S=-1$. In Sections 3, we reduce Theorem \[Finiteness\] to the following lemma: \[LemmePrincipal\] Let $(M,{\omega})$ be a symplectic $4$-manifold not diffeomorphic to ${{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2}$. Then, for any choice of ${\omega}$-tame almost complex structure $J$, all symplectic exceptional classes of minimal symplectic area are represented by an embedded $J$-holomorphic sphere. The problem of classisfying maximal tori in $\operatorname{Ham}(M,{\omega})$ up to Hamiltonian conjugation is much harder and remains largely open. Nevertheless, Theorem \[Finiteness\] allows us to prove the following partial result: \[HamConjugacyClasses\] Let $(M,{\omega})$ be a toric $4$-manifold. Then the set of Hamiltonian conjugacy classes of maximal $2$-tori in $\operatorname{Ham}(M,{\omega})$ is finite iff $\pi_{0}(\operatorname{Symp}(M,{\omega}))$ is finite. Because two compact subgroups in $\operatorname{Ham}(M,{\omega})$ are Hamiltonian conjugate iff they define homotopic actions, this theorem indicates that the homotopy theoretic properties of maximal tori in $\operatorname{Ham}(M,{\omega})$ are rather subtle. Interestingly, Lemma \[LemmePrincipal\] turns out to be a key ingredient in the following generalization of a result of Kedra [@Ke] on the rational homotopy type of symplectomorphism groups of $4$-manifolds: \[KedraGeneralise\] Let $(M,{\omega})$ be a compact simply connected $4$-manifold with $b_{2}(M)\geq 3$. Let $\imath:B_{\delta}{\hookrightarrow}M$ be a symplectic embedding of the standard ball of capacity $\delta$ and denote by $(\widetilde{M}_{\delta},\widetilde{{\omega}}_{\delta})$ the corresponding symplectic blow-up. Then, for $\delta$ sufficiently small, the rational cohomology of the Hamiltonian group of $(\widetilde{M}_{\delta},\widetilde{{\omega}}_{\delta})$ is infinitely generated as an algebra. Recently McDuff [@McDuff-ExtensionKedraResult] recast Kedra’s result in a more general framework which allows for various generalizations, some of them valid in higher dimensions. It would be interesting to know if, similarly, a version of Theorem \[Finiteness\] holds in dimensions greater than $4$. Note that since there is no hope to classify Hamiltonian torus actions in every dimensions, any general proof of such a theorem would have to rely solely on intrinsic properties of symplectic and Hamiltonian groups. In another direction, we do not know if there exist symplectic $4$-manifolds with infinitely many inequivalent symplecic, non-Hamiltonian actions. In view of the recent work of Pelayo [@Pe], this problem seems now accessible. It is interesting to note that there is no analog of Theorem \[Finiteness\] in the (full) contact category. Indeed, Lerman constructed [@Le:ContactCuts] an example of an overtwisted contact $3$-manifold whose contactomorphism group contains infinitely many distinct conjugacy classes of maximal tori. To our knowledge, however, no examples with tight contact structures are known. Finally, we note that Bloch, El Hadrami, Flaschka, and Ratiu drew very interesting analogies between compact Lie groups and Hamiltonian groups by studying, from an analytical point of view, centralizers of maximal tori in the Hamiltonian groups of two-dimensional annuli $A(r)\subset{\mathbb{R}}^{2}$ and toric manifolds. In particular, their results suggest[^2] that a version of the Schur-Horn-Kostant convexity theorem may hold for the Hamiltonian group of toric manifolds. The author would like to thank Yael Karshon and Dusa McDuff for their interest in this work and for many useful comments and suggestions, and the Fields Institute for providing a stimulating environment. This research was partially funded by NSERC Grant BP-301203-2004. Exceptional curves in symplectic $4$-manifolds ============================================== In this section, we prove Lemma \[LemmePrincipal\]. We start by reviewing the symplectic blow-up construction. Then we recall some facts from Taubes-Seiberg-Witten theory of Gromov invariants and from the classification of extremal rays on projective surfaces. Throughout this section, we assume the reader familiar with the theory of $J$-holomorphic curves in symplectic manifolds as explained in [@AL], [@Gr], and [@MS:J-HolomorphicCurves2]. Exceptional curves and symplectic blow-ups ------------------------------------------ Let $(M,{\omega})$ be a symplectic $4$-manifold and let ${\mathcal{E}}={\mathcal{E}}(M,{\omega})$ be the set of all symplectic exceptional homology classes $E\in H_{2}(M,{\mathbb{Z}})$, that is, classes that can represented by an embedded symplectic sphere of self-intersection $-1$. By the adjunction formula, any such class must satisfy $c_{1}(E)=1$, where $c_{1}$ is the first Chern class of $TM$ with respect to any tame almost complex structure $J$. Moreover, \[SetOfExceptionalClasses\] Let $(M,{\omega})$ be a symplectic $4$-manifold. 1. The set ${\mathcal{E}}$ depends only on the deformation class of ${\omega}$. 2. Given an exceptional class $E\in{\mathcal{E}}$, the subset ${\mathcal{J}}_{E}\subset{\mathcal{J}}(M,{\omega})$ of tamed almost complex structures for which there exist an embedded holomorphic sphere in class $E$ is open, dense and path-connected. 3. The subset ${\mathcal{J}}_{{\mathcal{E}}}$ of $J$’s for which all classes in ${\mathcal{E}}$ are represented by embedded $J$-holomorphic spheres is of second category in ${\mathcal{J}}$. 4. Given an arbitrary $J\in{\mathcal{J}}$, each symplectic exceptional class is represented by either an embedded $J$-holomorphic sphere or by a $J$-holomorphic cusp-curve[^3] of the form $$S=m_{1}S_{1}\cup\cdots\cup m_{n}S_{n}\text{,\hspace{5mm}}n\geq 2$$ where each $m_{i}S_{i}$ stands for a simple spherical component occurring with multiplicity $m_{i}\geq 1$. 5. Any two symplectic exceptional classes $E,E'\in{\mathcal{E}}$ intersect non-negatively. Statements (1) and (2) are proved in [@MD:Structure]. Statement (3) follows from (2) and Baire’s theorem. Statement (4) is a corollary of (3) together with Gromov’s compactness theorem. The last statement follows from (3) and from positivity of intersections for $J$-holomorphic curves. The following lemma also follows from Gromov’s compactness theorem. \[Properness\] The period map $P_{{\omega}}:{\mathcal{E}}\to {\mathbb{R}}_{+}$ which associates to a class $E$ its symplectic area ${\omega}(E)$ is proper. Consequently, $P_{{\omega}}$ has discrete image and reaches a minimal value $\varepsilon(M,{\omega})>0$. In particular, the set $${\mathcal{E}}_{\min}=\{E\in{\mathcal{E}}~\|~{\omega}(E)=\varepsilon(M,{\omega})\}$$ of smallest exceptional classes is finite and nonempty whenever ${\mathcal{E}}$ is nonempty. For now on, we will always denote by $\varepsilon(M,{\omega})$ the minimal area of an exceptional symplectic sphere in $(M,{\omega})$. \[SoftProperness\] When $b_{2}^{+}(M)=1$, Lemma \[Properness\] follows from a purely algebraic argument. Indeed, the intersection form defines a metric of signature $(1,b_{2}-1)$ on $H_{2}(M;{\mathbb{R}})$. It is then easy to show that given a homology class $A\in H_{2}(M;{\mathbb{R}})$ such that $A\cdot A\geq 0$, and given any real numbers $a\leq b$ and $0<p\leq q$, the set $$N(A,a,b,p,q)= \{ B\in H_{2}(M;{\mathbb{R}})~:~-q \leq B\cdot B \leq -p\text{~~and~~}a\leq A\cdot B\leq b\}$$ is compact. Lemma \[Properness\] follows by setting $A=\operatorname{PD}({\omega})$ and $p=q=1$. The compactness of the sets $N(A,a,b,p,q)$ has another interesting corollary: the group $\operatorname{Aut}_{[{\omega}]}$ of automorphisms of $H_{2}(M;{\mathbb{R}})$ which preserves the lattice $H_{2}(M;{\mathbb{Z}})$ and which fixes the homology class $[{\omega}]$ is always finite. Indeed, any automorphism $\phi\in\operatorname{Aut}_{[{\omega}]}$ is determined by its action on some orthogonal basis $\{A_{1},\ldots,A_{n}\}$ of $H_{2}(M;{\mathbb{Z}})$. Now, given any $1\leq i\leq k$, the set $${\mathcal{A}}_{i}=\{B\in H_{2}(M;{\mathbb{Z}})~\|~B\cdot B=A_{i}\cdot A_{i}\text{~and~}{\omega}(B)={\omega}(A_{i})\}$$ is compact, hence finite. The finiteness of $\operatorname{Aut}_{[{\omega}]}$ follows from the fact that any $\phi\in\operatorname{Aut}_{[{\omega}]}$ acts by permutations on the sets ${\mathcal{A}}_{i}$. Recall that given a symplectic embedding $\imath:B^{4}_{\delta}{\hookrightarrow}(M,{\omega})$ of the standard ball of radius $r$ and capacity $\delta=\pi r^{2}$ in a $4$-manifold $(M,{\omega})$, one can perform a symplectic blow-up of size $\delta$ centered at $m=\imath(0)$ by removing the image of the ball and identifying a neighborhood of the boundary with a neighborhood of the zero section $\Sigma$ of the universal bundle over ${\mathbb{CP}}^{1}$. This operation produces a new symplectic manifold $(\widetilde{M}_{\imath},\tilde{{\omega}}_{\imath})$ whose homology is naturally identified with $H_{}(M)\oplus{\mathbb{Z}}[\Sigma]$, and in which the exceptional divisor $\Sigma$ is a symplectic exceptional sphere of symplectic area $\delta$. Since the construction depends on a connected family of parameters, the symplectic blow-up is well defined up to isotopy[^4] and hence up to symplectomorphisms. The inverse process is the symplectic blow-down and consists in replacing a neighborhood of a symplectic exceptional sphere $S$ by a ball of capacity ${\omega}(S)$. Again, the resulting symplectic manifold is well defined up to isotopy. By analogy with the complex category, we say that two symplectic manifolds are rationally equivalent if one can be obtained from the other by a sequence of symplectic blow-ups and blow-downs. It is often useful to interpret the symplectic blow-up in a slightly different way, namely, as a procedure which defines, out of any symplectic embedding $\imath:B^{4}_{\delta}{\hookrightarrow}(M,{\omega})$, a symplectic form on a fixed smooth blow-up $\widetilde{M}$ of $M$. As explained in [@MP], this approach allows comparison of blow-ups obtained from different embeddings: 1. Given two embeddings $\imath_{i}:B^{4}_{\delta_{i}}\to (M,{\omega})$ of arbitrary sizes $\delta_{1}$ and $\delta_{2}$, the corresponding blow-ups are deformation equivalent. 2. Given a family of embeddings $\imath_{t}:B^{4}_{\delta}\to (M,{\omega})$ of same capacity $\delta$, the blow-up manifolds $(\widetilde{M},\tilde{{\omega}}_{t})$ are symplectomorphic. Moreover, if the embeddings are all centered at the same point $m\in M$, then the blow-up manifolds are isotopic. No general statement about the homotopy type of embeddings spaces of symplectic balls of fixed capacity is known at the moment. However, in the special case of manifolds rationally equivalent to ruled $4$-manifolds, McDuff proved the following result: \[UniquenessBlowups\] Suppose $(M,{\omega})$ is rationally equivalent to ${\mathbb{CP}}^{2}$ or to a ruled manifold $S^{2}\to N\to \Sigma_{g}$. Let $p$ be a point of $M$ and let $B^{4}_{\delta}\subset{\mathbb{R}}^{4}$ be the standard ball of capacity $\delta$. Then the space of centered embeddings $\imath:(B^{4}_{\delta},0){\hookrightarrow}(M,p)$ is path-connected. Consequently, all symplectic blow-ups of $(M,{\omega})$ of same capacities are isotopic. The previous proposition does not imply that all cohomologous symplectic forms on a blow-up $\tilde{M}$ are isotopic. Only those symplectic forms obtained from isotopic forms on $M$ are known to be isotopic. Because symplectic exceptional spheres in the same homology class are symplectically isotopic, the blow-down process is easier to analyse: \[UniquenessBlowdown\] 1. The manifold obtained from a symplectic $4$-manifold $(M,{\omega})$ by blowing down a symplectic exceptional sphere $S$ is determined up to symplectic isotopy by the homology classe $[S]\in H_{2}(M;{\mathbb{Z}})$. 2. Every symplectic $4$-manifold $(M,{\omega})$ covers a minimal symplectic manifold which may be obtained from $M$ by blowing down a maximal collection of disjoint exceptional spheres. Moreover, the symplectomorphism class of the minimal model is unique unless $(M,{\omega})$ is the $k$-fold blow-up of ${\mathbb{CP}}^{2}$ or of a ruled manifold. For manifolds rationaly equivalent to ${\mathbb{CP}}^{2}$ or to a ruled manifold the symplectic minimal model is often not unique. Worse, the blow-down of different sets of exceptional spheres may lead to the same minimal symplectic manifold. For instance, every rational manifold $M_{k}={\mathbb{CP}}^{2}{\#\,k\overline{{\mathbb{CP}}}\,\!^2}$ with $k\geq 9$ has infinitely many non-symplectomorphic minimal models. Fortunately, the ambiguity in the choice of a symplectic minimal model can always be reduced to a finite number of choices by considering blow-downs along spheres representing exeptional classes of minimal area, namely, \[ChainOfMinimalBlowdowns\] Given a symplectic manifold $(M,{\omega})$ rationaly equivalent to ${\mathbb{CP}}^{2}$ or to a ruled manifold, there exist finitely many maximal families of orthogonal exceptional classes $\{E_{k},\ldots ,E_{1}\}\subset H_{2}(M;{\mathbb{Z}})$ such that the corresponding chain of blow-ups manifolds $$(M,{\omega})=(M_{k},{\omega}_{k})\stackrel{E_{k}}{\to}(M_{k-1},{\omega}_{k-1})\stackrel{E_{k-1}}{\to} \cdots \stackrel{E_{1}}{\to}(M_{0},{\omega}_{0})$$ verify $E_{i}\in{\mathcal{E}}_{\min}(M_{i},{\omega}_{i})$. Let $(M_{k},{\omega}_{k})=(M,{\omega})$ and choose a symplectic exceptional class of minimal area $E_{k}\in{\mathcal{E}}_{\min}(M_{k},{\omega}_{k})$. By Lemma \[Properness\], there are only finitely many such classes. Now, Proposition \[SetOfExceptionalClasses\] implies that $E_{k}$ is represented by an embedded symplectic sphere $S$ which can be symplectically blow-down. By Proposition \[UniquenessBlowdown\], the symplectomorphism type of the resulting symplectic manifold only depends on ${\omega}_{k}$ and on the choice of the class $E_{k}$. The results follows by induction. In the following, we will say that a blow-up (resp. a blow-down) is minimal if the exceptional divisor (resp. the blow-down sphere) represents an exceptional class of minimal area. Although the choice of exceptional classes $\{E_{k},\ldots ,E_{1}\}\subset H_{2}(M;{\mathbb{Z}})$ in Corollary \[ChainOfMinimalBlowdowns\] is usualy not unique, their areas ${\omega}(E_{k})\geq\cdots\geq{\omega}(E_{1})$ only depend on the initial manifold $(M,{\omega})$. Moreover, for a sufficiently generic symplectic form ${\omega}$ on $M$, for example, when the periods of ${\omega}$ are linearly independent over ${\mathbb{Q}}$, each set ${\mathcal{E}}_{\min}(M_{i},{\omega}_{i})$ contains a unique class. In that case, the chain of minimal blow-ups is completely canonical. Gromov invariants ----------------- Recall that for a 4-dimensional symplectic manifold $(M,{\omega})$ and a class $A\in H_{2}(X;{\mathbb{Z}})$ such that $k(A)=\frac{1}{2}(A\cdot A + c_{1}(A))\geq 0$, the Gromov invariant of $A$ as defined by Taubes[^5] in [@Ta] counts, for a generic almost complex structure $J$ tamed by ${\omega}$, the algebraic number of embedded $J$-holomorphic curves in class $A$ passing through $k(A)$ generic points. In general, these curves may be disconnected and, in that case, components of negative self-intersection are exceptional spheres. An important corollary of Gromov’s compactness theorem is that, for any choice of tamed $J$, a class $A$ with non zero Gromov invariant $\operatorname{Gr}_{{\omega}}(A)$ is always represented by a union of $J$-holomorphic cusp-curves. In particular $\operatorname{Gr}_{{\omega}}(A)\neq 0$ implies ${\omega}(A)>0$. The Gromov invariants only depend on the deformation class of ${\omega}$, that is, $\operatorname{Gr}_{{\omega}_{t}}(A)$ is constant along any $1$-parameter family ${\omega}_{t}$ of symplectic forms. In fact, in dimension four, it follows from the work of Taubes that these invariants are smooth invariants of $M$. Now suppose that $(M,{\omega})$ is the $k$-fold blow-up of a ruled manifold $$S^{2}\to M_{0}\to\Sigma_{g}$$ Let $K_{M}$ stands for the canonical divisor of $(M,{\omega})$, that is, $K_{M}=-\operatorname{PD}(c_{1})$. Since $b_{2}^{+}=1$, the wall-crossing formula of Li-Liu for Seiberg-Witten invariants (see [@LL], Corollary 1.4) implies that for every class $A\in H_{2}(M,{\mathbb{Z}})$ verifying the condition $k(A)=\frac{1}{2}(A\cdot A + c_{1}(A))\geq 0$ we have $$\label{WallCrossing} \operatorname{Gr}(A)\pm \operatorname{Gr}(K_{M}-A) = \pm (1+F\cdot A)^{g}$$ where $F$ denotes the image in $H_{2}(M;{\mathbb{Z}})$ of a fiber in $M_{0}$. \[NonZeroGr\] Suppose $(M,{\omega})$ can be obtained from a ruled manifold $(M_{0},{\omega}_{0})$ by a sequence of symplectic blow-ups. Then any homology class $A\in H_{2}(M;{\mathbb{Z}})$ verifying $A\cdot A\geq 0$, ${\omega}(A)>0$, and $k(A)=\frac{1}{2}(A\cdot A + c_{1}(A))\geq 0$ has nonzero Gromov invariant. Let $F_{0}\in H_{2}(M,{\mathbb{Z}})$ be the image of a fiber of $(M_{0},{\omega}_{0})$ under the natural inclusion $H_{2}(M_{0},{\mathbb{Z}})\subset H_{2}(M,{\mathbb{Z}})$ determined by the blow-up sequence defining $(M,{\omega})$. Note that we necessarily have ${\omega}(F_{0})>0$. Because $b_{2}^{+}(M)=1$, the set $\overline{{\mathcal{P}}}\in H_{2}(M;{\mathbb{Z}})$ of integral homology classes with nonnegative self-intersection is a cone and $\overline{{\mathcal{P}}}-\{0\}$ has exactly 2 components. By the light-cone lemma, two classes in $\overline{{\mathcal{P}}}-\{0\}$ are in the same component if and only if they intersect nonnegatively. Since we have $\operatorname{PD}({\omega})^{2}>0$, $F_{0}^{2}=0$, ${\omega}(F_{0})>0$, and ${\omega}(A)>0$, the classes $F_{0}$ and $A$ must lie in the component containing $\operatorname{PD}({\omega})$. Consequently, $F_{0}\cdot C\geq 0$. Because $k(A)\geq 0$, we can now apply the wall-crossing formula (\[WallCrossing\]) for Gromov invariants to deduce that $ \operatorname{Gr}(A)\pm \operatorname{Gr}(K-A) = \pm(1+F_{0}\cdot A) \neq 0$. In particular, $\operatorname{Gr}(A)$ and $\operatorname{Gr}(K-A)$ cannot be both equal to zero. We claim that $\operatorname{Gr}(K-A)= 0$. To see this, note that the invariance of Gromov invariants along deformations allows us to replace ${\omega}$ by any convenient symplectic form in the same path-component. Note also that we can write the canonical class has $K=-\operatorname{PD}(c_{1})=-(B_{0}-E_{1}-\cdots -E_{k})$ where $B_{0}\in H_{2}(M_{0};{\mathbb{Z}})$ is a class with positive symplectic area ${\omega}_{0}(B)>0$ and where the $E_{i}$ are the exceptional classes defined by the blow-up sequence $M\to\cdots\to M_{0}$. By considering blow-ups of sufficiently small sizes, it is clear that we can find a deformation equivalent symplectic form ${\omega}'$ for which the symplectic area of $K$ is strictly negative. Moreover, because ${\omega}$ and ${\omega}'$ are in the same component of $\overline{{\mathcal{P}}}-\{0\}$, any such form verifies ${\omega}'(A)\geq 0$. Hence ${\omega}'(K-A)<0$ and the Gromov invariant of $(K-A)$ must be zero. Consequently, $\operatorname{Gr}(A)=\pm(1+F_{0}\cdot A)^{g}\neq 0$. Extremal curves on surfaces --------------------------- Let us now explain some fundamental results from algebraic geometry. Recall that given an (integrable) complex structure $J$ on an algebraic variety $V$, the effective cone $NE^{J}\subset H_{2}(V,{\mathbb{R}})$ is defined by setting $$NE^{J}=\left\{A = \sum_{i}a_{i}[C_{i}]~~:~~C_{i}\text{~is a $J$-curve and $a_{i}\in{\mathbb{R}}_{+}$}\right\}$$ where ${\mathbb{R}}_{+}$ denotes the set of nonnegative real numbers. In particular, on a Kähler $4$-manifold $(M,{\omega},J)$, Lemma \[SetOfExceptionalClasses\] implies that ${\mathcal{E}}(M,{\omega})\subset\overline{NE}^{J}$. More generally, as was first noted by Ruan [@Ru], any class $A$ whose Gromov invariant is non zero must be contained in $\overline{NE}^{J}$. We say that an element $A\in NE^{J}$ is extremal if whenever $A$ can be written as a sum $A=B+C$ for $A, B\in NE^{J}$, then $B$ and $C$ are multiples of $A$. For smooth algebraic varieties, the structure of $NE^{J}(V)$ can be described in terms of extremal classes. To this end, given any cohomology class $\Gamma\in H^{1}(V;{\mathbb{Z}})$, we define $(\Gamma)_{\leq 0}$ by setting $(\Gamma)_{\leq0}=\left\{ A\in H_{2}(M;{\mathbb{R}}) \| \Gamma(A)\leq 0 \right\}$. Then, \[ConeTheorem\] Let $V$ be a smooth $n$-dimensional complex projective variety with first Chern class $c_{1}$. Then there exists a set of rational curves $l_{v}$ with $0<c_{1}(l_{i})\leq n+1$, called Mori extremal rational curves, such that 1. $\overline{NE}^{J}(V)=\left(\sum{\mathbb{R}}_{+}[l_{i}]\right)+\overline{NE}^{J}(V)\cap(c_{1})_{\leq 0}$ 2. Each $l_{i}$ is extremal. The classification of extremal curves on complex projective surfaces is classical and can be traced back to the work of Castelnuovo and Enriques: \[ClassificationExtremalCurves\] An irreducible curve C on an algebraic surface $V$ is an extremal rational curve if and only if one of the following conditions is satisfied: 1. $C$ is an exceptional curve, 2. $V$ has the structure of a ruled manifold such that $C$ is one of the fibers, 3. $V=\mathbb{P}^{2}$ and $C$ is a projective line. Exceptional classes of minimal symplectic area ---------------------------------------------- The key fact in the proof of Lemma \[LemmePrincipal\] is the following simple observation: \[LemmeClassesPositives\] Let $(M,{\omega})$ be a non-minimal symplectic $4$-manifold rationally equivalent to a ruled manifold but not diffeomorphic to ${{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2}$. Then any class $A\in H_{2}(M;{\mathbb{Z}})$ verifying $c_{1}(A)\geq 1$, $A\cdot A\geq 0$, and ${\omega}(A)>0$, must also satisfy ${\omega}(A)\geq \varepsilon(M,{\omega})$. Because $k(A)\geq 1$, $A\cdot A\geq 0$, and ${\omega}>0$, we can apply Lemma \[NonZeroGr\] to deduce that the class $A$ has nonzero Gromov invariant. It follows that, for any choice of ${\omega}$-tamed $J$, the class $A$ must be in the effective cone $NE^{J}$. If the symplectic form ${\omega}$ is integral, $(M,{\omega})$ can be realized as a complex projective variety $(M,J_{0},{\omega})\subset{\mathbb{CP}}^{N}$. Because $A\in \overline{NE}^{j_{0}}$, theorem \[ConeTheorem\] implies that $A$ can be written as a sum of the form $$A=\sum a_{i}[l_{i}] + \text{~classes with non positive first Chern number}$$ But since $c_{1}(A)\geq 1$, at least one $J_{0}$-extremal class $[l_{i}]$ must enter this linear combination with strictly positive coefficient $a_{i}$. Because $(M,{\omega})$ is the blow-up of a ruled manifold, and because $(M,{\omega})$ is not itself ruled, theorem \[ClassificationExtremalCurves\] implies that this extremal class must be exceptional. It follows that ${\omega}(A)>\varepsilon(M,{\omega})$. Because any rational class can be made integral by an appropriate rescaling, this finishes the proof in the case $[{\omega}]$ is rational. To extend this argument to the case of a non rational symplectic form ${\omega}$, it is sufficient to construct, for any $\delta>0$, a deformation ${\omega}_{t}$ starting at ${\omega}={\omega}_{0}$ such that 1. ${\omega}_{1}$ is rational, 2. $\|{\omega}_{1}-{\omega}_{0}\|_{C^{\infty}}<\delta$, and 3. the minimal area $\varepsilon_{t}=\varepsilon(M,{\omega}_{t})$ of the symplectic exceptional spheres in $(M,{\omega}_{t})$ varies continuously with $t$. To show that such a deformation always exists, choose $E_{\min}\in{\mathcal{E}}_{\min}({\omega})$ and consider a maximal set of pairwise orthogonal exceptional classes $\{E_{\min},E_{1},\ldots,E_{k}\}$. By theorem \[UniquenessBlowdown\], $(M,{\omega})$ is obtained by blowing up a minimal surface $(M_{0},{\omega}_{0})$ along embedded balls of capacities $${\omega}(E_{min}),{\omega}(E_{1}),\ldots,{\omega}(E_{k})$$ It follows that any exceptional class $E\in{\mathcal{E}}(M,{\omega})$ can be uniquely written as $$E=e_{B}B-e_{\min}E_{\min}-\sum_{i=0}^{k} e_{i}E_{i}$$ where $B\in H_{2}(M_{0},{\mathbb{Z}})$ and $e_{B},e_{\min},e_{1},\ldots,e_{k}\in{\mathbb{Z}}$. Because distinct symplectic exceptional classes always intersect positively, the coefficients $e_{\min},e_{1},\ldots,e_{k}$ are all non-negative as long as $E$ is not equal to $E_{\min}$ or to one of the $E_{i}$. It follows that the symplectic area of such an exceptional class never decreases along a deformation of ${\omega}$ in which the areas ${\omega}_{t}(E_{\min})$ and ${\omega}_{t}(E_{i})$ decrease. Therefore, one can achieve the desired deformation ${\omega}_{t}$ by reducing the sizes of the blow-ups appropriately. Let $(M,{\omega})$ be a non-minimal symplectic $4$-manifold not diffeomorphic to the ruled manifold ${{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2}$. Suppose that for some almost complex structure $J\in{\mathcal{J}}({\omega})$, a class $E_{\min}\in {\mathcal{E}}_{\min}({\omega})$ has no embedded $J$-holomorphic representative. By Gromov’s compactness, $E_{\min}$ must be represented by a $J$-holomorphic cusp-curve having at least 2 distinct components. Because $c_{1}(E_{\min})=1$, there must be a spherical simple component representing a homology class $C$ with $c_{1}(C)\geq 1$ and such that ${\omega}(C)<{\omega}(E_{\min})$. If $c_{1}(C)=1$, the adjunction inequality $$0\leq 1+\frac{C\cdot C-c_{1}(C)}{2}\leq\frac{C\cdot C+1}{2}$$ implies that $C\cdot C\geq -1$, with equality if and only if $C$ is an exceptional class. The area of $E_{\min}$ being minimal among all symplectic exceptional classes, $C$ itself cannot be exceptional and hence $C\cdot C\geq 0$. Now, by McDuff [@MD:Immersions] Corollary 4.2.1, we can always perturb $J$ and assume that the $J$-holomorphic representative of the class $C$ is immersed. Consequently, we can apply [@MD:ImmersedSpheres] Theorem 1.4 to conclude that $(M,{\omega})$ is the $k$-fold symplectic blow-up of a ruled manifold $S^{2}\to M_{0}\to\Sigma_{g}$. Hence, we can apply Lemma \[LemmeClassesPositives\] to conclude that ${\omega}(C)\geq\varepsilon(M,{\omega})$. This contradiction shows that $E_{\min}$ can only be represented by embedded $J$-holomorphic curves. Consider the Hirzebruch surface of degree $2i+1\geq 3$, that is $$W_{2i+1}=\{ ([z_{0}:z_{1}],[w_{0}:w_{1}:w_{2}])\in{\mathbb{CP}}^{1}\times{\mathbb{CP}}^{2}~:~z_{0}^{2i+1}w_{1}=z_{1}^{2i+1}w_{0}\}$$ The projection ${\mathbb{CP}}^1\times{\mathbb{CP}}^2\to{\mathbb{CP}}^1$ endows $W_{2i+1}$ with the structure of a Kähler ${\mathbb{CP}}^1$-bundle over ${\mathbb{CP}}^1$ which is diffeomorphic to ${{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2}$ and in which the symplectic area of the exceptional divisor is larger than the area of the fiber class $F$. In these Kähler manifolds, the exceptional class $E$ is only represented by a holomorphic cusp-curve containing the zero section $s_{0}=\{ ([z_{0}:z_{1}],[0,0,1])\}$. Consequently, neither Lemma \[LemmePrincipal\] nor Lemma \[LemmeClassesPositives\] hold for ${{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2}$. In many circumstances, the set ${\mathcal{E}}_{\min}(M,{\omega})$ can be described. For instance, \[MinimalExceptionalClass\] Let $(M,{\omega})$ be a $4$-manifold and let $\widetilde{M}_{\delta}$ be a symplectic blow-up of capacity $\delta$. Let $\Sigma$ denote the exceptional divisor. Then, there exists a constant $\delta_{0}>0$ such that, for all $\delta< \delta_{0}$, the set ${\mathcal{E}}_{min}(\widetilde{M}_{\delta})$ contains only the class $[\Sigma]$. It is sufficient to show that the symplectic area of an exceptional class $E\neq [\Sigma]$ never decreases as $\delta={\omega}(\Sigma)$ decreases. To see this, first write the class $E$ as a linear combination $E=A-s[\Sigma]$, where $A\in H_{2}(M,{\mathbb{Z}})$ and $s\in{\mathbb{Z}}$. The symplectic area of $E$ is then equal to ${\omega}(A)-s\delta$. But distinct symplectic exceptional classes always intersect positively, so that $0\leq E\cdot [\Sigma] = s$. The result follows readily. \[OneThirdLemma\] Consider ${\mathbb{CP}}^{2}$ with its standard Fubini-Study form normalized so that the symplectic area of a line is $1$. Let $(M,{\omega})$ be obtained from ${\mathbb{CP}}^{2}$ by $k$ blow-ups of sizes $1/3\geq \delta_{1}\geq\cdots\geq\delta_{k}$. Let $E_{1},\ldots,E_{k}$ be the homology classes of the corresponding exceptional divisors. Then a chain of minimal blow-ups is given by $$(M,{\omega})\stackrel{E_{k}}{\to}\cdots\stackrel{E_{2}}{\to}{{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2}\stackrel{E_{1}}{\to} {\mathbb{CP}}^{2}$$ Let $L$ be the class of a line. By positivity of intersections, every exceptional class $E$ in ${\mathcal{E}}(M,{\omega})\setminus \{E_{1},\ldots,E_{k}\} $ can be written $$E=e_{0}L-\sum e_{i}E_{i}$$ where the coefficients $e_{i}$ are nonnegative. Consequently, $${\omega}(E)=e_{0}-\sum e_{i}\delta_{i} \geq e_{0}-\sum_{i}\frac{e_{i}}{3} = \frac{c_{1}(E)}{3}=\frac{1}{3}$$ By induction, this proves that $E_{i}\in{\mathcal{E}}_{\min}(M_{i},{\omega}_{i})$. Moreover, because ${\omega}(E_{1})<1/2$, the exceptional class in $(M_{1},{\omega}_{1})\simeq {{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2}$ is symplectically indecomposable, proving that we can extend the chain one step further to reach ${\mathbb{CP}}^{2}$. Conjugacy classes of maximal tori ================================= In this section, we prove the Finiteness theorem \[Finiteness\]. We first briefly recall standard results about Hamiltonian torus actions on $4$-manifolds. Again, we shall not give any details and we refer the interested reader to the papers [@Au], [@De], [@Ka-Classification], and the references therein. Effective Hamiltonian torus actions on $4$-manifolds ---------------------------------------------------- Let suppose that a compact torus $T$ acts effectively on a symplectic $4$-manifold $(M,{\omega})$ in a Hamiltonian fashion. Since the orbits of a Hamiltonian $T$-action are isotropic and, on an open dense set, orbits are free, $T$ is either the circle $S^{1}$ or the two dimensional torus $T^{2}$. In particular, any effective Hamiltonian $T^{2}$-action on a $4$-manifold is maximal. Such an action is called toric. We say that two torus actions are equivalent if they differ only by the composition of a reparametrization $T^{k}\to T^{k}$ with a symplectomorphism $M\to M$. Note that the symplectomorphism group $\operatorname{Symp}(M,{\omega})$ acts by conjugation on the set of maximal tori in $\operatorname{Ham}(M,{\omega})$ and that two actions $\rho,\rho':T\to\operatorname{Ham}$ are equivalent if, and only if, their image subgroups belong to the same orbit. For now on, we will always assume our torus actions both effective and Hamiltonian. Effective Hamiltonian $T^{2}$-actions ------------------------------------- Recall that given a Hamiltonian $T^{2}$-action on $M$, the image of the moment map $\mu:M\to\mathfrak{t}^{}\simeq{\mathbb{R}}^{2}$ is a convex polygon $\Delta$. The moment map preimage of an interior point is always a free orbit, while the preimage of an edge $e_{i}$ and of a vertex $v_{j}$ are, respectively, an invariant embedded symplectic sphere $S_{i}$ and a fixed point $p_{j}$. The moment polygon always satisfy the Delzant condition: the two edges emanating from any vertex can be generated by a ${\mathbb{Z}}$-basis of the lattice ${\mathbb{Z}}^{2}$ (equivalently, the slopes of its edges are either rational or infinite, and every two consecutive edges have integral outward normal vectors $(x,y)$ and $(x',y')$ with $xy'-yx'=1$). Moreover, The moment polygon determines the Hamiltonian space up to equivariant symplectomorphism. Conversely, given any rational polygon $\Delta$, one can construct a simply connected Kähler manifold $(M_\Delta,\omega_\Delta)$ with a Hamiltonian $T^{2}$-action which preserves the complex structure and whose moment map image is $\Delta$. Since the moment map is only defined up to an additive constant, the polygons $\Delta$ and $\Delta+C$ define the same toric manifold. Moreover, if two actions differ only by an automorphism of the torus, then their moment polygons are congruent modulo an element of the affine linear group $\operatorname{AGL}(2,{\mathbb{Z}})=\operatorname{GL}(2,{\mathbb{Z}})\ltimes{\mathbb{R}}^{2}$. Consequently, we have the following “Delzant correspondence”: \[DelzantCorrespondance\] The set of Hamiltonian $T^{2}$-actions on $4$-manifolds, up to symplectic conjugation and reparametrization of $T^{2}$, is in bijection with the orbits of the action of $\operatorname{AGL}(2,{\mathbb{Z}})$ on the set of Delzant polygons $\Delta\subset{\mathbb{R}}^{2}$. This correspondence reduces, in principle, the study of Hamiltonian toric $4$-manifolds to the combinatorics of Delzant polygons. In particular, the symplectic conjugacy classes of maximal $2$-dimensional tori in the group $\operatorname{Ham}(M,{\omega})$ can be enumerated provided we can identify which Delzant polygons, modulo AGL(2,Z)-congruence, give rise to manifolds symplectomorphic to $(M,{\omega})$. The homological properties of the toric space $(M_{\Delta},{\omega}_{\Delta})$ can be read directly on the moment polygon $\Delta$ once its edges has been cyclically ordered counterclockwise. To this end, let the size* of an edge be measured by its rational length, which is characterized by being invariant under $\operatorname{AGL}(2,{\mathbb{Z}})$-congruence and by being equal to the standard euclidean length along the coordinate axes. Then, \[HomologyDelzantPolygons\] 1. The number of edges of $\Delta$ is equal to $N=2+\operatorname{rk}H_2(M_\Delta)$ and the classes $[S_{1}],\ldots,[S_{N}]$ generate $H_{2}(M_{\Delta},{\mathbb{Z}})$ over ${\mathbb{Z}}$. 2. For $i\neq j$, the intersection number $[S_{i}]\cdot [S_{j}]$ is equal to the geometric intersection of the edges $e_{i}$ and $e_{j}$, while the self-intersection number of an invariant sphere is given by the formula $$[S_{i}]\cdot [S_{i}]=\det\begin{bmatrix}n_{i+1}\\ n_{i-1}\end{bmatrix}$$ where $n_{i}$ denotes the primitive outward normal vector to the $i^{\text{th}}$ edge. By the adjunction formula, it follows that the sum $\sum_{i}[S_{i}]$ is equal to $\operatorname{PD}[c_{1}(TM_{\Delta})]$. 3. The symplectic area of the sphere $[S_{i}]$ is $2\pi$ times the rational length of the edge $e_{i}$. In particular, the perimeter of $\Delta$ (measured in rational length) is equal to the pairing $\langle\omega_\Delta,c_{1}\rangle/2\pi$. 4. The symplectic volume of $M_\Delta$ is equal to the euclidean area of $\Delta$. The Delzant condition together with the self-intersection formula of Proposition \[HomologyDelzantPolygons\](2) imply that a Delzant polygon $\Delta$ with at least $5$ edges contains at least one edge, say $e_{i}$, whose moment map preimage is an invariant embedded sphere $S_{i}$ of self-intersection $-1$. The equivariant blow-down of such a sphere gives another toric manifold whose moment polygon $\Delta_{e_{i}}$ is obtained from $\Delta$ by gluing a triangle of area $ {\omega}(S_{i})$ along the edge $e_{i}$. Therefore, any Delzant polygon with at least 5 edges can be equivariantly blow-down to a Delzant triangle or to a Delzant quadrilateral. The inverse process is the equivariant blow-up and consists in replacing an embedded and invariant symplectic ball $B$ of capacity $\delta$ centered at a fixed point $p\in M$ by an invariant exceptional sphere of area $\delta$. The moment map image of the resulting toric manifold $\tilde{M}_{\delta}$ is the polygon $\tilde{\Delta}_{\delta}$ obtained from $\Delta$ by removing a triangle of area $\delta/2\pi$ based at the vertex $\mu(p)$. Delzant’s theorem implies \[EquivariantBlowups\] The equivariant symplectomorphism class of a $T^{2}$-equivariant blow-down depends only on the homology class of the blow-down sphere. Similarly, the equivariant symplectomorphism class of a $T^{2}$-equivariant blow-up depends only on its size $\delta$ and on the choice of the fixed point $p$. Consequently, up to equivariant symplectomorphisms, there are only finitely many ways to perform an equivariant blow-up of given capacity $\delta$ on a toric manifold. A Delzant triangle must have all its sides of equal rational length $\lambda$. Such a triangle corresponds to ${\mathbb{CP}}^{2}$, with its standard Fubini-Study form scaled such that ${\omega}_{\lambda}({\mathbb{CP}}^{1})=2\pi\lambda$, and equipped with the standard Hamiltonian action of $T^{2}$. Any Delzant quadrilaterals is $\operatorname{AGL}(2,{\mathbb{Z}})$-equivalent to a Hirzebruch trapezoid. Each Hirzebruch trapezoid is described by three positive numbers: its width $a\in{\mathbb{R}}_{+}$, its height $b\in{\mathbb{R}}_{+}$, and its slope $m\in{\mathbb{N}}$. These numbers must satisfy $a\geq b$ and $a>\frac{mb}{2}$. Conversely, for each triple $(a,b,m)$ verifying these inequalities corresponds a Hirzebruch trapezoid $\Delta(a,b,m)$. When $m$ is even, the manifold $X_{\Delta(a,b,m)}$ is symplectomorphic to $(S^{2}\times S^{2},{\omega}_{ab})$, where ${\omega}_{ab}$ is the product form for which the two $S^{2}$ factors have area $a$ and $b$ respectively. When $m$ is odd, $X_{\Delta(a,b,m)}$ is symplectomorphic to $({{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2},{\omega}_{ab})$ where ${\omega}_{ab}$ is a form giving area $a$ to a projective line and area $b$ to the exceptional divisor. If we denote by $\lceil x\rceil$ the smallest integer greater or equal to $x$, the previous discussion can be summarized as follows: \[ClassificationBasicActions\] 1. A Delzant polygon is either a Delzant triangle or is obtained from a Hirzebruch trapezoid by a finite sequence of blow-ups. Consequently, a four-dimensional symplectic toric manifold is symplectomorphic to ${\mathbb{CP}}^2$ or is obtained from a Hirzebruch surface by a sequence of equivariant symplectic blow-ups. 2. The standard Hamiltonian $T^{2}$-action on ${\mathbb{CP}}^{2}$ is unique up to equivalence. 3. The symplectic ruled manifolds $(S^{2}\times S^{2},{\omega}_{ab})$ and $({{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2},{\omega}_{ab})$ admit respectively $\lceil a/b\rceil$ and $\lceil a/b-1/2\rceil$ inequivalent Hamiltonian $T^{2}$-actions. Effective Hamiltonian $S^{1}$-actions {#SectionS1Actions} ------------------------------------- An effective Hamiltonian $S^{1}$-actions on a symplectic $4$-manifold is characterized, up to equivariant symplectomorphisms, by a labelled planar graph $\Gamma$ which encodes the essential information about the orbit types of the action and the cohomology class of the symplectic form. More precisely, suppose the circle $S^{1}$ acts effectively on a symplectic $4$-manifold $(M,{\omega})$ in a Hamiltonian way. If the stabilizer $G_{p}$ of a point $p\in M$ is not trivial, then $G_{p}$ is either the circle $S^{1}$ or a cyclic subgroup ${\mathbb{Z}}_{k}$ of order $k\geq 2$. Consider the closure $O(G)$ of the set of points whose stabilizer is isomorphic to the subgroup $G\subset S^{1}$. From Morse theory applied to the moment map we know that: 1. Each component of the fixed point set $O(S^{1})$ is either a single point or a symplectic surface. The maximum and minimum of the moment map are each attained on exactly one component of the fixed point set, while fixed points on which the moment map is not extremal are isolated. In particular, the fixed point set contains at most two disjoint surfaces. 2. If $G={\mathbb{Z}}_{k}$, $k\geq 2$, each connected component of $O(G)$ is a symplectic two-sphere on which the quotient circle $S^{1}/{\mathbb{Z}}_{k}$ acts with two fixed points. The graph associated to an effective Hamiltonian $S^{1}$-action on $(M,{\omega})$ is defined in the following way: - To each component of the fixed point set corresponds a unique vertex. - Two vertices are connected by an edge if and only if the corresponding isolated fixed points are connected by a ${\mathbb{Z}}_{k}$-sphere. - Each vertex is labelled by the value of the moment map on the corresponding fixed point component. If an extremal vertex correspond to a symplectic surface $S$, two additional labels are attached: the genus of that surface, and its normalized symplectic area $\frac{1}{2\pi}\int_{S}{\omega}$. - Each edge is labelled by the isotropy weight $k$ of the corresponding ${\mathbb{Z}}_{k}$-sphere. It follows from the definition of the graph that - The fixed points are ordered by their moment map labels. If a ${\mathbb{Z}}_{k}$-sphere connects two fixed points $p$ and $q$ with moment map labels $\mu(p)>\mu(q)$, we call $p$ its north pole and $q$ its south pole. - For $k\geq 2$, a fixed point has isotropy weight $-k$ (resp. $k$) iff it is the north (resp. south) pole of a ${\mathbb{Z}}_{k}$-sphere. - Any vertex is reached by at most two edges. Because the action is effective, the two isotropy weights at a fixed point $p$ are relatively prime. In particular, if an extremal vertex corresponds to a fixed surface, then it is not reached by any edge. - A non-free orbit is either a fixed point or belongs to a ${\mathbb{Z}}_{k}$-sphere. Because the moment map of a circle action is only defined up to a constant, and because the reparametrization group of $S^{1}$ is isomorphic to ${\mathbb{Z}}_{2}$, the graph associated to a circle subgroup $S^{1}\subset\operatorname{Ham}(M,{\omega})$ is well-defined up to the action of $\operatorname{AGL}(1;{\mathbb{Z}})\simeq {\mathbb{R}}\ltimes{\mathbb{Z}}_{2}$ on the moment map data. The first part of Karshon’s classification of Hamiltonian circle actions on $4$-manifolds is the following uniqueness theorem: The graph determines the Hamiltonian circle action up to equivariant symplectomorphism. Obvious examples of circle actions are obtained by restricting an Hamiltonian $T^{2}$-action to a subcircle. The corresponding graph is obtained from the moment polygon $\Delta$ by projecting it on the corresponding subalgebra $\operatorname{Lie}(S^{1})\subset\operatorname{Lie}(T^{2})$, keeping track of the fixed points components, the moment map data, and the possible ${\mathbb{Z}}_{k}$-spheres. An other class of examples is provided by considering effective $S^{1}$-actions on ruled manifolds ${\mathbb{CP}}^{1}\to M_{g}\to \Sigma_{g}$, where $g\geq 1$ (in the case $g=0$, these are Hirzebruch surfaces). These actions are constructed by thinking of a ruled manifold $M_{g}$ as the projectivization of a bundle $L\oplus {\mathbb{C}}$, where $L\to\Sigma_{g}$ is a line bundle of degree $k$. The diffeomorphism type of the resulting manifold $M_{g}$ depends only on the parity of $k$: $M_{g}$ is diffeomorphic to the product $\Sigma_{g}\times S^{2}$ if $k$ is even, and to the non-trivial bundle $S^{2}\ltimes \Sigma_{g}$ if $k$ is odd. In all cases, the $S^{1}$-action induced by the standard action on the trivial factor ${\mathbb{C}}\to\Sigma_{g}$ defines a maximal Hamiltonian action on ${\mathbb{P}}(L\oplus {\mathbb{C}})$ whose fixed point set consists in two holomorphic sections of self intersection $\pm k$. As in the toric case, a Hamiltonian circle action can be equivariantly blow-up by some amount $\delta>0$ at a fixed point $p$, creating an invariant exceptional sphere $\Sigma$ of area $\delta$. Let $m\geq n$ be the isotropy weights at $p$. There are three cases to consider: - If $m-n\geq 1$, and $mn\neq 0$, then $\Sigma$ is a ${\mathbb{Z}}_{m-n}$-sphere. - If $m=n=\pm 1$, then $p$ is an extremal isolated fixed point and $\Sigma$ is a fixed extremal surface. - If $p$ belongs to an extremal surface $S$ of area $\alpha$, the proper transform $\tilde{S}$ is a fixed surface of area $\alpha-\delta$ and $\Sigma$ is an invariant sphere on which $S^{1}$ acts freely. Because the graph determines the action up to equivariant symplectomorphisms, we have \[S1EquivariantBlowups\] An $S^{1}$-equivariant blow-up depends only on its size $\delta>0$ and on the fixed point component where it is performed. Consequently, given a symplectic $4$-manifold endowed with a Hamiltonian circle action, there exist only finitely many ways to perform a equivariant blow-up of size $\delta$ on it. Moreover, we know from the graph if an equivariant blow-up of given size $\delta$ can be performed: \[ConditionsS1BlowUps\] An equivariant blow-up of size $\delta$ can be performed at a fixed point $p$ if, and only, if the following conditions are satisfied: 1. If $p$ belongs to a fixed extremal surface $S$, then ${\omega}(S)>\delta$. 2. If $p$ belongs to a $Z_{k}$-sphere $S$, then then ${\omega}(S)>\delta$. 3. If $p$ is an interior fixed point, then $\min \mu<\mu(p)-\delta<\mu(p)+\delta<\max \mu$. 4. If $p$ is extremal and isolated then, for any other fixed point $q\neq p$, we have $\|\mu(p)-\mu(q)\|>\delta$. The second part of Karshon’s classification is the following characterization of action graphs: \[KarshonExistence\] A planar graph $\Gamma$ is the graph of a Hamiltonian circle action on a symplectic $4$-manifold if, and only if, it can be obtained from the graph of an action defined on ${\mathbb{CP}}^{2}$ or on a ruled manifold $M_{g}$ by performing a sequence of equivariant blow-ups. The next proposition characterizes the graphs of non-maximal circle actions on rational $4$-manifolds. \[NonMaximalActions\] An Hamiltonian $S^{1}$-action extends to a toric action if and only if each fixed surface has genus zero and each nonextremal level set for the moment map contains at most two non-free orbits. Moreover, any circle action whose fixed points are isolated extends to a toric action. Using this characterization of non-maximal actions, one can easily prove that \[NoMaximalCircleActions\] On ${\mathbb{CP}}^{2}$, $(S^{2}\times S^{2},{\omega}_{ab})$, and $({{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2},{\omega}_{ab})$, every Hamiltonian circle action extends to a toric action. Finally, Karshon’s Classification Theorem allows a complete understanding of maximal circle actions on minimal irrational ruled manifolds. Let normalize the symplectic form on a ruled manifold $M_{g}$ by fixing the symplectic area of its fiber $F$ to $1$ and by setting ${\omega}(B)=\mu>0$, where $B$ is a section of self-intersection $0$ in the case of the trivial bundle, or a section of self-intersection $-1$ in the case of the non-trivial bundle. By the work of Lalonde-McDuff [@LM], it is known that, up to rescaling, every ruled $4$-manifold is symplectomorphic to one of these $M_{g,\mu}$. \[S1ActionsOnRuledManifolds\] Let $M_{g,\mu}$ be a normalized ruled $4$-manifold with $g\geq 1$. Then $\operatorname{Ham}(M_{g,\mu})$ contains exactly $\lceil\mu\rceil$ inequivalent maximal $S^{1}$-actions, where $\lceil\mu\rceil$ is the smallest integer greater or equal to $\mu$. See the proof of Lemme 6.15 in Karshon [@Ka-Classification]. Proof of Theorem \[Finiteness\] ------------------------------- The classification theorems of Delzant and Karshon describe Hamiltonian torus actions up to equivariant symplectomorphisms. As such, they do not give the list of all maximal Hamiltonian torus actions on a given symplectic $4$-manifold. However, \[CanonicalModel\] Let $(M,{\omega})$ be the $k$-fold blow-up, $k\geq 1$, of a symplectic ruled $4$-manifold. Then there exists a symplectic ruled manifold $(M_{0},{\omega}_{0})$ and an ordered set of $k$ disjoint exceptional classes $\{E_{1},\ldots,E_{k}\}\subset {\mathcal{E}}(M,{\omega})$ of areas $\delta_{1}\geq\ldots\geq\delta_{k}$ such that every effective Hamiltonian torus action on $(M,{\omega})$ is equivalent to an action obtained from an action on $(M_{0},{\omega}_{0})$ through a sequence of $k$ equivariant blow-ups of capacities $\delta_{1},\ldots,\delta_{k}$. By Corollary \[ChainOfMinimalBlowdowns\], we know that there exists a chain of minimal symplectic blow-downs $$(M,{\omega})=(M_{k},{\omega}_{k})\stackrel{E_{k}}{\to}(M_{k-1},{\omega}_{k-1})\stackrel{E_{k-1}}{\to} \cdots \stackrel{E_{1}}{\to}(M_{0},{\omega}_{0})$$ ending with a ruled manifold $(M_{0},{\omega}_{0})$ and for which the symplectic areas $\delta_{k}={\omega}(E_{k})\leq\cdots\leq\delta_{1}={\omega}(E_{1})$ only depend on $(M,{\omega})$. Now, if $(M_{k},{\omega}_{k})$ comes equipped with an effective Hamiltonian torus action $\rho_{k}$, choose a compatible almost-complex structure $J_{k}\in{\mathcal{J}}({\omega}_{k})$ and consider the $\rho_{k}$-invariant almost complex structure $J_{k}'$ obtained by averaging the adapted metric $g_{k}(x,y)={\omega}_{k}(x,J_{k}y)$. By Lemma \[LemmePrincipal\], the class $E_{k}$ is represented by an invariant $J_{k}'$-holomorphic sphere which can be equivariantly blow-down. By Lemma \[EquivariantBlowups\], this operation defines a Hamiltonian action $\rho_{k-1}$ on $(M_{k-1},{\omega}_{k-1})$ whose conjugacy class in $\operatorname{Symp}(M_{k-1},{\omega}_{k-1})$ depends only on $\rho_{k}$. The result follows by induction. \[SmallBlowUpsOfCP2\] Consider ${\mathbb{CP}}^{2}$ with its standard Fubini-Study form normalized so that the symplectic area of a line is $1$. Let $(M,{\omega})$ be obtained from ${\mathbb{CP}}^{2}$ by $k$ blow-ups of sizes $1/3\geq \delta_{1}\geq\cdots\geq\delta_{k}$. Then all actions on $(M,{\omega})$ come from ${\mathbb{CP}}^{2}$ through equivariant blow-ups of sizes $\delta_{1}\geq\cdots\geq\delta_{k}$. This follows readily from Lemma \[OneThirdLemma\]. \[FinitenessToricCase\] There are only finitely many inequivalent toric actions on any given symplectic $4$-manifold. Let $(M,{\omega})$ be a toric manifold. By Proposition \[ClassificationBasicActions\] (1), $(M,{\omega})$ is symplectomorphic to a rational manifold. By Proposition \[CanonicalModel\], every toric action on $(M,{\omega})$ comes from a single ruled manifold $(M_{0},{\omega}_{0})$ through a sequence of $k$ equivariant blow-ups of fixed capacities $\delta_{1},\ldots,\delta_{k}$. By Proposition \[ClassificationBasicActions\] (3), $(M_{0},{\omega}_{0})$ admits only finitely many inequivalent toric actions, each of which corresponding to a Hirzebruch trapezoid. Then, Lemma \[EquivariantBlowups\] implies that there exist only finitely many ways to perform $k$ equivariant blow-ups of sizes $\delta_{1},\ldots,\delta_{k}$ on each of these trapezoids. Finally, Proposition \[UniquenessBlowups\] implies that all these $k$-fold blow-ups define toric manifolds symplectomorphic to $(M,{\omega})$. \[PreuveAlternative\] As mentioned in the Introduction, Corollary \[FinitenessToricCase\] can be proven by “soft” means, that is, without any references to $J$-holomorphic techniques. The main point is that any invariant exceptional sphere corresponds to a side of the polygon and, consequently, its symplectic area is bounded by $2\pi\langle{\omega},c_{1}\rangle$. Then, we can use Remark \[SoftProperness\] to prove that a toric $4$-manifold can be equivariantly blow-down to finitely many distinct Hirzebruch surfaces, see [@KKP] for more details. One drawback of this simpler proof is that it does not lead to an efficient algorithm to count the actual number of inequivalent toric actions on a given manifold. More importantly, this argument cannot be used in the case of $S^{1}$-actions since one can easily construct examples of rational $S^{1}$-manifolds for which the pairing $\langle{\omega},c_{1}\rangle$ is strictly negative. In the case of Hamiltonian $S^{1}$-actions, observe that in a chain of minimal blow-ups, a maximal $S^{1}$-action on $(M_{i},{\omega}_{i})$ may well come from the equivariant blow-up of a $S^{1}$-action on $(M_{i-1},{\omega}_{i-1})$ which is not maximal (in fact, Lemma \[NoMaximalCircleActions\] implies that this is always the case whenever $(M,{\omega})$ is rational). However, \[BlowUpsNonMaximalActions\] Let $(M,{\omega})$ be a symplectic $4$-manifold and denote by $\tilde{M}_{\delta}=(\tilde{M},\tilde{{\omega}})$ its symplectic blow-up of size $\delta$. Then, up to equivalences, there exist at most finitely many non-maximal Hamiltonian $S^{1}$-actions on $(M,{\omega})$ which induce maximal $S^{1}$-actions on $\tilde{M}_{\delta}$. First, observe that if a $4$-manifold $(M,{\omega})$ admits a non-maximal Hamiltonian $S^{1}$-action, then it is rational. By Corollary \[FinitenessToricCase\], there exist only finitely many inequivalent Hamiltonian $T^{2}$-actions on $(M,{\omega})$. Consequently, it is sufficient to show that at most finitely many maximal $S^{1}$-actions on $\tilde{M}_{\delta}$ comes from subcircles of a given toric action on $(M,{\omega})$. Second, because a Hamiltonian $S^{1}$-action with only isolated fixed points always extends to a toric action, a maximal $S^{1}$-action must have at least one symplectic surface in its fixed point components. So, if we assume a maximal $S^{1}$-action $\tilde{\rho}$ is the equivariant blow-up of a non-maximal action $\rho$, then the description of $S^{1}$-equivariant blow-ups in section \[SectionS1Actions\] implies that either 1. $\rho$ already has non-isolated fixed points, or 2. a sphere of fixed points is created by blowing-up $\rho$ at an extremal isolated fixed point of equal weights $\pm 1$. In the first case, note that a subcircle $S^{1}\subset T^{2}$ has non-isolated fixed points if and only if it fixes pointwise one of the spheres corresponding to the sides of the moment polygon $\Delta$, that is, if and only if its Lie algebra $\operatorname{Lie}(S^{1})\subset\mathfrak{t}^{2}$ is perpendicular to one of the edges of $\Delta$. Clearly, only finitely many such subcircles exist, and Lemma \[S1EquivariantBlowups\] implies that each of them gives rise to at most finitely many maximal $S^{1}$-actions on $\tilde{M}_{\delta}$. In the second case, note that the resulting $S^{1}$-action extends to a toric action since the blow-up operation itself can be made equivariant with respect to the whole torus $T^{2}$. Indeed, since we are blowing up at an extremal fixed point which is isolated, we can suppose that this fixed point is also extremal for the toric action and we only need to check that the capacity of the $S^{1}$-equivariant blow-up is the same as the capacity of the toric blow-up. But this follows from the fact that this fixed point has equal weights $\pm 1$. The proof of the finiteness theorem in the case of maximal Hamiltonian $S^{1}$-actions now proceeds, mutatis mutandis, as in the toric case: use Proposition \[CanonicalModel\] and Lemma \[BlowUpsNonMaximalActions\], together with Karshon’s classification theorem. This gives \[FinitenessS1Case\] A symplectic $4$-manifolds admits only finitely many inequivalent, maximal, Hamiltonian $S^{1}$-actions. This concludes the proof of Theorem \[Finiteness\]. Counting conjugacy classes of maximal tori ------------------------------------------ Our proof of Theorem \[Finiteness\] gives a inductive procedure to count the number of distinct symplectic conjugacy classes of maximal tori in the Hamiltonian group of many symplectic $4$-manifolds $(M,{\omega})$: 1. Find a chain of minimal blow-ups $(M,{\omega})\stackrel{E_{k}}{\to}\cdots\stackrel{E_{1}}{\to} (M_{0},{\omega}_{0})$. 2. List all maximal actions on $(M_{0},{\omega}_{0})$ and perform all possible equivariant blow-ups of size $\delta_{1}$. Repeat for all other blow-ups of sizes $\delta_{2},\ldots, \delta_{k}$. 3. Eliminate redundancies in the resulting list of actions. Note that the first step can be easily completed if one starts with either a toric manifold whose Delzant polygon is known, or with the $k$-fold blow-up of a ruled manifold which is not rational. Indeed, in the former case, the minimal exceptional classes always correspond to sides of the polygon, while in the later case, the set of exceptional classes is always finite. Applying this procedure to equal blow-ups of ${\mathbb{CP}}^{2}$, we can generalize a result of Karshon and Kessler [@KK]: \[ManifoldsWithoutActions\] Consider ${\mathbb{CP}}^{2}$ with its standard Fubini-Study form normalized so that the symplectic area of a line is $1$. Let $(M,{\omega})$ be obtained from ${\mathbb{CP}}^{2}$ by $k$ blow-ups of size $\delta$. 1. If $k\geq 4$, $(M,{\omega})$ does not admit any Hamiltonian $T^{2}$-action. 2. If $(k-1)\delta\geq 1$, $(M,{\omega})$ does not admit any Hamiltonian $S^{1}$-action. To prove Proposition \[ManifoldsWithoutActions\], we need the following lemma which follows easily from the description of $T^{2}$-equivariant blow-ups and from Proposition \[ConditionsS1BlowUps\]: \[NecessaryConditions\] Let normalize ${\mathbb{CP}}^{2}$ so that ${\omega}({\mathbb{CP}}^{1})=1$. 1. ${\mathbb{CP}}^{2}$ admits $k$ $T^{2}$-equivariant blow-ups of size $\delta>0$ iff $k\leq 3$ and $\delta<1/3$. 2. ${\mathbb{CP}}^{2}$ admits $k$ $S^{1}$-equivariant blow-ups of size $\delta>0$ iff $(k-1)\delta<1$. In view of Lemma \[NecessaryConditions\], it is sufficient to show that, under the hypotheses of the Proposition, all actions on $(M,{\omega})$, if any, must come from ${\mathbb{CP}}^{2}$ thought equivariant blow-ups. For this to be true, we only need to find a chain of minimal blow-ups ending with ${\mathbb{CP}}^{2}$. There are several cases to consider depending on $k$. If $k\geq 9$, the volume condition $0<[{\omega}]\cdot[{\omega}]=1-k\delta^{2}$ implies that $\delta\leq 1/3$ and the conclusion follows from Lemma \[SmallBlowUpsOfCP2\]. For $1\leq k\leq 8$, the set ${\mathcal{E}}(M,{\omega})$ is finite and can be explicitely described in terms of the natural basis $\{L,E_{1},\ldots,E_{k}\}$ defined by the blow-ups, see [@De], For $4\leq k\leq 8$, the conclusion follows from a case-by-case study of exceptional classes. For $2\leq k\leq 3$, the conditions of statements (1) and (2) are empty because of Gromov’s packing inequality (see [@MP]), while there is nothing to prove for $k=1$. Proof of Theorem \[HamConjugacyClasses\] ---------------------------------------- We now consider Hamiltonian conjugacy classes of $2$-tori in $\operatorname{Ham}(M,{\omega})$. Consider an effective toric action $\rho:T^{n}{\hookrightarrow}\operatorname{Symp}(M^{2n},{\omega})$ with moment map $\mu:M\to\mathfrak{t}\simeq{\mathbb{R}}^{n}$. Denote by $T$ the corresponding torus in $\operatorname{Symp}(M,{\omega})$, by $C_{T}$ its centralizer, by $N_{T}$ its normalizer, and by $W_{T}=N_{T} / C_{T}$ its Weyl group. 1. The normalizer $C_{T}$ is equal to the group of all symplectomorphisms $\phi$ that preserve the moment map, that is, such that $\mu\circ\phi=\mu$. Moreover, $C_{T}$ is connected and $C_{T}\subset\operatorname{Ham}(M,{\omega})$. 2. The normalizer $N_{T}$ is equal to the group of all symplectomorphisms $\psi$ such that $\mu\circ\psi=\Lambda\circ\mu$ for some $\Lambda\in\operatorname{AGL}(n;{\mathbb{Z}})$. In particular, the Weyl group $W_{T}$ is finite. \(1) Let $\phi$ be an element of $C_{T}$ and let $t$ be an element of $T$ generated by an Hamiltonian function $H:M\to{\mathbb{R}}$. Then, $\phi t \phi^{-1}$ is generated by $H\circ \phi^{-1}$. It follows that the moment map of the conjugated action $\phi\rho\phi^{-1}$ is $\mu\circ\phi^{-1}$. Since $\phi\rho\phi^{-1}=\rho$, there exists a constant $C\in{\mathbb{R}}^{n}$ such that $\mu\circ\phi^{-1}=\mu+C$. But because $M$ is compact, $C=0$, and $\phi$ preserves the moment map. Because the action is toric, the fibers of the moment map are orbits of the action. It follows from a theorem of Schwartz [@Sch] that $\phi$ can be written in terms of the action as $\phi(m)=\overline{f}(m)\cdot m$, where $\overline{f}:M\to T^{n}$ is a smooth function which is constant on each fiber. By Corollary 4.2 of [@KL], it follows that $\overline{f}$ is the pullback by $\mu$ of a smooth function $\Delta\to T^{n}$, and because $\Delta$ is contractible, there exists a smooth function $f:\Delta\to\mathfrak{t}$ such that $\overline{f}=\exp [f\circ\mu]$ (here, $\exp:\mathfrak{t}\to T^{2}$ denotes the exponential map). Now, the Hamiltonian flow of the real function $F$ defined by the pairing $F(m)=\langle \mu(m),f\circ \mu(m) \rangle$ is $$\phi_{s}(m)=\exp[ s (f\circ\mu(m))]\cdot m$$ Since $\phi_{s}\in C_{T}$ and $\phi=\phi_{1}$, this shows that $C_{T}$ is connected and that $C_{T}\subset\operatorname{Ham}$. \(2) Every element of the normalizer $N_{T}$ acts on $T$ by automorphisms. Consequently, given a parametrization of the torus, there is an exact sequence $$1\to C_{T}\to N_{T}\to W_{T}\to 1$$ in which $W_{T}\subset \operatorname{GL}(n;{\mathbb{Z}})$ is discrete. Given $\psi\in N_{T}$ this implies that the moment map of $\psi\rho \psi^{-1}$ equals $\Lambda\circ \mu$ for some $\Lambda\in\operatorname{AGL}(n;{\mathbb{Z}})$. On the other hand, the moment map of $\psi\rho\psi^{-1}$ is also equal to $\mu\circ \psi$, showing that $$\label{EqnNormalizer} \mu\circ\psi=\Lambda\circ\mu$$ The finiteness of $W_{T}$ follows from the fact that, up to translations, there exist only finitely many $\Lambda$ verifying (\[EqnNormalizer\]). Given a rational symplectic $4$-manifold $(M,{\omega})$, let ${\mathcal{T}}$ denotes the set of all $2$-tori in $\operatorname{Ham}(M,{\omega})$. Then, Theorem \[HamConjugacyClasses\] states that for any toric $4$-manifold $(M,{\omega})$, the set ${\mathcal{T}}/\operatorname{Ham}$ is finite iff $\pi_{0}(\operatorname{Symp})$ is finite. Suppose the set ${\mathcal{T}}/\operatorname{Ham}$ contains $n<\infty$ conjugacy classes. Because toric $4$-manifolds are simply connected, $\pi_{0}(\operatorname{Symp})=\operatorname{Symp}/\operatorname{Ham}$. Consequently, the group $\pi_{0}(\operatorname{Symp})$ acts on the set ${\mathcal{T}}/\operatorname{Ham}$ and there is an induced homomorphism $c:\pi_{0}(\operatorname{Symp})\to\mathbb{S}_{n}$. Let $\pi_{T}$ be the isotropy subgroup of $[T]\in{\mathcal{T}}/\operatorname{Ham}$, that is $$\begin{aligned} \pi_{T}&=&\{[g]\in\pi_{0}(\operatorname{Symp})~\|~\exists\, h\in\operatorname{Ham}\text{~such that~} hg\in N_{T}\}\\ &=&{\raisebox{1pt}{\ensuremath{N_{T}}}/\raisebox{-1pt}{\ensuremath{N_{T}\cap\operatorname{Ham}}}}\end{aligned}$$ Because $C_{T}\subset\operatorname{Ham}$, there is a quotient map $W_{T}\to{\raisebox{1pt}{\ensuremath{N_{T}}}/\raisebox{-1pt}{\ensuremath{N_{T}\cap\operatorname{Ham}}}}$. Consequently, $\pi_{T}$ is finite. The finiteness of $\pi_{0}(\operatorname{Symp})$ now follows from the exact sequence $1\to\ker(c)=\bigcap_{T} \pi_{T}\to\pi_{0}(\operatorname{Symp})\to\mathbb{S}_{n}$. Conversely, if ${\mathcal{T}}/\operatorname{Ham}$ is infinite, the map ${\mathcal{T}}/\operatorname{Ham}\to{\mathcal{T}}/\operatorname{Symp}$ has at least one infinite fiber consisting of distincts classes $[T_{i}]$, $i\in{\mathbb{N}}$, for which there exist $g_{i}\in\operatorname{Symp}$ such that $g_{i}T_{i}g_{i}^{-1}=T_{0}$. Assume that for two indices $i\neq j$ we have $[g_{i}]=[g_{j}]$ in $\pi_{0}(\operatorname{Symp})$. Then there exists $h\in\operatorname{Ham}$ such that $hg_{i}=g_{j} $ and $(g_{i}^{-1}h^{-1}g_{i})T_{i}(g_{i}^{-1}h^{-1}g_{i})^{-1}=T_{j}$. Since $(g_{i}^{-1}h^{-1}g_{i})\in\operatorname{Ham}$ this is impossible, showing that the isotopy classes $[g_{i}]\in\pi_{0}(\operatorname{Symp})$ are all distinct. Let $\operatorname{Symp}_{h}(M,{\omega})$ denote the subgroup of symplectomorphisms that act as the identity on $H_{}(M;{\mathbb{Z}})$. Because for symplectic $4$-manifolds with Hamiltonian circle actions the group $\operatorname{Aut}_{[{\omega}]}$ is finite (see the second part of Remark \[SoftProperness\]), the group $\pi_{0}(\operatorname{Symp}(M,{\omega}))$ is finite iff $\pi_{0}(\operatorname{Symp}_{h}(M,{\omega}))$ is finite. For $(M,{\omega})$ diffeomorphic to ${\mathbb{CP}}^{2}$, $S^{2}\times S^{2}$, ${{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2}$, or ${\mathbb{CP}}^{2}{\#\,2\overline{{\mathbb{CP}}}\,\!^2}$, we know from [@AM; @Gr; @Pi] that $\operatorname{Ham}(M,{\omega})=\operatorname{Symp}_{h}(M,{\omega})$. Because there are no maximal $S^{1}$-actions on these manifolds, this gives \[CorHamClasses\] There are at most finitely many Hamiltonian conjugacy classes of maximal tori in the Hamiltonian groups of a symplectic manifold diffeomorphic to ${\mathbb{CP}}^{2}$, $S^{2}\times S^{2}$, ${{\mathbb{CP}}^2\#\,\overline{{\mathbb{CP}}}\,\!^2}$, or ${\mathbb{CP}}^{2}{\#\,2\overline{{\mathbb{CP}}}\,\!^2}$. It is likely that Corollary \[CorHamClasses\] also holds for ${\mathbb{CP}}^{2}{\#\,3\overline{{\mathbb{CP}}}\,\!^2}$ and ${\mathbb{CP}}^{2}{\#\,4\overline{{\mathbb{CP}}}\,\!^2}$. Presently we do not know a single toric $4$-manifold $(M,{\omega})$ for which $\pi_{0}(\operatorname{Symp})$ is infinite, although we believe that such manifolds should be quite common. The reason is that the only examples of rational $4$-manifolds for which $\pi_{0}(\operatorname{Symp})$ is known to be infinite consist of monotone* ${\mathbb{CP}}^{2}{\#\,k\overline{{\mathbb{CP}}}\,\!^2}$ with $k\geq 5$ and, by Proposition \[ManifoldsWithoutActions\], these manifolds do not admit any hamiltonian circle action. These results are due to Seidel [@Se] who found a necessary condition for iterated symplectic Denh twists along Lagragian spheres to be symplectically isotopic to the identity. This condition involves the algebraic structure of the quantum ring $QH_{}(M;\Lambda_{{\omega}}^{\text{univ}})$ with coefficients in the universal Novikov ring $\Lambda_{{\omega}}^{\text{univ}}$, and it gives an effective criteria only when $(M,{\omega})$ is monotone. Although it is still not clear whether this limitation can be removed, it would be interesting to study some simple cases, such as the $5$-fold blow-up of ${\mathbb{CP}}^{2}$ with sizes $\{1/3,1/3,1/3,\epsilon,\epsilon\}$, where $\epsilon<1/3$. Rational cohomology of symplectomorphism groups =============================================== In [@Ke], Kedra proved Theorem \[KedraGeneralise\] in the case $(M,{\omega})$ is not the $k$-fold blow-up of a rational surface. His main result can be restated in the following way: \[Kedra\] Let $(M,{\omega})$ be a compact simply connected $4$-manifold with $b_{2}(M)\geq 3$. Let $\imath:B_{\delta}{\hookrightarrow}M$ be a symplectic embedding of the standard ball of capacity $\delta$ and denote by $(\widetilde{M}_{\delta},\widetilde{{\omega}}_{\delta})$ the corresponding symplectic blow-up. Suppose that for small enough $\delta$ the exceptional divisor $\Sigma\subset (\widetilde{M}_{\delta},\widetilde{{\omega}}_{\delta})$ is always represented by a $\tilde{J}$-holomorphic representative, for any choice of tamed almost-complex structure $\tilde{J}$ on $(\widetilde{M}_{\delta},\widetilde{{\omega}}_{\delta})$. Then, for $\delta$ sufficiently small, the rational cohomology of the Hamiltonian group of $(\widetilde{M}_{\delta},\widetilde{{\omega}}_{\delta})$ is infinitely generated as an algebra. The proof, which we outline below, combines known facts from homotopy theory together with elementary properties of symplectic balls in $4$-manifold, see [@Ke] and [@LP-Duke] for more details. The first step is to apply rational homotopy techniques and Gottlieb theory of evaluation subgroups to prove the following topological results: - If a topological group acts transitively on a simply connected rationally hyperbolic[^6] space $M$, then the rational cohomology of the isotropy subgroup of a point is infinitely generated as an algebra. - Every simply connected $4$-manifold $M$ for which $\dim H_{2}(M;{\mathbb{Q}})\geq 3$ is rationally hyperbolic. In particular, given any symplectic, simply connected $4$-manifold $(M,{\omega})$ with $b_{2}\geq 3$, the rational cohomology of the stabilizer $\operatorname{Symp}(M, p;{\omega})$ is infinitely generated as an algebra. Now consider the subgroup $\operatorname{Symp}^{\operatorname{U}(2)}(M,B;{\omega})$ of symplectomorphisms that restrict to a unitary transformation on a given standard symplectic ball $B_{\delta}$ of capacity $\delta$ centered at $p$. The second step is to show that, if $\delta$ is small enough, the inclusion induces a map $\pi_{}(\operatorname{Symp}^{\operatorname{U}(2)}(M,B;{\omega}))\otimes{\mathbb{Q}}\to\pi_{}(\operatorname{Symp}(M, p;{\omega}))\otimes{\mathbb{Q}}$ whose image has infinite rank. Finally, because every element of $\operatorname{Symp}^{\operatorname{U}(2)}(M,B;{\omega})$ lifts to the blow up $(\tilde{M_{\delta}},\tilde{{\omega}}_{\delta}) $, the group $\operatorname{Symp}^{\operatorname{U}(2)}(M,B;{\omega})$ maps into the subgroup $\operatorname{Symp}(\tilde{M_{\delta}},\Sigma;\tilde{{\omega}}_{\delta})$ of symplectomorphisms that fix the exceptional divisor $\Sigma$. The last step then consists in showing that if the exceptional class $[\Sigma]$ is symplectically indecomposable, each map in the sequence $$\operatorname{Symp}^{\operatorname{U}(2)}(M,B;{\omega})\simeq \operatorname{Symp}(\tilde{M_{\delta}},\Sigma;\tilde{{\omega}}_{\delta})\simeq \operatorname{Symp}(\tilde{M_{\delta}};\tilde{{\omega}}_{\delta})$$ is a homotopy equivalences, see [@LP-Duke]. This shows that the rank of the module $\pi_{}(\operatorname{Symp}(\tilde{M_{\delta}};\tilde{{\omega}}_{\delta}))\otimes {\mathbb{Q}}$ is infinite and that the rational cohomology of $\operatorname{Symp}(\widetilde{M}_{\delta},\widetilde{{\omega}}_{\delta})$ is infinitely generated as an algebra. Assuming Theorem \[Kedra\], the proof of Theorem \[KedraGeneralise\] reduces to showing that on any compact simply connected $4$-manifold with $b_{2}(M)\geq 3$, and for small enough $\delta$, the exceptional divisor $\Sigma\subset (\widetilde{M}_{\delta},\widetilde{{\omega}}_{\delta})$ is always represented by a $\tilde{J}$-holomorphic representative, for any choice of tamed almost-complex structure $\tilde{J}$. By lemma \[LemmePrincipal\], we only need to prove that for $\delta$ sufficiently small, $[\Sigma]\in{\mathcal{E}}_{\min}(\widetilde{M}_{\delta})$. But this follows from Lemma \[MinimalExceptionalClass\]. Combining Proposition \[ManifoldsWithoutActions\] with Theorem \[KedraGeneralise\], one gets examples of rational symplectic $4$-manifolds $(M,{\omega})$ which do not admit any Hamiltonian action by tori, but whose symplectomorphism groups have infinitely generated rational cohomology. 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--- abstract: 'Faraday tomography is a novel method to probe 3-dimensional structure of magnetic fields of polarized radio sources. In this paper, we investigate Faraday dispersion function (FDF) of disk galaxies extending a simple analytic model of galactic magnetic fields developed in [@Ideguchi2017]. The model consists of axisymmetric coherent fields and turbulent fields and we consider the effects of inclination, relative amplitude of coherent and turbulent magnetic fields and pitch angle of coherent fields. Our simple model makes it easy to obtain physical interpretation of FDFs and helps understanding observational results. We find FDFs have two peaks when galaxies are observed with non zero inclination. The gap and relative height of two peaks are dependent on the inclination angle and pitch angle but are not affected by relative amplitude of coherent and turbulent magnetic fields so much. These findings give us an important caution that two peaks in observed FDFs do not necessarily imply the presence of two separate radio sources within a beam.' author: - Haruya Eguchi - Masaki Suzuki - Yoshimitsu Miyashita - Shinsuke Ideguchi - Keitaro Takahashi --- Introduction ============ Magnetic fields play important roles in many of astrophysical systems. In studies on such objects, 3-dimensional structure of magnetic fields is a key to understand the dynamics of magnetized plasma. Faraday tomography is a technique to reconstruct the distribution of magnetic fields of polarized radio sources along the line of sight (LOS) and a combination with 2-dimensional imaging allows us to probe the 3-dimensional structure [@Burn1966; @Brentjens2005]. A key quantity in Faraday tomography is Faraday depth defined as $$\phi(x) = K \int_{0}^{x} n_e(x') B_\parallel(x') dx' ~~~ [{\rm rad \ m^{-2}}] \label{phi}$$ where $n_e(x)~[{\rm cm}^{-3}]$, $B_\parallel(x)~[\mu{\rm G}]$ and $x~[{\rm pc}]$ are the thermal electron density and magnetic fields parallel to the LOS and the coordinate along the LOS. The coefficient is $K=0.81$ when the above units for $n_e$, $B_\parallel$ and $x$ are adopted. Then, the Faraday dispersion function (FDF), the complex polarization intensity in Faraday depth space, is related to the polarization spectrum as, $$P(\lambda^2) = \frac{1}{\pi} \int_{-\infty}^{\infty} F(\phi) e^{2 i \phi \lambda^2} d\phi. \label{eq:P}$$ The FDF is the quantity which can be obtained from Faraday tomography and contains information on the LOS distribution of magnetic fields, high-energy electrons which emit polarized radio waves and thermal electrons. Although the distribution which is given by the FDF is in Faraday depth space, rather than physical space, it gives us much more information than the conventional rotation measure [@schnitzeler2007; @Govoni2010; @Mao2010; @Wolleben2010; @Akahori2014; @Anderson2016; @O'Sullivan2018]. However, there are two major difficulties concerning Faraday tomography. The first is the precise reconstruction of the FDF from observed polarization spectrum. Although Eq.(\[eq:P\]) is mathematically equivalent to Fourier transform, the inverse transformation to obtain the FDF is incomplete because the observational band is limited to a finite range of $\lambda^2 (> 0)$. A lot of algorithms have been developed to extract as much information as possible from observed polarization spectrum: RM CLEAN [@Heald2009; @Miyashita2016], QU fitting [@O'Sullivan2012; @Miyashita2019] and sparse modeling [@Li2011; @Andrecut2012; @Akiyama2019]. The second one, which is directly related to the current work, is the interpretation of the FDF. As we stated above, the FDF is the distribution of polarized emission in Faraday depth space, not physical space. Thus, the interpretation of the FDF is not straightforward, even if it was reconstructed precisely somehow. In the case of galaxies, previous works have emphasized that the presence of turbulent magnetic fields is important to understand the shape of the FDF [@Beck2009; @Bell2011; @Frick2011; @Ideguchi2014]. Generally, the turbulent fields induce randomness in the FDF and make the FDF difficult to interpret. In our previous study [@Ideguchi2017], we developed a simple analytic model of disk galaxies with global (coherent) and turbulent (random) magnetic fields. It was found that randomness of the FDF is significantly reduced when the FDFs of areas which are much larger than the typical scale of turbulence are summed over. The resultant FDF reflects the feature of global fields and the statistical properties of turbulence. In this paper, we extend the model of galaxies developed in @Ideguchi2017 to consider the effects of inclination of observation, relative amplitude between global and turbulent magnetic fields and pitch angle of global fields. The structure of this paper is as follows. In section 2, we describe our model of galactic magnetic fields and method to calculate FDFs. The results are given in section 3. Section 4 is devoted to the summary and discussion. Model of disk galaxies ====================== Galaxy Model in @Ideguchi2017 ----------------------------- In our previous work [@Ideguchi2017], we studied the characteristic properties of the FDF of a small portion, $\sim(100 {\rm pc})^2$, of “face-on” spiral galaxies by employing simple models for the magnetic field. For simplicity, we ignored galactic halos (or thick disks) and considered only galactic disk which has a scale height of $L_{\rm sh}$ and is filled with the constant thermal electron density, $n_e$. As a representative case, we adopted $L_{\rm sh}=1$ kpc and $n_e=0.02$ cm$^{-2}$. The intensity and angle of polarization is assumed to be constant within the computational domain. The random magnetic field is also assumed, but the emission from it is ignored and only LOS component of it is considered. The magnetic fields which is parallel with respect to the line of sight (LOS) is decomposed into random and coherent components such that $$B'_{\parallel} = B'_{\rm rand} + B'_{\rm coh}.$$ We assigned the physical quantities described above to cubic cells of size $L_{\rm cell}$, which is adopted as 10 pc as a representative. This is equivalent to make a simple assumption where $B'_{\rm rand}$ has a single coherence length. Under these assumptions, we calculated the FDF numerically by stacking up the cells for $[-L_{\rm sh},L_{\rm sh}]$. Thus, each LOS includes $N=2L_{\rm sh}/L_{\rm cell}$ cells, or “layers” since we also considered number of cells in the plane perpendicular to the LOS which corresponds to the observational region. We found that the “shape” of the FDF becomes smooth when multiple LOSs cover a region of $\gtrsim ({\rm 10~coherence~length})^2$. As the next step, we derived the FDF analytically in the limit of large number of LOSs. We found that the FDF can be expressed as the sum of many Gaussian functions with different means and variances such that $$F(\phi) \propto \sum_{j=1}^N P_j(\phi), \label{eq:FDF}$$ where $P_j(\phi)$ is the $j$th layer’s contribution to the FDF and is expressed as $$P_j(\phi) = \frac{1}{\sqrt{2\pi j}\sigma_\phi}\exp\left[ -\frac{(\phi-j\Delta\phi_{\rm coh})^2}{2j\sigma_\phi^2} \right]. \label{eq:layer-org}$$ Here $\Delta\phi_{\rm coh}=Kn_eB'_{\rm coh}L_{\rm cell}$ shows the slide of the peak location of the emission from each layer in $\phi$ space due to the coherent field, and $\sigma_\phi^2=K^2n_e^2\sigma_B^2L_{\rm cell}^2$ with $\langle {B'_{\rm rand}}^2 \rangle = \sigma_B^2$ shows the variance of emission from each layer in $\phi$ space due to the random field within the layer. Note that we do not consider the absolute emissivity of radiation and thus the intensity of $F(\phi)$ is in arbitrary unit, since we only study the shape of $F(\phi)$. We also demonstrated that the simulated FDF approaches to the analytical solution as the number of LOSs increases. Extension of Galaxy Model ------------------------- In this work, we follow the basic concepts and parameters of the analytical study of our previous work in @Ideguchi2017, but extend it by introducing the inclination of the spiral galaxies as well as the pitch angle of the spiral magnetic field, and study how these additional parameters make an impact on the shape of the FDF. The values of $n_e=0.02$ cm$^{-3}$, $L_{\rm sh}=1$ kpc, and $L_{\rm cell}=10$ pc follow our previous work. As in Figure \[jou\], we use Cartesian coordinates $(x,y,z)$, where the $x\mathchar`-y$ plane coincides with the galactic midplane with $z$ penetrating the midplane, and the origin, $(x,y,z)=(0,0,0)$, being the same as the galactic center. The inclination angle, $\theta$, represents the angle between the LOS and $z$ axis, and is set as 40 degrees as a representative. The LOS is assumed to be always parallel to $x\mathchar`-z$ plane, and the location of intersection of LOS and galactic plane (at $z=0$) is difined as $(x,y) = (\beta,y)$ which is set as $(0,200)$ pc as a representative (Figure \[model\_top\]). As the global magnetic field, $B_{\rm coh}$, we assume the concentric magnetic field centered on the galactic center and that it is parallel to the galactic ($x\mathchar`-y$) plane. This ring magnetic field is set as 5 $\mu$G as a representative. The perpendicular ($B_\perp$) and parallel ($B_\parallel$) components of $B_{\rm coh}$ with respect to the LOS are expressed as $$B_\perp(z_j) = \frac{\sqrt{(z\tan\theta + \beta)^2 + y^2\cos^2\theta}}{\sqrt{(z_j\tan\theta+\beta)^2+y^2}}B_{\rm coh}, \label{eq:perp}$$ $$B_\parallel(z_j) = \frac{y\sin\theta}{\sqrt{(z_j\tan\theta+\beta)^2+y^2}}B_{\rm coh}, \label{eq:parallel}$$ where $$z_j = \left(j-\frac{N}{2}\right)L_{\rm cell}\cos\theta. \label{eq:z_j}$$ Here $N$ is the number of layers (see below). Though we do not consider the coherent (global and uniform) magnetic field which is perpendicular to the galactic plane as was considered in our previous work, we note that the concentric field generates the coherent component which is parallel to the LOS when $\theta$ is non-zero as in Equation \[eq:parallel\]. Regarding the synchrotron radiation, we need to consider the dependence of $B_\perp$ on it since the direction and strength of $B_\perp$ vary with each layer when both of $\theta$ and $y$ are non-zero as in Eq. (\[eq:perp\]), while both of which are assumed to be constant in our previous work. In this paper, for simplicity, we assume that the emissivity of synchrotron radiation is only proportional to $B_\perp^{1.8}$ and ignore the dependence on the Cosmic-ray electron density. The proportion of 1.8 power of $B_{\perp}$ corresponds to the frequency distribution $\propto \nu^{-0.8}$. This is reasonably close to that suggested by observations of the Milky Way [@Reich1988] and external radio galaxies [@Beck1996]. In addition, we do not consider the absolute value of emissivity and focus only on the shape of the FDF. Also, we ignore the change of direction of the concentric field within a layer, and assume that it is constant within the layer. The random magnetic field is also assumed. The polarization from its perpendicular component is ignored also in this study. While the contribution of emission from the random field to $F(\phi)$ is not trivial, we confirm that it basically affects only to the amplitude of $F(\phi)$ and hardly affects to the shape of it [@Miyashita_in_prep]. Thus, only LOS component of the field, $B_{\rm rand}$, is considered in this work, and it affects to the variance of emission from each layer in $\phi$ space. As a representative, $B_{\rm rand}$ is set as 1 $\mu$G, then $\sigma_B=1\mu$G. Under the assumptions described above, the FDF from $j$th layer can be expressed using Eq. (\[eq:layer-org\]) as $$P_j(\phi) = \frac{B_\perp(z_j)^{1.8}}{\sqrt{2\pi j}\sigma_\phi}\exp\left[ -\frac{(\phi-\phi_0(z_j))^2}{2j\sigma_\phi^2} \right]\times\exp[2\chi(z_j)], \label{eq:layer}$$ where $$\phi_0(z_j) = Kn_eL_{\rm cell}\sum_{k=1}^jB_\parallel(z_k), \label{eq:phi0}$$ $$\chi(z_j) = \cos^{-1}\left[\frac{z_j\tan\theta+\beta}{\sqrt{(z_j\tan\theta+\beta)^2+y^2\cos\theta}}\right], \label{eq:angle}$$ and $\sigma_\phi^2=K^2n_e^2\sigma_B^2L_{\rm cell}^2$ with $\langle B_{\rm rand}^2\rangle = \sigma_B^2$. Since we always consider the region for \[-$L_{\rm sh}$,$L_{\rm sh}$\] through $z$ axis, the number of layers depends on the inclination angle and is expressed as $$N = \frac{2L_{\rm sh}}{L_{\rm cell}\cos\theta}.$$ Finally, $F(\phi)$ is calculated using Equation \[eq:FDF\] and Eq. (\[eq:layer\]). When non-zero pitch angle of spiral field, $\theta'$, is considered, $F(\phi)$ is calculated in the same way by modifying Equation \[eq:perp\], \[eq:parallel\] and \[eq:angle\] as [$$B_\perp(z_k) = \frac{\sqrt{(z_j\tan\theta+\beta)^2 + y^2 - (y\cos\theta' + (z_j\tan\theta+\beta)\sin\theta')^2\sin^2 \theta}}{\sqrt{(z_j\tan\theta+\beta)^2+y^2}}B_{\rm coh},$$]{} $$B_\parallel(z_k) = \frac{(y\cos\theta'+(z_j\tan\theta+\beta)\sin\theta')\sin\theta}{\sqrt{(z_j\tan\theta+\beta)^2+y^2}}B_{\rm coh},$$ and [$$\begin{aligned} && \chi'(z_k) = \nonumber \\ && \cos^{-1}\left[\frac{(z_j\tan\theta+\beta)\cos\theta'-y\sin\theta'}{\sqrt{((z_j\tan\theta+\beta)^2+y^2)\cos^2\theta+(y\sin\theta'-(z_j\tan\theta+\beta)\cos\theta')^2\sin^2\theta}}\right]. \nonumber \\\end{aligned}$$]{} We start with the fiducial model where the turbulent field is assumed to be smaller than the global field for allowing us to simply interpret the results, and later in the next section, we consider the comparable or larger turbulent field with respect to the global field whose situation is suggested by observations [@Fletcher2010]. The model parameters and their fiducial values used in this paper are summarized in Table \[table:data\_type\]. ![Schematic picture of the galactic model used in this paper. []{data-label="jou"}](model2.eps){width="8cm"} ![This shows the pattern diagram of our galaxy model viewed from above.[]{data-label="model_top"}](model_top2.eps){width="8cm"} Symbol Physical Quantity Fiducial Value --------------- ------------------------------------------------- ---------------------- $\theta$ inclination angle 40 deg $\theta'$ pitch angle 0 deg y height of LOS 200 pc $\beta$ intersection of LOS and galactic plane 0 pc $n_e$ thermal electron density $0.02~{\rm cm}^{-3}$ $B_{\rm coh}$ coherent magnetic field $5~\mu {\rm G}$ $B_{\rm ran}$ standard deviation of turbulent magnetic fields $1~\mu {\rm G}$ \[table1\] Results ======= basic results ------------- ![Absolute value of Faraday dispersion function for the case with $\theta = 0~{\rm deg}$ (face-on). Solid line corresponds to the total FDF. Contribution from groups of layers with $j = 1-50,51-100,101-150$ and $151-200$ are also shown. []{data-label="fig:FDF_face-on"}](base_absfsuiit0.eps){width="8cm"} First, let us see the case of $\theta=0~{\rm deg}$ (face-on) discussed in @Ideguchi2017. The absolute value of the FDF is shown in Fig. \[fig:FDF\_face-on\]. As explained in Section 2, the FDF is the sum of contribution from all layers along the line of sight. Because ring fields are perpendicular to the line of sight, only turbulent fields contribute to the parallel component of magnetic fields. In this case, the center of Gaussian function of each layer is located at $\phi = 0$, while the width is larger for larger $j$. Thus, the FDF is the sum of Gaussian functions with the same center ($\phi = 0$) and different widths and is not Gaussian itself. The width of the FDF reflects the magnitude of turbulent magnetic fields. @Ideguchi2017 further considered the case with coherent magnetic fields perpendicular to the galactic plane and found that the FDF is skewed. ![Same as Fig. \[fig:FDF\_face-on\] but for inclination $\theta = 40~{\rm deg}$.[]{data-label="fig:FDF_inc40"}](base_absfsuii.eps){width="8cm"} ![Distribution of $B_\perp$ (top) and $B_\parallel$ (middle), and Faraday depth (bottom) along a line of sight with different inclination $\theta =$ 20, 40 and 60 degree for the fiducial model.[]{data-label="fig:distribution"}](base_physics2.eps){width="8cm"} Fig. \[fig:FDF\_inc40\] represents the FDF with inclination $\theta = 40~{\rm deg}$ and offset from the galactic center of $y = 200~{\rm pc}$ and $\beta = 0~{\rm pc}$. Two peaks can be seen at $\phi = 0$ and $65~{\rm rad/m^2}$ and the left peak is sharp compared with the other. These features were not seen in the situation considered in [@Ideguchi2017]. To understand these features, we plot the distribution of magnetic field components and $\phi(z)$ along the line of sight in Fig. \[fig:distribution\]. The top panel represents the perpendicular component of coherent fields and it is small around the galactic plane ($z \sim 0$). Because the emissivity of synchrotron radiation is assumed to be proportional to $B_{\perp}^{1.8}$, the emissivity around the galactic plane is consequently relatively low. On the other hand, the distribution of the parallel component is shown in the middle panel and the resulting Faraday depth is plotted in the bottom panel as a function of $z$. Since $\phi(z)$ is monotonically increasing because $B_\parallel > 0$ everywhere, Gaussian functions move rightward with increasing $j$. This can also be seen in Fig. \[fig:FDF\_inc40\], where the contributions from 4 groups of layers are also shown. Thus, we can see that the peak at $\phi = 0$ is mostly contributed from emission from the near side of the galaxy, while the other is from the far side. The sharpness of the peaks can also be explained by the fact that Gaussian functions of the far side have larger widths. It should be noted that the dip between the two peaks around $\phi \sim 35$ corresponds to the galactic plane $z \sim 0$ (see the bottom panel of Fig. \[fig:distribution\]). There are two reasons for the existence of the dip. One is that a relatively few number of layers are contributing to the FDF at $\phi \sim 35$ since $B_\parallel$ is large around $z \sim 0$ and $\phi$ is rapidly increasing as can be seen in the middle and bottom panels of Fig. \[fig:distribution\]. Another reason is that the emissivity is relatively low there as stated above. ![Absolute value of Faraday dispersion function for the cases with $\theta =$ 20, 40 and 60 deg.[]{data-label="fig:FDF_inc"}](theta_fdf.eps){width="8cm"} Fig. \[fig:FDF\_inc\] shows the absolute value of the FDF for different inclination angles (20, 40 and 60 deg). As can be seen in the middle panel of Fig. \[fig:distribution\], $B_\parallel$ at the galactic plane is larger for a larger inclination angle, while the mutual relation is reversed off the plane. On the other hand, the path length within the galaxy along the line of sight is longer for a larger inclination. Here, note that the horizontal axis of Fig. \[fig:distribution\] is $z$, rather than the path length. As a result of these factors, Faraday depth turns out to be larger for a larger inclination and, consequently, the gap between the two peaks is larger. ![Same as Fig. \[fig:FDF\_inc\] but for $y$ = 100, 200 and 400 pc. The inclination angle $\theta$ is fixed to 40 deg.[]{data-label="fig:FDF_offset-y"}](y0_absf.eps){width="8cm"} ![Distribution of $B_\parallel$ (Top), and Faraday depth (Bottom) along a line of sight with different offset $y$ = 100, 200 and 400 pc. The inclination angle $\theta$ is fixed to 40 deg.[]{data-label="fig:distribution_offset"}](y0_physics.eps){width="8cm"} ![FDF distribution in the case of $\beta$ = 100, 200 and 400 pc.[]{data-label="fig:FDF_offset-beta"}](beta_fdf.eps){width="8cm"} Finally, let us see the variation of the FDF when the offset of the line of sight from the galactic center is changed. Fig. \[fig:FDF\_offset-y\] shows the absolute value of the FDF for $y =$ 100, 200 and 400 pc, fixing $\theta =$ 40 deg and $\beta =$ 0 pc. The distribution of $B_\parallel$ and Faraday depth along the line of sight are shown in Fig. \[fig:distribution\_offset\]. The offset does not affect the maximum value of $B_\parallel$ but its variation is smaller for a larger offset. Thus, Faraday depth grow more rapidly and the gap between the two peaks is larger for a larger offset. On the other hand, significant change cannot be seen in Fig. \[fig:FDF\_offset-beta\], where $\beta$ is shifted to 100, 200 and 400 pc with fixed values of $y =$ 200 pc and $\theta =$ 40 degree. variant models -------------- In the previous sebsection, a simple model of a galaxy was adopted to understand the behavior of the FDF easily. We extend the fiducial model in several ways to consider more variations of observed galaxies. ### turbulent magnetic fields In most galaxies, turbulent magnetic fields are comparable or stronger than coherent fields. In this subsection, we investigate the FDFs varying the amplitude of turbulent fields and fixing that of coherent fields. Fig. \[ran\_absf\] shows the absolute value of the FDFs with turbulent magnetic fields of 1,3, 5 and $10~\mu {\rm G}$. For stronger turbulent fields, the Gaussian function of each layer widens so that the peaks become smooth. On the other hand, the centers of Gaussian functions are unchanged because the magnitude of coherent fields is fixed, so the posiiton of peaks do not change so much either. Thus, these FDFs have most of the features of the FDF of the fiducial model and the discussion given in the previous subsection applies to the current case as well. ![Absolute value of the FDFs with turbulent magnetic fields of 1,3, 5 and $10~\mu {\rm G}$.[]{data-label="ran_absf"}](ran_fdf.eps){width="8cm"} ### pitch angle Here we consider coherent magnetic fields with pitch angle by rotating the ring field at each position as stated in section 2. Fig. \[physics4\] shows the line-of sight distribution of perpendicular and parallel components of magnetic fields and $\phi(z)$ for pitch angle $\theta' = 0, 20, 40$ and $60~{\rm deg}$ with fixed inclination $\theta = 40~{\rm deg}$. It is seen that, for non zero pitch angle, the distribution of magnetic fields is not symmetric with respect to the galactic plane ($z = 0$) as opposed to the case with ring fields. The parallel component starts from a negative value and becomes positive at somewhere in the near side ($z < 0~{\rm pc}$). The transition point is closer to the galactic plane for a larger pitch angle. Consequently, Faraday depth first decreases from zero, takes the minimum value at the transition point and becomes positive in the end. Resultant FDFs are shown in Fig. \[fdftheta40\]. Because Faraday depth becomes negative for non zero pitch angle, the left peak moves to negative $\phi$ region and is located at the minimum $\phi$. Therefore, the Faraday depth of the peak is smaller (larger $\|\phi\|$) for a larger pitch angle. On the other hand, the right peak has a lower height and is located at a smaller $\phi$ for a larger pitch angle. This is because only a narrow range of $z$ contributes to the right peak for a large pitch angle (see the bottom panel of Fig. \[physics4\]). The magnitude of perpendicular component shown in the top panel of Fig. \[physics4\] also affects the peak height, but this factor is very minor. It should be noted that the vertical axis of Fig. \[fdftheta40\] is in logarithmic scale, while that of Fig. \[fig:distribution\] is in linear scale. ![Same as Fig. \[fig:distribution\] but with pitch angle $\theta'$ = 0, 20, 40 and 60 deg and fixed inclination $\theta = 40~{\rm deg}$. []{data-label="physics4"}](physics4.eps){width="8cm"} ![Same as Fig. \[fig:FDF\_face-on\] but with pitch angle $\theta'$ = 0, 20, 40 and 60 deg and fixed inclination $\theta = 40~{\rm deg}$.[]{data-label="fdftheta40"}](FDFtheta40.eps){width="8cm"} Figs. \[phisics3\] and \[fdf30\] are comparison between different inclination angles ($\theta$ = 0, 20 and 40 deg) for fixed pitch angle of $\theta' = 30~{\rm deg}$. We can see that the effects of non zero pitch angle such as asymmetry of magnetic fields with respect to the galactic plane are more significant for a larger inclination angle. ![Same as Fig. \[fig:distribution\] but with inclination $\theta$ = 0, 20 and 40 deg and fixed pitch angle $\theta' = 30~{\rm deg}$.[]{data-label="phisics3"}](physics3.eps){width="8cm"} ![Same as Fig. \[fig:FDF\_face-on\] but with inclination $\theta$ = 0, 20 and 40 deg and fixed pitch angle $\theta' = 30~{\rm deg}$.[]{data-label="fdf30"}](FDF30.eps){width="8cm"} Summary and discussion ====================== In this paper, we investigated Faraday dispersion function (FDF) of disk galaxies extending a simple analytic model of galactic magnetic fields developed in [@Ideguchi2017], which has axisymmetric coherent fields and turbulent fields. We studied the effects of inclination, relative amplitude of coherent and turbulent fields and pitch angle of coherent fields. Our model was rather simple but allowed us to understand the behavior of FDFs. First, we found FDFs have two peaks when galaxies are observed with non zero inclination. The gap and relative height of two peaks are dependent on the inclination angle and pitch angle. The relative amplitude of coherent and turbulent magnetic fields does not affect the qualitative features so much and widens the peaks. It is rather surprising that FDFs exhibit a wide variety of shape just by varying the inclination and pitch angles. Especially, the fact that non zero inclination leads to two peaks in the FDF gives us an important caution. In practical observations, emission between the two peaks may not be identified due to thermal and systematic noises. Therefore, one may recognize that the FDF is contributed from two separate radio sources within the beam. Simulations of reconstruction of FDFs studied here with observational errors will be given elsewhere. Combining Faraday tomography and 2-dimensional imaging, we will be able to obtain Faraday cube, that is, polarization emissivity as a function of Faraday depth and position in the sky. This will allow us probe 3-dimensional structure of galactic magnetic fields, such as topology of global fields and the position dependence of the strength of turbulent fields. Investigation of Faraday cube is also our future work. Acknowledgement {#acknowledgement .unnumbered} =============== KT is partially supported by JSPS KAKENHI Grant Numbers JP15H05896, JP16H05999 and JP17H01110, and Bilateral Joint Research Projects of JSPS. YM is supported in part by Grand-in-Aid from JSPS Research Fellow, No. 17J06936. [99]{} Akahori, T., Kumazaki, K., Takahashi, K., and Ryu, D.: 2014, [Publications of the Astronomical Society of Japan]{} [66]{}, 65. Akiyama, K., Akahori, T., Miyashita, Y., Ideguchi, S., Yamaguchi, R., Ikeda, S., and Takahashi, K., 2019, submitted to [Astrophys. J.]{} Anderson, C.S., Gaensler, B.M., and Feain, I.J.: 2016, [Astrophys. J.]{} [825]{}, 59. Andrecut, M., Stil, J. M., $\&$ Taylor, A. R. 2012, AJ, 143, 33 1996ARA$\&$A..34..155 Beck, R., Brandenburg, A., Moss, D., Shukurov, A., Sokoloff, D., 1996, ARA$\&$A, 34, 155 Beck, R. 2009, , 320, 77 Bell, M. R., Junklewitz, H., $\&$ Ensslin, T. A. 2011, A$\&$A, 535, A85 Brentjens, M. A., $\&$ de Bruyn, A. G. 2005, A$\&$A, 441, 1217 Burn, B. J. 1966, MNRAS, 133, 67 Fletcher, A., 2010, ASPC, 438, 197 Frick, P., Sokoloff, D., Stepanov, R., $\&$ Beck, R. 2011, MNRAS, 414, 2540 Govoni, F., Dolag, K., Murgia, M., Feretti, L., Schindler, S., Giovannini, G., Boschin, W., Vacca, V., and Bonafede, A.: 2010, [Astron. Astroph.]{} [522]{}, A105. Heald, G., Braun, R., and Edmonds, R.: 2009, [Astron. Astroph.]{} [503]{}, 409. Ideguchi, S., Tashiro, Y., Akahori, T., Takahashi, K., $\&$ Ryu, D. 2014, [Astrophys. J.]{} [792]{}, 51. 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--- abstract: \| For each finite group $G$, we define the [Grothendieck-Teichmüller group]{} of $G$, denoted ${\mathcal{GT}}(G)$, and explore its properties. The theory of dessins d’enfants shows that the inverse limit of ${\mathcal{GT}}(G)$ as $G$ varies can be identified with a group defined by Drinfeld and containing ${\operatorname{Gal}({ {\overline{{\mathbb{Q}}}} }/ {\mathbb{Q}})}$. We give in particular an identification of ${\mathcal{GT}}(G)$, in the case when $G$ is simple and non-abelian, with a certain very explicit group of permutations that can be analyzed easily. With the help of a computer, we obtain precise information for $G= PSL_2({\mathbb{F}}_q)$ when $q \in \{ 4,$ $ 7,$ $ 8,$ $ 9,$ $ 11,$ $ 13,$ $ 16,$ $ 17,$ $ 19 \}$, and we treat $A_7$, $PSL_3({\mathbb{F}}_3)$ and $M_{11}$. In the rest of the paper we give a conceptual explanation for the technique which we use in our calculations. It turns out that the classical action of the Grothendieck-Teichmüller group on dessins d’enfants can be refined to an action on “$G$-dessins”, which we define, and this elucidates much of the first part. [Status:]{} this version should be nearly identical to that which is to appear in the proceedings volume [Symmetry in Graphs, Maps and Polytopes]{} (SIGMAP), Springer. (Precise reference not available just now.) address: \| Université de Strasbourg & CNRS\ Institut de Recherche Mathématique Avancée\ 7 Rue René Descartes\ 67084 Strasbourg, France author: - Pierre Guillot bibliography: - 'myrefs.bib' title: \| The Grothendieck-Teichmüller group\ of a finite group and $G$-dessins d’enfants ---
--- abstract: \| Feedback Control Systems, ME C134/EE C128, is an introductory control systems course at UC Berkeley. Over the entire course, students gain practical experience by implementing various control schemes and designing observers in an effort to ultimately stabilize an inverted pendulum on a linear track. Throughout this learning process, frequent mishaps occur where improper controller implementation damages hardware. A simple example concerns the student’s controller driving the cart into the wall at full speed. To offset the financial burden placed on the university in light of these mishaps, we designed a streamlined adaptive control system using set theory. We utilized lab-provided plant models to generate an $O_\infty$ set, attenuated the vertices to generate a safe, sub-region $S_\infty$, and attenuated in such a manner as to ensure an evolution of the vertices of $S_\infty$ remained within $O_\infty$ for at least one time step. Afterwards, we constructed a single Simulink block for students to easily implement within their own control schemes. This block consistently checks to see whether the system state remains within $S_\infty$. If that check is true, our controller does nothing. If it returns false, our controller takes over, drives the system to a prescribed safe-point, and shuts the system down. Overall, our process assumes perfect plant modelling, though our insistence on an evolution of $S_\infty$ remaining within $O_\infty$ resulted in considerable robustness to disturbances. In the end we were successful in implementing this real-time adaptive system and will provide it to the department for use in future labs.\ Video Link: <https://drive.google.com/file/d/1kPw5FTVs3hOcy_yhZ8gXrCNN5lqBWi0v/view?usp=sharing> author: - 'Prithvi Akella, Sean Anderson, David Lovell' title: 'Set-Based Adaptive Safety Control' --- Introduction ============ In an introductory controls class at Berkeley, students attempt to stabilize an inverted pendulum on a linear track [@lab_pend]. Often, students implement unstable or otherwise flawed controllers that cause a variety of issues. This can cause the cart to drive into one of the side-walls subsequently destroying hardware, or cause the system to act wildly posing a risk to the students. Damage to this commercial lab equipment represents a non-trivial financial cost to the university, and injuries to students are unacceptable. To mitigate these issues, we created an adaptive control scheme that monitors and overrides the student’s controller in the event of unsafe actions. In that effort, we note that the action of the cart running into the wall can be extrapolated from system dynamics. That is to say, given a specific braking scheme, we can use set theory to identify all possible states for which the aforementioned scheme can feasibly prevent unsafe actions. Since the actuation of the system occurs from a single motor voltage input that directly controls the position of the cart on the track, it sufficient to only check if the cart position leaves a region defined as safe. Provided that region is a subset of the larger, calculated region, our controller will still function to bring the system back to a predefined safe point. Background Information ====================== Stable Controller Development ----------------------------- To clarify, $\mathcal{O}_{\infty}$ is a positive invariant set defined as follows [@borrelli]: $$\begin{aligned} & \forall x(0) \in O \quad x(t) \in O \\ & \forall t \in [0,\infty] \quad \| \quad x(k+1) = f(x(k)). \label{O_infty}\end{aligned}$$ Note that $f(...)$ in the above equation represents the linearized evolutionary scheme we developed by Euler-discretization of a lab-developed transfer function. More specifically, for $$\begin{aligned} & y^{(n)} + \alpha_1y^{(n-1)} + ... = u^{(p)} + \beta_1u^{(p-1)} + ..., \\ & y_{k+1} = y_k + \Delta t \Dot{y}(k) \quad \Dot{y}_{k+1} = \Dot{y}_k + \Delta t \Ddot{y}(k) \quad ....\end{aligned}$$ The successive Euler-discretization steps can be reformatted into a matrix equation as $$y_{k+1} = Ay_k + Bu_k = f(x_k,u_k). \label{linear_evolution}$$ Note that the $y_{k+1}$ in the above equation corresponds to a vector whose elements are $y_k$, $\Dot{y}_k$, and so forth. However, O’s functional dependence is only restricted to functions of x. To remedy that, we develop a specific, stable controller such that each $u_k = -K x_k$. Stability in the discrete scenario implies that the closed-loop evolution, $A-BK$ decays to $0$ after multiple iterations. That is, $$(A-BK)^N \rightarrow 0 \quad \textrm{as} \quad N \rightarrow \infty.$$ It is also critical to mention that the aforementioned stability implies that the poles of the Closed Loop System, $A_{cl} = A-BK$ all lie within the unit circle on the imaginary plane. $MATLAB$ employs that last criteria to generate stable controllers through the place function as shown below: K = place(A,B,[p_1,p_2,...]). Note that each of the $p_i$ correspond to the a pole location for the closed-loop system. Choosing each $\|p_i\| < 1$ identifies a stable controller as per our definitions earlier. Calculation of $\mathcal{O}_{\infty}$ ------------------------------------- $\mathcal{O}_{\infty}$ is found through a recursive process detailed below: O = X; numiterations = 500; for i = 1:numiterations Pre_O = Pre(O); if Pre_O.intersect(O) == O Oinf = O; break; else O = Pre_O.intersect(O); end end To clarify, the initial line setting $\mathcal{O}=\mathcal{X}$ concerns initializing the invariant set to be the initial, invariant, state-constraint set, $\mathcal{X}$. As a result, intersecting the set each iteration ensures that the resulting $\mathcal{O}_{\infty} \in \mathcal{X}$. Furthermore, the $Pre$ operation is defined as follows: $$Pre(G) = S = \{x \, \| \, y = f(x) \, \forall y \in G \}.$$ Note that $\mathcal{G}$ is some predefined set, and for our purposes, $f(...)$ is the closed-loop evolutionary scheme we developed with $K$. The number of iterations is limited to prevent against infinite loop calculations. It is important to qualify that both of the aforementioned processes, generating $K$ and calculating $\mathcal{O}_{\infty}$ are model-specific processes. As such, this process as stated, assumes a perfect model with no noise or disturbances. To protect our process against faults in those assumptions, we implement a set attenuation and boundary evolution process detailed in the following implementation section. Implementation ============== Set Generation -------------- For our specific system, our continuous time matrices were, $$A = \begin{bmatrix} 0 & 1 \\ 0 & -7.2 \end{bmatrix} \quad B = \begin{bmatrix} 0 \\ 1.6 \end{bmatrix},$$ which, given a sample time of 0.002 seconds, transformed to the following discrete time matrices: $$A = \begin{bmatrix} 1 & \Delta t \\ 0 & 1-7.2\Delta t \end{bmatrix} \quad B = \begin{bmatrix} 0 \\ 1.6\Delta t \end{bmatrix}.$$ Afterwards, we placed our desired closed loop poles at $ \lambda_1 = 0.99$ and $ \lambda_2 = 0.985$, which generated the following K controller: $$K = \begin{bmatrix} 23.3 & 3.3 \end{bmatrix}$$ In addition, our only state constraints concerned the position limitations on the cart, while the velocity could feasibly be any real-value. In reality, the physical constraints, predefined evolution, and specific controller should limit velocity, but we had no constraints a priori. Noting that, our initial $\mathcal{X}$ set was defined as, $$\mathcal{X} = \{ x \, \| -0.4<=x(1)<= 0.4 \quad \forall \, x(2) \in \mathbb{R} \}.$$ After following the aforementioned process to create $\mathcal{O}_{\infty}$ we decided that the two main limitations inherent to our system could be addressed by implementing a buffer region. Namely, this system is only valid with respect to our specific plant model, and that we could not use the calculated $\mathcal{O}_{\infty}$ as a boundary for the prescribed safe zone. The second limitation arises if you consider an event where the system state left $\mathcal{O}_\infty$. In this event, no controller would be able to control the system back to a safe region, as the state is already outside its region of attraction. To remedy both, the vertices of $\mathcal{O}_{\infty}$ were found and scaled down to create $\mathcal{S}_{\infty}$. This set represents the the set of states that if violated give adequate response time for the system to be prevented from exiting $\mathcal{O}_{\infty}$. The entire system can be distributed to the students as a single Simulink sub-system block which is placed in the feedback loop between the student’s controller and the input to the plant. The simplicity of this application is shown in Figure \[controller\]. The inputs and outputs to the subsystem are labeled for ease of use. ![The background controller can easily be incorporated into the Simulink model by the student.[]{data-label="controller"}](images/Control_Loop.JPG){width="\linewidth"} The contents of the sub-system is shown in Figure \[subsystem\]. It contains two MATLAB function blocks, a variable switch, and a switch logic block. The function, $bound2D$, which can be found in the appendix, implements the real time monitoring of the system and produces a flag if the system will violate constraints. If a flag is thrown, the switch logic prevents the student’s input from feeding to the motor and instead feeds the signal through the $forestfires$ function block to drive the system to the defined zero point. The adaptive switch subsystem counts the number of flags and terminates the Simulink model at a defined time. If there is no flag, the student’s input is passed through after being saturated to prevent damage to the physical system from large inputs. ![The subsystem implementing the background controller mainly consists of two user-defined functions.[]{data-label="subsystem"}](images/Override_Controller.JPG){width="\linewidth"} Robustness Measures ------------------- To ensure that our controller would always bring the cart back to a safe region and that our system was relatively robust to model errors, noise, and disturbances, we attenuated $\mathcal{O}_{\infty}$ by scaling the vertices of the set by a scalar less than 1. In doing so, we generated a subset of $\mathcal{O}_{\infty}$, $\mathcal{S}_{\infty}$ while ensuring that the evolution of every vertex of $\mathcal{S}_{\infty}$, with respect to the worst possible input for that state, remained within $\mathcal{O}_{\infty}$. This method created a “gray-zone” that is the difference between the sets. This gray-zone represents combinations of states that are outside of the predefined “safe” region, but still remain controllable. The process that was used to develop the gray-zone, evolving the boundary with the worst case input, provided the largest possible buffer region that simultaneously did not overly limit the working space along the track. If the working space on the track was too severely limited, it would effect the student’s ability to conduct lab exercises. This gray-zone hedges against disturbances and modelling errors insofar as it provides the largest area to “catch” errors of the kind that would cause a danger to the lab equipment or students. In addition, this process resulted in a system so robust that when the result of the 2D (cart-only) system was tested on the 4D (cart-and-pendulum) system, it proved to be sufficient. Results ======= The outlined process worked across all testing scenarios without failure. $\mathcal{S}_{\infty}$ sets were calculated for both the 2D and 4D systems, however, the 4D process restricted the usable track length too severely. Rather than construct a different method for the monitoring of the 4D system, the 2D system was tested to determine its effectiveness. These tests proved that the 2D system was fully capable of monitoring and overriding the 4D system. A characteristic test result of the 2D system acting on both systems are shown in Figure \[result\]. ![The upper image depicts $\mathcal{O}_{\infty}$, $S_{\infty}$, and the trace of one instance of the 2-D system with $S_{\infty}$ violation. The lower image uses the same set but the trace is for the cart-pendulum self-erecting scenario. []{data-label="result"}](images/Trajectory_trace_labeled.png){width="\linewidth"} In Figure \[result\], $S_{\infty}$ is identified as the green area, with $\mathcal{O}_{\infty}$ as the larger gray area. The difference in areas, the ’grey-zone’, is the set of states that violate the constraints of $\mathcal{S}_{\infty}$, but remain within $\mathcal{O}_{\infty}$. This ’grey-zone’ gives the override controller sufficient time to physically stop the system from violating $\mathcal{O}_{\infty}$ by driving the cart away from the wall. In the top image of Figure \[result\], the 2D system is shown. The red line indicates the state trajectory of the cart during a step input which places it at the boundary of $\mathcal{S}_{\infty}$. At this point, a disturbance is applied to the cart which causes the system to leave $\mathcal{S}_{\infty}$. The monitoring system identifies this violation and initiates the override controller to bring the cart back to the defined zero point. Note that this proves an instance of our controller’s robustness to disturbances. Even though the disturbance that caused the violation was not factored into our set calculation, our controller still acted properly in the event of the failure and brought the system to safety. Additionally, the lower image of Figure \[result\], exhibits a characteristic response of the 2D monitoring and override system acting on the cart with the pendulum added. In this scenario the pendulum started at rest in the vertically down position. The axes of this plot are the position and velocity of the cart, as such, the trajectory of states for the pendulum are not shown. As the student’s control system initiates a pumping sequence, the position and velocity of the cart oscillate from positive to negative until the pendulum is erected and stabilized near the left boundary of the track. This process verifies that $\mathcal{S}_{\infty}$ does not overly restrict our available region. That is, there is more than sufficient track length for the pendulum to be erected and balanced. After the cart stabilized the pendulum, a disturbance was applied to the pendulum. The cart then acted to attempt to regain balance of the pendulum and in the process the $\mathcal{S}_{\infty}$ set was violated. Again, it is interesting to note that the disturbance to the pendulum was responsible for the violation of $\mathcal{S}_{\infty}$, but was not considered in the calculation. Still, the override system responded correctly to prevent the cart from colliding with the wall. In this way, the addition of the pendulum to the 2D system can be interpreted simply as a disturbance. This disturbance can cause the cart to violate $\mathcal{S}_{\infty}$, which the monitoring system then identifies causing the override system to respond. Also of note is that the unpredictable inertia of the swinging pendulum does not prevent the override system from recovering the cart before a violation of $\mathcal{O}_{\infty}$ occurs. This further proves the robustness of our system. Conclusion ========== We successfully created a simple Simulink block that monitors and overrides a student’s controller in the event of a hardware failure or safety concern. The application of set theory provided a means to accurately determine the safe operating length of the track. The resulting matrices allowed for real-time identification of state violations even with a controller operating in millisecond sample periods. This system also proved to be robust to the extent that a high degree of modifications to the mass and inertia of the plant were successfully tolerated. Appendices ========== Main Code --------- bounds2D -------- forestfires ----------- Acknowledgment {#acknowledgment .unnumbered} ============== $Pre$ function and set theory were sourced from ME C231A-EECS C220B, UC Berkeley, Fall 2017 course material. System dynamics, gains, and state matrices for the lab hardware were sourced from ME C134-EECS C128, UC Berkeley, Fall 2017 course material. [19]{} Staff ME C134/ EECS 128. "Lab 6d: Self-erecting inverted pendulum (seip), mec134/ eecs c128,” April 2017. F. Borrelli, “Lecture notes: Model predictive control reachability and invariance, eecs c220b/me c231a,” November 2017.
--- abstract: \| Almost every biological, economic and social system is a complex adaptive system (CAS). Mathematical and computer models are relevant to CAS. Some approaches to modeling CAS are given. Applications in vaccination and the immune system are studied. Mathematical topics motivated by CAS are discussed.\ \ Keywords: Complex adaptive systems; The immune system; Cellular automata; Game theory; Complex networks; Multi-objective optimization. author: - \| E. Ahmed$^{1}$, A. S. Elgazzar$^{2,3}$ and A. S. Hegazi$^{1}$\ $^{1.}$Mathematics Department, Faculty of Science\ 35516 Mansoura, Egypt\ $^{2.}$Mathematics Department, Faculty of Education\ 45111 El-Arish, Egypt\ $^{3.}$Mathematics Department, Faculty of Science - Al-Jouf\ King Saud University, Kingdom of Saudi Arabia\ E-mail: elgazzar@mans.edu.eg title: An Overview of Complex Adaptive Systems --- Basics of complex adaptive systems (CAS) ======================================== Definition (1): A complex adaptive system consists of inhomogeneous, interacting adaptive agents. Adaptive means capable of learning.Definition (2): An emergent property of a CAS is a property of the system as a whole which does not exist at the individual elements (agents) level.\ \ Typical examples are the brain, the immune system, the economy, social systems, ecology, insects swarm, etc.. Therefore to understand a complex system one has to study the system as a whole and not to decompose it into its constituents. This totalistic approach is against the standard reductionist one, which tries to decompose any system to its constituents and hopes that by understanding the elements one can understand the whole system. Why should we study complex adaptive systems? ============================================= Most of living systems are CAS. Moreover they have intrinsic unpredictability which causes some ”seemingly wise” decisions to have harmful side effects. Therefore we should try to understand CAS to try to minimize such side effects. Here we give two examples of these side effects. Mathematical models have played important roles in understanding the impact of vaccination programs. The complications of infectious diseases spread make the problem of predicting the impact of vaccinations a nonlinear problem. Sometimes a counter-intuitive result appears e.g. the threshold phenomena \[Edelstein-Keshet 1988\]. Here another example will be mentioned. Several vaccination programs are known e.g. mass vaccination where all population is vaccinated, target vaccination where only a certain group is vaccinated. If one tries to understand the expected impact of a vaccination program one should take the following points into account: 1. Vaccination is not perfect hence a probability of vaccination failure should be assumed. 2. Sometimes vaccination takes time to be effective. 3. Immunity is waning i.e. may be lost with time. 4. Long range contacts can play a significant role e.g. SARS (severe acute respiratory syndrome) has been transmitted between countries via air travellers. Rubella is a mild viral infectious disease. Typically it is most dangerous when infecting a pregnant female where it has severe effects on the fetus. Once one gets it he (she) gets a life long immunity. There are several vaccination strategies for rubella \[Vynnycky et al 2003\]. The US policy is to vaccinate all two years old children. The UK policy is to vaccinate only 14-years old girls. Another strategy which is adopted in some underdeveloped countries is not to vaccinate at all. It has been found \[Jazbec et al 2003\] that in most cases the UK strategy is equal or better than the US one despite being cheaper. An interesting situation arose when some countries adopted a private sector vaccination to MMR (Measles, Mumps and Rubella) \[Vynnycky et al 2003\]. It was expected that the number of Congenital Rubella Syndrome (CRS) will decrease. However it did not and in some countries (e.g. Greece and Costs Rica) it increased. The reason can be understood as follows: This vaccination to part of the population decreases the probability of contracting the disease at young age. Hence the number of susceptible individuals at adulthood increases. Consequently the probability of contracting the disease at adulthood increases. This is an example of the counterintuitive effects of some vaccination programs. Another example for bad side effects is Lake Victoria \[Chu et al 2003\] where a new species called Nile perch was introduced expecting that it is more economically profitable. Yet the following results have appeared: 1. The local fishermen’s tools were not suitable for the new fish hence only large corporations benefited. 2. Due to its higher price the locals were unavailable to buy the new type. 3. The original fish used to eat the larva of mosquitoes but now mosquitoes’ numbers have increased significantly thus the quality of life of the locals have deteriorated!! There are at least two sources for unpredictability in CAS. The first is the nonlinear interactions between its agents \[West 1990\]. The second is that CAS are open systems hence perturbation to one system may affect another related one e.g. perturbation to Lake Victoria affected the number of mosquitoes. How to model a CAS? =================== The standard approaches are 1. Ordinary differential equations (ODE), difference equations and partial differential equations (PDE). 2. Cellular automata (CA) \[Ilachinski 2001\]. 3. Evolutionary game theory \[Hofbauer and Sigmund 1998\]. 4. Agent based models. 5. Networks \[Watts and Strogatz 1998\] etc.. 6. Fractional calculus \[Stanislavsky 2000\]. Some of these approaches are included in \[Boccara 2004\]. The ODE and PDE approaches have some difficulties as follows \[Louzon et al 2003\]: 1. ODE and PDE assumes that local fluctuations have been smoothed out. 2. Typically they neglect correlations between movements of different species. 3. They assume instantaneous results of interactions. Most biological systems show delay and do not satisfy the above assumptions. They concluded that a cellular automata (CA) \[Ilachinski 2001\] type system called microscopic simulation is more suitable to model complex biological systems. We agree that CA type systems are more suitable to model complex biological systems but such systems suffer from a main drawback namely the difficulty of obtaining analytical results. The known analytical results about CA type systems are very few compared to the known results about ODE and PDE. Some mathematical results about CA are given in the appendix. Now we present a compromise i.e. a PDE which avoids the delay and the correlations drawbacks. It is called telegraph reaction diffusion equations \[Ahmed and Hassan 2000\]. To overcome the non-delay weakness in Fick’s law it is replaced by $$J(x,t)+\tau \frac{\partial J(x,t)}{\partial t}=-D\frac{\partial c}{\partial x% },$$ where the flux $J(x,t)$ relaxes, with some given characteristic time constant $\tau $ and $c$ is the concentration of the diffusing substance. Combining Eq. (1) with the equation of continuity, one obtains the modified diffusion equation or the Telegraph equation: $$\frac{\partial c}{\partial t}+\tau \frac{\partial ^{2}c}{\partial x^{2}}=D% \frac{\partial ^{2}c}{\partial x^{2}}.$$ The corresponding Telegraph reaction diffusion (TRD) is given by $$\tau \frac{\partial ^{2}c}{\partial t^{2}}+\left( 1-\frac{\mathrm{d}f(c)}{% \mathrm{d}c}\right) \frac{\partial c}{\partial t}=D\frac{\partial ^{2}c}{% \partial x^{2}}+f(c),$$ where $f(c)$ is a polynomial in $c$. Another motivation for TRD comes from media with memory where the flux $J$ is related to the density $c(x,t)$ through a relaxation function $K(t)$ as follows $$J(x,t)=-\int_{0}^{t}K(t-\acute{t})\frac{\partial c(x,\acute{t})}{\partial x}% \mathrm{d}\acute{t}.$$ It can be shown \[Compte & Metzler 1997\] that, with a suitable choice for the kernel $K(t)$, the standard Telegraph equation is obtained. A third motivation is that starting from discrete space time one does not obtain the standard diffusion equation but the telegraph equation \[Chopard and Droz 1991\]. Moreover it is known that TRD results from correlated random walk \[Diekmann et al, 2000\]. This supports the conclusion that Telegraph reaction diffusion equation is more suitable for modeling complex systems than the usual diffusion one. The immune system as a complex system \[Segel and Cohen 2001, Ahmed and Hashish 2004\] ====================================================================================== The emergent properties of the immune system (IS) included: 1. The ability to distinguish any substance (typically called antigen Ag) and determine whether it is damaging or not. If Ag is non-damaging (damaging) then, typically, IS tolerates it (responds to it). 2. If it decides to respond to it then IS determines whether to eradicate it or to contain it. 3. The ability to memorize most previously encountered Ag, which enables it to mount a more effective reaction in any future encounters. This is the basis of vaccination processes. 4. IS is complex thus it has a network structure. 5. The immune network is not homogeneous since there are effectors with many connections and others with low number of connections. 6. The Ag, which enters our bodies, has extremely wide diversity. Thus mechanisms have to exist to produce immune effectors with constantly changing random specificity to be able to recognize these Ag. Consequently IS is an adaptive complex system. 7. Having said that, one should notice that the wide diversity of IS contains the danger of autoimmunity (attacking the body). Thus mechanisms that limit autoimmunity should exist. 8. In addition to the primary clonal deletion mechanism, two further brilliant mechanisms exist: The first is that the IS network is a threshold or “window” one i.e. no activation exists if the Ag quantity is too low or too high (This is called low and high zone tolerance). 9. Thus an auto reactive immune effector (i.e. an immune effector that attacks the body to which it belongs) will face so many self-antigens that it has to be suppressed due to the high zone tolerance mechanism. 10. Another mechanism against autoimmunity is the second signal given by antigen presenting cells (APC). If the immune effector is self reactive then, in most cases, it does not receive the second signal thus it becomes anergic. 11. Also long term memory can be explained by the phenomena of high and low zone tolerance where IS tolerates Ag if its quantity is too high or too low. So persisting Ag is possible and continuous activation of immune effectors may occur. 12. There is another possible explanation for long term memory using the immune system (Extremal Dynamics). 13. Thus design principles of IS can explain important phenomena of IS. An interesting example is given by Matzinger \[Matzinger 2002\] where she argued that to prevent transplant rejection it may be more useful to design drugs that blocks signal II and not signal I (which the present drugs do). The reason is blocking signal II make the effectors (which originally were capable of recognizing the transplant) anergic while leaving the other immune effectors intact. Conclusions =========== 1. CAS should be studied as a whole hence reductionist point of view may not be reliable in some cases. 2. CAS are open with nonlinear local interactions hence: 1. Long range prediction is highly unlikely \[Strogatz 2000, Holmgren 1996\]. 2. When studying a CAS take into consideration the effects of its perturbation on related systems e.g. perturbation of lake Victoria has affected mosquitoes’ numbers hence the locals quality of life. This is also relevant to the case of natural disasters where an earthquake at a city can cause a widespread power failure at other cities. 3. Expect side effects to any ”WISE” decision. 4. Mathematical and computer models may be helpful in reducing such side effects. 3. Optimization in CAS should be multi-objective and not single objective \[Collette and Siarry 2003\]. 4. CAS are very difficult to control. Interference at highly connected sites may be a useful approach \[Dorogovtsev and Mendez 2004\]. The interlinked nature of CAS elements complicates both the unpredictability and controllability problems. It also plays an important role in innovations spread. 5. Memory effects should not be neglected in CAS. This lends more support for the proposed telegraph reaction diffusion Eq. (3). Also memory games have been studied \[Smale 1980, Ahmed and Hegazi 2000\]. Also delay and fractional calculus are relevant to CAS. 6. Mathematical topics motivated by CAS include ODE and PDE (non-autonomous, delayed, periodic coefficients, stability and persistence), multi-objective optimization (including biologically motivated methods e.g. Ant colony optimization, Extremal optimization, Genetic algorithm etc), difference equations, cellular automata, networks, fractional calculus, control (e.g. bounded delayed control of distributed systems), game theory, nonlinear dynamics and fuzzy mathematics. Some of the mathematics motivated by CAS will be reviewed in the appendices. Acknowledgments {#acknowledgments .unnumbered} =============== One of the authors (A. S. Hegazi) acknowledge the financial support of a Mansoura University grant. References {#references .unnumbered} ========== 1. Ahmed E. and Hassan S.Z. (2000), “On Diffusion in Some Biological and Economic Systems”, Z. Naturforsch. 55a, 669. 2. Ahmed E. and Hashish A.H. (2004), “On Modeling the immune system as a complex system”, Theor. BioSci. (Accepted). 3. Ahmed E. and Hegazi A.S.“On Discrete Dynamical Systems Associated to Games”, Adv. Complex. Sys. 2, 423-429 (2000). 4. Ahmed E., Hegazi A.S. “On Persistence and Stability of Some Spatially Inhomogeneous Systems”, J. Math. Analys. Appl. 268, 74-88 (2002). 5. Ahmed E., Hegazi A.S., Elgazzar A.S. and Yehia H.M., “On Synchronization, Persistence and Seasonality in Some Inhomogeneous Models in Epidemics and Ecology”, Physica A 322, 155-168 (2003). 6. Ahmed E. Hegazi A.S. and Abdel-Hafiz A.T., “On Multiobjective oligopoly” Nonlin. Dyn. Psych. Life Sci. 7, 203-217 (2003). 7. Ahmed, E. and Elgazzar, A.S., “On Coordination and Continuous Hawk-Dove Games on Small-World Networks”, Eur. Phys. J. B 18, 159 (2000 a). 8. Ahmed, E. and Elgazzar, A.S., “On the Dynamics of Local Hawk-Dove Game”, Int. J. Mod. Phys. C 11, 607 (2000 b). 9. Ahmed, E., Hegazi, A.S. and Elgazzar, A.S., “An Epidemic Model on Small-World Networks and Ring Vaccination”, Int. J. Mod. Phys. C 13, 189 (2002). 10. Albert, R. and Barabási, A.L., “Statistical mechanics of complex networks”, Rev. Mod. Phys. 74, 47 (2002). 11. Bagnoli F., Boccara N. and Rechtman R.(2002), “Nature of phase transitions in probabilistic cellular automata with two absorbing states”, Lect. notes Computer sci. 2493, 249-258. 12. Barnett S. (1990),“Matrices”,Oxford Univ. Press, U.K. 13. Boccara N. (2004), “Modeling complex systems”, Springer Publ., Berlin. 14. Chopard B. and Droz M. (1991), “Cellular automata model for the diffusion equation”, J. Stat. Phys. 64, 859. 15. Chu D., Strand R. and Fjelland R. (2003), “Theories of Complexity”, Complexity 8, 19. 16. Collette Y. and Siarry P. (2003), “Multiobjective Optimization”, Springer, Berlin. 17. Compte A. and Metzle R.(1997), “The generalized Cattaneo equation for the description of anomalous transport processes” J. Phys. A 30, 7277. 18. Diekmann O., R. Durrett, K. P. Hadeler, P. Maini, H. L. Smith, V. Capasso (eds.) (2000), “Mathematics inspired by biology”, Springer Verlag, Berlin. 19. Dorogovtsev S.N. and Mendes J.F (2004), “The shortest path to complex networks”, Cond-mat 0404593. 20. Domany E. and Kinzel W.(1984), Equivalence of CA and Ising models and directed percolation, Phys.Rev.Lett.53, 311. 21. Edelstein-Keshet L. (1988), “Introduction to mathematical biology”, Random House, N.Y. 22. Elgazzar, A.S., “Application of the Sznajd Sociophysics Model to Small-World Networks”, Int. J. Mod. Phys. C 12, 1537 (2001). 23. Elgazzar, A.S., “A Model for the Evolution of Economic Systems in Social Networks”, Physica A 303, 543 (2002). 24. Erdös, P. and Rényi, A., “On the Evolution of Random Graphs”, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960). 25. Holmgren R. (1996), “A first course in discrete dynamical systems”, Springer, Berlin. 26. Hofbauer J. and Sigmund K. (1998), “Evolutionary games and population dynamics”, Cambridge Univ. Press, U.K. 27. Ilachinski A.(2001), “Cellular automata”, World Scientific Publ., Singapore. 28. Jazbec A., Delimar M. and Vrzic V.S. (2003), “Simulation model of Rubella”, Appl. Math. Comp. (To Appear). 29. Jen E.(1990), “Aperiodicity in 1-dimensional cellular automata”, Physica D 45, 3. 30. Kaneko K. (1993), “Theory and application of coupled map lattices,” Wiley Pub. 31. Louzoun Y., Solomon S., Atlan H. and Cohen I.R. (2003), “Proliferation and competition in discrete biological systems”, Bull. Math. Biol. 65, 375. 32. Martin O., Odlyzko A.M. and Wolfram S. (1984), “Algebraic properties of cellular automata”, Comm. Math. Phys. 93, 219. 33. Matzinger P. (2002), “The danger model”, Science 296, 12, pp. 301. 34. Milgram, S., “The Small World Problem”, Psychol. Today 2, 60 (1967). 35. Rocco A. and West B.J. (1999), “Fractional Calculus and the Evolution of Fractal Phenomena”, Physica A 265, 535. 36. Schonfisch B. and De Roos A.(1999), “Synchronous and asynchronous updating in cellular automata”, Biosys. 51, 123-143. 37. Segel L.A. and Cohen I.R.(eds.)(2001), “Design principles for the immune system and other distributed autonomous systems”, Oxford Univ. Press, U.K. 38. Smale S.(1980). “The prisoners’ dilemma and dynamical systems associated to games”, Econometrica 48, 1617-1634. 39. Smith J.B.(2003), “Complex systems”, CS 0303020. 40. Stanislavsky A.A. (2000), “Memory effects and macroscopic manifestation of randomness”, Phys. Rev. E 61, 4752. 41. Stevens J.G., Rosenzweig R.E. and Cerkanowicz A.E.(1993), “Transient and cyclic behavior of cellular automata with null boundary conditions”, J. Stat. Phys. 73, 159-174. 42. Strogatz S. (2001), “Nonlinear dynamics and chaos”, Perseus Books Group 43. Tadaki S. (1994), "Periodicity of cylindrical cellular automata, Nonlin. 9412003 44. Vynnycky E., Gay N.J. and Cutts F.T. (2003), “The predicted impact of private sector MMR vaccination on the burden of Congenital Rubella Syndrome”, Vaccine 21, 2708. 45. Watts D.J. and Strogatz S.H. (1998), “Collective dynamics of small world network”, Nature 393, 440. 46. West B.J. (1991), “Fractal Physiology and chaos in medicine”, World Scientific Publ., Singapore. Appendix (1): Some mathematical results for one-dimensional cellular automata {#appendix-1-some-mathematical-results-for-one-dimensional-cellular-automata .unnumbered} ============================================================================= Definition (3): A cellular automata consists of 4 components: A graph $G$, a set of states such that each site (vertex) of the graph has one of the possible states, a neighborhood set which assigns to each vertex a certain neighborhood and a transition function $f$ which defines the evolution of the state of each site as a function of the states of that site and those in its neighborhood.We choose the set of possible states to be the ring $Z(p)$ i.e. the set of integers $0,1,2,...,p-1$, where addition is defined $\mathrm{mod\;}p$. The total number of sites is denoted by $N$. In most of the cases, we choose $% N,\;p$ to be relatively prime. The set of states of the sites at a given time is called a configuration. We now restrict us to a one-dimensional space. Let $x(j,t)$ be the state of site $j$ at time $t$.Definition (4): A finite initial configuration is one such that there are two natural numbers $L,\;R$ such that $0<L<R<N$, and $x(j,0)=0$ if $j<L$ or $j>R$.Theorem (1) \[Jen 1990\]: If $x(i,t),\;x(j,t),\;i<j$ are two periodic sequences i.e. $x(i,t)=x(i,t+p(i)),\;x(j,t)=x(j,t+p(j))$, then for every $k$ such that $i<k<j$ then $x(k,t)$ is periodic.Corollary (1) \[Jen 1990\]: If CA evolves according to the rule $$x(i,t)=f(x(i-1,t),x(i,t),x(i+1,t))\;\mathrm{mod\;}2,$$ such that $000\rightarrow 0,100\rightarrow 0,001\rightarrow 0$, then for any finite initial configuration the system is temporarily periodic i.e. the sequence $(x(i,t))$ is periodic for all $i$ such that $0<i\leq N,0<T<t$.Proof. The fact that $100\rightarrow 0$ implies that $x(i,t)=0$ for $i>R$, similarly $x(i,t)=0$ for $i<L$ for all $t>0$. Applying theorem (1) the result is proved. In the case that $f$ in Eq. (4) is linear, one can use the methods of \[Stevens et al 1993, Tadaki 1994\] to get useful information about possible periodicity’s of the system. In this case the system can be written as $$X(t+1)=UX(t),$$ where $U$ is called the evolution matrix. Then $X(t)=U^{t}X(0)$. In this case the asymptotic behavior of the system is governed by the characteristic polynomial of $U$ on the field $Z(p)$. Assuming periodic boundary conditions, the matrix $U$ is circulant matrix \[Barnett 1990\]. Let $P(N,\lambda )$ be the characteristic polynomial of the system (5) with $% N$ sites, then typically it has the form $$P(n,\lambda )=\lambda ^{a}d(n,\lambda ),\;d(n,0)=1.$$ If $a>0$, then the systems tends to a fixed configuration (which corresponds to a fixed point for discrete time continuous state dynamical systems). Reducing $d(n,\lambda )$ to its irreducible factors on the field of states then in most cases a cycle of length $p^{k}-1$ exist for the system where $k$ is the degree of the irreducible factors. As an example consider rule 90 \[Martin et al 1984\] $$x(i,t+1)=x(i-1,t)+x(i+1,t)\;\mathrm{mod\;}2.$$ For $N=5$, we have $P(5,\lambda )=\lambda (\lambda ^{2}+\lambda +1)^{2}\;% \mathrm{mod\;}2$, hence the system may evolve to a fixed configuration (e,g, $x(i,t)=0$ for all $t>T>0$, for all $0<i\leq N$). It can also evolve to a cycle of period $3\;(=2^{2}-1)$. Similarly for $N=9$, $P(9,\lambda )=\lambda (\lambda +1)^{2}(\lambda ^{3}+l+1)^{2}$ on $Z(2)$. Hence this system may evolve to a fixed configuration or to a periodic one with period 7. For $N=13$, similar study implies that $P(13,\lambda )=\lambda (\lambda ^{6}+\lambda ^{5}+\lambda ^{4}+\lambda +1)^{2}$ on $Z(2)$. Hence fixed configurations and periodic ones with period 63 are expected. Such long periods may not be easy to find numerically. These results can be obtained using more elaborate methods \[Martin et al 1984\]; but the simplicity of the present approach is appealing. Moreover it is directly applicable to nonlocal cases which have gained much attention after the pioneering work of Watts and Strogatz on small world network (SWN) \[Watts and Strogatz 1998\]. As an example consider the following system $$x(i,t+1)=x(i-1,t)+x(i+1,t)+x(i+k,t)\;\mathrm{mod\;}2,$$ where $k$ is fixed. Some of the characteristic polynomials P(N,$\lambda $,k) are: $$\begin{aligned} P(11,0,\lambda ) &=&\lambda ^{11}+\lambda ^{10}+\lambda ^{5}+\lambda ^{4}+\lambda +1, \nonumber \\ P(11,3,\lambda ) &=&\lambda ^{11}+\lambda ^{9}+\lambda ^{7}+\lambda ^{6}+\lambda ^{5}+\lambda ^{4}+\lambda +1, \nonumber \\ P(11,1,\lambda ) &=&\lambda ^{11}+\lambda ^{8}+\lambda ^{7}+\lambda ^{5}+\lambda ^{2}+1,\end{aligned}$$ Hence we have the following proposition:Proposition (1): a) The system (8) depends on $k$.b) The asymptotic behavior of (8) contains the following: For $N=11,\;k=3$, no fixed configuration but a periodic one with period 1023.Proof. a) For $N=11,\;k=5$, a homogeneous configuration is expected. This is not the case for $N=11,\;k=0$ or $k=3$.b) Use the procedure explained before. Typically updating of CA is synchronous. It is important to notice that other types of updating e.g. a uniform random asynchronous one (where only one site is chosen randomly and updated at each time step) gives other patterns \[Schonfisch and de Roos 1999\]. The following lemma is usefulLemma (1): a) States which are stationary under synchronous updating are also stationary under asynchronous one.b) If there is a site $j$ which is not updated for all time $t>T>0$ then stationary configuration with respect to asynchronous updating may not be so under synchronous one.Proof. a) If $f(x(1),x(2),...,x(N))=(x(1),x(2),...,x(N))$ then $% f(j,x(j))=x(j)$. This proves part a). Since site $j$ is not updated for $% t>T>0$ then $f(j,x(j))\neq x(j)$ can still belong to a homogeneous configuration for the asynchronous updating but not the homogeneous one. This proves b). Loosely speaking patterns present in asynchronous updating are mostly present in synchronous one. Motivated by these results we study sequential CA e.g. the sequential rule 90 is $$x(j,t+1)=x(j-1,t+1)+x(j+1,t)\;\mathrm{mod\;}2.$$ This can be written in the following equivalent form $$x(j,t+1)=\sum_{k=2}^{j+2}x(k,t)\;\mathrm{mod\;}2,$$ where free periodic boundary conditions are assumed. The characteristic polynomials of the system (7) are: $$\begin{aligned} P(5,\lambda ) &=&\lambda ^{5}+\lambda ^{3},\;P(6,\lambda )=\lambda ^{6}+\lambda ^{5}+\lambda ^{3},\;P(7,\lambda )=\lambda ^{7}, \nonumber \\ P(13,\lambda ) &=&\lambda ^{7}(\lambda ^{3}+\lambda ^{2}+1)^{2}. \nonumber\end{aligned}$$ Hence homogeneous configurations are expected for $N=5,6,7$. For $N=13$ a periodic configuration with period 7 is expected. Studying the system (10) numerically showed that chaos (in the sense of sensitive dependence on initial conditions which is sometimes called damage spread) exists.Proposition (2): Every initially finite configuration will evolve under the CA $$x(j,t+1)=x(j-1,t+1)\;x(j+1,t)\;\mathrm{mod\;}2,$$ into the zero configuration $x(j,t)=0$ for all $j$, $0<j\leq N$, for all time $t>T>0$ where $T<N$.Proof. We have $$x(R,1)=x(R+1,0)\;x(R-1,1).$$ But $x(R+1,0)=0$ by definition of initially finite configuration thus $% x(R,1)=0$. Repeating for $x(R-1,2)$, one gets $x(R-1,2)=0$ and continue. Now the above results are applied to two known examples. The first is Domany-Kinzel (DK) model \[Kinzel and Domany 1984\], which is given by: $$\begin{aligned} \mathrm{If\;}x(j-1,t)+x(j+1,t) &=&0\;\mathrm{then\;}x(j,t+1)=0. \nonumber \\ \mathrm{If\;}x(j-1,t)+x(j+1,t) &=&1\;\mathrm{then\;}x(j,t+1)=1, \nonumber \\ \mathrm{with}\;\mathrm{probability\;}p_{1}. \\ \mathrm{If\;}x(j-1,t)+x(j+1,t) &=&2\;\mathrm{then\;}x(j,t+1)=1, \nonumber \\ \mathrm{with\;probability\;}p_{2}. \nonumber\end{aligned}$$ where $x(j,t)$ are Boolean variables. For $p_{1}\rightarrow 1$, $% p_{2}\rightarrow 1$, the system (13) corresponds to the CA $$x(j,t+1)=x(j-1,t)+x(j+1,t)+x(j-1,t)\;x(j+1,t)\;\mathrm{mod\;}2.$$ Proposition (3): Any finite initial configuration with two consecutive ones will tend to the homogeneous configuration $x(j,t)=1$ for all $0\leq j\leq N$, $0<T<t$, $T$ is sufficiently large under the CA (12). Consequently the region $p_{1}\rightarrow 1$, $p_{2}\rightarrow 1$ in the DK CA does not show chaos (damage spread).Proof. Assume that $x(j,0)=x(j+1,0)=1$. Then the system (14) implies $$x(j-1,1)=x(j,1)=x(j+1,1)=x(j+2,1)=1.$$ Continue one gets after $t$ time steps $x(k,t)=1$, where $j-t\leq k\leq j+t+1 $. This proves the first part. Now since the CA (14) will tend to $% x(j,t)=1$ for all $0\leq j\leq N$, $0<T<t$, then any change in the initial conditions that preserves the condition $x(j,0)=x(j+1,0)=1$ for some $j$ will not affect the asymptotic behavior of the CA (14). This completes the proof. The case $p_{1}\rightarrow 0$ in the DK model corresponds to the CA $$x(j,t+1)=x(j-1,t)\;x(j+1,t)\;\mathrm{mod\;}2.$$ Following similar steps as those in proposition (3) one can prove the following:Proposition (4): Any finite initial configuration with two consecutive zeros will tend to the homogeneous configuration $x(j,t)=0$ for all $0\leq j\leq N$, $0<T<t$, $T$ is sufficiently large under the CA (15). Consequently the region $p_{1}\rightarrow 1,\;p_{2}\rightarrow 1$ in the DK CA does not show chaos (damage spread) or periodic configurations.In the limit $p_{2}\rightarrow 0,\;p_{1}\rightarrow 1$, DK model corresponds to rule 90 $$x(j,t+1)=x(j-1,t)+x(j+1,t)\;\mathrm{mod\;}2,$$ which is known to be chaotic. All of the above results agree with numerical simulations. Bagnoli et al model \[Bagnoli et al 2002\] is given by $$\begin{aligned} \mathrm{If}\;x(j-1,t)+x(j,t)+x(j+1,t) &=&0\;\mathrm{then\;}x(j,t+1)=0. \nonumber \\ \mathrm{If\;}x(j-1,t)+x(j,t)+x(j+1,t) &=&1, \nonumber \\ \mathrm{then\;}x(j,t+1) &=&1\;\mathrm{with\;probability\;}p_{1}. \nonumber \\ \mathrm{If\;}x(j-1,t)+x(j,t)+x(j+1,t) &=&2, \\ \mathrm{then\;}x(j,t+1) &=&1\;\mathrm{with\;probability\;}p_{2}. \nonumber \\ \mathrm{If\;}x(j-1,t)+x(j,t)+x(j+1,t) &=&3\;\mathrm{then\;}x(j,t+1)=1. \nonumber\end{aligned}$$ where $x(j,t)$ are Boolean variables. The limit $p_{1}\rightarrow 1,\;p_{2}\rightarrow 1$ corresponds to the CA $$\begin{aligned} x(j,t+1) &=&x(j-1,t)+x(j+1,t)+x(j,t)+x(j-1,t)\;x(j+1,t)+ \nonumber \\ &&x(j,t)\;x(j+1,t)+x(j-1,t)\;x(j,t)+ \\ &&x(j-1,t)\;x(j,t)\;x(j+1,t)\;\mathrm{mod\;}2. \nonumber\end{aligned}$$ The limit $p_{1}\rightarrow 0,\;p_{2}\rightarrow 0$ corresponds to the CA $$x(j,t+1)=x(j-1,t)\;x(j,t)\;x(j+1,t)\;\mathrm{mod\;}2.$$ Proposition (5): a) Any nonzero finite initial configuration will evolve under the CA (17) into the homogeneous configuration $x(j,t)=1$ for all $0\leq j\leq N$, for $t$ is sufficiently large. Hence the limit $% p_{1}\rightarrow 1,\;p_{2}\rightarrow 1$ in Bagnoli et al model does not show chaos or periodic configurations.b) Any finite initial configuration containing at least one zero site will evolve under the CA (18) into the homogeneous configuration $x(j,t)=0$ for all $0\leq j\leq N$, for $t$ is sufficiently large. Hence the limit $% p_{1}\rightarrow 0,\;p_{2}\rightarrow 0$ in Bagnoli et al model does not show chaos or periodic configurations.Proof. similar to proposition (3). The limit $p_{1}\rightarrow 1,\;p_{2}\rightarrow 0$ corresponds to the CA $$\begin{aligned} x(j,t+1) &=&x(j-1,t)+x(j+1,t)+x(j,t)+ \nonumber \\ &&2x(j-1,t)\;x(j,t)\;x(j+1,t)\;\mathrm{mod\;}2\end{aligned}$$ which is similar to rule 150 $x(j,t+1)=x(j-1,t)+x(j+1,t)+x(j,t)$, hence chaos is expected in Bagnoli et al model in this limit. All of the above results agree with numerical simulations. It is interesting how CA unite polynomials on finite fields, circulant matrices, graph theory techniques and many other branches of mathematics into one branch which is important both mathematically and from the point of view of applications in complex systems. Appendix (2): Overview of networks in CAS {#appendix-2-overview-of-networks-in-cas .unnumbered} ========================================= Complex systems are often modeled as graphs where agents are the vertices and the interactions form the edges of the graph. Typically graphs are either regular lattices (e.g. square or cubic), random or scale free where the probability that a vertex has degree $k$ is $p(k)\approx k^{-\gamma }$. Most of the real networks are of the scale free type. Some proposed mechanisms for this fat tailed distribution \[Dorogovtsev and Mendes 2004\] are self organization (c.f. biological systems) and optimization involving many agents (c.f. economy). Random graphs were first studied by the mathematicians Erdös and Rényi \[Erdös and Rényi 1960\]. Their model consists of $N$ nodes, such that every pair of nodes is connected by a bond with probability $p$. The recent increase in computing power and the appearance of interdisciplinary sciences has lead to a better understanding of the properties of complex networks. Two main properties of complex networks are clustering and small world effect. Small-world effect means the average shortest node to node (vertex to vertex) distance is very short compared with the whole size of the system (total number of vertices). For social networks, the social psychologist Milgram \[Milgram 1967\] concluded that the average length of the path of acquaintances connecting each pair of people in the United States is six. This concept is known as the six degrees of separation. Such an effect makes it easier for an effect (e.g. an epidemic) to spread throughout the network. In a regular 1-dimensional lattice of size $N$, the average shortest path connecting any two vertices $l$ increases linearly with the system size. So regular lattices do not display small-world effect. On the other hand for a random graph, with coordination number $z$, one has $z$ first (nearest) neighbors, $z^{2}$ second neighbors and so on. This means that the total number of vertices $N=z^{l}$, this gives $$l=\frac{\mathrm{Ln}(N)}{\mathrm{Ln}(z)}.$$ The logarithmic increase with the size of the lattice allows the distance $l$ to be very short even for large $N$. Then random graphs display the small-world effect. Clustering is a common property of complex networks. It means that every vertex has a group of connected nearest neighbours (NN) (collaborators, friends), some of them will often be a connected NN to another vertex. As a measure for the clustering property, a clustering coefficient $C$ is defined as the probability that connected pairs of NN of a vertex are also connected to each others. For a random graph, $C=z/N$ which goes to zero for large $N$. So random graphs do not display clustering property. On the other hand, a fully connected regular lattice itself forms a cluster, then its cluster coefficient is equal to $1$. Complex networks display a small-world effect like random graphs, and they have large clustering coefficient as regular lattices. For a review on many real-world examples, see \[Dorogovtsev and Mendes 2004\]. A small-world network (SWN) proposed initially by Watts and Strogatz \[Watts and Strogatz 1998\] is a superposition of a regular lattice (with high clustering coefficient) and a random graph (with the small world effect). SWN satisfy the main properties of social networks. Also, the structure of SWN combines between both local and nonlocal interactions which is observed in many real systems. For example epidemic spreading show nonlocal interactions e.g SARS. The concept of SWN has been applied successfully in modelling many CAS, e.g, some games \[Ahmed and Elgazzar 2000 a\], epidemics \[Ahmed et. al. 2002\], economic systems \[Elgazzar 2002\], and opinion dynamics \[Elgazzar 2001\]. An important property related to disease spread in a network is the second moment of the degree distribution i.e. $\langle k^{2}\rangle $. If it is divergent then on average a vertex has an infinite number of second nearest neighbors thus if a single vertex is infected the disease will spread in the whole network. This explains the results that disease spread on scale free networks has zero threshold (contrary to the ODE and PDE models). However one should realize that real networks are finite hence a kind of threshold is expected. Scale-free networks \[Albert and Barabási 2002\] are another class of complex networks. A scale-free network does not have a certain scale. Some nodes have a huge number of connections to other nodes, whereas most nodes have only a few, following a power law distribution. Appendix (3): Basics of game theory {#appendix-3-basics-of-game-theory .unnumbered} =================================== Game theory \[Hofbauer and Sigmund 1998\] is the study of the ways in which strategic interactions among rational players produce outcomes (profits) with respect to the preferences of the players. Each player in a game faces a choice among two or more possible strategies. A strategy is a predetermined program of play that tells the player what actions to take in response to every possible strategy other players may use. A basic property of game theory is that one’s payoff depends on the others’ decisions as well as his. The mathematical framework of the game theory was initiated by von Neumann and Morgenstern in 1944. Also they had suggested the max-min solution for games which is calculated as follows: Consider two players A and B are playing against each other. Two strategies $S_{1}$, $S_{2}$ are allowed for both of them. This game is called two-player, two-strategy game. Assume that the constants $a,b,c$ and $d$ represent the payoffs (profits) such that, if the two players use the same strategy $S_{1}(S_{2})$, their payoff is $a(d)$. When a player with strategy $S_{1}$ plays against another one with strategy $S_{2}$, the payoff of the $S_{1}$-player is $b$ and the payoff of the $S_{2}$-player is $c$ and so on. This is summarized in the payoff matrix as follows: $$\begin{array}{ccc} & S_{1} & S_{2} \\ S_{1} & a & b \\ S_{2} & c & d \end{array} .$$ The max-min solution of von Neumann and Morgenstern is for the first player to choose max{min$(a,b)$,[min]{}$(c,d)$}. The second player chooses [min]{}{[max]{}$(a,c)$, [max]{}$(b,d)$}. If both quantities are equal then the game is stable. Otherwise use mixed strategies. A weakness of this formalism has been pointed out by Maynard Smith in the hawk-dove (HD) game whose payoff matrix is $$\Pi = \begin{array}{ccc} & \mathrm{H} & \mathrm{D} \\ \mathrm{H} & \frac{1}{2}(v-c) & v \\ \mathrm{D} & 0 & \frac{v}{2} \end{array} .$$ The max-min solution implies (for $v<c$) that the solution is D yet as he pointed out this solution is unstable since if one of the players adopts H in a population of D he will have a very large payoff which will make other players switch to H and so on till number of H is large enough that they play each other frequently and get the low payoff $(v-c)/2$. Thus the stable solution is that the fraction of hawks should be nonzero. To quantify this concept one may use the replicator equation which intuitively means that the rate of change of the fraction of players adopting strategy $i$ is proportional to the difference between their payoff and the average payoff of the population i.e. $$\frac{\mathrm{d}x_{i}}{\mathrm{d}t}=x_{i}\left[ (\Pi x)_{i}-x\Pi x\right] ,\;i=1,2,...,n,\;\sum_{i=1}^{n}x_{i}=1,$$ where $x_{i}$ is the fraction of players adopting strategy $i$, and $\Pi $ is the payoff matrix. Applying Eq. (20) to the HD game, one gets that the asymptotically stable equilibrium solution is $x=v/c$, where $x$ is the fraction of hawks in the population. For asymmetric game the replicator dynamics equation is $$\frac{\mathrm{d}x_{i}}{\mathrm{d}t}=x_{i}\left[ (\Pi _{1}y)_{i}-x\Pi _{1}y\right] ,\;\frac{\mathrm{d}y_{i}}{\mathrm{d}t}=y_{i}\left[ (\Pi _{2}x)_{i}-y\Pi _{2}x\right] ,\;i=1,2,...,n.$$ A basic drawback of normal game theory is the assumption that all players interact globally. It is more realistic to study local games \[Ahmed and Elgazzar 2000 b\] e.g. games on a lattice where players interact only with their nearest neighbors. Also there are several modifications for game formulations. Appendix (4): Unpredictability in CAS {#appendix-4-unpredictability-in-cas .unnumbered} ===================================== There are at least two sources for unpredictability in CAS. The first is that CAS are open systems hence perturbing a CAS may affect another related one e.g. the insect population affected by the perturbation of Lake Victoria. Another reason is the nonlinear interactions \[Strogatz 2000\] between the elements of the CAS. The scientific and mathematical study of Chaos Theory contains many overlaps with the study of Complex Systems, but with differences related to method: Chaos Theory can be used to study Complex Systems, but is not restricted to the study of these systems. Chaos Theory ”deals with deterministic systems whose trajectories diverge exponentially over time” (Bar Yam, NECSI website). It has been used to study Complex Systems, because these systems can be generally defined as a ”deterministic system that is difficult to predict”. On the other hand, complexity deals with systems composed of many interacting agents” The point being that Chaos Theory is one of many tools and methods that can be applied to the study of Complex Systems, but is not specifically devoted to the way these systems are designed, developed, studied, and modeled. That being stated, the famous example of the ”Butterfly Effect” in a chaotic system is an example of an agent (a butterfly) evoking a non-linear response (the storm in New England) within a Complex System (Global Weather System). A simple example of nonlinear interactions is the logistic difference equation $$x_{t+1}=rx_{t}(1-x_{t}),\;t=0,1,2,...,n,\;r>0.$$ This equation has two equilibrium solutions $x=0,\;x=1-1/r\;(r>1)$ which are asymptotically stable if $r<1$ or $1<r<3$ respectively. If $3<r<3.6$ then cycles appears and if $r>3.6$ chaos sets in. Intuitively chaos is sensitive dependence on initial conditions (for more mathematical definition see \[Holmgren 1996\]). Hence in chaotic systems one cannot make long range predictions c.f. weather. A useful measure of chaos are Lyapunov exponents $$\lambda =\frac{1}{n}\sum_{t=0}^{n-1}\mathrm{Ln}\left\| \acute{f}% (x_{t})\right\| .$$ Since CAS consists of several interacting agents one studies coupled systems e.g. coupled map lattices \[Kaneko 1993\] given by $$x_{i}^{t+1}=(1-D)f(x_{i}^{t})+\frac{D}{2}\left[ f(x_{i-1}^{t})+f(x_{i+1}^{t})\right] ,\;i=1,2,...,n.$$ The homogeneous equilibrium is given by $x=f(x)$ and it is asymptotically stable if \[Ahmed and Hegazi 2002\] $$\left\| \acute{f}(x)\left[ (1-D)+D\cos (\frac{k\pi }{n})\right] \right\| <1,\;k=0,1,...,n-1.$$ The more realistic case is to assume that the map depends on the agents e.g. $$x_{i}^{t+1}=(1-D)f_{i}(x_{i}^{t})+\frac{D}{2}\left[ f_{i-1}(x_{i-1}^{t})+f_{i+1}(x_{i+1}^{t})\right] ,\;i=1,2,...,n.$$ But analytic studies for Eq. (25) are more difficult. These systems shed some light on how to control (synchronize) some CAS \[Ahmed et al 2003\]. One may increase the coupling constant $D$. Also if the network of the agents is more connected (e.g. SWN), then the system is easier to synchronize. Finally external control can be applied preferably at highly connected sites. Appendix (5): Elements of multi-objective optimization {#appendix-5-elements-of-multi-objective-optimization .unnumbered} ====================================================== Almost every real life problem is multi-objective (MOB) \[Collette and Siarry 2003\]. Methods for MOB optimization are mostly intuitive.Definition (5): A MOB problem is: $$\mathrm{Minimize\;(min)}Z_{i}(\underline{x}),\;i=1,2,...,k,\;\mathrm{% subject\;to}\;\underline{g}(\underline{x})\leq 0,\;\underline{h}(\underline{x% })\leq 0.$$ Definition (6): A vector $\underline{x}^{}$ dominates $\underline{% \acute{x}}$ if $Z_{i}(\underline{x})\leq Z_{i}(\underline{\acute{x}})\forall i=1,2,...,k$ with strict inequality for at least one $i$, given that all constraints are satisfied for both vectors. A non-dominated solution $\underline{x}^{}$ is called Pareto optimal and the corresponding vector $Z_{i}(\underline{x}^{}),\;i=1,2,...,k$ is called efficient. The set of such solutions is called a Pareto set. Now we discuss some methods for solving MOB problems: The first method is the lexicographic method. In this method objectives are ordered according to their importance. Then a single objective problem is solved while completing the problem gradually with constraints i.e. $$\begin{array}{l} \min \;Z_{1}\;\mathrm{subject\;to} \\ \underline{g}(\underline{x})\leq 0,\;\underline{h}(\underline{x})=0 \end{array} ,$$ then if $\mathrm{ZMIN}(1)$ is the solution, the second step is $\min \;Z_{2}$ subject to $Z_{1}=\mathrm{ZMIN}(1)$, and the constraints in Eq. (26), and so on. A famous application is in university admittance where students with highest grades are allowed in any college they choose. The second best group is allowed only the remaining places and so on. This method is useful but in some cases it is not applicable.\ \ Proposition (6):* An optimal solution for the lexicographic problem is Pareto optimal.\ \ Proof. Let $\underline{x}^{}$ be the solution to the Lexicographic problem $% P_{l}$. Thus $$\underline{x}\neq \underline{x}^{},\;\mathrm{then\;}Z_{i}(\underline{x}% )=Z_{i}(\underline{x}^{}),\;i=1,2,...,l-1\;\mathrm{and\;}Z_{l}(\underline{x}% ^{})<Z_{l}(\underline{x}).$$ Thus $\underline{x}^{}$ is not dominated.\ The second method is the method of weights. Assume that it is required to minimize the objectives $Z(j),j=1,2,...,n$. (The problem of maximization is obtained via replacing $Z(j)$ by $-Z(j)$. Define $$Z=\sum_{i=1}^{k}Z_{i}w(i),\;0\leq w(i)\leq 1,\;\sum_{i=1}^{k}w(i)=1.$$ Then the problem becomes to minimize $Z$ subject to the constraints. This method is easy to implement but it has several weaknesses. The first is that it is not applicable if the feasible set is not convex. The second difficulty of this method is that it is difficult to apply for large number of objectives. However it is quite effective for multiobjective problems with discrete parameters since in this case Pareto optimal set is discrete not a continuous curve. The third method is the compromise method (sometimes called $\varepsilon -$constr-aint method $P_{\varepsilon }(k)$. In this case one minimizes only one objective while setting the other objectives as constraints e.g. minimize $% Z(k)$ subject to $Z(j)\leq a(j),\;j=2,3,...,k-1,k+1,...,n$, where $a(j)$ are parameters to be gradually decreased till no solution is found. The problem with this method is the choice of the thresholds $a(j)$. If the solution is unique, then this method is guaranteed to give a Pareto optimal solution.\ \ Proposition (7):* If the solution is unique, then the $\varepsilon -$constraint method is guaranteed to give a Pareto optimal solution.\ \ Proof. Let $\underline{x}^{}$ be the optimal solution for the $\varepsilon - $constraint method then $$\forall \;\underline{x}\neq \underline{x}^{},\;\mathrm{then\;}Z_{k}(% \underline{x}^{})<Z_{k}(\underline{x}),$$ hence $\underline{x}^{}$ is Pareto optimal. If $\underline{x}^{}$ is not unique, then it is weakly Pareto i.e. there is no $\underline{x}\neq \underline{x}^{}$ such that $Z_{i}(\underline{x}^{*})<Z_{i}(\underline{x}% )\forall \;i=1,2,...,n$.\ A fourth method using fuzzy logic is to study each objective individually and find its maximum and minimum say $\mathrm{ZMAX}(j)$, $\mathrm{ZMIN}(j)$, respectively. Then determine a membership $m(j)=(\mathrm{ZMAX}(j)-Z(j))/(% \mathrm{ZMAX}(j)-\mathrm{ZMIN}(j))$. Thus $0\leq m(j)\leq 1$. Then apply $% \max \{\min \{m(j),j=1,2,,n\}\}$. Again this method is guaranteed to give a Pareto optimal solution provided that the solution is unique otherwise it is weakly Pareto. This method is a bit difficult to apply for large number of objectives. A fifth method is Keeney-Raiffa method which uses the product of objective functions to build an equivalent single objective one. Appendix (6): Fractional calculus in CAS {#appendix-6-fractional-calculus-in-cas .unnumbered} ======================================== Recently \[Stanislavsky 2000\] it became apparent that fractional equations solve some of the above mentioned problems for the PDE approach. To see this consider the following evolution equation $$\frac{\mathrm{d}f(t)}{\mathrm{d}t}=-\lambda ^{2}\int_{0}^{t}k(t-\acute{t})f(% \acute{t})\mathrm{d}\acute{t}.$$ If the system has no memory then $k(t-\acute{t})=\delta (t-\acute{t})$ and one gets $f(t)=f_{0}\exp (-\lambda ^{2}t)$. If the system has an ideal memory, then $$k(t-\acute{t})=\left\{ \begin{array}{l} 1,\;\;\;t\geq \acute{t} \\ 0,\;\;\;t<\acute{t} \end{array} ,\right.$$ hence $f\approx f_{0}\cos (\lambda t)$. Using Laplace transform $$L[f]=\int_{0}^{\infty }f(t)\exp (-st)\mathrm{d}t,$$ one gets $L[f]=1$ if there is no memory and $L[f]=1/s$ if there is ideal memory hence the case of non-ideal memory is expected to be given by $% L[f]=1/s^{\alpha },\;0<\alpha <1$. In this case Eq. (28) becomes $$\frac{\mathrm{d}f(t)}{\mathrm{d}t}=\int_{0}^{t}\frac{(t-\acute{t})^{\alpha -1}f(\acute{t})\mathrm{d}\acute{t}}{\Gamma (\alpha )},$$ where $\Gamma (\alpha )$ is the Gamma function. This system has the following solution $$f(t)=f_{0}E_{\alpha +1}(-\lambda ^{2}t^{\alpha +1}),$$ where $E_{\alpha }(z)$ is the Mittag Leffler function given by $$E_{\alpha }(z)=\sum_{k=0}^{\infty }\frac{z^{k}}{\Gamma (\alpha k+1)}.$$ It is direct to see that $E_{1}(z)=\exp (z),\;E_{2}(z)=\cos (z)$ Following a similar procedure to study a random process with memory, one obtains the following fractional evolution equation $$\frac{\partial ^{\alpha +1}P(x,t)}{\partial t^{\alpha +1}}=\sum_{n}\frac{% (-1)^{n}}{n!}\frac{\partial ^{n}[K_{n}(x)P(x,t)]}{\partial x^{n}},\;0<\alpha <1,$$ where $P(x,t)$ is a measure of the probability to find a particle at time $t$ at position $x$. We expect that Eq. (30) will be relevant to many complex adaptive systems and to systems where fractal structures are relevant since it is argued that there is a relevance between fractals and fractional differentiation \[Rocco and West 1999\]. For the case of fractional diffusion equation the results are $$\begin{aligned} \frac{\partial ^{\alpha +1}P(x,t)}{\partial t^{\alpha +1}} &=&D\frac{% \partial ^{2}P(x,t)}{\partial x^{2}},\;P(x,0)=\delta (x),\frac{\partial P(x,0)}{\partial t}=0\;\Rightarrow \nonumber \\ P &=&\frac{1}{2\sqrt{D}t^{\beta }}M\left( \frac{\left\| x\right\| }{\sqrt{D}% t^{\beta }};\beta \right) ,\;\beta =\frac{\alpha +1}{2}, \\ M(z;\beta ) &=&\sum_{n=0}^{\infty }\frac{(-1)^{n}z^{n}}{n!\;\Gamma (-\beta n+1-\beta )}. \nonumber\end{aligned}$$ For the case of no memory $\alpha =0\Rightarrow M(z,1/2)=\exp (-z^{2}/4)$.
--- abstract: \| The ratio $R_{\eta}=\Gamma(\eta \to \pi^+\pi^-\gamma)/\Gamma(\eta \to \pi^+\pi^-\pi^0)$ has been measured by analysing 22 million $\phi \to \eta \gamma$ decays collected by the KLOE experiment at DA$\Phi$NE, corresponding to an integrated luminosity of 558 pb$^{-1}$. The $\eta \to \pi^+\pi^-\gamma$ proceeds both via the $\rho$ resonant contribution, and possibly a non-resonant direct term, connected to the box anomaly. Our result, $R_{\eta}= 0.1856\pm 0.0005_{\text {stat}} \pm 0.0028_{\text {syst}}$, points out a sizable contribution of the direct term to the total width. The di-pion invariant mass for the $\eta \to \pi^+\pi^-\gamma$ decay could be described in a model-independent approach in terms of a single free parameter, $\alpha$. The determined value of the parameter $\alpha$ is $\alpha = (1.32 \pm 0.08_{\text {stat}}$$^{+0.10} _{-0.09}$$_{\text {syst}}$$\pm 0.02_{\text {theo}})$ GeV$^{-2}$. address: - '[Dipartimento di Fisica dell’Università di Bari, Bari, Italy.]{}' - '[INFN Sezione di Bari, Bari, Italy.]{}' - 'Institute of High Energy Physics of Academica Sinica, Beijing, China.' - 'Centro Siciliano di Fisica Nucleare e Struttura della Materia, Catania, Italy.' - '[INFN Sezione di Catania, Catania, Italy.]{}' - '[Dipartimento di Fisica dell’Università della Calabria, Cosenza, Italy.]{}' - 'INFN Gruppo collegato di Cosenza, Cosenza, Italy.' - 'Institute of Physics, Jagiellonian University, Cracow, Poland.' - 'Laboratori Nazionali di Frascati dell’INFN, Frascati, Italy.' - 'Dipartimento di Fisica e Scienze della Terra dell’Università di Messina, Messina, Italy.' - 'Institute for Theoretical and Experimental Physics (ITEP), Moscow, Russia.' - '[INFN Sezione di Napoli, Napoli, Italy.]{}' - 'Dipartimento di Scienze di Base ed Applicate per l’Ingegneria dell’Università “Sapienza”, Roma, Italy.' - 'Dipartimento di Scienze e Tecnologie applicate, Università “Guglielmo Marconi", Roma, Italy.' - '[Dipartimento di Fisica dell’Università “Sapienza”, Roma, Italy.]{}' - '[INFN Sezione di Roma, Roma, Italy.]{}' - '[Dipartimento di Fisica dell’Università “Tor Vergata”, Roma, Italy.]{}' - '[INFN Sezione di Roma Tor Vergata, Roma, Italy.]{}' - '[Dipartimento di Fisica dell’Università “Roma Tre”, Roma, Italy.]{}' - '[INFN Sezione di Roma Tre, Roma, Italy.]{}' - 'Physics Department, State University of New York at Stony Brook, USA.' - 'Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden.' - 'National Centre for Nuclear Research, Warsaw, Poland.' - 'Present Address: CERN, CH-1211 Geneva 23, Switzerland.' author: - 'D. Babusci' - 'D. Badoni' - 'I. Balwierz-Pytko' - 'G. Bencivenni' - 'C. Bini' - 'C. Bloise' - 'V. Bocci' - 'F. Bossi' - 'P. Branchini' - 'A. Budano' - 'L. Caldeira Balkeståhl' - 'G. Capon' - 'F. Ceradini' - 'P. Ciambrone' - 'E. Czerwiński' - 'E. Dané' - 'E. De Lucia' - 'G. De Robertis' - 'A. De Santis' - 'P. De Simone' - 'A. Di Domenico' - 'C. Di Donato' - 'B. Di Micco' - 'R. Di Salvo' - 'D. Domenici' - 'O. Erriquez' - 'G. Fanizzi' - 'A. Fantini' - 'G. Felici' - 'S. Fiore' - 'P. Franzini' - 'P. Gauzzi' - 'G. Giardina' - 'S. Giovannella' - 'F. Gonnella' - 'E. Graziani' - 'F. Happacher' - 'B. Höistad' - 'L. Iafolla' - 'M. Jacewicz' - 'T. Johansson' - 'A. Kupsc' - 'J. Lee-Franzini' - 'B. Leverington' - 'F. Loddo' - 'S. Loffredo' - 'G. Mandaglio' - 'M. Martemianov' - 'M. Martini' - 'M. Mascolo' - 'R. Messi' - 'S. Miscetti' - 'G. Morello' - 'D. Moricciani' - 'P. Moskal' - 'F. Nguyen' - 'A. Passeri' - 'V. Patera' - 'I. Prado Longhi' - 'A. Ranieri' - 'C. F. Redmer' - 'P. Santangelo' - 'I. Sarra' - 'M. Schioppa' - 'B. Sciascia' - 'M. Silarski' - 'C. Taccini' - 'L. Tortora' - 'G. Venanzoni' - 'R. Versaci' - 'W. Wiślicki' - 'M. Wolke' - 'G. Xu' - 'J. Zdebik' title: \| Measurement of $\Gamma(\eta \to \pi^+\pi^-\gamma)/\Gamma(\eta \to \pi^+\pi^-\pi^0)$ with the KLOE Detector --- , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , $e^{+}e^{-}$ collisions ,$\eta$ decays, light mesons, chiral perturbation theory Introduction {#sec:introduction} ============ The Chiral Perturbation Theory (ChPT) provides an accurate description of interactions and decays of light mesons [@GasserLut]. The Wess-Zumino-Witten (WZW) term in the ChPT Lagrangian accounts for anomalous decays involving an odd number of pseudoscalar mesons. The triangle anomaly is responsible for the two-photon decays of the $\pi^0 / \eta / {\eta^{\prime}}$ mesons. Both triangle and box anomalies should contribute to the $\eta^{(\prime)} \to \pi^+ \pi^- \gamma$ decays. Since the kinematic region of the decays is far from the chiral limit, the amplitude of the $\pi^+\pi^-$ final state interaction has to be properly included. The decays are therefore often described by a resonant contribution due to the $\rho$-meson exchange using the Vector Meson Dominance (VMD) model, and an additional contact term (CT), whose strength is constrained by the requirement to obtain a total contribution consistent with the WZW term in the chiral limit. In the case of $\eta \to \pi^+ \pi^- \gamma$ the resonant $\rho$ contribution is sub-dominant, making the partial decay width sensitive to the CT, while for the $\eta^{\prime} \to \pi^+ \pi^- \gamma$ decay the partial width is dominated by the resonance but the direct term will influence the shape of the di-pion invariant mass distribution. The present world average of the $\eta \to \pi^+ \pi^- \gamma$ partial width, $\Gamma(\eta \to \pi^+ \pi^- \gamma)$ = (60 $\pm$ 4) eV [@PDG12], provides strong evidence of the CT in the box anomaly when compared with the values obtained with and without the direct term, (56.3 $\pm$ 1.7) eV and (100.9 $\pm$ 2.8) eV, respectively [@Ben2003]. Various approaches have been used to describe the final state interaction in these decays: the Hidden Local Symmetry (HLS) model [@Ben2003], the chiral unitary approach [@Borasoy] and the Omnes function encoding pion-pion interaction [@Venugopal]. A model-independent approach, based on a combination of ChPT and dispersion theory, has been recently proposed, where a parametrisation of the experimental pion vector form factor is used instead of VMD [@Stoll].\ Recently, CLEO [@Lopez07] has measured the ratio $R_{\eta}=\Gamma(\eta \to \pi^+ \pi^- \gamma)/\Gamma(\eta \to \pi^+ \pi^- \pi^0)= 0.175\pm 0.007_{\text {stat}} \pm 0.006_{\text {syst}}$, which differs by more than $3\sigma$ from the average of previous measurements [@Gormley; @Thaler], $R_{\eta}= 0.207\pm 0.004$ [@PDG06]. We present a new measurement of $R_{\eta}$ with smaller statistical and systematic errors, together with the fit of the $M_{\pi\pi}$ distribution according to the model-independent approach presented in [@Stoll]. The KLOE detector at DA$\Phi$NE {#sec:detector} =============================== The KLOE experiment operated at the Frascati $\phi$-factory, DA$\Phi$NE, an $e^+e^-$ collider running at a center-of-mass energy of $\sim 1020$ MeV, the mass of the $\phi$ meson. The beams collide at a crossing angle of ($\pi$ - 0.025) rad, producing $\phi$ mesons with a small momentum in the horizontal plane, $p_{\phi} = 12.5$ MeV. The detector consists of a large cylindrical Drift Chamber (DC), surrounded by a lead-scintillating fiber electromagnetic calorimeter (EMC) and a superconducting coil around the EMC providing a 0.52 T field. The DC [@DCH], 4 m in diameter and 3.3 m long, has 12,582 all-stereo tungsten sense wires and 37,746 aluminum field wires. The chamber shell is made of carbon fiber-epoxy composite with an internal wall of 1.1 mm thickness, the gas used is a 90% helium, 10% isobutane mixture. The spatial resolutions are $\sigma_{xy} \sim 150\ \mu$m and $\sigma_z \sim$ 2 mm and the momentum resolution is $\sigma(p_{\perp})/p_{\perp}\approx 0.4\%$. The EMC [@EMC] consists of a barrel and two endcaps, for a total of 88 modules, and covers 98% of the solid angle. The modules are read out at both ends by photomultipliers, both in amplitude and time. The readout granularity is $\sim$(4.4 $\times$ 4.4) cm$^2$, for a total of 2440 cells arranged in five layers. The energy deposits are obtained from the signal amplitude, while particle position along fiber direction is obtained from the arrival time difference of the signals to the photo-multipliers at the ends of each calorimeter cell. Signals of calorimeter cells close in time and space are grouped into clusters and the cluster energy $E$ is the sum of the cell energies. The cluster time $T$ and position $\vec{R}$ are energy-weighted averages. Energy and time resolutions are, $\sigma_E/E = 5.7\%/\sqrt{E\ {\rm(GeV)}}$, and $\sigma_t = 57\ {\rm ps}/\sqrt{E\ {\rm(GeV)}} \oplus100\ {\rm ps}$, respectively. The trigger [@TRG] uses both calorimeter and chamber information. Data are then analysed by an event classification filter (EVCL), that organises data in different output files, according to their particle content [@NIMOffline]. Event selection {#sec:eventselection} =============== The analysis has been performed using 558 pb$^{-1}$, collected at $\sqrt{s} \simeq 1020$ MeV, which correspond to about $22 \times 10^{6}\ \eta$-mesons produced. KLOE Monte Carlo (MC) program [@NIMOffline] is used to simulate the final states produced in $e^+e^-$ collisions, taking into account machine parameters and beam-related background on run-by-run basis. At KLOE, the $\eta$ mesons are produced together with a monochromatic recoil photon, $E_{\gamma}=363$ MeV, through the radiative decay ${\phi \rightarrow \eta \gamma}$. The final state under study is $\pi^+\pi^-\gamma\gamma$ with the main background coming from $\phi \to \pi^+\pi^-\pi^0, \pi^0 \to \gamma \gamma$. Another important background is $\eta$ decay $\phi \to \eta \gamma\to \pi^+\pi^-\pi^0 \gamma \to \pi^+ \pi^- 3\gamma$ with one undetected photon. In the MC generator the signal is simulated using a matrix element $$\|M\|^2\simeq k^2 \sin^2\theta \left(\frac{M_{\pi\pi}}{q}\right) \frac{\Gamma}{(M_{\rho}^2-M_{\pi\pi}^2)^2+M_{\rho}^2\Gamma^2}$$ where, $k$ is the photon momentum in the $\eta$ rest frame, $\theta$ is the angle between the $\pi^+$ and the photon in the di-pion rest frame, $q$ is the momentum of both pions in the di-pion rest frame and $\Gamma=$124$\cdot( q/q_0)^3$ MeV with $q_0$ being the value of $q$ at $\rho$-meson resonance [@Gormley].\ After the EVCL filter, a preselection is performed, requiring at least two tracks with opposite charge pointing to the interaction point (IP) and at least two clusters in time[^1], not associated to any track, having energy $E_{clu} \ge 10$ MeV and a polar angle in the range $(23^\circ -157^\circ)$. Tracks are sorted according to the distance of the point of closest approach from the IP. The first two tracks with opposite charge are selected as pion candidates. $\eta \to \pi^+ \pi^- \gamma$ selection {#signal} ------------------------------------------- We require that the most energetic cluster has an energy $E_{clu}>250$ MeV and we identify it as the photon ($\gamma_{\phi}$) recoiling against the $\eta$ in the $\phi \to \eta \gamma$ decay. Moreover, we ask that the $\gamma_{\phi}$ is inside the calorimeter barrel (with a polar angle in the range $55^{\circ}-125^{\circ}$), to reject events with cluster split between barrel and endcap. To reject electrons, cuts on cluster-track association and identification by time of flight (TOF) are used. These cuts reject Bhabha scattering background and other processes with electrons in the final state. We exploit the $\phi \to \eta \gamma$ decay kinematics, to evaluate the $\gamma_{\phi}$ energy: $$\vec{p}_{\phi}=\vec{p}_{\eta}+\vec{p}_{\gamma_{\phi}} \qquad E_{\gamma_{\phi}}= \frac{M_{\phi}^2-M_{\eta}^2}{2(E_{\phi}-\|\vec{p}_{\phi}\|\cos\vartheta)}$$ where, $\vartheta$ is the angle between $\gamma_{\phi}$ and the $\phi$ meson momentum, $\vec{p}_{\phi}$, measured run by run with high accuracy using Bhabha scattering events. This allows us to improve the energy measurement accuracy of the recoil photon to $0.1\%$. Using $\phi$ and $\pi$-mesons momenta, we determine the direction of the photon ($\gamma_{\eta}$) from $\eta$ decay: $$\vec{p}_{\gamma_{\eta}} = \vec{p}_{\phi}- \vec{p}_{\pi^+} - \vec{p}_{\pi^-}-\vec{p}_{\gamma_{\phi}}$$ the $\gamma_{\eta}$ photon direction is then compared with the direction of each neutral cluster: $\Delta \varphi = \varphi_{\mathrm{clu}} - \varphi_{\gamma_{\eta}}$ (here, and in the following, the angles are evaluated using variables in the transverse plane[^2]). If no clusters with $\Delta \varphi < 8.5^{\circ}$ are found, the event is rejected. The cluster with the minimum value of $\Delta\varphi$ is identified with $\gamma_{\eta}$. In order to reject the $\phi \to \pi^+ \pi^- \pi^0$ background, the angle between the two photons in the $\pi^0$ reference frame, evaluated using the $\phi$ and the $\pi$-meson momenta, is calculated and it is required to be smaller than $165^{\circ}$. The $\pi^+ \pi^- \gamma$ mass spectrum is shown in Fig. \[pppresults1\]. The candidate events are selected requiring $539.5$ MeV $< M_{\pi^+ \pi^- \gamma} < 554.5$ MeV. ![The $\pi^+ \pi^- \gamma_{\eta}$ invariant mass distribution. Crosses are experimental points, the hashed area is the MC signal ${\eta \rightarrow \pi^+ \pi^- \gamma}$, the filled region represents the total MC. Relevant background is due to $\phi\to\pi^+\pi^-\pi^0$ events and much smaller contribution from $\phi$-meson decay into kaons for higher masses as well as $\phi\to\eta\gamma$ events for the masses below the signal peak.[]{data-label="pppresults1"}](Mppg_fit2_zoom.eps){width="65.00000%"} $\eta \to \pi^+ \pi^- \pi^0$ selection {#sec:ppp} ------------------------------------------ The process $\phi \to \eta \gamma$ with $\eta \to \pi^+ \pi^- \pi^0$ represents a good control sample, having a topology similar to the signal. Moreover, in the ratio $\Gamma(\eta \to \pi^+ \pi^- \gamma)/\Gamma(\eta \to \pi^+ \pi^-\pi^0)$ the luminosity, the $\phi$ production cross section and the $BR(\phi \to \eta \gamma)$ cancel out. We use the same preselection as for the $\eta \to \pi^+ \pi^- \gamma$ signal and calculate the missing four-momentum: $$\bf{P}_{\rm miss}=\bf{P}_{\phi}-\bf{P}_{\pi^+}-\bf{P}_{\pi^-}-\bf{P}_{\gamma_{\phi}}$$ where the variables in the formula represent the four-momenta of the $\phi$ meson and of the decay products. For the $\eta \to \pi^+ \pi^- \pi^0$ sample, the missing mass peaks at the $\pi^0$ mass value and we select events with $\|M_{\rm miss}-M_{\pi^0}\|<15$ MeV. The remaining background is rejected by an angular cut applied to the two photons in the $\pi^0$ rest frame, $\varphi_{\gamma\gamma}^{3\pi}>165^{\circ}$. Figure \[figPPP\] shows the distribution of the missing mass and $\varphi_{\gamma\gamma}^{3\pi}$. ![Normalisation sample $\eta \to \pi^+ \pi^- \pi^0$. Left - $\pi^+ \pi^- \gamma_{\phi}$ missing mass distribution. Right - event distribution for the angle between prompt neutral clusters in the $\pi^0$ rest frame evaluated in the transverse plane, $\varphi_{\gamma\gamma}^{3\pi}$. Crosses are experimental points, the hashed area is the MC $\eta \to \pi^+ \pi^- \pi^0$, the filled region represents the total MC, where the only relevant background contribution is due to $\phi\to\pi^+\pi^-\pi^0$ events.[]{data-label="figPPP"}](Mmiss_fit1.eps "fig:"){width="52.00000%"}![Normalisation sample $\eta \to \pi^+ \pi^- \pi^0$. Left - $\pi^+ \pi^- \gamma_{\phi}$ missing mass distribution. Right - event distribution for the angle between prompt neutral clusters in the $\pi^0$ rest frame evaluated in the transverse plane, $\varphi_{\gamma\gamma}^{3\pi}$. Crosses are experimental points, the hashed area is the MC $\eta \to \pi^+ \pi^- \pi^0$, the filled region represents the total MC, where the only relevant background contribution is due to $\phi\to\pi^+\pi^-\pi^0$ events.[]{data-label="figPPP"}](opan_fit1_new1.eps "fig:"){width="52.00000%"} The two cuts select $N(\eta \to \pi^+\pi^-\pi^0)= 1.116 \cdot 10^3$ events. The global selection efficiency is $\varepsilon = 0.2276\pm0.0002$ with residual background contamination of $0.65\%$. Results ======= The ratio $\Gamma(\eta \to \pi^+\pi^-\gamma)/\Gamma(\eta \to \pi^+\pi^-\pi^0)$ {#sec:fit-invmass} ------------------------------------------------------------------------------- The total selection efficiency for the signal $\eta \to \pi^+\pi^-\gamma$ is $\varepsilon = 0.2131 \pm 0.0004$. In the final sample, the relative weights of signal and background components are evaluated with a fit to the $E_{\rm miss}-P_{\rm miss}$ distribution of the $\pi^{+} \pi^{-} \gamma_{\phi}$ system, with the MC shapes of the remaining background and signal MC, Fig. \[EPfit\]. Signal events are counted in the range $\|E_{\rm miss}-P_{\rm miss}\|<10$ MeV. We find $N(\eta \to \pi^+\pi^-\gamma)= 204950 \pm 497$ events, with a background contamination at level of $10\%$. The analysis has been repeated on an independent sample selected without EVCL filter to evaluate any bias due to the event classification. An overall correction factor is used to account for data/MC difference related to event classification: $K_{EVCL}= \frac{\varepsilon^{MC}_{\pi^+\pi^-\gamma} \cdot \varepsilon^{data}_{\pi^+\pi^-\pi^0} } {\varepsilon^{data}_{\pi^+\pi^-\gamma} \cdot \varepsilon^{MC}_{\pi^+\pi^-\pi^0} } = 1.010\pm0.009$. Combining the results we obtain the ratio: $$R_{\eta}=\frac{\Gamma(\eta \to \pi^+ \pi^- \gamma)}{\Gamma(\eta \to \pi^+ \pi^- \pi^0)}= 0.1856\pm0.0005_{\text {stat}} \pm 0.0028_{\text {syst}}$$ This result is in agreement with the recent CLEO measurement [@Lopez07], while improving the accuracy of a factor better than three, thus confirming a smaller value for $R_{\eta}$ with respect to previous evaluations [@Gormley; @Thaler]. ![$E_{\rm miss}-P_{\rm miss}$ distribution for the $\pi^+ \pi^- \gamma_{\phi}$. The sample has been selected applying all the cuts described in Sec.\[signal\]. The event counting is performed in the region $\|E_{\rm miss}-P_{\rm miss}\|< 10$ MeV.[]{data-label="EPfit"}](ep_new_bw1.eps){width="75.00000%"} The systematic uncertainties due to analysis cuts have been evaluated by varying the cuts and re-evaluating the value of $R_{\eta}$. Each cut is moved $\pm 2 \sigma$ with respect to the value used in the analysis, where $\sigma$ is the resolution on the reconstructed variable. The corresponding error for each source of systematics is reported in Table \[tab:systematics\]. The total error is taken as the quadratic sum of all of the contributions. Source of Uncertainty Cut Value Window cut Fractional Error on $R_{\eta}$ --------------------------------------------- -------------------- ------------------- -------------------------------- $\varphi_{\gamma\gamma}^{\pi^+\pi^-\gamma}$ $ < 165^{\circ}$ $\pm 2^{\circ}$ $\pm0.6\%$ $\Delta \varphi$ $ < 8.5^{\circ} $ $ \pm 2^{\circ}$ $\pm 0.4\%$ $\|M_{\pi^+ \pi^-\gamma}-M_{\eta}\| $ $< 7.5$ MeV $\pm 2$ MeV $\pm 0.6\%$ $E_{min}^{\gamma}$ $ > 10$ MeV $ \pm 2$ MeV $\pm 0.1\%$ $E_{clu}^{\gamma_{\phi}}$ $ > 250 $ MeV $ \pm 4$ MeV $\pm 0.1\%$ $\|M_{\rm miss} - M_{\pi^0}\|$ $ < 15 $ MeV $ \pm 4$ MeV $\pm 0.4\%$ $\varphi_{\gamma\gamma}^{3\pi}$ $ > 165^{\circ} $ $ \pm 2^{\circ}$ $\pm 0.1\%$ EVCL $\pm0.9\%$ Fit $E_{\rm miss}-P_{\rm miss}$ $\pm0.6\%$ Total $1.5\%$ : Summary table of systematic uncertainties on $R_{\eta}$.[]{data-label="tab:systematics"} Fit to di-pion invariant mass {#sec:fit-invmass} ----------------------------- The $M_{\pi\pi}$ dependence of the decay amplitude has been studied in several frameworks. The HLS model, in particular, has been investigated in [@Picciotto92] and more recently in [@Ben2003]. In this approach, the relative strength of the CT and the resonance contribution from VMD are fixed. The model-independent approach in [@Stoll], based on ChPT and dispersive analysis, does not fix this relative strength and parametrises the CT via a process-specific term. We use the last method to fit the di-pion invariant mass distribution. The function describing the partial width as a function of $s_{\pi\pi} = M_{\pi\pi}^2$ is the following: $$\frac{d\Gamma(\eta \to\pi^+\pi^-\gamma)}{d s_{\pi\pi}} = \left\|AP(s_{\pi\pi})F_V(s_{\pi\pi})\right \|^2 \Gamma_0(s_{\pi\pi}) \label{eqStoll}$$ where, $A$ is a normalisation factor and $$\Gamma_0(s_{\pi\pi})=\frac{1}{3\cdot2^{11}\cdot \pi^3 M_{\eta}^3}\left (M_{\eta}^2-s_{\pi\pi}\right )^3s_{\pi\pi} \cdot \beta_{\pi}^3$$ is the simplest gauge-invariant matrix element multiplied by the phase-space term with $\beta_{\pi}=\sqrt{1-4M_{\pi}^2/s_{\pi\pi}}$. $F_V(s_{\pi\pi})$ is the pion vector form factor, approximated in the energy range of interest by the polynomial $\|F_V(s_{\pi\pi})\|=1+(2.12\pm0.01)s_{\pi\pi}+ (2.13\pm0.01)s^2_{\pi\pi}+(13.80\pm0.14)s^3_{\pi\pi}$, where $s_{\pi\pi}$ is expressed in GeV$^{2}$ [@Stoll]. The $P(s_{\pi\pi})$ function, a process-specific part, can be treated perturbatively in the frame of ChPT, for the decay of light mesons. Taylor expansion around $s_{\pi\pi}=0$ gives $P(s_{\pi\pi})= 1+\alpha \cdot s_{\pi\pi}+\mathcal{O}(s^2_{\pi\pi})$. We fit the $M_{\pi\pi}$ distribution by minimising the variable: $$\label{chi2} \chi^2 = \Sigma_{i}^{Nbin}\frac{(N_{i}^{data}-\Sigma_{j}^{Nbin}N_{j}^{Teo}\varepsilon_{j}S_{ij})^2}{\sigma_{i}^{2}}$$ where, $N_{i}^{data}$ is the content of $i-$th bin after background subtraction, $N_{j}^{Teo}$ is the content of $j-$th bin of the expected $M_{\pi\pi}$ spectrum as from Eq.(\[eqStoll\]), $\varepsilon_{j}$ is the efficiency, $S_{ij}$ is the smearing matrix and $\sigma_i^{2} = \sigma^2_{N_i^{data}}+\sigma^2_{N^{Teo}_i}$, with $\sigma^2_{N^{Teo}_i}= \Sigma_{j}^{Nbin}(N_{j}^{Teo})^2(\sigma^2_{\varepsilon_{j}}S^2_{ij}+ \varepsilon^2_{j} \sigma^2_{S_{ij}})$. Figure \[fig:Mpp\] shows the measured distribution compared with results of the fit taking into account efficiency and smearing. ![Distribution of $M_{\pi\pi}$ after background subtraction (black markers). Histogram is the fit of Eq. (\[eqStoll\]), corrected for acceptance and experimental resolution.[]{data-label="fig:Mpp"}](Mpp_bw_5.eps){width="65.00000%"} Minimising the function in Eq. (\[chi2\]) we get $$\alpha =\left(1.32\pm0.08_{\text {stat}} \, ^{+0.10} _{-0.09} \,_{\text {syst}} \pm 0.02_{\text {theo}}\right ) {\text GeV}^{-2}$$ with $\chi^2 / Ndf =61/64$. The theoretical error $0.02$ GeV$^{-2}$ accounts for uncertainty due to vector form factor parametrisation, and is determined mainly by the accuracy of the existing $e^+e^- \to \pi^+\pi^-$ data. The fit is insensitive to the addition of a quadratic term in $P(s_{\pi\pi})$. The contributions to systematic uncertainty on $\alpha$ are listed in Table \[tab:systematics2\]. Source of Uncertainty Cut Value Window cut $ \Delta \alpha$ (GeV$^{-2}$) --------------------------------------------- ------------------- ------------------- ------------------------------- $\varphi_{\gamma\gamma}^{\pi^+\pi^-\gamma}$ $ < 165^{\circ}$ $\pm 2^{\circ}$ $+0.07/-0.03$ $\Delta \varphi$ $ < 8.5^{\circ} $ $ \pm 2^{\circ}$ $+0.05/-0.06$ $\|M_{\pi^+ \pi^-\gamma}-M_{\eta}\| $ $< 7.5$ MeV $\pm 2$ MeV $+0.04/-0.04$ $E_{min}^{\gamma}$ $ > 10$ MeV $ \pm 2$ MeV $+0.01/-0.04$ Total $+0.10/-0.9$ : Summary table of systematic uncertainties on $\alpha$ parameter.[]{data-label="tab:systematics2"} The value of $\alpha$ is in agreement with the result of the WASA Collaboration obtained from the fit to the $\gamma_{\eta}$ spectrum giving $\alpha = (1.89 \pm 0.25_{\text {stat}} \pm 0.59_{\text {syst}}\pm 0.02_{\text {theo}})$ GeV$^{-2}$ [@WASA]. Conclusions {#sec:conclusions} =========== Using a data sample corresponding to an integrated luminosity of 558 pb$^{-1}$, we select about $205000$ $\eta \to \pi^+ \pi^- \gamma$ and $1116000$ ${\eta \rightarrow \pi^+ \pi^- \pi^0}$ events from the $\phi \to \eta \gamma$ decays. We obtain the ratio of the partial widths: $$\Gamma(\eta \to \pi^+\pi^-\gamma)/\Gamma(\eta \to \pi^+\pi^-\pi^0)= 0.1856\pm0.0005_{\text {stat}} \pm 0.0028_{\text {syst}}$$ in agreement with the most recent result from CLEO [@Lopez07].\ Combining our measurement with the world average value $\Gamma(\eta \to \pi^+\pi^-\pi^0)= (295 \pm 16)$ eV [@PDG12], we find $\Gamma(\eta \to \pi^+\pi^-\gamma)=(54.7 \pm 3.1)$ eV, which is in agreement with the value expected in the HLS context including the contact-term contribution [@Ben2003].\ We have measured the di-pion invariant mass distribution and performed a fit using the model-independent approach of Ref. [@Stoll]. The fit gives $\alpha = (1.32 \pm 0.08_{\text {stat}}$$^{+0.10} _{-0.09}$$_{\text {syst}}$$\pm 0.02_{\text {theo}})$ GeV$^{-2}$. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the DA$\Phi$NE team for their efforts in maintaining low background running conditions and their collaboration during all data taking. We want to thank our technical staff: G.F. Fortugno and F. Sborzacchi for their dedication in ensuring efficient operation of the KLOE computing facilities; M. Anelli for his continuous attention to the gas system and detector safety; A. Balla, M. Gatta, G. Corradi and G. Papalino for electronics maintenance; M. Santoni, G. Paoluzzi and R. Rosellini for general detector support; C. Piscitelli for his help during major maintenance periods. This work was supported in part by the EU Integrated Infrastructure Initiative Hadron Physics Project under contract number RII3-CT- 2004-506078; by the European Commission under the 7th Framework Programme through the ‘Research Infrastructures’ action of the ‘Capacities’ Programme, Call: FP7-INFRASTRUCTURES-2008-1, Grant Agreement No. 227431; by the Polish National Science Centre through the Grants No. 0469/B/H03/2009/37, 0309/B/H03/2011/40, DEC-2011/03/N/ST2/02641, 2011/01/D/ST2/00748 and by the Foundation for Polish Science through the MPD programme and the project HOMING PLUS BIS/2011-4/3 [00]{} J. Gasser, H. Leutwyler, Nucl. Phys. B [250]{} (1985) 465. J. Beringer et al. (Particle Data Group), Phys. Rev. D [86]{} (2012) 010001. M. Benayoun et al., Eur. Phys. J. C [31]{} (2003) 525. B. Borasoy, R. Nissler, Nucl. Phys. A [740]{} (2004) 362 E.P. Venugopal and B.R. Holstein, Phys. Rev. D [57]{} (1998) 4397 F. Stollenwerk et al., Phys. Lett. B [707]{} (2012) 184. A. Lopez et al., CLEO Collaboration, Phys. Rev. Lett. [99]{} (2007) 122001. M. Gormley et al., Phys. Rev. D [2]{} (1970) 501. J. J. Thaler et al., Phys. Rev. D [7]{} (1973) 2569. W.-M. Yao et al. (Particle Data Group), J. Phys. G [33]{} (2006) 1. M. Adinolfi et al., KLOE Collaboration, Nucl. Inst. and Meth. A [488]{} (2002) 51. M. Adinolfi et al., KLOE Collaboration, Nucl. Inst. and Meth. A [482]{} (2002) 364. M. Adinolfi et al., KLOE Collaboration, Nucl. Inst. and Meth. A [492]{} (2002) 134. F. Ambrosino et al., KLOE Collaboration, Nucl. Inst. and Meth. A [534]{} (2004) 403. C. Picciotto, Phys. Rev. D [45]{} (1992) 1569. P. Adlarson et al., WASA-at-COSY Collaboration, Phys. Lett. B [707]{} (2012) 243. [^1]: We require for each cluster $\|T_{clu}-R_{clu}/c\|< 5\sigma_{T_{clu}}$, where $T_{clu}$ is the arrival time at the EMC, $R_{clu}$ is the distance of the cluster from the beam interaction point, and $c$ is the speed of light. [^2]: The azimuthal angle of the cluster is measured with an angular resolution of 6 mrad using the position of the calorimeter cell. The polar angle is instead determined by the time difference of the signals at each side of the barrel and is affected by larger uncertainty. The use of azimuthal angle reduces the systematics.
--- abstract: 'The mechanism of Stokes shift in semiconductor quantum dots is investigated by calculating the energy of the excitonic states. We have taken into account all possible contributions to the total electronic energy in the dot, i.e, dielectric mismatch between dot and surrounding medium, the effects of finite barrier height and electron-hole exchange interaction. The Stokes shift is calculated as a function of radius of dot and compared with experimental data on two different semiconductor based quantum dots. These results provide evidence for exchange splitting of excitonic states, as the mechanism of Stokes shift in quantum dot' author: - 'Anjana Bagga$^{\star}$, P. K. Chattopadhyay$^{\dag}$ and Subhasis Ghosh$^{\star} $' title: The Origin of Stokes Shift in Semiconductor Quantum Dots --- The red shift of the emission spectra with respect to absorption spectra is known as Stokes shift, which is commonly observed in semiconductor quantum dots(QDs) and the most important quantity that determines the optical properties of QDs. However the underlying mechanism of Stokes shift is still controversial. In case of semiconductor QDs, this shift decreases with the increase in radius of the dot and disappears beyond a certain radius. The red shift, principally, occurs if either the top of the valence band is an optically passive state (say P state) or if the electron and the hole are in a triplet state. Absorption of a photon from the top of the valence band in such cases is not allowed and is possible only from an optically active state lying deeper in the valence band. The exciton, once formed after absorption, can not decay to the top of the valence band by direct dipole transition and hence is called “dark exciton”[@MN95]. Deexcitation eventually takes place with the help of phonons thus giving rise to red shifted photons. Two different mechanisms[@JL00]$^,$[@ALE96] have been proposed to explain the origin of Stokes shift in semiconductor QDs, as shown in Fig. 1. Absorption takes place from the S-state which lies deeper than the P-state in the valence band to form an exciton in the singlet state of the electron and hole. According to the first mechanism, (1) of Fig. 1, deexcitation takes place into the P state with the help of phonons giving rise to a Stokes shift by an amount $\Delta E_{SP}$ representing the difference in energy between the exciton states formed with S and P hole states(Ref.2). According to the second mechanism the exciton in the singlet state first thermalizes into a triplet state from where it deexcites with the help of phonons either (a) to the deeper lying original S state, (2) of Fig. 1, giving rise to a Stokes shift by an amount $\Delta E_{ST}$ representing the singlet triplet splitting(Ref. 3), or (b) to the P state at the top of the valence band, (3) of Fig. 1, with the Stokes shift given by $\Delta E_{SP}+\Delta E_{ST}$. In addition to this, the red shift is also observed if the final excited state has a different atomic configuration than the initial ground state. Such a red shift, known as Frank Condon shift, is commonly observed in molecules and point defects in solids and may be relevant in ultra small QDs[@AF03]. In this Letter we investigate the origin of Stokes shift in III-V(InAs) and II-VI(CdSe) semiconductor based QDs by calculating the energies of excitonic states. We have taken into account the dielectric mismatch effects, the effect of finite barrier height and the effect of the electron hole exchange interaction and then compared the results with experimental data. InAs and CdSe are chosen because of their (i) different crystal structures, (ii) different crystal field effects which is zero in zinc blende structures(i.e InAs) but significantly large in wurtzite structure(CdSe), (iii) different effective masses which is small in InAs, but large for CdSe and (iv) different nature of bonds. It has been observed experimentally[@MN95]$^,$[@ALE96]$^,$[@UB97] that the red shift decreases with radius and disappears beyond a certain radius. Essentially, the strong dependence of Stokes shift on the radius of the QD, is used to investigate the origin of Stokes shift in QDs based on these semiconductors(InAs, CdSe). Optical transitions from the electron state $\psi_{e}\left(\bf{r}\right) =j_{0}(\frac{\alpha_{1}^{0}}{R}r)Y_{00}\left(\theta,\phi\right)\left\|S\alpha\right\rangle$, where $ \left\|S\alpha\right\rangle$ are the Bloch functions of the conduction band, are possible only to hole states in the valence band which have the S-state as a component because [$ \int Y_{Lm}Y_{00} d\Omega=\delta_{L,o}\delta_{m,o}$.]{} The optical transition probability from the bottom of the conduction band to a hole state in the valence band is given by [@ALE92] $P_{\alpha\beta}=\left\|\int dr r^2 j_{0}(\frac{\alpha_{1}^{0}}{R}r) \left(cR_o(r)\right)\right\|^{2} \left\|\left\langle u_{\mu}\beta\left\|e\widehat{\bf{p}}\right\|S\alpha\right\rangle\right\|^{2}$, where c is the component of the S-state $R_{o}(r)$ in the hole wavefunction, $\beta$ is the spin projection of the removed electron in the hole state and $u_{\mu}$ are the Bloch wavefunctions of the hole states at the valence band top. $P_{\alpha\beta}$ is zero if $\alpha\neq\beta$. Optical transitions from the bottom of the conduction band to the top of the valence band is not possible if either (1) the hole state does not have a component of the S-state (i.e if c=0), or (2) the spin projections of the removed electron in the hole state and the electron in the conduction band are not equal, $\alpha\neq\beta$. Since the hole spin is obtained by flipping the spin of the removed electron, it follows that optical transitions are not possible if the spins of the electron and the hole are parallel i.e if they are in a triplet state. Thus, if the top of the valence band is a P-state, or the electron and the hole are in a triplet state, a [dark exciton]{} is formed. The exciton states are obtained from the Hamiltonian $$\label{h1} H=H_{e}+H_{h}+H_{so}+V_{e-h}+V_{Pol-s}+V_{Pol-eh}$$ where $H_{e}, H_{h}, H_{so}, V_{e-h}$ are the electronic Hamiltonian, the hole Hamiltonian, the spin-orbit interaction and the e-h coulomb interaction respectively given in Ref.2 and 7. The last two terms represent the surface polarization energies arising due to the difference in the dielectric constants between the semiconductor quantum dot and the surrounding medium[@TT93]. $V_{Pol-s}$ is the self energy of the electron and hole due to their image charges and $V_{Pol-eh}$ is the mutual interaction energy between the electron and hole via image charges. The excitonic states obtained from Eq.(\[h1\]) will be split further due to exchange interaction[@ALE96] $H_{ex}$ between the electron and hole. Since the nature of the state at the top of the valence band plays a pivotal role in the formation of dark and bright excitons, we first investigate the hole states for both the zinc-blende and wurtzite structures using the [k.p]{} hole Hamiltonian $H_{h}+H_{so}$ given in Ref.9. The energy eigenvalues of the hole states without inclusion of excitonic effects as a function of dot radius for InAs(zinc-blende structures) and CdSe(wurtzite structures) are shown in Figs. 2(a) and 3(a) respectively. These eigenvalues get modified when coulomb interaction between electron in the conduction band and hole in the valence band are taken into account. The corresponding results including excitonic effects are shown in Figs. 2(b) and 3(b) respectively. From Fig. 2(a) we observe that in InAs the optically passive $P_{\frac{3}{2}}$ is the ground state which lies below the first optically active state $S_{\frac{3}{2}}$ state. The states are labeled $S_{\frac{3}{2}}$, $P_{\frac{3}{2}}$ etc where the capital letters correspond to the lowest $\bf{L}$ present and the subscripts gives the total angular momentum $\bf{F^{\prime}}$=$\bf{L}+\bf{I}+\bf{S}$, where $\bf{L}$ is the orbital angular momentum, $\bf{I}$(=1) is angular momentum of the Bloch wavefunction at the valence band top and $\bf{S}$ is the spin of the hole. The difference in energy between the optically passive $P_{\frac{3}{2}}$ state and the optically active $S_{\frac{3}{2}}$ state decreases as the radius of the dot increases and eventually $S_{\frac{3}{2}}$ becomes the ground state at around $R \sim 50$. Hence dark exciton formation and associated Stokes shift will be observed below this radius. For the wurtzite structure, the Hamiltonian is axially symmetric and only the z-component $M=L_{z}+I_{z}+S_{z}$ of the total angular momentum is conserved. From Fig.3(a) it is observed that for CdSe the optically passive $\left\|P_{x}\uparrow\right\rangle$ with $M=\frac{1}{2}$ state is the ground state at $R \sim 10$. The energy of $\left\|P_{x}\uparrow\right\rangle$ in units of $\epsilon_{o}=\frac{\gamma_{1}}{2m_{o}}\left(\frac{\hbar}{R}\right)^{2}$ ($\gamma_{1}$ is a Luttinger parameter) has an upward slope as a function of radius and crosses the relatively flat $\left\|S_{x}\uparrow\right\rangle$ state with $M=\frac{3}{2}$ at $R \sim 28$. Dark exciton formation and Stokes shift is predicted for CdSe below $R \sim 28$. The trend of S-state and P state can be explained by observing that $\left\|S_{x}\uparrow\right\rangle$ with $M=\frac{3}{2}$ is predominantly made from only one state with composition [$L=0, I_{z}=+1, S_{z}=+\frac{1}{2}$]{} whose energy decreases as $\frac{1}{R^{2}}$ but in units of $\epsilon_{o}$ is independent of R and hence the state is relatively flat. But the optically passive P state with $M=\frac{1}{2}$ can be made either from the state with composition [$L_{z}=-1, I_{z}=+1, S_{z}=+\frac{1}{2} (J_{z}=I_{z}+S_{z}=+\frac{3}{2}) $]{} or the state with the composition [$L_{z}=+1, I_{z}=-1, S_{z}=+\frac{1}{2} (J_{z}=-\frac{1}{2})$]{}. These two states are separated from each other by spin-orbit interaction. At very low radii spin orbit effects are negligible relative to kinetic energy terms due to confinement. Hence the two states are almost degenerate at $R \sim 10$. The coupling between these two states give rise to a splitting of the states with the lower P state lying below the S state. As the radius increases the energies of both the S and P states decrease. The energy of S state decreases as $\frac{1}{R^{2}}$ but the energy of the P state decreases more slowly because of the presence of the spin-orbit interaction energy which becomes important as the kinetic energy starts decreasing with the increase of radius. Eventually the two levels cross. In Figs. 2 and 3 the energies have been shown in the units of $\epsilon_{o}$. When the coulomb interaction between the electron in $1S_{e}$ state and the hole is taken into account, the energies decrease. The decrease in energy of S-state is more than that of P state, the overlap between $1S_{e}$ state and S state of the hole being greater. This causes the crossing between S and P states to occur at lower values of R as can be seen from comparison of Figs. 2(b) and 3(b) with the corresponding Figs. 2(a) and 3(a). The differences between the excitonic S and P states $ \Delta E_{SP}$ as a function of R, are shown in the insets to Figs. 4 and 5. The surface polarization charges arising due to the dielectric mismatch between the dot and the surroundings also affect the excitonic energies. The contributions of self energy $V_{Pol-s}$ and attractive energy $V_{Pol-eh}$ are opposite in sign and comparable in magnitude[@VAF02]. Hence the dielectric mismatch effects are greatly reduced with the net effect being an increase in energy depending on the amount of dielectric mismatch between the dot and the surrounding. The increase in energy for the P state is more than that for the S state resulting in a crossover to occur at a lower dot radius. This can be seen from Figs.2(b) and 3(b) which gives the comparison between zero and finite dielectric mismatch, i.e $\epsilon=1$ and $\epsilon=3$. The self energy of a charge confined to a spherical dielectric is minimum[@LEB83] at r=0. Since $\left\langle r^{2}\right\rangle$ for the P-state is larger compared to the S-state, the self energy $V_{Pol-s}$ is higher for the P-state. But the attractive energy $V_{Pol-eh}$ is higher for the S state compared to the P state. This leads to a higher net increase of polarization contribution $\left[V_{Pol-s} + V_{Pol-eh}\right]$ for the P-state as compared to the S-state and hence to a shifting of the S-P crossover to lower values of R as can be seen from Figs. 2(b)and 3(b). Another cause of dark exciton formation and the consequent Stokes shift is the splitting of the active exciton states due to electron hole exchange interaction[@ALE96] $ H_{ex}= \left(-\frac{2}{3}\right) \epsilon_{ex}\left(a_{o}\right)^{3}\delta\left(\vec{r_{e}-\vec{r_{h}}}\right)\bf{\sigma}.\bf{J}$ where $\epsilon_{ex}$ is the exchange strength constant, $a_{o}$ the lattice constant, $\bf{\sigma}$ the Pauli spin matrices representing the electron spin and $\bf{J}$ is the hole spin matrix. The exchange interaction gives rise to a splitting between triplet and singlet state of the electron and hole, with the triplet state lying lower in energy. The triplet state being passive, excitation takes place to the higher singlet state. Deexcitation can take place from the triplet state reached by thermalization, thus giving rise to a red shift of the emmision spectra with respect to the absorption spectra. In InAs the first optically active S state (Fig. 2(a)) has total anguar momentum $ \bf{F^{\prime}}$ $ =\frac{3}{2}$($\bf{L}$ = 0, $\bf{I}$=1, $\bf{S}$=$\frac{1}{2}$) and is fourfold degenerate. The $1S_{e}$ electron state in the conduction band being doubly degenerate, the exciton states are eightfold degenerate in the absence of exchange interaction. The exchange interaction splits these degenerate levels into two: the lower optically passive triplet state with $ \bf{F}$=2($\bf{F}=\bf{F^{\prime}}+\bf{S_{e}}$) and the upper optically active singlet state with $ \bf{F}$=1 (Fig. 2(c)). The difference in energies $\Delta E_{ST}$ between the singlet and triplet states as a function of dot radius is shown in Fig. 4. $\Delta E_{ST}$ decreases with R and vanishes at $R \sim 35$. The exchange interaction is enhanced with decrease in radius because of the increase in the overlap of electron and hole wavefunctions due to confinement. In CdSe (Fig. 3(c)) the ground state with $F_{z}=\pm2$ can be obtained by coupling a hole with $M=\pm\frac{3}{2}$ with the electron state with $\left(S_{e}\right)_{z}=\pm\frac{1}{2}$. Since these combinations form triplet states, the ground state with $F_{z}=\pm2$ is optically passive and forms a dark exciton. The first excited state with $F_{z}$=$\pm 1^{L}$ can be obtained either by coupling $M$=$\pm\frac{3}{2}$, $(S_{e})_{z}$=$\mp\frac{1}{2}$ or alternatively by $M$=$\pm\frac{1}{2}$ with $(S_{e})_{z}$=$\pm\frac{1}{2}$. The state $\pm1^{L}$ will be a mixture of singlet and triplet states and will be optically active. The difference in energies $\Delta E_{ST}$ between the optically passive ground state and the first optically active excited state as function of R for CdSe is shown in Fig. 5. As in the case of InAs, $\Delta E_{ST}$ for CdSe also decreases with R and vanishes around $R \sim 50$. As seen from Figs. 2, 3, 4 and 5 the curves corresponding to finite barrier case lie lower than infinite barrier case. Both the exchange interaction and coulomb interaction depend on the amount of overlap of electron and hole wavefunctions, which is reduced in the finite barrier case due to penetration of the substantial amount of electron wavefunction outside the dot resulting in the decrease of exchange energy and coulomb binding energy. Having theoretically calculated $\Delta E_{SP}$ and $\Delta E_{ST}$ we are in a position to investigate the various mechanisms for Stokes shift given in Fig. 1 by comparison with experimental data. In Figs. 4 and 5 are shown the plots of $\Delta E_{ST}$ as a function of the dot radius together with the experimentally observed Stokes shift for CdSe and InAs respectively. $\Delta E_{SP}$ as a function of R for these cases are shown in the insets of the corresponding figures. It is observed that $\Delta E_{SP}$ is much larger than the experimental red shift for InAs. For CdSe, as seen from the inset of Fig. 5 at $R \sim 10$ $\Delta E_{SP}$ is almost the same as the experimental Stokes shift but for larger values of R the experimental values are much higher. It is seen from Figs. 4 and 5 that $\Delta E_{ST}$ fits the experimental data rather well. We observed that the fit for CdSe corresponds to an infinite barrier at the dot radius but for InAs a good fit is obtained for a finite barrier height. This can be explained by observing that the effective mass of electron in InAs is very small and hence the lowest eigenvalue lies near the top of the barrier height and there is substantial probability of finding the electron outside the dot. For CdSe, the effective mass of electron is large and the lowest eigenvalue lies near the bottom of the potential well, making the infinite barrier effects predominant. In Fig. 5 we have also plotted the theoretical curve for Stokes shift obtained by Efros et al[@ALE96] in which the coulomb interaction between the electron and hole was not taken into account. We observe that better agreement with experimental Stokes shift is obtained by our theoretical curve which takes into account the coulomb interaction. From a comparision of the theoretical results with experimental data we conclude that the Stokes shift observed in QDs is caused by the splitting of the exciton states by the electron hole exchange interaction through the deexcitation mechanism (2) in Fig. 1. [99]{} M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros, and M. Rosen, Phys. Rev. Lett. [75]{}, 3728 (1995). J. Li and J. B. Xia, Phys. Rev. B. [61]{}, 15880 (2000). Al. L. Efros, M. Rosen, M. Kuno, M. Nirmal, D. J. Norris and M. G. Bawendi, Phys. Rev. B. [54]{}, 4843 (1996). U. Banin, J. C. Lee, A. A. Guzelian, A. V. Kadavanich and A. P. Alivisatos, Superlattices Microstruct. [22]{}, 559 (1997) A. Franceschetti and S. T. Pantelides, Phys. Rev. B. [68]{}, 033313 (2003). Al. L. Efros, Phys. Rev. B. [46]{}, 7448 (1992). A.Bagga, P. K. Chattopadhyay and S.Ghosh, Phys. Rev. B. (in press). T. Takagahara, Phys. Rev. B. [47]{}, 4569 (1993). A. Bagga, P. K. Chattopadhyay and S. Ghosh, Phys. Rev. B. [68]{}, 155331 (2003). V. A. Fonoberov, E. P. Pokatilov and A. A. Balandin, Phys. Rev. B. [66]{}, 085310 (2002). L. E. Brus, J. Chem. Phys [79]{}, 5566 (1983). Figure Captions {#figure-captions .unnumbered} =============== Fig.1: The schematic representation of excitation-deexcitation processes taking place at the band edges. Fig.2: The energy levels in units of $\epsilon_{0}$=$\frac{\gamma_{1}}{2m_{o}}\left(\frac{\hbar}{R}\right)^{2}$ for zinc blende InAs QDs as a function of R in the (a) absence and (b) presence of e-h coulomb interaction(without exchange). The energy reference is taken as the top of the valence band without the spin orbit effects included. Eigenvalues in the absence($\epsilon=1$) as well as presence($\epsilon=3$ of dielectric mismatch effects are also shown in (b) for both the cases of infinite and finite barrier where $\epsilon=\frac{\epsilon_{dot}}{\epsilon_{surrounding}}$. (c) eigenvalues including e-h coulomb and exchange interactions. The states are labelled with $F_{z}$ with superscripts standing for upper and lower state. The midpoint of the exciton energies with $M$=$\pm \frac{3}{2}$ and $M$=$\pm \frac{1}{2}$ is taken as the reference energy (zero of energy). Fig.3: The energy levels in units of $ \epsilon_{o} $ for wurtzite CdSe QDs as a function of dot radius R in the (a) absence and (b) presence of e-h coulomb interaction and dielectric mismatch effects(without exchange) for both the cases of infinite and finite barrier. The energy reference is taken as the top of the valence band with the spin orbit effects included. The states are labelled as $S_{x}\downarrow, P_{x}\uparrow$ etc, where capital letters represent the dominant $\bf{L}$ present and the subscripts stand for X-, Y-like or Z-like states, the arrow indicating the spin state. (c) eigenvalues including the e-h coulomb and exchange interactions. Fig.4: Singlet triplet splitting in InAs in the absence $\epsilon=1$ and presence $\epsilon=3$ of dielectric mismatch effects for both the cases of infinite and finite barrier. The inset gives the energy differences of S and P states. Experimental data(empty circle and square) are taken from Ref.4. Fig.5: The singlet triplet splitting in CdSe in the absence and presence of dielectric mismatch effects for both the cases of infinite and finite barrier. The inset gives the energy difference of S and P states with $M=\frac{3}{2}$ and $\frac{1}{2}$ respectively. Experimental data(empty circle) are taken from Ref.3.
=1 Introduction {#Intro} ============ For most physical situations, we have the comfort of a background space-time geometry – be it the space and time of the non-relativistic regime or the Minkowski space-time of special relativistic regimes or even more exotic black hole space-times. Even to have a well defined, causally well behaved and deterministic classical theory of fields, the background space-times are limited to the class of [globally hyperbolic]{} geometries. Quantum field theories can be defined in all such space-times [@WaldQFT]. However, the familiar Fock representation or the particle interpretation can be defined [uniquely]{} only for a sub-class of space-times namely those that are [stationary]{} or at least asymptotically (in future and past) stationary. This is because in relativistic theories, the Fock representation depends crucially on having a notion of positive (negative) energy solutions, i.e. a notion of preferred time [@WaldCST]. However, in regimes of early universe or evaporating black holes, we loose this comfort of having a fixed background space-time. In these cases, not only do we have to face ‘quantum gravity’ but also face the issue of quantization of other matter quantum fields [in the absence of a background space-time]{}. One strategy to construct quantum theories is to use the Gelfand–Naimark–Segal theory [@GNS]. Given a commutative $C^$ algebra, for every positive linear functional on it, there is a cyclic, unitary representation. The strategy is to look for a suitable sub-algebra of the Poisson bracket algebra of classical observables, identify its commutative sub-algebra and construct various representation. Choose among these, those on which the full sub-algebra is suitably represented. Under favourable conditions, a unique representation is picked out. A well known example of this is the unique, weakly continuous representation of the Weyl–Heisenberg $C^$ algebra (non-commutative) [@WeylAlgebra; @WeylAlgebra2]. The other example is the holonomy-flux representation of Loop Quantum Gravity (LQG) [@UniqueLQG2; @UniqueLQG]. =1 A natural commutative $C^$ algebra is constructed from suitable [configuration space variables]{}. For a linear configuration space such as $\mathbb{R}^{N}$, the coordinates themselves form suitable variables. For a more general non-linear configuration space, coordinates have only local meaning and are not suitable variables. Instead, one constructs a set of [functions]{} on the configuration space which are invariant under change of local coordinates. These will be more in number than the dimension of the configuration space (manifold) and may possibly have some relations among them. For example, for the configuration space $S^2$, we can choose 3 functions on it – namely $X^i(\theta, \phi)$, $i = 1, 2, 3$ satisfying $\sum_i (X^i)^2 = 1$ (say). The chosen class of functions is required to constitute a [separating set of functions]{} so that these functions serve to distinguish different points of the configuration space. A set of functions $\{f_i\}$ on a space $Q$ forms a separating set for $Q$ if, for every pair of distinct points, there are at-least two functions in the set which have distinct values at these points. For field theories, there is a further feature. The configuration space is heuristically described in terms of (tensor etc) fields on a manifold. The ‘delta function’ in the canonical brackets, means that the canonical coordinates are to be understood as suitably smeared fields. For example, for scalar field $\phi$, $F_f(\phi) := \int_{\Sigma}d^3x f(x) \phi(x)$, for all scalar densities of weight 1, $f$’s, defines functions on the configuration space. For a gauge field $T_iA^i_{a}dx^{a}$, the holonomies $h_{e}(A) := {\rm Pexp}(\int_e A)$, is another example of such functions. It turns out that suitable representation of the Poisson bracket algebra can also be constructed using a special class of functions on the configuration space, the so called [cylindrical functions]{}, in conjunction with projective techniques [@ALReview]. When applied to formulation of gravity in terms of the real SU(2) connection, this constructs the [kinematical Hilbert space]{} of LQG[^1]. On this Hilbert space, geometrical operators have discrete spectra and their eigenstates are associated with appropriate class of graphs. This is paraphrased by saying that quantum geometry lives on [graphs]{}. The quantum excitations of matter are thus also expected to be associated with [vertices and edges]{} of graphs. Construction of such a representation constitutes only a first step in the construction of a quantum theory of gravity. Due to the diffeomorphism invariance, the system has [gauge invariances]{} and the (Dirac) quantization procedure is split into [three]{} stages: construction of the [kinematical Hilbert space]{} followed by (distributional) solutions of the constraints followed by specification of physical Hilbert space. =1 In this review we will discuss the construction of kinematical Hilbert space. Our discussion is confined to [four]{} space-time dimensions wherein the matter fields can be [scalars, spinors and $1$-form gauge fields]{}. We will not discuss the gravitational sector (discussed in [@ALReview] for instance) except to use the real SU(2) variables – connection and densitized triad – when needed in the matter sector. Our discussion is within a canonical framework leaving out discussion of matter in a spin foam framework. Each type of matter is discussed in a separate section. Section \[Scalars\] discusses loop quantization of scalars and some of its implications. Both cases of scalar fields ranging over a compact set (Higgs scalars in the adjoint representation of a gauge group [@ThiemannScalar]) and those ranging over the full $\mathbb{R}$ [@BohrQuantized] are discussed. Section \[Fermions\] describes fermions, their Hamiltonian formulation and loop quantization while Maxwell and Yang–Mills fields are discussed in Section \[YM\]. Propagation of matter waves on quantum geometry is briefly discussed in Section \[MatterWaves\]. Finally, in Section \[Summary\] we make a few remarks. Scalars {#Scalars} ======= In loop quantum gravity approach, the scalar fields, like other matter fields, remain comparatively under-studied. Often the scalar field has been used to play a secondary role for describing dynamics of gravitational degrees of freedom. In particular, it has been used as a ‘clock’ variable in loop quantum cosmology. Nevertheless, there have been some notable efforts for studying quantum scalar fields themselves. In this section, we survey these different efforts involving quantum scalar fields. Scalar field as a ‘clock’ {#ScalarClock} ------------------------- We begin the review for scalar fields by considering the situations where scalar fields have been used as ‘clock’ variables. In these studies, mostly for cosmological scenario, scalar fields have been quantized using standard quantization. Of course, one should use loop quantization also for scalar fields, as done for gravitational degrees of freedom. In a general background the equation of motion of a massless, free* scalar field $\phi$ is $$\begin{gathered} \label{ScalarLEoMGeneral} \partial_{\mu} \big( \sqrt{-g} g^{\mu\nu} \partial_{\nu}\phi \big) = 0 ,\end{gathered}$$ where $g$ is the determinant of the metric $g_{\mu\nu}$. For cosmological backgrounds, i.e. in a spatially homogeneous space-time, the equation of motion (\[ScalarLEoMGeneral\]) reduces to $$\begin{gathered} \label{ScalarLEoMCosmo} \partial_{t} \big( \sqrt{q(t)} N^{-1} \partial_{t} \phi \big) = 0.\end{gathered}$$ Now if one makes the choice of lapse function $N$ to be $\sqrt{q(t)}$ where $q$ is the determinant of the spatial metric then the equation (\[ScalarLEoMCosmo\]) implies scalar field $\phi \propto t$. This suggests that one may choose a massless scalar field as an ‘internal clock’ to describe the dynamics of the remaining degrees of freedom. Use of ‘internal time’ to describe the dynamics, instead of the coordinate time $t$, has the advantage of avoiding the so-called ‘problem of time’ in quantum gravity. By choosing such a lapse function, the total Hamiltonian density for a cosmological system with a massless free scalar field can be written as ($N = \sqrt{q}$ chosen) $$\begin{gathered} H = H_{g} + \frac{1}{2} \pi_{\phi}^2 ,\end{gathered}$$ where $H_{g}$ represents gravitational Hamiltonian. Now imposition of Hamiltonian constraint on the Hilbert space, i.e. $\hat{H} \Psi = 0$ implies $$\begin{gathered} \partial_{\phi}^2 \Psi = 2 \hat{H}_{g} \Psi,\end{gathered}$$ where we have used $\hat{\pi}_{\phi} = -i \partial_{\phi}$. Thus, the positive and negative frequency solutions can be viewed as satisfying a set of first order evolution equations of the form $$\begin{gathered} \pm i \partial_{\phi} \Psi = \sqrt{-2 \hat{H}_{g}} \Psi =: \hat{\mathbb{H}} \Psi .\end{gathered}$$ By identifying the field $\phi$ to play the role of time and $\hat{\mathbb{H}}$ that of a proper Hamiltonian, one may note that the positive or negative frequency solutions satisfy appropriate Schrödinger equation with respect to the internal time $\phi$. This approach of considering scalar field as an internal time has been used to describe the quantum dynamics of isotropic cosmological models such as Friedmann–Robertson–Walker space-time [@APS:QuanBBAnaNum] as well as anisotropic cosmology such as Bianchi I space-time [@AW:LQCBI]. It has been shown for isotropic models that universe undergoes a bounce [@APS:Improved] when quantum dynamics is described with respect to an internal time. Compact scalars {#Compact} --------------- The Higgs scalars plays a very important role in standard model of particle physics. It provides a mechanism for generation of masses for different elementary particles. Naturally, in any approach to quantum gravity, one would expect to have an appropriate representation of the Higgs scalars in the given quantum theory. In the standard model of particle physics, while constructing the quantum framework one specifically uses the fact that the background space-time is a Minkowski space-time. A natural kinematical measure available in constructing a suitable Hilbert space in a such space-time is Gaussian measure. Use of this measure leads to the standard Fock Hilbert space. However, one cannot use Gaussian measure in a background independent construction of the Hilbert space such as those used in loop quantum gravity. This is because Gaussian measure is not diffeomorphism invariant. In order to avoid this problem, Thiemann proposes [@ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4] to use a set of bounded variables for quantizing Higgs scalars. In particular, bounded variables of the form $U(x) = e^{\phi_I(x) \tau^I}$ are used as the configuration space variables rather than the scalars $\phi_I$ themselves. Here, scalars $\phi_I$ have been made dimensionless with the help of a constant scale of $\sqrt{8\pi G}$ where $G$ is the Newton’s constant of gravitation. These variables are the analogues of the holonomies of connection variables. Unlike the gauge holonomies which are labelled by curves, these are labelled by [points]{} (no smearing) and are referred to as ‘point-holonomies’. These are valued in $\mathcal G$ since the $\phi_I(x)\tau^I$ are valued in its Lie algebra. Note that $U(x)$ is a matrix representing a [group element]{} in a certain representation determined by the generators $\tau^I$. Matrix elements of $U(x)$ therefore determine functions of the Higgs scalars. If one considers a single real scalar field, valued in a compact interval $[0, 2\pi]$ (say), then the basic variables can be chosen to be $U(x) = e^{i\phi(x)}$ which are valued in $U(1)$. For constructing the kinematical Hilbert space, one considers products of finitely many such matrix elements labelled by a finite set of points of the spatial manifold together with unitary representations of $\mathcal G$ associated with them. Finite linear combinations of these functions generate the (quantum) configuration space $\overline{\mathcal{U}}$ and the kinematical Hilbert space is then obtained as the space of square integrable functions $L_2(\overline{\mathcal{U}}, d\mu_H)$ where $d\mu_H$ is naturally available Haar measure on the group $\mathcal G$. On this, the point holonomies, but not the scalar field, are well defined (multiplicative) operators. The momentum is a density weight 1 object and its integral over three-dimensional regions are analogues of the flux variables of the gravity sector. Their action on the polynomials needs careful definition for which we refer to [@ThiemannScalar]. All other operators of interest are to be constructed using these basic operators. We may recall that the classical dynamics of a real scalar field is governed by the Hamiltonian $$\begin{gathered} \label{ScalarHamiltonian} H_{\phi} = \int d^3x N \left[ \frac{\pi_{\phi}^2}{2\sqrt{q}} + \frac{1}{2} \sqrt{q} q^{ab} \nabla_a \phi \nabla_b \phi + \sqrt{q}V(\phi) \right].\end{gathered}$$ Formally, one can construct the quantum Hamiltonian operator corresponding to the classical Hamiltonian (\[ScalarHamiltonian\]). Regularization of this quantum Hamiltonian operator is performed by using the triangulation of manifold which is analogous to the construction of gravity sector operators. While formal construction of the scalar Hamiltonian has been performed by Thiemann, understanding it’s physical consequences remains an open issue. Non-compact scalars {#Noncompact} ------------------- In Thiemann’s construction [@ThiemannScalar], the configuration space variables are essentially periodic functions of the scalar field. This is sufficient to describe the scalar fields which range over a compact interval. However, such treatment of scalars is inadequate to describe the Klein–Gordon type real-valued scalar field which takes values in $\mathbb{R}^1$ rather than in $U(1)$. To circumvent this problem, Ashtekar, Lewandowski and Sahlmann propose the use of almost periodic functions as the configuration space variables [@BohrQuantized]. The formal construction of the kinematical Hilbert space for a single non-compact scalar is carried out as follows: First one considers finite sets of points in $\mathbb{R}^3$ referred to as the vertex sets. For a given vertex set $V = \{x_1, \dots, x_n\}$, the corresponding vector space is generated by linear combinations of the functions of the form $$\begin{gathered} \psi_{V, \vec{\lambda}}(\phi) = e^{i \sum_j \lambda_j \phi(x_j)} ,\end{gathered}$$ where $\lambda_j$’s are real numbers. After considering all possible sets of vertices, the union of corresponding vector spaces and together with its completion, one construct the quantum configuration space required for loop representation. If one denotes this space as $\overline{\mathcal{A}}$ then the corresponding Hilbert space is given by $L_2(\overline{\mathcal{A}}, d\mu)$. The measure $d\mu$ is defined as follows $$\begin{gathered} \label{KGmeasure} \int_{\bar{\mathcal{A}}} d\mu \psi_{V, \vec{\lambda}} = \begin{cases} 1, & \text{if $\lambda_j = 0 \ \forall \, j$,} \\ 0, & \text{otherwise.} \end{cases}\end{gathered}$$ The elementary configuration operators are analogous of the holonomy operator for connection variables and they act by multiplications as $$\begin{gathered} \hat{h}(x,\lambda) \Psi = e^{i\lambda \phi(x)} \Psi .\end{gathered}$$ Use of the measure (\[KGmeasure\]), makes the elementary holonomy operator $h(x,\lambda) = e^{i\lambda\phi(x)}$ not to be weakly continuous in the parameter $\lambda$. This implies that one cannot construct an operator corresponding to the scalar field $\phi(x)$ itself. The conjugate variable to the holonomy which is promoted as an operator in the quantum theory, is taken to be the smeared field momentum $$\begin{gathered} P(f) = \int dx^3 \pi_{\phi}(x) f(x) ,\end{gathered}$$ where $\pi_{\phi}$ is conjugate momentum field and $f(x)$ is a test function. Their Poisson bracket is given by $$\begin{gathered} \label{KGPBSmeared} \{h(x,\lambda), P(f) \} = i\lambda f(x) h(x,\lambda).\end{gathered}$$ Subsequently, one looks for a representation of the Poisson bracket (\[KGPBSmeared\]) in the quantum theory as a commutator of the elementary operators. The non-trivial commutator bracket of the elementary operators is $$\begin{gathered} [ \hat{h}(x,\lambda), \hat{P}(f) ] = i\lambda f(x) \hat{h}(x,\lambda),\end{gathered}$$ as holonomy operators and smeared momentum operators commute with themselves. After having the kinematical framework ready, one may construct the scalar Hamiltonian operator to understand its quantum dynamics. Parametrized scalar fields {#PFT} -------------------------- Within loop approach, scalar fields have been studied in the context of parametrized field theory (PFT). In the PFT approach, scalar field theory on a flat space-time is recast in the form as if the fields reside in a generally covariant background. In the references [@LM:PFT1; @LM:PFT2], Laddha and Varadarajan have studied parametrized field theory for a free scalar field $\phi$ in $2$-dimensional flat space-time. The action for such a scalar field with respect to an inertial frame is $$\begin{gathered} \label{LVPFTAction0} S[{\phi}] = \int d^2X \eta^{AB} \partial_A \phi \partial_B \phi ,\end{gathered}$$ where metric $\eta_{AB} = {\rm diag}(-1,1)$ and $A=0,1$. Now one may choose to use an arbitrary coordinate system involving $x^a$ with $a=0,1$ such that $X^A$ can be viewed as ‘parametrized’ by $x^a$ as $X^A = X^A (x^a)$. Then the action (\[LVPFTAction0\]) can be rewritten as $$\begin{gathered} S[{\phi}] = \int d^2x \sqrt{g} g^{ab} \partial_a \phi \partial_a \phi ,\end{gathered}$$ where $g_{ab} = \eta_{ab} \partial_a X^A \partial_b X^B$ and $g$ is the absolute value of the determinant of the metric $g_{ab}$. In the parametrized field theory, the action $S[\phi]$ is treated as not only a functional of $\phi$, but also a functional of $X^A$’s. In other words the action for parametrized field theory is taken to be $$\begin{gathered} \label{LVPFTAction2} S[{\phi}, X^A] = \int d^2x \sqrt{g(X)} g^{ab}(X) \partial_a \phi \partial_a \phi .\end{gathered}$$ The variation of the action (\[LVPFTAction2\]) with respect to the field $\phi$ leads to $\partial_a (\sqrt{g} g^{ab} \phi \partial_a \phi ) = 0$. This equation is basically the same as the flat space equation of motion for $\phi$, i.e. $ \eta^{AB} \partial_A \partial_B \phi=0$ but written in $x^a$ coordinates. The variation with respect to $X^A$ leads to the equations which are automatically satisfied as long as $ \eta^{AB} \partial_A \partial_B \phi=0$. In other words, choice of the new coordinates remains undetermined as expected. For canonical formulation of this theory, it is convenient to choose the light-cone coordinates defined as $X^{\pm} \equiv X^0 \pm X^1$. One may denote the corresponding conjugate momenta by $\Pi_{\pm}$. In terms of these coordinates the action (\[LVPFTAction2\]) can be expressed as $$\begin{gathered} S = \int dt \int dx \big[ \pi_{\phi} {\dot \phi} + \Pi_+ {\dot X^+} + \Pi_- {\dot X^-} - N^+ H_+ - N^- H_- \big ] ,\end{gathered}$$ where $N^{\pm}$ are the Lagrange multipliers for the respective constraints $H_{\pm}$. As constructed, the non-trivial Poisson brackets are $$\begin{gathered} \label{LMPFTPB} \{\phi(x), \pi_{\phi}(x')\} = \delta(x,x') , \{X^{\pm}(x), \Pi_{\pm}(x')\} = \delta(x,x') .\end{gathered}$$ In the standard Fock quantization, one seeks an appropriate representation of the Poisson brackets (\[LMPFTPB\]) as commutator brackets of the operators in the quantum theory. To proceed to polymer quantization, one notes that in 1-space dimension any tensor can be viewed as a density of weight equal to its covariant rank minus its contravariant rank. Conversely, a density weight 1 object can be viewed as a 1-form. Hence $\Pi_{\pm}, Y_{\pm} := \pi_f \pm \partial_x\phi$ are all 1-forms which are integrated along intervals. Exponentials of the integrals are the holonomies ([not point holonomies]{}). The polymer quantization of the system consists of choosing holonomies of $\Pi_{\pm}$ as multiplicative operators for the $X^{\pm}$, $\Pi_{\pm}$ sector and choosing holonomies of both $Y_{\pm}$ in the matter sector as the basic variables (non-commuting). Details should be seen in the references cited above. Using this quantization it is shown that one can construct a suitable state in the polymer Hilbert space which reproduces the Fock space two points function in the large wavelength limit. However, understanding the behaviour of two-point function in the strong polymer regime remains an open issue. Polymer quantized scalars in Fourier space {#PolymerScalar} ------------------------------------------ A different study of scalar fields using loop quantum gravity techniques has done by Hossain, Husain and Seahra [@HHS:Propagator]. In their approach, polymer quantization is performed for the Fourier modes of the free scalar field in flat space-time. This has the advantage of dealing with a known Hamiltonian which is a sum of decoupled harmonic oscillators. One then computes $2$-point correlation function directly in the energy eigenfunction basis. The phase space variables for the free scalar field are the canonical pair ($\phi(\x,t), \pi(\x,t)$) satisfying the Poisson bracket $$\begin{gathered} \{\phi(t,\x), \pi(t,\y)\} = \delta^{(3)}(\x-\y).\end{gathered}$$ The Hamiltonian is $$\begin{gathered} \label{SFHamGen} H_{\phi} = \int d^3\x \left[ \frac{\pi^2}{2} + \frac{1}{2} \eta^{ab} \partial_a\phi \partial_b\phi \right],\end{gathered}$$ where the space-time metric is $ds^2 = -dt^2 + \eta_{ab}dx^adx^b$. It is convenient though not essential to put the system in a box by restricting to a finite region of flat $3$-space so that the volume $V = \int d^3x \sqrt{\eta}$ is finite. One can then perform Fourier expansion of the field into the $3$-momentum space as $$\begin{gathered} \phi(t,\x) = \frac{1}{\sqrt{V}} \sum_{\k}{\phi}_{\k}(t) e^{i {\k}\cdot{\x}}, \qquad {\phi}_\k(t) = \frac1{\sqrt{V}}\int d^3x\ e^{-i\k\cdot x} \phi(\x,t),$$ along with a similar expansion for $\pi(\x,t)$. After a suitable redefinition of the Fourier modes, one can express scalar field Hamiltonian (\[SFHamGen\]) in terms of real-valued $\phi_{\k}$ and $\pi_{\k}$ as $$\begin{gathered} H_{\phi} = \sum_{\k} H_{\k} = \sum_{\k} \left[ \frac{\pi_{\k}^2}{2} + \frac{1}{2} k^2 \phi_{\k}^2 \right],\end{gathered}$$ with the Poisson bracket being $\{\phi_\k, \pi_{\k'}\} = \delta_{\k\k'}$. Clearly, the Hamiltonian is a sum of decoupled harmonic oscillators. The polymer quantization of each mode then follows that of a quantum oscillator. In particular, the variables used in polymer quantization are $\phi_\k$ and $U_{\lambda\k }= e^{i\lambda \pi_k}$ which satisfy the Poisson bracket $\{ \phi_\k, U_{\lambda\k}\} = i \lambda U_{\lambda\k }$. The parameter $\lambda$ has dimensions of $(\text{length})^{1/2}$. As in the case of polymer quantization, $\pi_k$ itself cannot be a well-defined operator in the quantum theory as the action of $U_{\lambda\k}$ is not weakly continuous with respect to $\lambda$. To represent momentum operator, instead one uses $\pi_k^{\star} = (U_{\lambda_{\star}\k } - U_{\lambda_{\star}\k }^{\dagger})/2i\lambda_{\star}$. It may be emphasized that the polymer quantization method introduces a new length scale (which is $\lambda_{\star}$ here) in addition to Planck’s constant into the quantum theory. One can define a dimensionless parameter as $$\begin{gathered} g=\lambda_\star^2 \|\k\| \equiv \frac{\|\k\|}{M_\star} \propto \frac{\text{polymer length scale}}{\text{spatial wavelength}},\end{gathered}$$ where $M_{\star}^{-1}$ is the fundamental length scale associated with the polymer quantization of $\phi$. Clearly, $g=0$ should recover the results that one get from Fock quantization. The standard $2$-point correlation function is defined as $$\begin{gathered} \langle 0\| \hat{\phi}(\x,t) \hat{\phi}(\x',t')\|0\rangle \equiv \frac{1}{V} \sum_{\k} e^{i {\k}\cdot(\x-\x')}D_{\k}(t-t') ,\end{gathered}$$ where $\|0\rangle=\Pi_{\k}\otimes \|0_\k\rangle$ is the vacuum state and the matrix element is $$\begin{gathered} \label{pprop-def} D_{\k}(t-t') = \langle 0_{\k}\| e^{i\hat{H}_{\k}t} \hat{\phi}_{\k} e^{-i\hat{H}_{\k}t} e^{i\hat{H}_{\k}t'} \hat{\phi}_{\k} e^{-i\hat{H}_{\k}t'} \|0_{\k}\rangle,\end{gathered}$$ where $\hat{H}_\k$ is the Hamiltonian operator. The matrix element (\[pprop-def\]) can be computed using the polymer oscillator spectrum $\hat{H}_{\k}\|n_{\k}\rangle = E_n^{(\k)}\|n_{\k}\rangle$, and the expansion of the state $\hat{\phi}_{\k} \|0_{\k}\rangle$ in the energy eigenstates as $\hat{\phi}_{\k} \|0_{\k}\rangle = \sum_{n} c_n \|n_{\k}\rangle$. By defining $4$-momentum $p \equiv(\omega,\k)$, one can write the momentum space propagator as $$\begin{gathered} \label{KGPropagatorMomentum} D_{p} = \sum_{n} \frac{2i\Delta E_n \|c_n\|^2 }{p^2 + \Delta E_n^2 - \|\k\|^2 - i\epsilon},\end{gathered}$$ where $p^2=-\omega^2 +\|\k\|^2$ and choice of the sign of $i\epsilon$ corresponds to the Feynman propagator. One can recover the expected Fock space result $D_{p} = i/(p^2 - i\epsilon)$ by using properties of the Schrödinger oscillator, i.e. $c_n = \delta_{1,n}/\sqrt{2\|\k\|}$ and $\Delta E_n = n \|\k\|$. The propagator (\[KGPropagatorMomentum\]) with polymer corrections in the infrared limit ($g\ll 1$) can be written as $$\begin{gathered} D_p = \frac{i(1-2g) }{p^2 - g\|\k\|^2 - i\epsilon } + {\cal O}\big(g^2\big).\end{gathered}$$ The pole of the propagator implies an effective dispersion relation $$\begin{gathered} \omega^2 =\|\k\|^2(1 -\|\k\|/M_\star),\end{gathered}$$ which violates Lorentz symmetry. On the other hand in the ultraviolet limit ($g \gg 1$), the propagator with leading order corrections is $$\begin{gathered} D_p = \frac{i/8g^2}{p^2 + 4 g^2\|\k\|^2-i\epsilon} +{\cal O}\left(\frac{1}{g^6}\right).\end{gathered}$$ The corresponding dispersion relation $$\begin{gathered} \omega^2 = 4\|\k\|^4/M_\star^2\end{gathered}$$ also violates Lorentz invariance. One may also note that the propagation amplitude at high momentum is suppressed by a factor $1/g^2$. Polymer quantization introduces a length scale $\lambda^2 = M_^{-1}$. This is unavoidable. Such a scale could be provided by an underlying quantum geometry. However at the present level of understanding, it is to be treated as an arbitrary parameter. Current observational limits on Lorentz violation may be then used to constrain its value. Other studies of scalar fields {#ScalarApp} ------------------------------ Ultraviolet behaviour of polymer quantized scalar fields has been studied by Husain and Kreienbuehl by constructing Fock-like states in flat space-time [@Husain:UVBehavior]. Instead of Fourier space, as discussed in section (\[PolymerScalar\]), here one applies polymer quantization in real space. It is shown that the vacuum expectation values of the commutator and anti-commutator of the creation and annihilation operators become energy dependent, and tends to show some sort of fermionic behavior at high energy. Furthermore, the modified dispersion relation that arises leads to violation of Lorentz invariance. In the context of spherically symmetry, Gambini, Pullin, and Rastgoo have studied polymer quantization of scalar field [@Gambini:2011mw] coupled to gravity. Their study is performed in the midi-superspace context where even after imposition of the given symmetry, field theoretical nature of scalar fields, i.e. with infinitely many degrees of freedom, is retained. After choosing a quantum geometry state corresponding to locally flat geometry, quantization of the spherically symmetric scalar field is carried out using a discretization followed by a ‘polymerization’ done in two different ways (the field and the momentum). Their notable conclusion is that the propagator obtained is not Lorentz invariant. Polymer quantized free massless scalar field in a homogeneous and isotropic cosmological space-time has been studied by Hossain, Husain and Seahra [@HHS:CosmoPM]. Use of semi-classical Friedman equation yields a non-singular and non-bouncing universe, without quantum gravity. The system exhibits an early de Sitter-like inflationary phase with sufficient expansion to resolve the horizon and entropy problems, and a built in mechanism for a graceful exit from inflation. The propagation of polymer quantized scalar field in flat space-times has been studied in [@HHS:WaveProp]. Polymer quantization is performed in real space there unlike in Fourier space as discussed in section (\[PolymerScalar\]). Using semi-classical states, it has been shown that effective wave equation is both nonlinear and Lorentz invariance violating. Furthermore, it is demonstrated that polymer effects tend to accumulate with time for plane-symmetric waveforms. Possibility of measuring deviations from the Klein–Gordon equation in particle accelerators or astrophysical observations is also discussed. Fermions {#Fermions} ======== Fermions coupled to gravity has been discussed in the literature [@FermionHistory7; @FermionHistory2; @FermionHistory3; @FermionHistory4; @FermionHistory5; @FermionHistory6; @FermionHistory], at both classical and quantum level. In the formulation of general relativity in terms of real SU(2) connection, Thiemann discussed loop quantization of standard model fields [@ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4]. The fermions were treated in the second order form, i.e. fermions couple to gravity through the spin connection (torsion free Lorentz connection). Perez and Rovelli returned to fermions in presence of the Holst term and found that the Barbero–Immirzi parameter, $\gamma$, inverse of the coefficient of the Holst term, becomes classically observable [@PerezRovelli]. Mercuri [@Mercuri; @Mercuri2] discovered that with a further addition of suitable non-minimal fermionic couplings, $\gamma$ can be made classically unobservable. He also noted that the added terms (Holst plus non-minimal) can be expressed as the Nieh–Yan topological term after using the connection equation of motion. This strategy of adding non-minimal couplings to keep $\gamma$ classically unobservable was followed for $N = 1, 2$ and 4 supergravities also [@Kaul]. Canonical analysis and loop quantization of fermions with non-minimal couplings was discussed by Bojowald and Das [@BojowaldDas; @BojowaldDas2]. It was subsequently realised that $\gamma$ will automatically be classically unobservable provided it is the (inverse) of the coefficient of the Nieh–Yan term (a total divergence) in the Lagrangian density. Thus, instead of the Holst terms alone, if the Nieh–Yan term (Holst + (torsion)$^2$ piece) is used in conjunction with the Hilbert–Palatini, then for [arbitrary]{} matter and their couplings, the $\gamma$ will drop out of the classical equations of motion[^2]. But now, since the action is modified, it was not obvious that the real SU(2) formulation will result from the new action. It turned out that it is possible to systematically derive the real SU(2) Hamiltonian formulation from such an action [@DateKaulSengupta; @KaulSenguptaSugra]. Since it is in presence of fermions that non-trivial torsion results from the equation of motion of the Lorentz connection, fermions were also included in the canonical analysis and real SU(2) formulation was seen to emerge. The canonical analysis leading to real SU(2) formulation has since been extended to include the other two topological terms namely the Pontryagin and the Euler classes [@KaulSenguptaTop; @PerezRezende]. It is straight forward to derive the real SU(2) formulation from the Hilbert–Palatini action with $\gamma^{-1}$ times the Nieh–Yan term [@DateNew; @DateKaulSengupta]. Its main points may be summarised as follows. In the [time gauge parametrization]{}[^3], to begin with there are 13 components of the co-tetrad ($N, N^a, V_a^i$) and 24 components of the Lorentz connection ($\omega_t^{IJ}, \omega_a^{IJ}$). Of these, 10 variables – $N$, $N^a$, $\omega_t^{IJ}$ – occur as Lagrange multipliers and 18 variables – $E^a_i \sim V^a_i$ (densitized triad) and 9 combinations $A^i_a$ of $\omega_a^{IJ}$ – explicitly appear in the form $p\dot{q}$. The coefficients of the Lagrange multipliers form the primary constraints ${\cal H}_a$, ${\cal H}$, ${\cal G}^{0i}$ and ${\cal G}^{ij}$. The remaining 9 combinations of the Lorentz connections, have no velocities and lead to additional 9 primary constraints, say $\pi_i \approx 0 \approx \pi^{ij}$ (symmetric in $i$, $j$). Of these, the 3 primary constraints, $\pi_i \approx 0$ form a second class system with the primary constraints ${\cal G}^{0i}$ and are eliminated easily using Dirac brackets. Preservation of the remaining 6 primary constraints $\pi^{ij} \approx 0$ lead to further 6 secondary constraints $S_{ij} \approx 0$, with which they form a second class system. These are again eliminated using Dirac brackets. The left over system has the 18 phase space variables, ($A^i_a, E^a_i$), and 7 first class constraints and is the real SU(2) connection formulation. This counting and the steps in the constraint analysis remain the same when matter is included. This derivation also introduces factors of ${\mathrm{sgn}(e)}:= {\rm sign}(\det(e^I_{\mu}$)) $(= N\,{\rm sign}(\det(V^i_a))$ in the time gauge parametrization) in appropriate places. For instance, we are naturally lead to the definitions: $E^a_i := {\mathrm{sgn}(e)}\sqrt{q}V^a_i $ and $ A^i_a := \gamma^{-1} {\mathrm{sgn}(e)}\, K^i_a - \Gamma^i_a$ Under an [improper]{} orthogonal transformation $\Lambda^i_{~j} \in {\rm O}(3)$ acting on the index $i$, the triad changes its handedness and the ${\mathrm{sgn}(e)}$ factor changes sign leaving the handedness of $E^a_i$ unchanged. This is as it should be since the index $i$ on $E$ represents adjoint representation of SO(3) while on $V$ it represents the defining representation. For SO(3), both are equivalent but [not for]{} O(3). Under an [inversion]{}, $\Lambda^i_{~j} = - \delta^i_{~j}$, quantities in the defining representation change sign while those in the adjoint don’t. The same reasoning applies to the definition of the connection. Now the connection is also even under inversion. The ${\mathrm{sgn}(e)}$ factor also change the behaviour of $E^a_i$ and $A^i_a$ under the action of [orientation reversing diffeomorphisms]{}. These factors are relevant for discussion of ‘parity’ properties of the canonical formulation (discussed later). This is independent of matter couplings. When a Dirac fermion is included, the solution of the secondary, second class constraint $S_{ij} \approx 0$ changes since fermions couple to Lorentz connection apart from coupling to the triad. This solution leads to non-trivial Dirac brackets between the SU(2) connection and the fermions. One can however make natural shifts in the definition of the connection to recover the canonical brackets. This also simplifies the constraints. Four Fermi interaction terms however survive in the Hamiltonian and are signatures of first order formulation. In the second order formulation where fermions couple to the torsion free connection, there are no terms quartic in the fermions. Fermions are also tied with possible parity violations [@BojowaldDas; @BojowaldDas2; @Freidel]. There are two distinct notions of ‘parity’: one related to orientation of the space-time manifold ([parity]{}) and one related to the improper Lorentz transformation ([Lorentz parity]{}). Depending upon the definitions of the basic canonical variables (with or without the ${\mathrm{sgn}(e)}$ factors in this work (Section \[Parities\])), the canonical framework and the action are (non-)invariant under [one]{} of the notions of parity. These possibilities are distinguished and discussed in [@DateNew]. Bojowald and Das discuss the non-invariance under [Lorentz parity]{} [@BojowaldDas; @BojowaldDas2] in the context of non-minimally coupled fermion. We now turn to the classical, canonical form of a Dirac fermion minimally coupled to gravity in the first order formulation and discuss the loop quantization of fermions in the following sub-section. The Hamiltonian formulation {#CanonicalFormulation} --------------------------- =1 The starting point is a choice of (tensor/spinor) fields and a corresponding generally covariant, local action on 4-dimensional space-time $M \simeq \mathbb{R} \times \Sigma_3$. The next step is to carry out a 3+1 decomposition to identify the Lagrangian which is a function of (tensor) fields on $\Sigma_3$ together with their velocities with respect to the chosen time coordinate. The fields whose velocities appear in the Lagrangian are [potentially]{} the configuration space variables while those without velocities appearing in the Lagrangian are [Lagrange multipliers]{} whose coefficients will be [primary constraints]{}. This Lagrangian leads to the [kinematical phase space]{}. Now a constraint analysis a la Dirac is performed. If there are second class constraints, one may hope to simplify the analysis by solving the second class constraints. However, now one must use the Dirac brackets. These may not have the canonical form for the remaining variables (i.e. may not be Darboux coordinates) and a new choice of variables may be necessary. This is particularly relevant for Lagrangians which are [linear* ]{} in the velocities eg the Hilbert–Palatini–Nieh–Yan and the Dirac Lagrangians which typically do have second class constraints. The classical Hamiltonian formulation is completed when the action is expressed in the Hamiltonian form together with first class constraints. We also have fields coordinatizing the kinematical phase space (after the second class constraints are eliminated) with the configuration space coordinates identified. We begin with the Lagrangian 4-forms built from the basic fields the co-tetrad $e^I_{\mu}dx^{\mu}$ and the Lorentz connection $\omega^{IJ}_{\mu}dx^{\mu}$ $(\kappa := 8\pi G)$ $$\begin{gathered} {\cal L}_{\mathrm{HP}}(e, \omega) = \frac{1}{2\kappa}\left[ {\mathrm{sgn}(e)}\frac{1}{2}{\cal E}_{IJKL}R^{IJ}(\omega)\wedge e^K \wedge e^L\right], \nonumber\\ {\cal L}_{\mathrm{NY}}(e, \omega) = \left[T^I(e, \omega)\wedge T_I(e, \omega) - R_{IJ}(\omega)\wedge e^I\wedge e^J\right], \nonumber\\ {\cal L}_{\mathrm{grav}} := {\cal L}_{\mathrm{HP}} + \frac{\eta}{2\kappa}{\cal L}_{\mathrm{NY}} , \nonumber\\ {\cal L}_{\mathrm{Dirac}} = - \frac{i}{2} \|e\| \left[ \bar{\lambda} e^{\mu}_I \gamma^I D_{\mu}(\omega, A, \ldots) \lambda - {\overline{D_{\mu}(\omega, A, \ldots) \lambda}} e^{\mu}_I \gamma^I \lambda \right], \\ D_{\mu}(\omega)\lambda := \partial_{\mu} \lambda + \frac{1}{2}\omega_{\mu}^{\ IJ} \sigma_{IJ}\lambda + i e'A_{\mu}\lambda + \dots+ \lambda, \nonumber \\ \overline{D_{\mu}(\omega)\lambda} := \left\{\partial_{\mu} \lambda^{\dagger} + \frac{1}{2}\omega_{\mu}^{\ IJ} \lambda^{\dagger}\sigma^{\dagger}_{IJ} - i e' A_{\mu} \lambda^{\dagger} + \dots + \lambda^{\dagger} \right\}\gamma^0.\nonumber\end{gathered}$$ The $\ldots$ refer to possible couplings of the Dirac fermion to other gauge fields, e.g. the Maxwell field. These are suppressed in the following[^4]. Here $R := d\omega + \omega\wedge\omega$ and $T := de + \omega\wedge e$ are the usual curvature and torsion 2-forms. The factor of sgn($e$) is present because only then the Hilbert–Palatini Lagrangian matches with the $\sqrt{\|g\|} R(g)$. This arises from noting that determinant of the co-tetrad is given by, $e = {\mathrm{sgn}(e)}\sqrt{\|g\|}$. A 3+1 decomposition is carried out as usual by choosing a foliation defined by [a time function]{}, ${\cal T}:M \to \mathbb{R}$ and a vector field $t^{\mu}\partial_{\mu}$, transversal to its leaves. The vector field is normalised by $t\cdot\partial {\cal T} = 1$ so that the parameters of its integral curves, serve as the time coordinate. Given such a decomposition, we choose a parametrization of the co-tetrad which leads to the usual ADM parametrization in terms of the metric, $g_{\mu\nu} := e^I_{\mu} e^J_{\nu}\eta_{IJ}$. Both the co-tetrad and the corresponding tetrad in this parametrization are displayed below $$\begin{gathered} e^I_{t} = Nn^I + N^aV_a^I ,\qquad e^I_{a} = V^I_{a} , \qquad n^In_I = -1 , \qquad n^IV^J_{a} \eta_{IJ} = 0, \label{CoTetrad} \\ e_I^{t} = - N^{-1} n_I , \qquad e^{a}_I = N^{-1} n_I N^{a} + V_I^{a} ,\qquad n^I V_I^a = 0 , \nonumber \\ \mathrm{with} \quad V^a_I V^J_a = \delta_I^J + n_I n^J , \qquad V^a_I V^I_b = \delta^a_b . \label{Tetrad}\end{gathered}$$ We will now restrict to configurations such that $n_i = 0$, $n_0 = -1$. This also implies that $V_{a}^0 = 0 = V_0^{a}$ and that $V^a_i$ are invertible with $V_a^i$ as the inverse[^5]. We also define the 3-metric $q_{ab} := V_a^i V_b^j \delta_{ij}$ (which is positive definite in classical theory) and denote $q := \det(q_{ab})$. For future convenience we introduce $\Psi := q^{1/4}\lambda$, $\Psi^{\dagger} := q^{1/4}\lambda^{\dagger}$. This absorbs away the $\sqrt{q}$ factors in the Lagrangian as well as in the constraints. Note that the terms involving the derivatives of $\sqrt{q}$ cancel out. The $\lambda$ fermionic variables being of density weight zero, the $\Psi$ fermionic variables are of density weight 1/2. From now on we will use the half density variables. Substituting the 3+1 parametrization of the tetrad and using the time-gauge, the Lagrangian can be written as $$\begin{gathered} {\cal L}_{\mathrm{Dirac}} = \frac{i}{2} ( \Psi^{\dagger} \partial_t \Psi - \partial_t (\Psi^{\dagger}) \Psi) - \omega_{t0i} {\cal G}_F^{0i} - \frac{1}{2} \omega_{tij} {\cal E}^{ijk}{\cal G}_{k}^F - N^{a'} {\cal H}^F_{a'} - N {\cal H}_F,\end{gathered}$$ where $$\begin{gathered} {\cal G}^{0i}_F = 0 ,\qquad {\cal G}^F_i = - \frac{i}{2} {\cal E}_{ijk} \Psi^{\dagger}\sigma^{jk} \Psi = \Psi^{\dagger}\gamma_5\sigma_{0i}\Psi, \\ {\cal H}^F_{a'} = - \frac{i}{2}\left( \bar{\Psi}\gamma^0 D_{a'} \Psi - \overline{D_{a'} \Psi}\gamma^0 \Psi \right) ,\qquad {\cal H}_F = \frac{i}{2} V^a_i \left( \bar{\Psi}\gamma^i D_a \Psi - \overline{D_a \Psi}\gamma^i \Psi \right).\end{gathered}$$ For the fermions, the action being linear in velocities, we have primary constraints, $\pi_{\lambda} \sim \lambda^{\dagger}$, $\pi_{\lambda^{\dagger}} \sim \lambda$, which are second class. Also these variables fail to be Darboux coordinates – do not have vanishing Poisson brackets[^6] with the gravitational variables due to the $\sqrt{q}$ factor. The shift to $\Psi$, $\Psi^{\dagger}$ variables makes the matter and gravitational variables Poisson-commute. Defining Dirac brackets relative to these primary, second class constraints allows us to use $\Psi$, $\Psi^{\dagger}$ as basic variables with Dirac brackets given by $$\begin{gathered} \big\{ \Psi^{\alpha}(x), \Psi^{\dagger}_{\beta}(y) \big\} = - i\delta^{\alpha}_{\beta} \delta^3(x, y). $$ From the details given in [@DateNew], we have the final expressions: $$\begin{gathered} \label{FinalDefns} P^a_i := (\kappa\gamma)^{-1} E^a_i = (\kappa\gamma)^{-1}{\mathrm{sgn}(e)}V^a_i\sqrt{q},\qquad A^i_a = \gamma\,{\mathrm{sgn}(e)}K_a^i - \Gamma^i_a(V), \nonumber\\ \Gamma^i_a (V) := \frac{{\cal E}^{ijk}}{2}V_k^b\big\{\partial_b V_{aj} - \partial_a V_{bj} + V^c_j V^l_a \partial_b V_{cl}\big\}, \nonumber\\ \{ A^i_a(x) , P^b_j(y) \} = \delta^b_a\delta^i_j \delta^3(x, y) ,\qquad \{\Psi^{\alpha}(x) , \Psi^{\dagger}_{\beta}(y) \}_+ = - i \delta^{\alpha}_{\beta}\delta^3(x,y), \nonumber\\ {\cal G}_i = \partial_a P^a_i + {\cal E}_{ij}^{~k} A^j_a P^a_k - \frac{i}{2}{\cal E}_{ijk}\Psi^{\dagger}\sigma_{jk}\Psi ,\qquad \sigma_{jk} := \frac{1}{4}[\gamma_j, \gamma_k], \nonumber\\ {\cal H}_a = F^i_{ab}P^b_i - \frac{i}{2}\left(\bar{\Psi}\gamma^0 {\cal D}_a \Psi - \overline{{\cal D}_a\Psi} \gamma^0 \Psi\right) ,\qquad {\cal D}_a \Psi := \left(\partial_a -i A^i_a \gamma_5\sigma_{0i}\right) \Psi, \nonumber\\ {\cal H} := \kappa\gamma^2 \frac{1}{2}\frac{P^b_j P^c_k}{\sqrt{q}}\big\{ {\cal E}^{jk}_{~~l}F^l_{bc}({A}) - (1 + \gamma^2) \big( {K}^j_b {K}^k_c - {K}^j_c {K}^k_b\big) \big\} + \gamma\partial_a\left({\mathrm{sgn}(e)}V^a_i{\cal G}^i_{\rm vac}\right) \nonumber \\ \phantom{{\cal H} :=}{} + \frac{i}{2}\frac{\kappa\gamma\, {\mathrm{sgn}(e)}P^a_i}{\sqrt{q}}\left\{ \bar{\Psi}\gamma^i{\cal D}_{a}\Psi - \overline{{\cal D}_{a}\Psi}\gamma^i\Psi \right\}({A}) + \left[ \left(\frac{\kappa\gamma^2}{2 \sqrt{q}}{K}^i_aP^a_i\right) \big(\Psi^{\dagger}\gamma_5\Psi\big)\right] \nonumber \\ \phantom{{\cal H} :=}{} - \left[ \frac{3}{16}\kappa\frac{\big(\Psi^{\dagger}\gamma_5\Psi\big)^2}{\sqrt{q}} -~\kappa\frac{(\Psi^{\dagger}\gamma_5\sigma^{0i}\Psi)(\Psi^{\dagger}\gamma_5\sigma^{0}_{\ i}\Psi)}{\sqrt{q}} \right]. \label{FinalHam}\end{gathered}$$ In the above, ${\cal G}_i^{\rm vac} = \partial_aP^a_i + {\cal E}_{ij}^{~k}A^j_a P^a_k := {\cal G}^i - {\cal G}^i_F$. [Dimensionally]{}, $\kappa \sim L^2$, $(A, K, \partial) \sim L^{-1}$, $E \sim L^0$, $P \sim L^{-2}$, $(\Psi, \bar{\Psi}) \sim L^{-3/2}$, ${\cal G} \sim L^{-3}$, $({\cal H}_a, {\cal H}) \sim L^{-4}$. The [density weights are:]{} $( P, {\cal G}, {\cal H}_a, {\cal H} ) = +1$, $(\Psi, \bar{\Psi}) = 1/2$ and $(A, K, V, \Gamma) = 0$. Under[^7] [Lorentz parity:]{} $(V, K, {\mathrm{sgn}(e)})$ are odd, $(\Gamma, A, P, \Psi, \bar{\Psi})$ and the constraints are all even. Under [parity combined with $\gamma \to - \gamma$:]{} $(V, K, \Gamma, A, P, \gamma\,{\mathrm{sgn}(e)})$ and the constraints are all even. The $A$ and $K$ above correspond to the vacuum case for which the Thiemann identities hold. The inverse square root of $q$ and $K_a^i$ appearing above are manipulated exactly as in the vacuum case. Explicitly, the identities are $$\begin{gathered} {\mathrm{sgn}(e)}{\cal E}^{bca} V^i_a = {\cal E}^{ijk}\frac{E^b_j E_k^c}{\sqrt{\det (E^a_i)}} , \qquad q := (\det(V^i_a))^2 = \det(E^a_i), \\ \kappa\gamma\frac{{\mathrm{sgn}(e)}}{2}V^i_a(x) = \left\{{A}^i_a(x), \int d^3y \sqrt{q}\right\} \quad \Rightarrow \nonumber \\ {\cal E}^{ijk}\frac{E^b_j E_k^c}{\sqrt{\det(E^a_i)}} = \frac{2}{\kappa\gamma} {\cal E}^{bca} \left\{{A}^i_a(x), \int d^3y \sqrt{q}\right\}, \\ H_E(1) := \frac{\kappa\gamma^2}{2}\int \frac{P^b_j P^c_k}{\sqrt{q}}{\cal E}^{jk}_{~l}F^l_{bc} , \qquad \overline{{K}} := \int d^3y \, {\mathrm{sgn}(e)}{K}^i_a P^a_i \quad \Rightarrow \\ \overline{{K}} = (\kappa\gamma^3)^{-1}\left\{ H_E(1), \int d^3y \sqrt{q} \right\},\qquad {\mathrm{sgn}(e)}{K}^i_a(x) = \{{A}^i_a(x), {\overline{K}}\}.\end{gathered}$$ These identities suffice to derive a quantization the Hamiltonian constraint from that of the ‘Euclidean Hamiltonian constraint’ (the first term in the Hamiltonian constraint) and of the volume operator. This completes the classical canonical formulations as it follows from the action. Constraint algebra ------------------ It is easy to see that the gauge constraint generates correct gauge transformation of the basic fields. Specifically, with ${\cal G}(\Lambda) := \int_{\Sigma_3}d^3x \Lambda^i {\cal G}_i$, $$\begin{gathered} \{ A^i_a(x), {\cal G}(\Lambda)\} = - {\cal D}_a \Lambda^i = - \partial_a \Lambda^i - {\cal E}^i_{~jk}A^j_a\Lambda^k, \\ \{ P_i^a(x), {\cal G}(\Lambda)\} = + {\cal E}_{ij}^{~k}\Lambda^j P^a_k, \\ \{ \Psi^{\alpha}(x), {\cal G}(\Lambda)\} = -i \Lambda^i (\gamma_5\sigma_{0i}\Psi)^{\alpha}, \\ \{ \Psi^{\dagger}_{\alpha}(x), {\cal G}(\Lambda)\} = +i \Lambda^i (\Psi^{\dagger}\gamma_5\sigma_{0i})_{\alpha}.\end{gathered}$$ If we compute the infinitesimal action of the ${\cal H}_a$ constraint on the basic variables, we see that it equals the Lie derivatives of the basic variables only up to an SU(2) gauge transformation. We are however free to modify the constraints by adding suitable combinations of themselves. So we [define]{} the diffeomorphism constraint as $$\begin{gathered} {\cal C}(\vec{N}) := \int_{\Sigma_3}d^3x N^a {\cal C}_a \quad \mathrm{with} \nonumber \\ {\cal C}_a := {\cal H}_a - A^i_a {\cal G}_i = P^b_i\partial_a A^i_b - \partial_b(A^i_a P^b_i) + \frac{i}{2}\big(\Psi^{\dagger}\partial_a \Psi - \partial_a\Psi^{\dagger}\cdot\Psi\big),\end{gathered}$$ which leads to the infinitesimal transformations, $$\begin{gathered} \{ A^i_a(x), {\cal C}(\vec{N})\} = {\cal L}_{\vec{N}} A^i_a = \partial_a(N^bA^i_b) + N^b(\partial_b A^i_a - \partial_a A^i_b), \\ \{ P_i^a(x), {\cal C}(\vec{N})\} = {\cal L}_{\vec{N}} P_i^a = N^b\partial_b P^a_i - P_i^b \partial_b N^a + 1\cdot(\partial_b N^b) P^a_i, \\ \{ \Psi^{\alpha}(x), {\cal C}(\vec{N})\} = {\cal L}_{\vec{N}} \Psi^{\alpha} = N^b\partial_b \Psi^{\alpha} + \frac{1}{2}\cdot(\partial_b N^b)\Psi^{\alpha}, \\ \{ \Psi^{\dagger}_{\alpha}(x), {\cal C}(\vec{N})\} = {\cal L}_{\vec{N}} \Psi^{\dagger}_{\alpha} = N^b\partial_b \Psi^{\dagger}_{\alpha} + \frac{1}{2}\cdot(\partial_b N^b)\Psi^{\dagger}_{\alpha}.\end{gathered}$$ This implies that $\{{\rm var}, \int N^a C_a\} = {\cal L}_{N^a}({\rm var})$ for all variables. The Gauge constraint already generates the correct gauge transformation of the basic variables. By inspection, it follows that the gauge constraints (weakly) commute with the diffeomorphism and the Hamiltonian constraint, the gauge constraint and the diffeomorphism constraints form sub-algebras and the diffeomorphism constraint transforms the Hamiltonian constraint by the Lie derivative. The non-trivial bracket is the bracket of two Hamiltonian constraints. Before turning to loop quantization, we briefly draw attention to the invariance (or lack of it) under two distinct ‘parity’ operations. Parity and Lorentz parity {#Parities} ------------------------- Recall that we begin with the (co-)tetrad field $e^I_{\mu}$, the Lorentz connection $\omega^{IJ}_{\mu}$ and the fermion fields $\lambda$, $\bar{\lambda}$ (or $\Psi$, $\bar{\Psi}$) defined over a manifold $M \sim \mathbb{R}\times\Sigma_3$ which is assumed to be orientable. With the topology specified, $M$ can be taken to be [time-orientable]{} with respect to all the metric tensors constructed by the parametrization (\[CoTetrad\]), (\[Tetrad\]). Obviously, $\Sigma_3$ is orientable as well. There are two distinct sets of ‘parity’ transformations: orientation reversing diffeomorphism of $M$ and a ${\rm O}(1,3)$ transformation with determinant $= -1$. Note that these distinctions do not exist in flat space-time with Minkowski metric where the allowed diffeomorphisms are the isometries of the Minkowski metric which are the same as ${\rm O}(1,3)$ transformations. We will keep the time orientation fixed. Orientation reversing diffeomorphisms of $M$ will then be reversing the orientation of $\Sigma_3$. We will refer to these as [parity]{} transformations. The improper Lorentz transformations $\Lambda^I_J$, will also be taken so that $\det \Lambda = -1$ and $\Lambda^0_{~0} = 1$ and will be referred to as [Lorentz parity transformations]{}. After going to the canonical framework in the ‘time gauge’, we have the fields $A^i_a$, $P^a_i$, $\Psi$, $\bar{\Psi}$ defined on $\Sigma_3$. A parity transformation is now an orientation reversing diffeomorphism of $\Sigma_3$ while the improper ${\rm O}(3)$ transformation, [inversion]{}, will be taken as the Lorentz parity transformation. In the Lagrangian framework, the Hilbert–Palatini action is invariant under both sets of transformations while the Nieh–Yan (and the Pontryagin) actions [change signs under parity]{} but are [invariant under Lorentz parity.]{} Hence the combined action is invariant under Lorentz parity and [non]{}-invariant under parity. The ${\mathrm{sgn}(e)}$ factor in the Hilbert–Palatini action is crucial for this. The variables of the canonical framework are defined in terms of those of the Lagrangian framework. These definitions of the ${\rm SU}(2)$ connection in terms of $K$ and $\Gamma$ and the conjugate momentum in terms of the triad [are]{} consistent with the ${\rm SO}(3)$ gauge transformation extended to include the Lorentz parity. Thus the triad which transforms by the defining representation changes sign under Lorentz parity. The ‘densitised triad’ (or the conjugate momentum) transforms by the adjoint representation and should be invariant under Lorentz parity. The ${\mathrm{sgn}(e)}$ factor in their definitions precisely takes care of this. The same can be seen in the definition of the connection. It is easy to see that the symplectic structure and the constraints (vacuum) are [all invariant under Lorentz parity]{}. When fermions are included, these are [scalars]{} under orientation preserving diffeomorphisms[^8], and transform as $\Psi \to \gamma^0 \Psi$ under Lorentz parity. All the constraints including fermions are invariant under Lorentz parity. This is true in both the Lagrangian and the canonical frameworks. With regards to parity the situation is different. The action is not invariant under parity, due to the Nieh–Yan term. In the canonical framework, the connection is not simply even/odd under parity since the $K$ term changes sign while the $\Gamma$ does not. The ‘densitised triad’ also acquires an extra minus sign under parity (behaves as a ‘pseudo-vector of weight 1’). The symplectic structure thus is [not]{} invariant. The constraints also are not invariant under parity and this is consistent with the non-invariance of the action. The action [is]{} invariant under parity [combined with $\gamma \to - \gamma$]{}. Our definitions have the appropriate factors of $\gamma$ to restore the simple (even) behaviour of the basic canonical variable resulting also in the invariance of the Poisson brackets and constraints. In short, [Lorentz parity is an invariance of the action as well as the canonical framework while parity is not. However, parity combined with $\gamma \to - \gamma$ is an invariance of both action and the canonical framework.]{} We could consider a canonical formulation [ab initio]{}, say by making a canonical transformation from the ADM variables, without any reference to an action. We could then [define]{} of the basic canonical variables $A^i_a$, $E^a_i$, without the factors of ${\mathrm{sgn}(e)}$. This will restore the ‘densitized triad’ to its usual density weight 1 vector density status and the connection to its 1-form status. The canonical framework is then [invariant]{} under parity (without changing sign of $\gamma$). Under Lorentz parity, however, the connection does not have simple behaviour and the conjugate momentum is odd. The reason of course is that the internal index ‘$i$’ on the triad and on the extrinsic curvature refers to the defining representation while that on the connection and its conjugate refer to the adjoint representation and these are distinguished by the improper orthogonal transformations. This results in non-invariance of the canonical framework under Lorentz parity. If the sign of $\gamma$ is changed along with the Lorentz parity transformation, then the basic variables are even, the symplectic structure is invariant and so are the constraints [and the action]{}. Thus, the definitions without the ${\mathrm{sgn}(e)}$ factors, interchanges the role of Lorentz parity and parity appropriately combined with $\gamma \to - \gamma$. Which of these notions is ‘appropriate’? If we were to consider formulation in terms of the metric tensor, then the notion of Lorentz parity is not even definable as there is no internal Lorentz transformation. On the other hand, with fermionic matter we [have to]{} consider oriented manifolds and the tetrad formulation, introducing the possibility of Lorentz parity. If the orientation of the manifold is regarded as a fixed background structure, the parity transformations are excluded by definition and Lorentz parity alone remains. Which of these notions is [relevant]{} for experimental observation, is unclear and so is the issue of ‘gravitational parity violation/invariance’ in LQG. In the next subsection, we briefly summarise the loop quantization of fermions [@BojowaldDas; @BojowaldDas2; @ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4]. Loop quantization of fermions {#LoopQuantization} ----------------------------- The loop quantization of the gravitational fields as well as of the fermions has been already given by Thiemann and those arguments remain valid. We have already gone from $\lambda$ fermions to the half density $\Psi$ fermions. In our derivation from the action, this is simply seen as the choice of Darboux coordinates. We also do not work with the ‘conjugate’ variables $\pi_{\Psi}$, $\pi_{\bar{\Psi}}$. Instead we solve the primary second class constraints and use Dirac brackets relative to these. Thus we take, for a single Dirac fermion, $\Psi^A$, $\bar{\Psi}_A$ to be complex, Grassmann valued, half densities: $\bar{\Psi}_A := (\Psi^A)^$, $A = 1, \ldots, 4$. This can easily be generalized to many Dirac fermions or Weyl fermions[^9]. As mentioned in the introduction, the first step is to construct functions on the fermionic phase space. Here we discuss Thiemann’s proposal presented in a slightly different form. It is based on a particular, naturally available, class of smearing densities. Consider a decomposition of $\Sigma_3$ into cells $B_n$, closed subsets of $\Sigma_3$. Let $\epsilon^3 := \mu(B_n)$ be the [Lebesgue measure]{} (‘coordinate volume’) of the cell $B_n$ and let $v_n$ be the [centre]{} of $B_n$ (a marked point in the interior of $B_n$). Define the [indicator]{} or ([characteristic]{}) function: $$\begin{gathered} \chi_{\epsilon}(v_n, x) := \chi_n(x) := \begin{cases} 1, & \mathrm{if} ~ x \in \mathrm{Int}(B_n), \\ 0, & \mathrm{otherwise}. \end{cases}\end{gathered}$$ Then the quantity, $\chi_n(x)/\epsilon_n^{3w}$ has density weight $w$. This is because, the Lebesgue measure of a cell, transforms as a scalar density of weight $-1$ under a coordinate transformation and the indicator function is of course invariant. Note that eventually we are interested in the limit of infinite refinement of the cell decomposition, $\epsilon_n \to 0$, and the stated behaviour of the Lebesgue measure is valid in this limit. Using these quantities, labelled by the cells of a cell decomposition, we can smear any scalar density of weight $(1 - w)$ and construct a function on the configuration/phase space. For a Grassmann variable $\Psi$ (the spinorial index is suppressed) of weight 1/2, we define the Grassmann valued functions $$\begin{gathered} \theta_n (\Psi) := \int_{\Sigma_3} d^3x\, \frac{\chi_n(x)}{\epsilon_n^{3/2}} \Psi(x) = \int_{B_{n}} d^3x\, \frac{\chi_n(x)}{\epsilon_n^{3/2}} \Psi(x).\end{gathered}$$ Under a diffeomorphism: $B_n \to B'_n$, its Lebesgue measure $\epsilon^3_n \to \epsilon^{'3}_n = \|{{\textstyle \frac{\partial x'}{\partial x}}}\| \epsilon_n^3$, $d^3x' = \|{{\textstyle \frac{\partial x'}{\partial x}}}\| d^3x$ while $\Psi'(x') = \sqrt{\|{{\textstyle \frac{\partial x}{\partial x'}}}\|} \Psi(x)$. This implies that $$\begin{gathered} \theta_n' := \int_{B'_n}d^3x' \frac{\chi_n(x') \Psi'(x')}{\epsilon_n^{'3/2}} = \int_{B_n}d^3 x \frac{\chi_n(x)\Psi(x)\|{{\textstyle \frac{\partial x'}{\partial x}}}\|\|{{{\textstyle \frac{\partial x}{\partial x'}}}}\|^{1/2}}{\epsilon_n^{3/2}\|{{\textstyle \frac{\partial x'}{\partial x}}}\|^{1/2}} \nonumber \\ \phantom{\theta_n'}{} = \int_{B_n} d^3x \frac{\chi_n(x) \Psi(x)}{\epsilon^{3/2}_n} = \theta_n.\end{gathered}$$ Thus, the $\theta$ variables are indeed [invariant]{} under diffeomorphism[^10]. It is immediate that $\bar{\theta}_n = (\theta_n)^$. In the limit of finer cell decomposition, $\epsilon_n \to 0$, we get the following expressions: $$\begin{gathered} \lim_{\epsilon_n \to 0} \chi_{\epsilon_n}(v_n, x) = \delta_{v_n,x} \qquad (\mathrm{Kroneker~delta}), \\ \lim_{\epsilon_n \to 0} \frac{\chi_{\epsilon_n}(v_n, x)}{\epsilon_n^3} := \delta^3(v_n, x) \qquad (\delta\text{-distribution}).\end{gathered}$$ From these it follows that $$\begin{gathered} \{\theta_m, \bar{\theta}_n\} = \int_{B_m}d^3x\int_{B_n}d^3y \frac{\chi_m(x)}{\epsilon_m^{3/2}} \frac{\chi_n(y)}{\epsilon_n^{3/2}} \{ \Psi(x), \bar{\Psi}(y) \} \nonumber \\ \phantom{\{\theta_m, \bar{\theta}_n\}}{} = -i \int_{B_m}d^3x\int_{B_n}d^3y \frac{\chi_m(x)}{\epsilon_m^{3/2}} \frac{\chi_n(y)}{\epsilon_n^{3/2}} \delta^3(x,y) \nonumber \\ \phantom{\{\theta_m, \bar{\theta}_n\}}{} = - i \delta_{m,n} \int_{B_n}d^3x \frac{\chi_n(x)}{\epsilon_n^3} = -i \delta_{m,n} , \qquad \because \chi_m(x) \chi_n(x) = \delta_{mn}\chi_m(x) .\end{gathered}$$ With these, we obtain the Poisson brackets among the $\theta$ variables (functions on the phase space). The identity in the third line above holds because the $\chi_n(x) = 0$ unless $x$ is in the [interior]{} of the cell $B_n$. At finite but small $\epsilon_n$, we can write, $\theta_n \approx \epsilon_n^{3/2}\Psi(v_n)$. Notice that the $\theta_n$ variables are effectively associated with the points $v_n$. This is particularly useful in expressing the constraint expressions in terms of the $\theta$’s as illustrated below. Recall from equation (\[FinalHam\]) that the smeared Hamiltonian constraint has fermion bilinears with and without derivative as well as terms quartic in fermions. Using a cell decomposition, we can express the integral as a sum by restricting the integrals over the cells. In the limit of small $\epsilon_n$, these integrals can be approximated by using the mean value theorem. The factors of $\epsilon_n$’s available can be distributed with the fermions to go from $\Psi(v_n)$ to $\theta_n$ variables. The correct density weight of +1 ensures that no factors of $\epsilon_n$’s remain unabsorbed. In equations $$\begin{gathered} \int_{\Sigma_3}d^3x \bar{\Psi}_A(x) M^A_{~B}(x)\Psi^B(x) \approx \sum_n\epsilon_n^3\bar{\Psi}_A(v_n)M^{A}_{~B}(v_n) \Psi^B(v_n) \nonumber \\ \qquad{} = \sum_n\big(\epsilon_n^{3/2} \bar{\Psi}_A(v_n)\big)M^{A}_{~B}(v_n) \big(\epsilon_n^{3/2}\Psi^B(v_n)\big) \approx \sum_n\bar{\theta}_A(v_n)M^{A}_{~B}(v_n) \theta^B(v_n), \\ \int_{\Sigma_3}d^3x \bar{\Psi}_A(x) (\sigma^i f^a_i(x))^A_{~B}\frac{\partial\Psi^B(x)}{\partial x^a} \approx \sum_n\epsilon_n^3\bar{\Psi}_A(v_n)(\sigma^i f^a_i(v_n))^{A}_{~B} \frac{\partial \Psi^B(v_n)}{\partial v^a_n} \nonumber \\ \qquad{} \approx \sum_n\bar{\theta}_A(v_n)(\sigma^i f^a_i(v_n))^{A}_{~B} \frac{\partial \theta^B(v_n)}{\partial v^a_n}.\end{gathered}$$ This takes care of the fermionic bilinears. The quartic terms have a factor of $\sqrt{q}(x)$ in the denominator. Following [@ALReview; @BojowaldDas; @BojowaldDas2], we first express it as a Poisson bracket with only positive powers of the volume of the cell containing the point $x$. From the definition of the determinant and our conventions given in the footnote \[Notation\], it follows that $$\frac{{\mathrm{sgn}(e)}}{\sqrt{q}} = \frac{1}{6} {\cal E}^{abc}{\cal E}_{ijk} \frac{V^i_a V^j_b V^k_c}{q}.$$ Let ${\cal V}_n := \int_{B_n}d^3x \sqrt{q}(x)$. Then $$\begin{gathered} \{ A^i_a(x), {\cal V}_n^{1/3}\} = \frac{1}{3} {\cal V}_n^{-2/3}\{ A^i_a(x), {\cal V}_n\} = \begin{cases} \dfrac{\kappa\gamma}{6} V^i_a {\cal V}_n^{-2/3}, & \mbox{if $x \in B_n$},\\ 0, & \mbox{otherwise }, \end{cases} \nonumber \\ \therefore \frac{{\mathrm{sgn}(e)}}{\sqrt{q}(x)} = \frac{36}{(\kappa\gamma)^3}\{A^i_a, {\cal V}_n^{1/3}\}\{A^j_b, {\cal V}_n^{1/3}\}\{A^k_c, {\cal V}_n^{1/3}\} {\cal E}^{abc}{\cal E}_{ijk}\frac{{\cal V}^2_n}{q(x)}.\end{gathered}$$ For small $\epsilon_n$, and for non-zero answer ($x \in B_n$), ${\cal V}_n \approx \epsilon_n^3 \sqrt{q}(x)$ and therefore ${\cal V}^2_n/q(x) \approx \epsilon^6_n$. These factors of $\epsilon_n$ are neatly combined with the 4 $\Psi$, $\bar{\Psi}$ variables to go over to the $\theta$ variables. The $\epsilon^3_n$ from the integration measure combines with the three $A^i_a$ to produce the combinations ${\rm Tr}({\cal E}^{IJK}h^{-1}_I\{h_I,\dots\}h^{-1}_J\{h_J, \dots\}h^{-1}_K\{h_K, \dots\}$) with the ${\cal E}$ ensuring that the three edges are non-coplanar and the Tr ensuring the SU(2) invariance. Thus all integral expression in the constraints are expressible in terms of the $\theta$ variables. Since the transition from the $\Psi$ variables to the $\theta$ variables is a linear operation, the Grassmann properties of the $\theta$ variables are the same as those of the $\Psi$ variables. We have thus the classical phase space for Grassmann valued fields. To construct a quantum theory, we first construct the space of complex valued (and Grassmann valued) functions and define an inner product on it. The elementary $\theta$ variables, are to represented as multiplicative operators while the their complex conjugates (“momenta”) are to be represented by derivative operators. The procedure is standard [@Berezin; @Berezin2; @ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4]. The Grassmann nature restricts the functions to be polynomials eg for a single pair of complex Grassmann variables, $\theta$, $\bar{\theta}$, the most general function is: $f(\bar{\theta}, {\theta}) = a + b\theta + c\bar{\theta} + d\bar{\theta}{\theta}$ where $a$, $b$, $c$, $d$ are complex numbers. A [holomorphic]{} function, has no $\bar{\theta}$ dependence. For $k$ number of complex pairs, the vector space of holomorphic functions is $2^k$-dimensional. On this space one has the usual [Berezin]{} measure, $\int d\bar{\theta}d\theta f(\bar{\theta},{\theta}) = d$, however as pointed out by Thiemann, this is not positive definite and thus unsuitable for constructing a Hilbert space. The modification proposed by Thiemann is: $\mu(\bar{\theta}, \theta) := e^{\bar{\theta}\theta}d\bar{\theta}d\theta$. The inner product defined by $\langle f, g\rangle := \int\mu(\bar{\theta}, \theta)f^(\bar{\theta})g(\theta)$, turns the function space into a Hilbert space. On this, define the operators $$\begin{gathered} [\hat{\theta}f](\theta) := \theta f(\theta) ,\qquad [\hat{\bar{\theta}}f](\theta) := \hbar\frac{d f(\theta)}{d\theta}.\end{gathered}$$ These operators satisfy: $[\hat{\theta}, \hat{\bar{\theta}}]_+ = \hbar$ as desired by the general quantization rule, $\{A, B\} \to (-i/\hbar)[\hat{A}, \hat{B}]$, valid for the generalized Poisson bracket. It follows that $\hat{\bar{\theta}} = (\hat{\theta})^{\dagger} $ also holds reflecting the classical relation $\bar{\theta} = \theta^$ and $\mu(\bar{\theta}, \theta)$ is the unique normalised measure selected by the above adjointness property [@ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4]. Generalization to finitely many Grassmann variables is immediate. For fermionic fields (infinitely many variables) one proceeds via projective limit. For $d$-Weyl fermion fields, we will have $2d$ $\theta$ variables, $\theta^1, \theta^2, \ldots, \theta^{2d}$ at each point of $\Sigma_3$. The holomorphic functions are polynomials of maximal degree $2^{2d}$ in the $\theta$ variables at [each]{} point. Denote a [monomial]{} of degree $k$ at a point $v$ by $$\begin{gathered} F_{v, \{i\}_{k(v)}} := \theta^{i_1}(v)\theta^{i_2}(v)\cdots\theta^{i_{k(v)}}(v), \qquad i_1, \ldots, i_k \in [1, 2d], \qquad k(v) \in [0, 2^{2d}].\end{gathered}$$ For every finite set of points[^11], $\{\vec{v}\} = \{v_1, v_2, \ldots, v_n\}$ and a corresponding vector of labels, $\vec{I}_{\vec{k}} := \{\{i\}_{k(v_1)}, \{i\}_{k(v_2)}, \ldots, \{i\}_{k(v_n)} \}$, define the [elementary functions of $\theta$’s]{}, $$\begin{gathered} {\cal F}_{{\vec I}_{\vec k}} := \prod_{j = 1, n} F_{v_j, \{i\}_{k(v_j)}}.\end{gathered}$$ Finite linear combinations of these elementary functions are called (fermionic) cylindrical functions which are orthonormal with respect to the inner product defined point-wise. Closure of the set of these cylindrical functions defines the [Hilbert space of the loop quantization of fermions]{}. Quantization of more general observables proceeds by first expressing these observables in terms of the $\theta$ variables and promoting them to the corresponding operators. Implications for loop quantum cosmology --------------------------------------- Implications of fermionic matter has also been discussed in the context of homogeneous models by Bojowald and Das in [@BojowaldDas; @BojowaldDas2]. The symmetric fermionic fields are only restricted to be constants on the spatial manifold by homogeneity. Isotropy however requires these constants to be zero. One way to see this is to note that from the presumed non-zero $\Psi$’s we can construct spatial vectors as $V^a_i\bar{\Psi}\gamma^i\Psi$ and by isotropy there cannot be any non-zero constant vector. For homogeneous, anisotropic models however non-zero fermionic constants are allowed[^12]. This has a major implication for diagonal models. Recall that the homogeneous connection and triad are of the form: $A^i_a(x) = \Phi^i_I\omega^I_a(x)$, $E^a_i(x) = \sqrt{g_0} P^a_I X^a_I(x)$ where, $\omega^I_a$ are the Maurer–Cartan forms and $X^a_I$ their dual vector fields. The sub-class of [diagonal]{} models is defined by the restriction: $\Phi^i_I := c_I\Lambda^i_I$, $P^I_i := p^I\Lambda^I_i$ (no sum over $I$), where $\Lambda_i^I \Lambda^i_J = \delta^I_J$, $\Lambda^i_I\Lambda^j_J\Lambda^k_K{\cal E}_{ijk} = {\cal E}_{IJK}$. With this choice, the homogeneous, gravitational part of the Gauss Law constraint vanishes identically. With the possibility of non-zero, homogeneous fermionic degrees of freedom, the homogeneous ${\cal G}^i_F \neq 0$ and therefore the gravitational part [cannot]{} be zero. Consequently, the connection and the triad variables cannot be diagonalised simultaneously. A way out, suggested in [@BojowaldDas; @BojowaldDas2] is to use different $\Lambda$’s for connection and triad. The corresponding dynamical system is still complicated and a specific choice is made to get a simplified model, wherein the constant fermionic vector is taken to be along, 1-axis. This permits a identification of canonical coordinates, 4 in number. To explore parity invariance more explicitly, the model is further specialized to Bianchi I LRS class. The main conclusion drawn is that in this specific case, parity invariance at the level of underlying difference equation is violated only if there are parity violating interactions in the [matter sector]{}. However, the difference equation remains deterministic even when parity is violated [@BojowaldDas; @BojowaldDas2]. Gauge fields {#YM} ============ The construction of the kinematical Hilbert space for gauge fields proceeds exactly as in the case of the gravitational, SU(2) gauge fields. For identical logic, in a background independent context, the gauge fields are quantized through the use of the holonomies and fluxes of the electric fields. The new feature is the structure of the corresponding contributions to the Hamiltonian constraint and of course the new Gauss constraints corresponding to the additional gauge invariances. As remarked in the footnote \[FootNote1\], there is no contribution to the rotation constraint. The contribution to the diffeomorphism constraint is of the same form as in the gravitational sector. The couplings of these matter fields to the gravitational ones are through the triad variables for minimally coupled matter and there are also non-gravitational couplings among the matter fields. The underlying background independence which forces the use of holonomy and fluxes impacts the quantization of these interaction terms as well. Loop quantization of gauge fields has been discussed by Thiemann [@ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4] and Ashtekar–Lewandowski [@ALReview] (see also [@Alfaro]). The salient points are noted below. Maxwell field theory -------------------- The gauge group in the Maxwell case case is Abelian which simplifies some of the technical details. In particular, since the holonomies are simple numbers (not matrices), the Poisson bracket of holonomy with 3-dimensionally smeared electric field is proportional to the holonomy and the 3-smeared electric field maps cylindrical functions to cylindrical functions [@ALReview]. We will however use 2-dimensional smearing and the fluxes Poisson commute in both cases. The classical analysis beginning with the Maxwell action followed by canonical formulation using the time gauge parametrization is summarised below $$\begin{gathered} {\cal L}_{\mathrm {Maxwell}} := \frac{1}{16\pi}\sqrt{\|g\|} g^{\mu\alpha}g^{\nu\beta}\mathbb{F}_{\mu\nu}\mathbb{F}_{\alpha\beta}, \\ H(N, N^a, \Lambda) = \int \frac{N}{8\pi}\frac{q_{ab}}{\sqrt{q}}\big(\mathbb{P}^a\mathbb{P}^b + \mathbb{B}^a\mathbb{B}^b\big) + \frac{N^a}{8\pi}\mathbb{F}_{ab}\mathbb{P}^b + \frac{\Lambda}{8\pi}\partial_a\mathbb{P}^a, \\ \mathbb{B}^a := \frac{1}{2}{\cal E}^{abc}\mathbb{F}_{bc},\qquad \mathbb{E}^a := \mathbb{P}^a = - \sqrt{q}\ N\ \mathbb{F}^{ta} \nonumber\end{gathered}$$ with $\mathbb{A}_a$, $\mathbb{P}^b$ being the canonical coordinates. Note that $\mathbb{P}^a$, $\mathbb{B}^a$ are both of density weight 1. As usual, before quantization, we need to express the constraints (and other observables of interest), in terms of the holonomies and fluxes. We have the common factor of $q_{ab}/\sqrt{q}$. As before, introduce a cell decomposition with cells $B_n$, containing points $v_n$ and with Lebesgue measure $\epsilon_n^3$ and define ${\cal V}_n := \int_{B_n} d^3x \sqrt{q(x)}$. Noting that $$\begin{gathered} \{A^i_a(x), {\cal V}_n^l\} = l {\cal V}_n^{l - 1} \frac{\kappa\gamma}{2}{\mathrm{sgn}(e)}V^i_a ,\quad l \in (0, 1) ,\qquad {\cal V}_n \approx \epsilon_n^3\sqrt{q}(v_n) \quad \mathrm{for~small} \ \ \epsilon_n,\end{gathered}$$ we write $$\begin{gathered} \frac{q_{ab}}{\sqrt{q}}(x) \approx \frac{16}{(\kappa\gamma)^2}\epsilon_n^3 \delta_{ij}\big\{A_a^i(x), \sqrt{{\cal V}_n}\big\} \big\{A_b^j(x), \sqrt{{\cal V}_n}\big\}.\end{gathered}$$ Next, observe that for 2-smeared fluxes of $\mathbb{P}^a$, we have $\mathbb{\Phi}_S := \int_S \mathbb{P}^a {\cal E}_{abc} dS^{bc} \approx \mathbb{P}^a(v) n_a(S) \epsilon_S^2$ where $S$ is a (small) surface, $v$ is a point in it and $\epsilon^2_S$ (small) is its Lebesgue measure. The ‘normal’ to $S$ is given by, $n_a(S) := {\cal E}_{abc}{{\textstyle \frac{\partial x^b}{\partial \xi_{\alpha}}}}{{\textstyle \frac{\partial x^c}{\partial \xi_{\beta}}}}{\cal E}^{\alpha\beta}$ where $\xi^{\alpha}$, $\alpha = 1, 2$ denote the local coordinates [on]{} $S$ while $x^a(\xi)$ denote the embedding of the 2-surface in $\Sigma_3$. For a small triangular surface, we can also replace $n_a(S)\epsilon_S^2 \approx {{\textstyle \frac{1}{2}}}{\cal E}_{abc}\delta S^b_I\delta S^c_J$ where $\delta S^b_I$, $\delta S^c_J$ denote coordinate lengths of a pair of edges of the triangle. Likewise, for an infinitesimal curve $e$, $h_e := \mathrm{Pexp}(\int_e A) = \exp{(\int_e A)}$ implies $h^{-1}_e \{ h_e, \dots \} \approx \{\int_e A, \dots \} \approx \{\delta t \dot{e}^aA^i_a\tau_i , \dots \}$. =1 To regularise the Hamiltonian, we introduce a cell decomposition and approximate the integral by a sum-over-cells. This provides a $\epsilon^3$ and the $q_{ab}/\sqrt{q}$ expression provides another factor of $\epsilon^3$ for each term. The 6 factors of $\epsilon$ can be distributed as 2 each for the two $\mathbb{P}$’s and 1 each for the two gravitational connections in the Poisson brackets. To express the variables in terms of fluxes and holonomies, we need to choose two surfaces and two curves paying attention to the [contraction]{} of $a, b$ indices which is part of specification of gravitational coupling. We recall from the construction done in the gravitational sector. We choose a triangulation of $\Sigma_3$ by elementary tetrahedra $\Delta_n$ [@ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4]. In anticipation of the quantization step, we choose the edges of the tetrahedra to be analytic. [One]{} vertex of each tetrahedron is distinguished and at the most finitely many tetrahedra meet at such a vertex. Edges meeting at a vertex are taken to be out-going. The three edges of each tetrahedron, provide three linearly independent tangent vectors at its vertex $v_n$, $\dot{e}^a_{~I}$, $I = 1, 2, 3$, such that the $a^{\rm th}$ coordinate interval of the $I^{\rm th}$ edge $ = \delta t_I\dot{e}^a_I$, no sum over $I$ and $I$ is [not]{} to be confused with the Lorentz index which is no longer relevant now. $\delta t_I$ refers parametrization of the $I^{\rm th}$ edge. Each tetrahedron, also provides three non-coplanar surfaces $S_{IJ}$, bounded by the edges $e_{I}$, $e_{J}$ and the ‘opposite’ edge. We have thus a natural choice of surfaces and edges. For these surfaces and edges, we have, $\Phi_{IJ} \approx {{\textstyle \frac{1}{2}}}\mathbb{P}^a{\cal E}_{abc}(\dot{e}^b_I\delta t_I)(\dot{e}^c_J\delta t_J)$ and $h_{K}^{-1}\{h_K, \sqrt{{\cal V}_n}\} \approx \{\delta t_K \dot{e}^a_K A_a^i\tau_i, \sqrt{{\cal V}_n}\}$. Clearly $$\frac{1}{6}{\cal E}^{IJK} \Phi_{IJ} h_{K}^{-1}\big\{h_K, \sqrt{{\cal V}_n}\big\} \approx \frac{1}{12}{\cal E}^{IJK}\mathbb{P}^a{\cal E}_{abc}(\delta t_I\dot{e}_I^b)(\delta t_J\dot{e}_J^c)(\delta t_K \dot{e}^d_K) \big\{A^i_d\tau_i , \sqrt{{\cal V}_n}\big\}.$$ Now $${\cal E}^{IJK}\dot{e}^b_I\dot{e}^c_J\dot{e}^d_K = {\cal E}^{bcd}\det(\dot{e}^a_I),\qquad \det (\dot{e}^a_I)\delta t_1\delta t_2\delta t_3 =: \epsilon^3 ,\qquad {\cal E}_{abc}{\cal E}^{bcd} = 2\delta^d_a.$$ Using these, it follows $$\begin{gathered} \frac{1}{8\pi}\int_{\Sigma_3} N \frac{q_{ab}}{\sqrt{q}}\mathbb{P}^a\mathbb{P}^b \approx \frac{2}{\kappa^2\gamma^2\pi}\sum_n N(v_n)\sum_{\Delta_n} \delta_{ij} \big(\epsilon_n^3\mathbb{P}^c\delta^a_c \big\{A_a^i, \sqrt{{\cal V}_{\Delta}}\big\}\big) \big(\epsilon_n^3\mathbb{P}^d\ \delta^b_d\ \big\{A_b^j, \sqrt{{\cal V}_{\Delta}}\big\}\big) \nonumber \\ \hphantom{\frac{1}{8\pi}\int_{\Sigma_3} N \frac{q_{ab}}{\sqrt{q}}\mathbb{P}^a\mathbb{P}^b }{} = \left(- \frac{4}{\kappa^2\gamma^2\pi}\right) \sum_{n} N(v_n) \sum_{\Delta_n}\frac{{\cal E}^{IJK}}{6}\frac{{\cal E}^{I'J'K'}}{6} \mathbb{\Phi}_{IJ} \mathbb{\Phi}_{I'J'} \nonumber \\ \hphantom{\frac{1}{8\pi}\int_{\Sigma_3} N \frac{q_{ab}}{\sqrt{q}}\mathbb{P}^a\mathbb{P}^b =}{} \times\mathrm{Tr}\left[ h_{K}^{-1} \big\{ h_{K}, \sqrt{{\cal V}_{\Delta}}\big\} h_{K'}^{-1} \big\{ h_{K'}, \sqrt{{\cal V}_{\Delta}}\big\} \right].\end{gathered}$$ In the last equation, we have included $\mathrm{Tr}\,\tau_i\tau_j = - {{\textstyle \frac{1}{2}}}\delta_{ij}$. All the factors of $\epsilon$’s have been neatly absorbed, again thanks to density weight 1 of the Hamiltonian. The sum over $n$ is a sum over vertices of the triangulation. The sum over $\Delta_n$ includes those tetrahedra whose distinguished vertex is $v_n$ and this is a finite sum. Note that some vertices of the triangulation may not be distinguished vertices of [any]{} tetrahedron. Such vertices do not contribute to the sum. Therefore, the sums over $v_n$ and $\Delta_n$ together, denote sum over the cells of the decomposition. Notice that surfaces over which the [electric flux]{} is taken is correlated with the edge along which the [gravitational holonomy]{} is to be taken thanks to the ${\cal E}^{IJK}$ factors. In the passage to quantum theory, the triangulation is adapted to the underlying graph of a cylindrical function[^13] by aligning the edges of the elementary tetrahedra to be segments of corresponding edge of the graph and the distinguished vertex to be the vertex of the graph [@ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4]. Upon quantization, only those tetrahedra will contribute which have an edge overlapping with the edge of the holonomy. In each contributing cell (tetrahedron), the action of the electric flux operator, $\mathbb{\Phi}_{IJ}$ will be non-zero [only]{} on the Maxwell holonomy along the edge $e_{K}$ which is transversal to the surface $S_{IJ}$ and will give $\hbar q_K$. This also selects the edge $K$ in the gravitational holonomy, $h_{K}$. Here $q_K$, an integer, is the charge (in units of electric charge) in the Maxwell holonomy. On any cylindrical function, the action of the regulated quantum operator, results in a finite sum of terms since only vertices of the underlying graph contribute to the sum, regardless of refinement of the triangulation. We have deviated somewhat from the procedure given in [@ALReview; @ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4]. We have not done the point splitting and explicitly made passage to the flux operators absorbing the factors of $\epsilon$’s (in the limit of small $\epsilon$). The correlation between the edge in the gravitational holonomy and the surface in the Maxwell flux is preserved through the use of triplet of edges and corresponding unique transversal surfaces provided by an elementary tetrahedron. For the magnetic term, we can proceed identically by defining a flux $\mathbb{\Psi}_S := \int_S \mathbb{B}^a{\cal E}_{abc}dS^{bc}$ which will lead to an equation same as the one above with $\mathbb{\Phi}_S$ replaced by $\mathbb{\Psi}_S$. Unlike the electric flux however, this magnetic flux is [not]{} an elementary variable. For $$\begin{gathered} \mathbb{\Psi}_S := \int_S \frac{1}{2}{\cal E}_{abc}\mathbb{B}^adx^b\wedge dx^c = \int_S {\cal E}_{abc}\frac{{\cal E}^{aef}}{4}\mathbb{F}_{ef}dx^b\wedge dx^c \nonumber \\ \phantom{\mathbb{\Psi}_S }{} = \int_S (\delta^e_b \delta^f_c - \delta^e_c \delta^f_c)\frac{1}{4} \mathbb{F}_{ef} dx^b\wedge dx^c = \int_S \mathbb{F} \approx \frac{1}{2}\left(\mathbb{h}(e_S) - \mathbb{h}^{-1}(e_S)\right),\end{gathered}$$ where $e_S$ is a small close curve bounding the surface $S$. When the triangulation is adapted to the underlying graph of a cylindrical function, the curve $e_S$ may be taken to be one of the loops $\alpha_{IJ}$ of Thiemann [@ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4], which bounds the face $S_{IJ}$. Because of the presence of the Maxwell holonomies (with unit charge), the underlying graph of a cylindrical state is changed by adding these loops. But there are no gravitational holonomies along these loops and hence the spin labels of these extra loops are zero. We note in passing that if we have a spin and charge network state, i.e.a graph with both the spin and the charge labels for its edges some of which could be zero, then the action of the electric part of the Hamiltonian will be [non-zero only on those edges which have both the spin and the charge labels to be non-zero]{} – the volume operator will kill edges with no spin while the electric flux operator will kill the edge with no charge. This feature enables the operator to be well defined on the kinematical Hilbert space even in the limit of infinite refinement. By contrast, the magnetic part will [not]{} kill an edge with zero charge but will add loops with unit charge. This prevents the operator to be well defined on the kinematical Hilbert space in the limit of infinite refinement. It could however be defined on a subspace of the algebraic dual of the space of the cylindrical functions [@ALReview]. Yang–Mills field theory ----------------------- Apart from the non-Abelian nature of the gauge field, in form, the expressions proceed in the same manner as for the Maxwell case. The details are given in [@ThiemannScalar; @ThiemannQSD; @ThiemannQSD2; @ThiemannQSD4]. Propagation of matter waves on quantum geometry {#MatterWaves} =============================================== While there are good arguments to justify search for a quantum gravity, its observable signatures have been quite difficult to come by. Various different approaches to quantum gravity have contained a suggestion that space-time of quantum gravity is different from the continuum classical geometry be it discreteness and/or non-commutativity etc. Coupled with a heuristic idea that existence of a fundamental length scale may indicate a conflict with Lorentz invariance, a possible violation of Lorentz invariance has been considered as a possible signature of quantum gravity. The violation is thought to be manifested as a deviation from the Lorentz invariant [dispersion relation]{} for particle propagation on a ‘quantum space-time’. Its cumulative effect over cosmological distances are estimated to be detectable. From LQG Hamiltonian constraint, such deviation have been extracted [@Alfaro; @Morales; @Morales2; @Pullin] using the following set of assumptions[^14]. The Lorentz violations in the Polymer quantized scalar in Minkowski background has already been discussed in Section \[PolymerScalar\]. One begins by assuming a quantum state to be of a product form with a suitable state for geometry as well as for matter. For example, the early works, e.g. [@Pullin], the geometry state was a [weave]{} state [@Weaves; @Weaves3; @Weaves6; @Weaves7; @Weaves4; @Weaves5; @Weaves2] which provides a length scale $L_{\rm weave} \gg {\ell_{\mathrm P}}$ such that for larger scales the 3-geometry may be approximated by a continuum while the discrete structure manifests near the Planck scale. The scales relevant for observations of interest are assumed to be much larger than $L_{\rm weave}$. The matter state is chosen to be a coherent state so that the expectation values of the matter operators can be approximated by classical fields (fields obeying classical equations to leading order). These fields are assumed to be varying slowly over a length scale of $\lambda \gg L_{\rm weave}$. Taking expectation value of the Hamiltonian constraint, the geometry operators are replaced by the metric used in the construction of the weave plus corrections of the order of ${\ell_{\mathrm P}}/L_{\rm weave}$. The expectation values of the matter part of the Hamiltonian constraint is taken as defining an [effective Hamiltonian]{}. The [space-time metric]{} is taken to be a static metric determined by the 3-metric and a lapse $N$. The matter equations of motion following from the effective Hamiltonian contain the modifications implied by quantum gravity. In computing the effective Hamiltonian, one expands the classical fields at the vertices in a region of size $L_{\rm weave}$ around a central point of the region and averages over these vertices. The symmetries of the 3-geometry (eg rotational invariance of the flat geometry) is used to restrict the form of the averages. In turns out that there are corrections in the quadratic order in the Maxwell fields which imply polarization dependent speed of propagation suggesting birefringence. The computations are varied by different choice of states for expectation values eg more general semi-classical states rather than weaves or even using only qualitative properties of ‘would be semi-classical states’ [@Alfaro]. These computations also give additional corrections. Subsequently, the method is also applied to propagation of neutrinos [@Morales; @Morales2]. These early computations show how Lorentz invariance violations – as manifested by dispersion relations – could potentially arise in LQG. But do they [imply]{} that LQG violates Lorentz invariance? This is not easy to answer. The above computations are essentially at a kinematical level. This however need not be regarded as a strong drawback, because the very characterization of Lorentz violation pre-supposes a background space-time in which one can find locally inertial observers. Introducing a background is akin to a gauge fixing and heuristically at least the use of kinematical states could be justified. There are still not satisfactory semi-classical states of the geometry and matter system which will describe ‘test fields’ on a given space-time background. The proper framework for this would be quantum field theory on curved space-times. To obtain it from the full quantum theory of gravity and matter, is a challenging task. First steps in this direction are discussed by Sahlmann and Thiemann [@ThiemannSahlmann; @ThiemannSahlmann2]. A cautionary note on a different aspect, has also been sounded in [@BojowalMoralesSahlmann], pointing out that staying within Hamiltonian framework for computation including [approximation]{} leading to higher order derivatives, could be misleading because the corrections could be due to additional degrees of freedom with a Lorentz invariant dynamics. There is yet another aspect. Dispersion relation, relativistic or otherwise, need a background space-time. The presumed space-times in the first approaches have been static space-times with spatial geometry being described by a semi-classical state. Thus the ‘time’ part of the geometry does not have anything to do with the quantised spatial geometry. One has not used a quantum space-time, only quantum space. An extension to quantum space-time has been considered in the context of FLRW cosmology [@QFTonQG]. Such extensions are probably needed to discuss matter wave propagations on quantum space-time and its implications. Concluding remarks {#Summary} ================== We began by noting that non-Fock quantization of matter fields is necessary once ‘space-time’ is dynamical. However, the illusion of a fixed background space-time with Fock quantized fields on it, is also extremely persuasive. In these circumstances, gravitational interactions of matter fields are negligible compared to intra-matter interactions. Consequently, the observed semi-classicality of geometry itself would conceivably be sensitive to matter content as well as their interactions. Matter interactions must play a role in getting our ‘late universe’ with its essentially classical space-time emerge from the quantum gravity regime of the early universe. Study of (loop) quantized matter has the potential to provide selection criteria from the stability of semi-classical states of geometry. We are still quite far away from this goal. There are other ‘applications’ of matter fields. They can serve as probes for understanding properties of background independent quantization. For instance, in a fixed background space-time such as the Minkowski space-time, the short distance behaviour of quantum fields play a role in admitting the chiral (and the trace) anomalies. One of these, namely the axial anomaly, has played a historic role in understanding the $\pi^0 \to 2 \gamma$ process. Cancellation of gauge anomalies has also provided constraints on model building. In a background independent quantization, we loose the handle of short distance behaviour. How then are the anomalies to be understood? This is an old issue more recently mentioned by Nicolai et al. [@Nicolai], still awaiting an understanding. The focus of analysis of matter in quantum gravity has been to study its propagation in the regime of semi-classical gravity. These have indicated that one could expect dispersion relations not compliant with Lorentz invariance. This has been seen in the propagation of massless fermions (neutrinos), light as well as massless scalar fields. For fermions and light, quantum effects of spatial geometry are incorporated while space-time is constructed classically. The case of the scalar field discussed in Section \[PolymerScalar\], is qualitatively different from these. Here the relativistic (massless) scalar field is classically expressed as a collection of classical ‘harmonic oscillators’ with frequencies $\omega_{{\bf k}} \sim \|{\bf k}\|$ and there is no Lorentz violation at this stage. For the usual Fock quantization, the energy spectrum, in units of $\hbar\omega_{{\bf k}}$ is linearly spaced and the Lorentz invariant dispersion results from the propagator. With polymer quantization, the spectrum is not linearly spaced and leads to Lorentz violating dispersion relation. While one can construct Lorentz generators from the $\pi_{{\bf k}}$, $\phi_{\bf k}$ basic variables, one cannot do so with the $U_{\lambda{\bf k}} - U^{\dagger}_{\lambda{\bf k}}$ variables for $\lambda \neq 0$. Clearly, it is the polymer quantization which requires the use of $U_{\lambda{\bf k}}$ with non-zero $\lambda$ which is the source of this Lorentz violation at the level of ‘free field’. Importing the Lorentz violating dispersion relation into the usual computational scheme of perturbative quantum field theory based on [Fock]{} quantization with its UV divergences (especially the ‘quadratic divergences’), will lead to the ‘fine tuning’ problem discussed by Collins et al. [@CollinsPerezSudarsky] (see also [@JainRalston]). To check the viability of a polymer quantized field theory on a Minkowski background, one will need to develop a computational scheme for a polymer quantized, interacting theory and then test if at ‘low energies’ Lorentz violating effects are suppressed or not. 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Associate to each of its $V$ vertices, an intertwiner $c$ and finally take the trace (contract all indices). These functions of connections are naturally labelled by $(\gamma, j_1, \dots, j_E, c_1, \dots, c_V)$. Similar functions are defined for matter fields as well. [^2]: A necessary condition for a topological origin of $\gamma$ is thus satisfied. [^3]: $I, J = 0, 1, 2, 3$ are the Lorentz indices, $a = 1, 2, 3$ denote the spacial indices. The co-tetrad components are $e^0_t := N$, $e^i_t := N^aV^i_a$, $e^i_a := V^i_a$, $q_{ab} := V^i_a V^j_b \delta_{ij}$, $q := \det(q_{ab}) \neq 0 $. The Lorentz connection components are denoted as $\omega^{0i}_a =: K^i_a$, $\omega_a^{ij} =: {\cal E}^{ij}_{~\ k} \Gamma^k_a$ [@DateNew]. [^4]: Our conventions are: Greek letters denote space-time indices, lower case Latin letters denote space indices, upper case Latin ones denote Lorentz indices. Our signature is $(- + + +)$. The (metric independent) Levi-Civita symbols are ${\cal E}^{txyz} = +1 = {\cal E}_{0123}$. The determinant is defined by $ e\ {\cal E}^{IJKL} = - {\cal E}^{\mu\nu\alpha\beta}e^I_{\mu} e^J_{\nu} e^K_{\alpha} e^L_{\beta}.$ For the Dirac matrices: $2\eta^{IJ}\mathbb{1} = \gamma^I\gamma^J + \gamma^J\gamma^I$, $\sigma^{IJ} := \frac{1}{4}\left[ \gamma^I, \gamma^J\right]$, $\gamma_5 := i\gamma^0\gamma^1\gamma^2\gamma^3$, $\bar{\lambda} := \lambda^{\dagger}\gamma^0.$\[Notation\] [^5]: This would correspond to the choice of the so-called [time gauge]{} if we started without restricting the configurations a priori. [^6]: Strictly [Generalized Poisson brackets]{} [@HennauxTeitelbaum], due to the Grassmann nature of the fermions. [^7]: Discussed in Subsection \[Parities\]. [^8]: Definition of spinors depends on the orientation. Thus the action of parity on spinors needs a much more careful statement. We will assume for simplicity that the under orientation reversing diffeomorphism, the fermionic action remains invariant. [^9]: The overbar on a $\Psi$ with an index $A$, denotes the complex conjugate while one without an index denotes the Dirac conjugate: $\bar{\Psi} = \Psi^{\dagger}\gamma^0$. [^10]: In the units where $\hbar = 1 = c$, these variables are also [dimensionless]{}. This is because the action is dimensionless which implies $\Psi \sim L^{-3/2}$ and $\epsilon \sim L$. This is similar to the holonomy variables constructed from connections. For the similar variables corresponding to scalar fields, this is not so. [^11]: When matter and gravity are considered together, it is natural (sufficient) to take the set of points to be the [vertices]{} of graph labelling a spin-network function in the gravity sector (and matter gauge sector if present). [^12]: Only fermionic matter is sensitive to the action of the gravitational gauge group (subgroup of local Lorentz transformations). All other matter fields are [scalars]{} with respect to this action and thus do not contribute to the Gauss constraint.\[FootNote1\] [^13]: The cylindrical functions in this case are analogous to the gravitational spin network functions except for the replacement of the spin label by a [integer charge]{} label with charge conservation at each vertex and are called [charge network]{} functions [@Madhavan]. These have been called [flux networks]{} in [@ALReview]. [^14]: The subject of Lorentz invariance violation in all its aspects is rather large. We have included it here only to the extent of indicating how a computation of potentially observable signatures involving matter could be done. For review, we refer the reader to [@LorentzViolation].
--- abstract: 'This paper contains a rigorous mathematical example of direct derivation of the system of Euler hydrodynamic equations from Hamiltonian equations for $N$ point particle system as $N\to\infty$. Direct means that the following standard tools are not used in the proof: stochastic dynamics, thermodynamics, Boltzmann kinetic equations, correlation functions approach by N. N. Bogolyubov.' author: - 'Lykov A. A., Malyshev V. A. [^1]' title: 'From $N$-body problem to Euler equations' --- Key words: $N$-body problem, continuum mechanics, intersection of particle trajectories, Euler equations. Introduction ============ Classical mechanics, from mathematical point of view, is mostly developed in cases, where two extremely idealized forms of material objects are assumed - point particles (ordinary differential equations) and continuum media (partial differential equations). However, big difference exists in the ideology of these two theories: for point particles the model is defined by the choice of the interaction potential between particles, which is supposed to be known, but in the continuum mechanics the interaction is defined by the pressure, which is one of the unknown functions in the equations. Many papers - both mission proposals [@Bogoliubov; @Morrey] and concrete results for concrete models [@March_Pulvirenti; @Esposito1; @Esposito2; @Dobrushin] - discussed the connections between these two fields. We do not give here review of these papers as we do not use neither their results nor methods. Moreover, our approach is direct that is we do not use any of the following approaches: stochastic dynamics, thermodynamics, Boltzmann kinetic equations, correlation functions approach by N. N. Bogolyubov. Now we make the above claims more precise. Hamiltonian finite particle system is defined by the system of equations for the particle trajectories $x_{i}(t)\in R^{d},i=1,...,N$ $$m_{i}\frac{d^{2}x_{i}}{dt^{2}}=-\frac{\partial H}{\partial x_{i}}$$ with the Hamiltonian $H$. We define the continuum ($d$-dimensional) media as a bounded open subset $\Lambda\subset R^{d}$, the dynamics of this media is given by the system of such domains $\Lambda_{t},t\in[0,T),0<T\leq\infty,$ together with the system of diffeomorphisms $S^{t}:\Lambda=\Lambda_{0}\to\Lambda_{t},t\in[0,T)$, smooth also in $t$. The trajectory of the point (particle) $x\in\Lambda_{0}$ of the continuum media is the function $y(t,x)=S^{t}x$. The main unknown variable in the Euler equations is the velocity $u(t,y)$ of the particle, which at time $t$ is at point $y$. This definition of $u$ has sense iff such particle is unique, that is iff for any $t$ and any $x_{1}\neq x_{2}$ $$y(t,x_{1})\neq y(t,x_{2})$$ that is iff the trajectories (particles) $y(t,x)$ do not collide. This property obviously should be related to the similar property for $N$ particle system, if we want to obtain continuum media trajectories in the limit $N\to\infty$ (one could call this the ultralocal limit). We say that the $N$-particle system has no collisions, if for all $1\leq j<k\leq N$ and all $t\in[0,\infty)$ $$x_{j}(t)\neq x_{k}(t),$$ and has strong property of absence of collisions if $$\inf_{t\geqslant0}\inf_{j,k:j\neq k}\|(x_{j}(t)-x_{k}(t))\|>0\label{strong_nonintersection}$$ It is evident that there will not be any collisions if the repulsion between particles is sufficiently strong. However, for general Hamiltonian systems the following question is completely non trivial: for which initial conditions $x_{k}(0),\ \dot{x}_{k}(0),k=1,...,N,$ the system enjoys the absence of collisions property. In this paper the property (\[strong\_nonintersection\]) plays the central role. It is surprising that we did not find papers where this property is discussed in the derivation of continuum media equations. However, it was widely discussed in Celestial Mechanics (gravitation potential), see for example [@Siegel]. We consider the particle system on the real line with a particular Lennard-Jones type potential and prove that the particle trajectories of the $N$-particle system, for $N\to\infty$, converge, in the sense defined below, to the trajectories of the continuum particle system. Moreover, we get the system of 3 equations of the Euler type (which is considered in [@ChorinMarsden]) for the functions: $u(t,x)$ - the velocity, $p(t,x)$ - the pressure and $\rho(t,x)$ - the density $$\rho_{t}+u\rho_{x}+\rho u_{x}=0,\label{mass_conservation_1}$$ $$u_{t}+uu_{x}=-\frac{p_{x}}{\rho},\label{euler_1}$$ $$p=p(\rho),\label{equation_of_state_general}$$ In continuum mechanics these equations correspond to the conservation laws of mass, momentum and to the thermodynamic equation of state. In physics the first two equations are quite general. But the third one depends on the matter type and thermodynamic situation and should be given separately. In our derivation, all these equations and functions obtain simple and intuitive mechanical meaning (without probability theory and thermodynamics) for the $N$-particle system. In particular, the pressure can be considered as an analog of interaction potential in Hamiltonian mechanics. Main Results ============ #### The model We consider Hamiltonian system of $N$ particles (of unit mass) with coordinates $x_{1},...,x_{N}$ on $R$ and the Hamiltonian $$H=\sum_{k=1}^{N}\frac{v_{k}^{2}}{2}+U$$ The potential energy $U$ of the particle system with the coordinates $x_{1},x_{2},,...,x_{N}$ is defined by the interaction potential $$U=\sum_{1\leq k<l\leq N}\frac{\omega^{2}}{2}I(\|x_{k}-x_{l}\|),I\in\mathbf{I}(a,a_{1})$$ where $\mathbf{I}(a,a_{1}),0<a_{1}<a,$ is the class of functions $I(x)$ on $R_{+}$ with the following two properties 1\) $I(x)=(x-a)^{2}$ for $a-a_{1}<x<a+a_{1}$ with some constant $0<a_{1}<a$. 2\) $I(x)=const$ for $x\geq a+a_{1}$ 3\) $I(x)$ is arbitrary for $0<x\leq a-a_{1}$ #### Scaling Our system contains three parameters: $\omega,a$ and $a_{1}$. We could add also mass but the scaling of mass and/or time could be reduced to the scaling of $\omega$. If $N$ is large and all particles are situated on some finite interval then $a$ should be of order $N^{-1}$. We put $a=\frac{1}{N}$. Then the system will be in equilibrium (zero force on each particle) iff for all $k$ $x_{k+1}-x_{k}=\frac{1}{N}$. Correspondingly, we put $a_{1}=\frac{r}{N}$ for some $0<r<1$ not depending on $N$. The remaining parameter we choose as $$\omega=\omega'N\label{omega}$$ for some $\omega'>0$, not depending on $N$, #### Initial conditions We always assume the following initial conditions $$x_{1}(0)=0,\ \dot{x}_{1}(0)=v,\label{initial_1}$$ $$x_{k+1}(0)-x_{k}(0)=\frac{1}{N}X\left(\frac{k}{N}\right)>0,\quad\dot{x}_{k+1}(0)-\dot{x}_{k}(0)=\frac{1}{N}V\left(\frac{k}{N}\right),\ k=1,\ldots,N-1\label{initial_other}$$ for some $v\in\mathbb{R}$, and for some functions $X,V\in C^{4}([0,1]),$ where $X>0$. Thus, the functions $X$ and $V$ define smooth profile of the initial conditions. Then the kinetic and potential energies of the system will be of the order $O(N)$. It is convenient to assume also that $$X(0)=X(1)=1,\ V(0)=V(1)=0,\label{particles_1_N}$$ The second condition (\[particles\_1\_N\]) means that two leftmost (two rightmost) particles initially have almost (up to $O(N^{-2})$) equal velocities, and the first condition (\[particles\_1\_N\]) means that both boundary particles are subjected to almost zero force. Let $\Omega_{N}=\Omega_{N}(\gamma)$ be the domain of $R^{N}=\{(x_{1},...,x_{N})\}$, defined for some $0<\gamma<1$ by the estimates $$\frac{1-\gamma}{N}\leqslant x_{k+1}-x_{k}\leqslant\frac{1+\gamma}{N}$$ uniformly in $k=1,...,N$. We want to prove that if initially our system is in this region, then under certain conditions it will stay in $\Omega_{N}$ forever. The obvious corollary is that there will never be collisions between particles. It might seem that under such conditions there will not be interesting dynamics, but this is wrong, see pictures at the end of the paper. #### Absence of collisions for $N$-particle case The condition below allows to estimate distances between particles at any time moment. Denote $$\alpha=\int_{0}^{1}\|X''(y)\|\ dy,\,\,\,\beta=\int_{0}^{1}\|V''(y)\|\ dy.$$ Further on we will use the concrete value of $\gamma$, defined in terms of the main parameters of the system $$\gamma=\gamma(\alpha,\beta,\omega')=2\alpha+\frac{\beta N}{\omega}=2\alpha+\frac{\beta}{\omega'}\label{gamma_concrete}$$ Further on the condition $\gamma=\gamma(\alpha,\beta,\omega')<\min(r,\frac{1-r}{2})$ is always assumed. \[Th\_bound\] Assume that initially our system is in $\Omega_{N}$. Then it stays in $\Omega_{N}$ forever. That is for any $t\geqslant0$ and any $k=1,2,\ldots,N-1$ the following inequalities hold: $$\frac{1-\gamma}{N}\leqslant x_{k+1}(t)-x_{k}(t)\leqslant\frac{1+\gamma}{N}\label{gamma_bounds}$$ It follows that the particles never collide in the strong sense (\[strong\_nonintersection\]). Note that the scaling of $\omega$ is crucial to create the repulsion necessary for the particles did not collide. To understand the importance of the choice of $\alpha$, note that $X(y)-1$ characterizes the deviation of the chain from the equilibrium, $X'(y)$ characterizes the speed of change of this equilibrium, and $\alpha$ can be considered as the full variation. Thus the following simple statement is useful to estimate such deviation at initial time moment. \[X\_blizko\_1\] For any $y\in[0,1]$ $$1-\alpha\leqslant X(y)\leqslant1+\alpha$$ #### Strategy of the proof If we could prove Theorem \[Th\_bound\] for some potential in the class $\mathbf{I}(a,a_{1})$, then the $\gamma$-bounds (\[gamma\_bounds\]) indicate that it will also hold for any potential $I\in\mathbf{I}(a,a_{1}).$ In the following proofs we will use the simplest of such potential - the quadratic potential $$I(x_{k+1}-x_{k})=(x_{k+1}-x_{k}-a)^{2}$$ Even more, we assume the nearest neighbour interaction for this potential . Then the following system of linear differential equations holds $$\begin{aligned} \ddot{x}_{1}= & \ \omega^{2}(x_{2}-x_{1}-a),\label{mainEq1}\\ \ddot{x}_{k}= & \ \omega^{2}(x_{k+1}-x_{k}-a)-\omega^{2}(x_{k}-x_{k-1}-a),\ k=2,3,\ldots,N-1,\label{mainEqk}\\ \ddot{x}_{N}= & \ -\omega^{2}(x_{N}-x_{N-1}-a),\label{mainEqN}\end{aligned}$$ In this case we will also prove $\gamma$-bounds (\[gamma\_bounds\]). Now from these $\gamma$-bounds we want to show how from this the Theorem \[Th\_bound\] follows for any class of potentials $\mathbf{I}(\frac{1}{N},\frac{r}{N})$. It is sufficient to show that for any $t$ and $k$ the following inequalities hold $$\frac{1-r}{N}\leqslant x_{k+1}(t)-x_{k}(t)\leqslant\frac{1+r}{N},\label{r_1}$$ $$x_{k+2}(t)-x_{k}(t)>\frac{1+r}{N}\label{r_2}$$ Then (\[r\_1\]) obviously holds if $\gamma<r$. From the $\gamma$-bounds (\[gamma\_bounds\]) we have the estimate $$x_{k+2}(t)-x_{k}(t)\geq2\frac{1-\gamma}{N},$$ This estimate implies (\[r\_2\]) if $\gamma<\frac{1-r}{2}$. #### Convergence to continuous chain dynamics Denote $q(t,x)$ the solution of the wave equation $$q_{tt}=(\omega')^{2}q_{xx},$$ (here and below the lower indices define the derivatives in the corresponding variables) with fixed boundary conditions $$q(t,0)=q(t,1)=0,\label{q_wave_boundary}$$ and with the initial conditions: $$q(0,x)=X(x)-1,\ q_{t}(0,x)=V(x).\label{q_initial}$$ Let $x_{k}^{(N)}(t)$ be the solution of the main system (\[mainEq1\])-(\[mainEqN\]) for given $N$. For any fixed $t$ we want to define two functions (algorithms) which map the set of points $\Lambda_{t}$ of the continuous media to the set of particles $\{1,2,...,N\}$, that is to the set of particle coordinates $\{x_{k}^{(N)}(t)\}$ of the $N$-particle approximation. To do this, we will use two coordinate systems on the real intervals: $x$ and $z$, where $z(x)$, for $x\in(0,L),L=L(0)=\int_{0}^{1}X(y)dy,$ is uniquely defined from the equation: $$\int_{0}^{z(x)}X(x')dx'=x.\label{z_x}$$ In the first algorithm to any $z\in(0,1]$ correspond the particle with number $[zN]$ (the integer part of $zN\in\mathbb{R}$). Note that for any $0<z\leq1$ there exists $N(z)$ such that $N>N(z)$ we have $1<[zN]\leq N$. In the second algorithm to any point $x\in[0,L]$ corresponds the particle with number $k(x,N)=k(x,N,0)$ so that $$x_{k(x,N)}^{(N)}(0)\leqslant x<x_{k(x,N)+1}^{(N)}(0)\label{k_x_N_0}$$ Due to positivity of $X(y)$ such number is uniquely defined. Then it is natural to call the function $x_{k(x,N)}^{(N)}(t)$ the $N$-particle approximation of the trajectory of the particle $x\in[0,L]$ of the continuum media. By definition we put $x_{N+1}^{(N)}=\infty$. \[Th\_convergence\_1\] Let the conditions of the theorem \[Th\_bound\] hold, and assume also (\[omega\]). Then: 1\) for any $0<T<\infty$ uniformly in $t\in[0,T]$ and in $z\in(0,1]$ $$\lim_{N\rightarrow\infty}x_{1}^{(N)}(t)=G(t,0)=vt+(\omega')^{2}\int_{0}^{t}(t-s)q_{x}(s,0)\ ds,\label{G_t_0}$$ $$\lim_{N\rightarrow\infty}x_{[zN]}^{(N)}(t)=G(t,z)=G(t,0)+z+\int_{0}^{z}q(t,x')\ dx',\label{G_t_z}$$ 2) Let $0\leqslant z_{1}<z_{2}\leqslant1$. Then for any $t\geqslant0$ $$(1-\gamma)(z_{2}-z_{1})\leqslant G(t,z_{2})-G(t,z_{1})\leqslant(1+\gamma)(z_{2}-z_{1}),$$ where $\gamma$ is defined in (\[gamma\_concrete\]). Otherwise speaking, the continuum media particles do not collide; 3\) for any $T>0$ uniformly in $x\in[0,L]$ and in $t\in[0,T]$ $$\lim_{N\rightarrow\infty}x_{k(x,N)}^{(N)}(t)=y(t,x)=G(t,z(x)),\label{G_t_z_x}$$ 4) the function $G(t,z)$ satisfies the wave equation $$\frac{d^{2}G(t,z)}{dt^{2}}=(\omega')^{2}\frac{d^{2}G(t,z)}{dz^{2}}\label{G_wave}$$ with boundary conditions $G_{z}(t,0)=G_{z}(t,1)=1$ and initial conditions $$G(0,z)=\int_{0}^{z}X(x')dx',\quad G_{t}(0,z)=v+\int_{0}^{z}V(x')dx'.$$ An obvious corollary is that for any $T>0$, uniformly in $t\in[0,T]$, we have the following asymptotic limit for the length $L_{N}(t)=x_{N}(t)-x_{1}(t)$ of the chain $$\lim_{N\rightarrow\infty}L_{N}(t)=L(t)=1+\int_{0}^{1}q(t,x')dx'$$ #### Continuity equation (mass conservation law) Further on, the function $y(t,x)$ will be called the trajectory of the particle $x\in[0,L]$. Then the particles do not collide and one can unambiguously define the function $u(t,y)$ as the speed of the (unique) particle situated at time $t$ at the point $y$ , that is $$u(t,y(t,x))=\frac{dy(t,x)}{dt}.\label{yveq}$$ Also we will need the notation: $$Y_{0}(t)=y(t,0),\quad Y_{L}(t)=y(t,L).$$ For given $N$ we define the distribution function at time $t$: $$F^{(N)}(t,y)=\frac{1}{N}\|\{k\in\{1,2,\ldots,N\}:\ x_{k}^{(N)}(t)\leqslant y\}\|,\ y\in\mathbb{R}$$ where $\|\cdot\|$ is the number of particle in the set. \[Lemma\_distrFunc\] Denote $x(t,y)\in[0,L]$ the (unique) particle, which reached the point $y$ at time $t$, that is $$y(t,x(t,y))=y.\label{xtydef}$$ Then uniformly in $y\in[Y_{0}(t),Y_{L}(t)]$ and in $t\in[0,T]$, for any $T<\infty$, we have $$\lim_{N\rightarrow\infty}F^{(N)}(t,y)=z(x(t,y))=F(t,y),$$ where the (smooth) function $z(x)$ is the same as the one introduced in (\[z\_x\]). In connection with Lemma \[Lemma\_distrFunc\] define the density by the formula $$\rho(t,y)=\frac{dF(t,y)}{dy}=\frac{dz(x(t,y))}{dy}\label{density}$$ \[Th\_mass\_conservation\] For any $t\geqslant0,\ y\in[Y_{0}(t),Y_{L}(t)]$ $$\frac{\partial\rho(t,y)}{\partial t}+\frac{d}{dy}\left(u(t,y)\rho(t,y)\right)=0.\label{mass conservation_2}$$ #### Euler equation (momentum conservation) and equation of state \[Th\_Euler\] For any $t\geqslant0,\ y\in[Y_{0}(t),Y_{L}(t)]$ we have: $$\frac{\partial u(t,y)}{\partial t}+u(t,y)\frac{\partial u(t,y)}{\partial y}=-(\omega')^{2}\frac{1}{\rho(t,y)}\frac{d}{dy}\frac{1}{\rho(t,y)}=-\frac{1}{\rho(t,y)}\frac{dp(t,y)}{dy},\label{EulerEq_2}$$ if we put $$p(t,y)=-\frac{(\omega')^{2}}{\rho(t,y)}+C.\label{state_equation_2}$$ Constant $C$ can be chosen as $$C=(\omega')^{2},$$ so that at equilibrium (when $\rho=1$) the pressure were zero. ### Right side of the Euler equation as the limit of interaction forces {#right-side-of-the-euler-equation-as-the-limit-of-interaction-forces .unnumbered} For given $y$ and $t$ define the number $k(y,N,t)$ so that $$x_{k(y,N,t)}^{(N)}(t)\leqslant y<x_{k(y,N,t)+1}^{(N)}(t)\label{k_y_N_t}$$ Consider the point $y\in[Y_{0}(t),Y_{L}(t)]$ and the force acting on the particle with number $k(y,N,t)$: $$R^{(N)}(t,y)=\omega^{2}\left(x_{k(y,N,t)+1}^{(N)}-x_{k(y,N,t)}^{(N)}-\frac{1}{N}\right)-\omega^{2}\left(x_{k(y,N,t)}^{(N)}-x_{k(y,N,t)-1}^{(N)}-\frac{1}{N}\right)$$ \[Th\_force\] Let the conditions of the theorem \[Th\_convergence\_1\] hold. Then for any $0<T<\infty$, uniformly in $y\in[Y_{0}(t),Y_{L}(t)]$ and in $t\in[0,T]$ the following equality holds: $$\lim_{N\rightarrow\infty}R^{(N)}(t,y)=R(t,y)=-\frac{1}{\rho(t,y)}\frac{dp(y)}{dy},$$ where the functions $p,\rho$ are the same as in theorem \[Th\_Euler\]. Thus, the pressure can be considered as a continuous interaction potential for continuum media, an analog of interaction potentials in Hamiltonian particle mechanics. #### Limit of the energy Define the potential and kinetic energy of the particle with number $k(y,N,t)$ at time $t$ for $N$-particle approximation correspondingly as: $$\begin{aligned} U^{(N)}(t,y)= & \frac{1}{4}\omega^{2}\left(x_{k(y,N,t)+1}^{(N)}-x_{k(y,N,t)}^{(N)}-\frac{1}{N}\right)^{2}+\frac{1}{4}\omega^{2}\left(x_{k(y,N,t)}^{(N)}-x_{k(y,N,t)-1}^{(N)}-\frac{1}{N}\right)^{2},\\ T^{(N)}(t,y)= & \ \frac{1}{2}\left(\dot{x}_{k(y,N,t)}^{(N)}\right)^{2}.\end{aligned}$$ \[Th\_Energy\] For any $t,y$ (uniformly as in the previous theorem) the following limits hold: $$\begin{aligned} U(t,y)=\lim_{N\rightarrow\infty}U^{(N)}(t,y)= & \ \frac{1}{2}(\omega')^{2}\left(\frac{1}{\rho(t,y)}-1\right)^{2}=\frac{1}{2(\omega')^{2}}p^{2}(t,y),\\ T(t,y)=\lim_{N\rightarrow\infty}T^{(N)}(t,y)= & \ \frac{1}{2}u^{2}(t,y)\end{aligned}$$ Proofs ====== As we explained previously, we shall always use the system (\[mainEq1\])-(\[mainEqN\]). Proof of Theorem \[Th\_bound\] ------------------------------ #### Proof of Lemma \[X\_blizko\_1\] Put $f(y)=X(y)-1$ and use the following Lemma \[zeroEndsLemma\]. \[zeroEndsLemma\] Assume that $f\in C^{2}([0,1])$ and $f(0)=f(1)=0$. Then the following inequality holds: $$\sup_{y\in[0,1]}\|f(y)\|\leqslant\int_{0}^{1}\|f''(y)\|\ dy.$$ In fact, $$f(y)=\int_{0}^{y}f'(x)\ dx.$$ It follows that $$\sup_{y\in[0,1]}\|f(y)\|\leqslant\int_{0}^{1}\|f'(x)\|\ dx.$$ As $f(0)=f(1)=0$, then there exists point $x^{}\in(0,1)$ such that $$f'(x^{})=0.$$ Thus we have: $$\int_{0}^{1}\|f'(x)\|\ dx=\int_{0}^{1}\left\|\int_{x}^{x^{}}f''(u)du\right\|\ dx\leqslant\sup_{x\in[0,1]}\left\|\int_{x}^{x^{}}f''(u)du\right\|\leqslant\int_{0}^{1}\|f''(u)\|\ du.$$ This proves the Lemma. The set of functions $\Sigma$, satisfying the conditions of Lemma \[zeroEndsLemma\] is a linear space. Moreover, $$\alpha(f)=\int_{0}^{1}\|f''(y)\|\ dy.$$ defines a norm on this space. Lemma \[zeroEndsLemma\] states that the uniform norm does not exceed the norm $\alpha(\cdot)$. However, we want to note that these two norms are not equivalent. In fact, assume the contrary, i. e. that there exists constant $a>0$ such that for any function $f\in\Sigma$ $$\alpha(f)\leqslant a\sup_{y\in[0,1]}\|f(y)\|.$$ Then put $f_{k}(y)=\sin\pi ky$. Then $\alpha(f_{k})\rightarrow\infty$ as $k\rightarrow\infty$, but $\sup_{y\in[0,1]}\|f_{k}(y)\|=1$. This is a contradiction. #### Deviation variables Define the deviation variables $q_{k}(t)=x_{k+1}(t)-x_{k}(t)-a,\ k=1,\ldots,N-1$, and put by definition $q_{0}=q_{N}=0$, Then the functions $q_{k}$ satisfy the equations: $$\ddot{q}_{k}=\omega^{2}(q_{k+1}-q_{k})-\omega^{2}(q_{k}-q_{k-1})=\omega^{2}(q_{k+1}-2q_{k}+q_{k-1}),\ k=1,2,\ldots,N-1\label{diffeqk}$$ with initial conditions which follow from (\[initial\_other\]) $$q_{k}(0)=\frac{1}{N}X\left(\frac{k}{N}\right)-\frac{1}{N},\quad\dot{q}_{k}(0)=\frac{1}{N}V\left(\frac{k}{N}\right),\ k=0,1,\ldots,N$$ In fact, from equations (\[mainEq1\])-(\[mainEqN\]) we have $$\ddot{x}_{k}=\omega^{2}(q_{k}-q_{k-1}),\,\,\,k=1,...,N,$$ $$\ddot{x}_{k}=\ddot{x}_{k+1}-\ddot{q}_{k},k=1,...,N-1,$$ Then for $k=1,\ldots,N-1$ $$\omega^{2}(q_{k+1}-q_{k})-\ddot{q}_{k}=\omega^{2}(q_{k}-q_{k-1}).$$ The last equality is equivalent to (\[diffeqk\]). The inverse transformation is given by $$x_{k}(t)=x_{1}(t)+\frac{k-1}{N}+\sum_{i=1}^{k-1}q_{i}(t),\quad k=2,3,\ldots,N.\label{xkqk}$$ and for $x_{1}(t)$, by definition of $q_{1}$, we have the equation: $$\ddot{x}_{1}=\omega^{2}q_{1}.$$ It follows $$x_{1}(t)=x_{1}(0)+\dot{x}_{1}(0)t+\omega^{2}\int_{0}^{t}ds\int_{0}^{s}q_{1}(s')\ ds'=x_{1}(0)+\dot{x}_{1}(0)t+\omega^{2}\int_{0}^{t}(t-s)q_{1}(s)\ ds\label{x1formula}$$ The last equality follows from comparison of derivatives of both sides. One can rewrite the system (\[diffeqk\]) in the matrix form: $$\ddot{q}=-Wq,$$ where the matrix $W$ is a three diagonal non negative definite $(n\times n)$-matrix with $n=N-1$, and $q=(q_{1}(t),\ldots,q_{n}(t))^{T}$ is a column vector. #### Spectrum of the matrix $W$ We will show that $W$ is positive definite, and will find the basis $v_{1},\ldots,v_{n}$ of eigenvectors of $W$, with corresponding eigenvalues $\lambda_{1},\ldots,\lambda_{n}$: $Wv_{j}=\lambda_{j}v_{j},\ j=1,\ldots,n$. Let $e_{k}$ be the standard unit coordinate vectors in $\mathbb{R}^{n}$. For $j=1,\ldots,N-1$ define vectors $v_{j}$ by $$y_{j}(k)=(v_{j},e_{k})=\sqrt{\frac{2}{N}}\sin\frac{\pi jk}{N},\ k=1,2,\ldots,N-1.\label{specDef}$$ and the numbers $\lambda_{j}$ by $$\lambda_{j}=4\omega^{2}\sin^{2}\frac{\pi j}{2N}.$$ Let us prove that $v_{j}$ are eigenvectors of $W$ with eigenvalues $\lambda_{j}$. Note that if we define $y_{j}(k)$ from (\[specDef\]) also at the points $k=0$ and $k=N$, then we get $y_{j}(0)=y_{j}(N)=0$. Then for all $k=1,\ldots,N-1$ $$(Vv_{j},e_{k})=-\omega^{2}(y_{j}(k+1)-y_{j}(k))+\omega^{2}(y_{j}(k)-y_{j}(k-1))=$$ $$=2\omega^{2}\sqrt{\frac{2}{N}}\left(-\sin\frac{\pi j}{2N}\cos\frac{\pi j(2k+1)}{2N}+\sin\frac{\pi j}{2N}\cos\frac{\pi j(2k-1)}{2N}\right)$$ $$=-2\omega^{2}\sqrt{\frac{2}{N}}\sin\frac{\pi j}{2N}\left(\cos\frac{\pi j(2k+1)}{2N}-\cos\frac{\pi j(2k-1)}{2N}\right)=$$ $$=4\omega^{2}\sqrt{\frac{2}{N}}\sin\frac{\pi j}{2N}\sin\frac{\pi jk}{N}\sin\frac{\pi j}{2N}=\lambda_{j}y_{j}(k).$$ where $(,)$ is the standard scalar product in $\mathbb{R}^{n}$. As all $\lambda_{j}$ positive and different, then $W$ is positive definite. #### Dynamics of deviations Let for any $j=1,\ldots,N-1$ $$Q_{j}=\sqrt{\frac{2}{N}}\sum_{i=1}^{N-1}q_{i}(0)\sin\frac{\pi ij}{N},\quad P_{j}=\sqrt{\frac{2}{N}}\sum_{i=1}^{N-1}\dot{q}_{i}(0)\sin\frac{\pi ij}{N},\quad\omega_{j}=2\omega\sin\frac{\pi j}{2N}.$$ Then $$q_{k}(t)=\sqrt{\frac{2}{N}}\sum_{j=1}^{N-1}\left(Q_{j}\cos\omega_{j}t+P_{j}\frac{\sin\omega_{j}t}{\omega_{j}}\right)\sin\frac{\pi jk}{N},$$ Proof. Using the expansion of $q(t)$ in the basis $$q(t)=\sum_{j=1}^{n}Q_{j}(t)v_{j},\quad Q_{j}(t)=(q(t),v_{j}),$$ we will get equations for $Q_{j}(t)$: $$\ddot{Q}_{j}=-\lambda_{j}Q_{j},$$ This gives $$Q_{j}(t)=Q_{j}(0)\cos\omega_{j}t+\dot{Q}_{j}(0)\frac{\sin\omega_{j}t}{\omega_{j}},\quad\omega_{j}=\sqrt{\lambda_{j}}.$$ Then $$q_{k}(t)=(q(t),e_{k})=\sum_{j=1}^{n}Q_{j}(t)(v_{j},e_{k})=\sum_{j=1}^{n}\left(Q_{j}(0)\cos\omega_{j}t+\dot{Q}_{j}(0)\frac{\sin\omega_{j}t}{\omega_{j}}\right)(v_{j},e_{k}),$$ and the Lemma is proved. #### Estimate of the coefficients $Q_{j},P_{j}$ We have $$Q_{j}=\sqrt{\frac{2}{N}}\sum_{i=1}^{N-1}q_{i}(0)\sin\frac{\pi ij}{N}=(q(0),v_{j})=\frac{1}{\lambda_{j}}(q(0),Vv_{j})=\frac{1}{\lambda_{j}}(Vq(0),v_{j})=$$ $$=-\frac{\omega^{2}}{\lambda_{j}}\sqrt{\frac{2}{N}}\sum_{i=1}^{N-1}\left((q_{i+1}(0)-q_{i}(0))-(q_{i}(0)-q_{i-1}(0))\right)\sin\frac{\pi ij}{N}.$$ Then let us estimate the sum $$S_{N}=\sum_{i=1}^{N-1}\left((q_{i+1}(0)-q_{i}(0))-(q_{i}(0)-q_{i-1}(0))\right)\sin\frac{\pi ij}{N}$$ where $$q_{i}(0)=\frac{1}{N}X(\frac{i}{N})-\frac{1}{N},\,\,\,i=0,1,...,N$$ We have $$q_{i+1}(0)-q_{i}(0)=\frac{1}{N}\left(X\left(\frac{i+1}{N}\right)-X\left(\frac{i}{N}\right)\right)=\frac{1}{N^{2}}X'(\theta_{i}),$$ for some point $\theta_{i}\in(\frac{i}{N},\frac{i+1}{N})$. This gives $$\|S_{N}\|=\frac{1}{N^{2}}\left\|\sum_{i=1}^{N-1}(X'(\theta_{i})-X'(\theta_{i-1}))\sin\frac{\pi ij}{N}\right\|\leqslant\frac{1}{N^{2}}\sum_{i=1}^{N-1}\left\|X'(\theta_{i})-X'(\theta_{i-1}))\right\|\leqslant\frac{1}{N^{2}}\int_{0}^{1}\|X''(y)\|\ dy.$$ and thus $$\|Q_{j}\|\leqslant\frac{1}{4\sin^{2}\frac{\pi j}{2N}}\sqrt{\frac{2}{N}}\frac{\alpha}{N^{2}},\quad\alpha=\int_{0}^{1}\|X''(y)\|\ dy.$$ Taking into account the inequality $\omega_{j}=\omega\sin\frac{\pi j}{2N}\geqslant\omega\frac{j}{N}$, we get $$\|Q_{j}\|\leqslant\frac{1}{4}\sqrt{\frac{2}{N}}\frac{\alpha}{j^{2}}.$$ Similar estimates holds for $P_{j}$: $$\|P_{j}\|\leqslant\frac{1}{4}\sqrt{\frac{2}{N}}\frac{\beta}{j^{2}},\quad\beta=\int_{0}^{1}\|V''(y)\|\ dy.$$ This gives the final estimate $$\|q_{k}(t)\|\leqslant\frac{1}{4}\frac{2}{N}\sum_{k=1}^{N-1}(\frac{\alpha}{j^{2}}+\frac{\beta N}{2\omega j^{3}})\leqslant\frac{\gamma}{N},\quad\gamma=2\alpha+\frac{\beta N}{\omega}.$$ Proof of Theorem \[Th\_convergence\_1\] --------------------------------------- We will denote now $q_{k}(t)=q_{k}^{(N)}(t)$, emphasizing the dependence on $N$. \[qklimit\] Assume the conditions of Theorem \[Th\_convergence\_1\]. Then for any $T>0$ $$\max_{t\in[0,T]}\max_{k=1,\ldots,N-1}\left\|q_{k}^{(N)}(t)-\frac{1}{N}q(t,\frac{k}{N})\right\|\leqslant\frac{c\ln N}{N^{3}},$$ for some constant $c>0$ not depending on $N$. Proof. Consider the difference $$\Delta_{k}^{(N)}(t)=q_{k}^{(N)}(t)-\frac{1}{N}q(t,\frac{k}{N}),\ k=0,\ldots,N$$ For any $k=1,\ldots,N-1$ we have $$\ddot{\Delta}_{k}^{(N)}(t)=\omega^{2}(q_{k+1}^{(N)}-q_{k}^{(N)})-\omega^{2}(q_{k}^{(N)}-q_{k-1}^{(N)})-\frac{1}{N}(\omega')^{2}q_{xx}(t,\frac{k}{N})$$ Note that for all $k=1,\ldots,N-1$ $$(q(t,\frac{k+1}{N})-q(t,\frac{k}{N}))-(q(t,\frac{k}{N})-q(t,\frac{k-1}{N}))=\frac{1}{N^{2}}q_{xx}(t,\frac{k}{N})+r_{k}^{(N)}(t),$$ and moreover the remainder term can be estimated as $$\|r_{k}^{(N)}(t)\|\leqslant\frac{1}{12N^{4}}\max_{t\in[0,T]}\max_{x\in[0,1]}\left\|\frac{d^{4}q(t,x)}{dx^{4}}\right\|=\frac{c_{1}}{N^{4}}.$$ Then we have the equations $$\ddot{\Delta}_{k}^{(N)}(t)=\omega^{2}(\Delta_{k+1}^{(N)}-\Delta_{k}^{(N)})-\omega^{2}(\Delta_{k}^{(N)}-\Delta_{k-1}^{(N)})+r_{k}^{(N)}(t)(\omega')^{2}N$$ with initial conditions $$\Delta_{k}^{(N)}(0)=0,\quad\dot{\Delta}_{k}^{(N)}(0)=0,\,\,\,k=0,...,N$$ Introduce the vectors $$\Delta^{(N)}(t)=(\Delta_{1}^{(N)}(t),\ldots,\Delta_{N-1}^{(N)}(t))^{T},\quad r^{(N)}(t)=(r_{1}^{(N)}(t),\ldots,r_{N-1}^{(N)}(t))^{T}.$$ Then we have the equation $$\ddot{\Delta}^{(N)}=-W\Delta^{(N)}+r^{(N)}(t)(\omega')^{2}N,$$ where the matrix $W$ was introduced in the proof of Theorem \[Th\_bound\]. It is easy to see that the solution of this equation is $$\Delta^{(N)}(t)=(\omega')^{2}N\int_{0}^{t}(\sqrt{W})^{-1}\sin\sqrt{W}(t-s)r^{(N)}(s)\ ds,$$ where $\sqrt{W}$ is the positive definite square root of the matrix $W$. Thus $$\Delta_{k}^{(N)}(t)=(\Delta^{(N)}(t),e_{k})=(\omega')^{2}N\sum_{j=1}^{N-1}\frac{(v_{j},e_{k})}{\omega_{j}}\int_{0}^{t}\sin\omega_{j}(t-s)(r^{(N)}(s),v_{j})\ ds.$$ For all $s\in[0,t],\ j=1,\ldots,N-1$ we have the inequality: $$\|(r^{(N)}(s),v_{j})\|\leqslant\sqrt{\frac{2}{N}}\frac{c_{1}}{N^{3}}$$ The consequence is that for all $t\in[0,T],\ k=1,\ldots,N-1$ the following estimate holds: $$\|\Delta_{k}^{(N)}(t)\|\leqslant(\omega')^{2}N\frac{2}{N}\frac{c_{1}}{N^{3}}\sum_{j=1}^{N-1}\frac{T}{2N\omega'\sin\frac{\pi j}{2N}}\leqslant(\omega')^{2}N\frac{1}{N}\frac{Tc_{1}}{\omega'N^{3}}\sum_{j=1}^{N-1}\frac{1}{j}\leqslant\frac{c_{2}\ln N}{N^{3}},$$ for some constant $c_{2}>0$, not depending on $N$. The Lemma is proved. #### Proof of the assertion 1) of Theorem \[Th\_convergence\_1\] Note that for any $t\in[0,T]$ $$q(t,\frac{1}{N})=\frac{1}{N}q_{x}(t,0)+\frac{r(t)}{N^{2}},$$ where $\|r(t)\|\leqslant c$ for some constant $c>0$, not depending on $N$. That is why from the equality (\[x1formula\]) and Lemma \[qklimit\], we get that uniformly in $t\in[0,T]$ the following limiting equality holds $$\lim_{N\rightarrow\infty}x_{1}^{(N)}(t)=vt+(\omega')^{2}\int_{0}^{t}(t-s)q_{x}(s,0)\ ds.$$ Using the equality (\[xkqk\]) and Lemma \[qklimit\], we get: $$x_{[zN]}^{(N)}(t)=x_{1}(t)+z+\frac{1}{N}\sum_{k=1}^{[zN]-1}q(t,\frac{k}{N})+r^{(N)}(t,z),$$ and moreover, there exists constant $C>0$ such that $\|r^{(N)}(t,z)\|\leqslant\frac{C}{N}$ for all $z\in[0,1],\ t\in[0,T]$. Taking the limit in this equality we get the assertion of the Theorem. #### Proof of assertion 2) From evident equality $$x_{[z_{2}N]}(t)-x_{[z_{1}N]}(t)=\sum_{k=[z_{1}N]}^{[z_{2}N]-1}(x_{k+1}(t)-x_{k}(t))$$ and from Theorem \[Th\_bound\] we get the estimate: $$\frac{1-\gamma}{N}([z_{2}N]-[z_{1}N])\leqslant x_{[z_{2}N]}(t)-x_{[z_{1}N]}(t)\leqslant\frac{1+\gamma}{N}([z_{2}N]-[z_{1}N]).$$ Taking the limit here we get the assertion. #### Proof of assertion 3) Firstly, let us prove that for some constant $c>0$, not depending on $N$, for all $x\in[0,L]$ $$\left\|\frac{k(x,N)}{N}-z(x)\right\|\leqslant\frac{c}{N}\label{k_x_N_inequality}$$ Denote $$f(z)=\int_{0}^{z}X(x')dx'.$$ Then we have $f(z(x))=x$. On the other side, the integral can be calculated as follows $$f(\frac{k(x,N)}{N})=\frac{1}{N}\sum_{i=1}^{k(x,N)}X(\frac{i}{N})+r_{N}(x)=x_{k(x,N)+1}(0)+r_{N}(x),$$ where the remainder term enjoys the following estimate: $$\|r_{N}(x)\|\leqslant\frac{1}{N}\max_{y\in[0,1]}\|X'(y)\|=\frac{c_{1}}{N}.$$ By definition of $k(x,N)$ we have: $$x-\frac{c_{1}}{N}\leqslant f(\frac{k(x,N)}{N})<x+\frac{1}{N}X(\frac{k(x,N)}{N})+\frac{c_{1}}{N}<x+\frac{c_{2}}{N},\ c_{2}=c_{1}+\max_{y\in[0,1]}X(y).$$ The following inequality follows: $$\|f(\frac{k(x,N)}{N})-f(z(x))\|\leqslant\frac{c_{2}}{N}.$$ But also for some point $\theta\in[0,1]$ $$f(\frac{k(x,N)}{N})-f(z(x))=(\frac{k(x,N)}{N}-z(x))f'(\theta),$$ This gives $$\left\|\frac{k(x,N)}{N}-z(x)\right\|\leqslant\frac{1}{\min_{y\in[0,1]}X(y)}\frac{c_{2}}{N}=\frac{c}{N}.$$ From the proved inequality (\[k\_x\_N\_inequality\]) it follows that $$\|k(x,N)-[z(x)N]\|\leqslant c',\ c'=c+1\label{k_N_bound_1}$$ Then by Theorem \[Th\_bound\] $$\|x_{[z(x)N]}(t)-x_{k(x,N)}(t)\|\leqslant c\frac{1+\gamma}{N}$$ Taking the limit in the last inequality we get the assertion. #### Proof of Lemma 2 We will use the particle numbers $k(y,N,t)$, introduced in (\[k\_y\_N\_t\]). By definition we take $x_{0}^{(N)}(t)=Y_{0}(t),\ x_{N+1}^{(N)}(t)=Y_{L}(T)$. It is clear that $$F^{(N)}(t,y)=\frac{k(y,N,t)}{N}.$$ Further on for given $N$ we consider particle trajectories for the initial points $x_{k(y,N,t)}^{(N)}(0)$ and $x_{k(x(t,y),N)}^{(N)}(0)$. We want to prove that at time $t$ the distance between them does not exceed $c/N$. Using theorem \[Th\_bound\], we will show that $k(y,N,t)$ differs from $k(x(t,y),N)$ not more than on some constant. Lemma will follow from this. Now we give the formal proof. We use the inequalities: $$\|x_{k(x(t,y),N)}^{(N)}(t)-x_{k(y,N,t)}^{(N)}(t)\|\leqslant\|x_{k(x(t,y),N)}^{(N)}(t)-y(t,x(t,y))\|+\|x_{k(y,N,t)}^{(N)}(t)-y\|$$ By assertions 1), 2), 3) of Theorem 2, and its proof, we can conclude, that the following inequality holds: $$\|x_{k(x(t,y),N)}^{(N)}(t)-y(t,x(t,y)))\|\leqslant\frac{c_{1}}{N},$$ for some constant $c_{1}>0$ not depending on $N$ and $y$. Then by definition of $k(y,N,t)$ and Theorem \[Th\_bound\] we have the estimate for $0<k(y,N,t)<N$: $$\|x_{k(y,N,t)}^{(N)}(t)-y\|\leqslant\|x_{k(y,N,t)}^{(N)}(t)-x_{k(y,N,t)+1}^{(N)}(t)\|\leqslant\frac{c_{2}}{N},$$ for some constant $c_{2}>0$ not depending on $N,y$, . In cases $k(y,N,t)=N$ and $k(y,N,t)=0$ the latter inequality follows from Theorem \[Th\_convergence\_1\]. Then $$\|x_{k(x(t,y),N)}^{(N)}(t)-x_{k(y,N,t)}^{(N)}(t)\|\leqslant\frac{c}{N},\quad c=c_{1}+c_{2}.$$ From this inequality and Theorem \[Th\_bound\] we have $$\|k(x(t,y),N)-k(y,N,t)\|\leqslant c',\label{k_N_bound_2}$$ for some constant $c'>0$, not depending on $N,y$. We can conclude that $$\lim_{N\rightarrow\infty}\frac{k(y,N,t)}{N}=\lim_{N\rightarrow\infty}\frac{k(x(t,y),N)}{N}=z(x(t,y)),$$ where the latter equality follows from the proof of Theorem \[Lemma\_distrFunc\], assertion 3. The Lemma is thus proved. #### Proof of assertion 4) The simple calculation gives with (\[q\_wave\_boundary\]),(\[G\_t\_z\]) $$\frac{d^{2}G(t,z)}{dz^{2}}=q_{z}(t,z).$$ $$\frac{d^{2}G(t,z)}{dt^{2}}=\frac{d^{2}G(t,0)}{dt^{2}}+\int_{0}^{z}q_{tt}(t,z')dz'=(\omega')^{2}q_{z}(t,0)+(\omega')^{2}\int_{0}^{z}q_{zz}(t,z')dz'=$$ $$=(\omega')^{2}(q_{z}(t,z)-q_{z}(t,0))+(\omega')^{2}q_{z}(t,0)=(\omega')^{2}q_{z}(t,z)$$ The boundary and initial conditions can be easily found from the corresponding conditions on the function $q(t,x)$ Proof of Theorem \[Th\_mass\_conservation\] ------------------------------------------- By definition (\[density\]) we have $$\rho(t,y)=z'(x(t,y))x_{y}(t,y).$$ On the other side, differentiation in $y$ of the equality (\[xtydef\]) gives: $$x_{y}(t,y)=\frac{1}{y_{x}(t,x(t,y))}.\label{xderivy}$$ Hence, $$y_{x}(t,x(t,y))=\frac{z'(x(t,y))}{\rho(t,y)}.\label{yderivx}$$ By definition $$\frac{\partial\rho(t,y)}{\partial t}=\frac{d}{dy}\frac{dz(x(t,y)}{dt}=\frac{d}{dy}\left(x_{t}(t,y)z'(x(t,y))\right).$$ Differentiating in $t$ the equality (\[xtydef\]) we get: $$x_{t}(t,y)=-\frac{\frac{\partial y(t,x(t,y))}{\partial t}}{y_{x}(t,x(t,y))}=-\frac{u(t,y)}{y_{x}(t,x(t,y))}=-\frac{u(t,y)\rho(t,y)}{z'(x(t,y))}.$$ The theorem is thus proved. Proof of Theorem \[Th\_Euler\] ------------------------------ We need the following Lemma. For all $t\geqslant0,\ x\in[0,L]$ $$\frac{\partial u(t,y(t,x))}{\partial t}+u(t,y(t,x))\frac{\partial u(t,y(t,x))}{\partial y}=$$ $$=(\omega')^{2}G_{zz}(t,z(x))=(\omega')^{2}\frac{y_{xx}(t,x)-\frac{z''(x)}{z'(x)}y_{x}(t,x)}{[z'(x)]^{2}}.\label{velocityEqLemma}$$ Proof of the Lemma. The left hand side of the formula (\[velocityEqLemma\]) is the complete derivative of $u(t,y(t,x))$ in $t$, that is $\frac{du(t,y(t,x))}{dt}$. On the other side, we have by definition: $$\frac{du(t,y(t,x))}{dt}=\frac{d^{2}y(t,x)}{dt^{2}}=\frac{d^{2}G(t,z(x))}{dt^{2}}=(\omega')^{2}\frac{d^{2}G(t,z(x))}{dz^{2}}.$$ Moreover the following formulas hold: $$\begin{aligned} y_{x}(t,x)= & \ \frac{dy(t,x)}{dx}=\frac{dG(t,z(x))}{dx}=z'(x)\frac{dG(t,z(x))}{dz}=z'(x)G_{z}(t,z(x)),\label{y_po_x}\\ y_{xx}(t,x)= & \ \frac{d^{2}y(t,x)}{dx^{2}}=[z'(x)]^{2}\frac{d^{2}G(t,z(x))}{dz^{2}}+z''(x)\frac{dG(t,z(x))}{dz}=[z'(x)]^{2}G_{zz}(t,z(x))+z''(x)G_{z}(t,z(x)).\nonumber \end{aligned}$$ Then $$G_{zz}(t,z(x))=\frac{y_{xx}(t,x)-\frac{z''(x)}{z'(x)}y_{x}(t,x)}{[z'(x)]^{2}}.$$ and the Lemma is proved. Let us prove now theorem \[Th\_Euler\]. Putting $x=x(t,y)$ in the equation (\[velocityEqLemma\]) gives the following equation: $$\frac{\partial u(t,y)}{\partial t}+u(t,y)\frac{\partial u(t,y)}{\partial y}=R(t,y),$$ where we introduced the function: $$R(t,y)=(\omega')^{2}\frac{y_{xx}(t,x(t,y))-\frac{z''(x(t,y))}{z'(x(t,y))}y_{x}(t,x(t,y))}{[z'(x(t,y))]^{2}}.\label{R_t_y}$$ Differentiating the equality (\[yderivx\]) in $y$ and using (\[xderivy\]), we get: $$y_{xx}(t,x(t,y))=\frac{1}{x_{y}(t,y)}\frac{d}{dy}\frac{z'(x(t,y))}{\rho(t,y)}=\frac{z'(x(t,y))}{\rho(t,y)}\frac{d}{dy}\frac{z'(x(t,y))}{\rho(t,y)}=$$ $$=\frac{z'(x(t,y))}{\rho(t,y)}\left(\frac{z''(x(t,y))x_{y}(t,y)}{\rho(t,y)}-\frac{z'(x(t,y))\rho_{y}(t,y)}{\rho^{2}(t,y)}\right)=$$ $$=\frac{z'(x(t,y))}{\rho(t,y)}\left(\frac{z''(x(t,y))}{z'(x(t,y))}-\frac{z'(x(t,y))\rho_{y}(t,y)}{\rho^{2}(t,y)}\right)=\frac{1}{\rho(t,y)}\left(z''(x(t,y))-\frac{[z'(x(t,y))]^{2}\rho_{y}(t,y)}{\rho^{2}(t,y)}\right).$$ That is why the function $R(t,y)$ can be written in terms of the density $$R(t,y)=(\omega')^{2}\frac{1}{\rho(t,y)}\left(\frac{z''(x(t,y))-\frac{[z'(x(t,y))]^{2}\rho_{y}(t,y)}{\rho^{2}(t,y)}-z''(x(t,y))}{[z'(x(t,y))]^{2}}\right)=$$ $$=-(\omega')^{2}\frac{1}{\rho(t,y)}\frac{\rho_{y}(t,y)}{\rho^{2}(t,y)}=(\omega')^{2}\frac{1}{\rho(t,y)}\frac{d}{dy}\frac{1}{\rho(t,y)}$$ Thus all assertions of the theorem are proved. Proof of the theorem on the force and energy -------------------------------------------- #### Proof of the Theorem \[Th\_force\] Write down the force $R^{(N)}(t,y)$ in terms of $q$ variables, introduced in the proof of Theorem \[Th\_bound\] $$R^{(N)}(t,y)=\omega^{2}(q_{k(y,N,t)}-q_{k(y,N,t)-1}).$$ By Lemma \[qklimit\] we have: $$R^{(N)}(t,y)=N(\omega')^{2}\left(q\left(t,\frac{k(y,N,t)}{N}\right)-q\left(t,\frac{k(y,N,t)-1}{N}\right)\right)+O(\frac{\ln N}{N})=$$ $$=(\omega')^{2}q_{x}\left(t,\frac{k(y,N,t)}{N}\right)+O(\frac{\ln N}{N})$$ Using inequalities (\[k\_N\_bound\_2\]) and (\[k\_N\_bound\_1\]) we have the following estimate: $$\left\|\frac{k(y,N,t)}{N}-z(x(t,y))\right\|\leqslant\frac{c}{N}.$$ for some constant $c$, not depending on $N$. Then we can conclude that $$\lim_{N\rightarrow\infty}R^{(N)}(t,y)=R(t,y)=(\omega')^{2}q_{x}(t,z(x(t,y)))=(\omega')^{2}\frac{d^{2}G(t,z(x(t,y)))}{dz^{2}}.$$ Using formula (\[R\_t\_y\]), we get the proof. #### Proof of Theorem \[Th\_Energy\] Let us check the first equality. Rewrite the potential energy $U^{(N)}(t,y)$ in terms of the $q$ variables, which were introduced in the proof of Theorem \[Th\_bound\] $$U^{(N)}(t,y)=\frac{1}{4}\omega^{2}(q_{k_{N}(t,y)}^{2}+q_{k_{N}(t,y)-1}^{2}).$$ The same arguments as in the proof of Theorem \[Th\_force\] give $$U(t,y)=\lim_{N\rightarrow\infty}U^{(N)}(t,y)=\frac{1}{2}(\omega')^{2}q^{2}(t,z(x(t,y)))=\frac{1}{2}(\omega')^{2}\left(\frac{dG(t,z(x(t,y)))}{dz}-1\right)^{2}.$$ Using formulas (\[y\_po\_x\]) and (\[yderivx\]), we get: $$U(t,y)=\frac{1}{2}(\omega')^{2}\left(\frac{y_{x}(x(t,y))}{z'(x(t,y))}-1\right)^{2}=\frac{1}{2}(\omega')^{2}\left(\frac{1}{\rho(t,y)}-1\right)^{2}$$ The formula for the kinetic energy is obvious. The density dynamics ==================== On the three-dimensional $(t,x,z)\in R^{3}$ graph the surface $z=\rho(t,x)-1$ is presented, as the result of computer modelling with $N=200,\omega'=1$. Initial data were chosen as: $$X(x)=1+\epsilon S_{n}(x),\quad S_{n}(x)=\sum_{k=4}^{100}\frac{s_{k}}{k^{2}}\sin(\pi kx),V(x)=0$$ with random numbers $s_{k}\in[0,1]$ . $\epsilon$ is chosen so that there were no particle collisions, namely as $$\epsilon<\frac{1}{2s},\quad s=\int_{0}^{1}\|S''_{n}(x)\|dx.$$ ![image](LM_pic_1){width="15cm" height="7cm"} [1]{} Bogolyubov N. N. On some statistical methods in mathematical physics. 1845. Kiev. Acad of Science of USSR, 1945. Morrey C. On the derivation of the equations of hydrodynamics from statistical mechanics. Comm. Pure Appl. Math., 1955, 8, 279-326. Marchioro, M. Pulvirenti. Mathematical Theory of Incompressible Nonviscous Fluids. Springer. 1993. R. Esposito, J. Lebowitz, R. Marra. On the derivation of hydrodynamics from the Boltzmann equation. Phys. of Fluids, 1999, v. 11, 8, pp. 2354-2366. R. Esposito, R. Marra. Incompressible fluids on three levels: hydrodynamic, kinetic, microscopic. Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics , RIMS, Kyoto (1993). Boldrighini C., Dobrushin R. L., Sukhov Yu. M. One-dimensional hard rod caricature of hydrodynamics. Journal of Statistical Physics, 1983, 31, No, 3. Chorin A., Marsden J. A mathematical introduction to fluid mechanics. Third Ed. Springer. 2000. Siegel C,. Moser J. Lectures on Celestial Mechanics. Springer-Verlag. 1971. [^1]: Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Vorobyevy Gory 1, Moscow, 119991, Russia
--- abstract: 'A double quantum dot coupled to an $s$-wave superconductor and subject to an inhomogeneous magnetic field can host a pair of zero-energy Majorana fermions when the dot properties are tuned appropriately. Here, we demonstrate the possibility to generate a fractional $4\pi$ Josephson effect in two such double dots tunnel-coupled to each other. We discuss the robustness of this effect with respect to perturbations away from the special point in parameter space where the uncoupled double dots host Majorana fermions. We demonstrate the possibility to generate Josephson effects with a period of $8\pi$ and $12\pi$ in strongly-coupled double dots.' address: 'Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland' author: - 'Björn Sothmann, Jian Li and Markus Büttiker' title: Fractional Josephson effect in a quadruple quantum dot --- \[sec:introduction\]Introduction ================================ Currently, there is a rapidly increasing interest in the possibility to generate Majorana fermions in condensed-matter systems [@alicea_new_2012; @beenakker_search_2013]. This is triggered by the fact that Majorana fermions exhibit non-Abelian braiding properties and offer the possibility to realize topological qubits that are immune to local perturbations for quantum computing [@alicea_non-abelian_2011; @flensberg_non-abelian_2011; @van_heck_coulomb-assisted_2012]. By now, various theoretical proposals of how to realize Majorana fermions exist, e.g., by quasiparticles in the $\nu=5/2$ fractional quantum Hall state [@moore_nonabelions_1991], as edge states of a one-dimensional wire made from a $p$-wave superconductor [@kitaev_unpaired_2001] or in vortices of a $p$-wave superconductor [@das_sarma_proposal_2006]. Unfortunately, all of these proposal are very hard to realize experimentally. Recently, several more feasible setups have been suggested where the surface states of a three-dimensional topological insulator or edge states of a quantum spin Hall insulator are coupled to an $s$-wave superconductor [@fu_superconducting_2008; @fu_josephson_2009]. In addition, it was shown that a one-dimensional $p$-wave superconductor can effectively be induced in a single-channel nanowire with strong spin-orbit interaction subject to an external magnetic field and in proximity with a standard $s$-wave superconductor [@lutchyn_majorana_2010; @oreg_helical_2010]. Majorana fermions give rise to a number of characteristic transport signatures. First of all, they lead to a zero-bias peak of $2e^2/h$ in the differential conductance of a weakly coupled normal metal probe [@law_majorana_2009; @flensberg_tunneling_2010]. In addition, Majorana fermions also give rise to a fractional Josephson effect where the supercurrent is $4\pi$ periodic in the phase difference between the superconductors instead of being $2\pi$ periodic [@kitaev_unpaired_2001]. The fractional Josephson effect is a consequence of the crossing of Andreev bound state energies as a function of the phase difference. It has by now been studied theoretically in Josephson junctions where Majorana fermions are realized using quantum spin Hall insulator edge states [@fu_josephson_2009; @badiane_nonequilibrium_2011; @beenakker_fermion-parity_2013], surface states of a three-dimensional topological insulator [@ioselevich_anomalous_2011] and nanowires [@jiang_unconventional_2011]. The effects of many-channel wires [@law_robustness_2011], finite wire length [@san-jose_ac_2012] and Coulomb charging have been investigated [@van_heck_coulomb_2011]. The possibility to observe a fractional Josephson effect due to Landau-Zener tunneling in realistic systems where imperfections in the system turn the level crossing into an avoided crossing has been discussed [@dominguez_dynamical_2012; @pikulin_phenomenology_2012]. Landau-Zener tunneling has also been shown to yield a fractional Josephson effect even in Josephson junctions without Majorana fermions [@sau_possibility_2012]. Nanowire setups that are supposed to host Majorana fermions at their ends have by now been realized by several groups [@mourik_signatures_2012; @das_zero-bias_2012; @deng_anomalous_2012; @finck_anomalous_2013; @churchill_superconductor-nanowire_2013]. These experiments have indeed shown zero-bias conductance peaks in agreement with theoretical predictions. However, there exist other mechanisms like interference effects in the presence of disorder [@bagrets_class_2012] and weak antilocalization [@pikulin_zero-voltage_2012] that can yield similar zero-bias anomalies. Additionally, recent experiments have shown the occurrence of fractional Shapiro steps in Josephson junctions from InSb nanowires [@rokhinson_fractional_2012] as well as unconventional behaviour of a superconductor-topological insulator Josephson junction [@williams_unconventional_2012]. One of the biggest issues in clearly identifying signatures of Majorana fermions is the fact that in experiments the nanowire is buried beneath a superconducting electrode. Hence, its properties are not accessible to direct characterization. This problem can be circumvented by alternative setups based on quantum dots [@sau_realizing_2012; @leijnse_parity_2012; @fulga_adaptive_2013; @wright_localized_2013; @brunetti_anomalous_2013]. The simplest realization of such a system consists of a double quantum dot coupled to an $s$-wave superconductor and subject to an inhomogenous magnetic field [@leijnse_parity_2012]. The presence of Majorana fermions in such a system is indicated by characteristic features in the zero-bias conductance in analogy to the Majorana setups realized in nanowires. Here, we demonstrate the possibility to generate a fractional Josephson effect in a setup of two double dots and analyse the robustness of the effect with respect to level detunings and Coulomb interactions. Compared to a conductance measurement, the Josephson current offers the advantage of being a non-invasive probe as it does not require to couple any additional normal electrode to the system. Using quantum dots to investigate Majorana fermions offers a number of advantages. First of all, quantum dots are well-controlled systems with parameters like level positions that can be easily tuned in experiment. In addition, very similar setups have already been realized to act as Cooper pair splitter [@hofstetter_cooper_2009; @herrmann_carbon_2010; @hofstetter_finite-bias_2011; @das_high-efficiency_2012; @schindele_near-unity_2012]. Furthermore, supercurrents through quantum dots have been observed in a number of experiments as well [@van_dam_supercurrent_2006; @cleuziou_carbon_2006; @grove-rasmussen_kondo_2007; @winkelmann_superconductivity_2009; @eichler_tuning_2009; @kanai_electrical_2010; @lee_zero-bias_2012; @kim_transport_2013]. From a theoretical perspective, a double quantum dot represents the shortest Kitaev chain consisting of two links only. Hence, it is the simplest possible Majorana setup. Finally, quantum dots are an ideal playground to investigate the interplay between Majorana physics and strong Coulomb interactions. The paper is organized as follows. We introduce our model in Sec. \[sec:model\] and discuss how to evaluate the Josephson current from the quantum dot spectrum in Sec. \[sec:Josephson\]. In Sec. \[sec:energies\], we discuss the eigenenergies which are directly related to the Josephson current. Finally, we discuss the limit of strong coupling between the double dots where additional crossings occur in Sec. \[sec:strong\]. We finish with conclusions given in Sec. \[sec:conclusions\]. \[sec:model\]Model ================== ![\[fig:model\]Sketch of the quadruple quantum dot. Quantum dots L1 and L2 as well as quantum dots R1 and R2 form double quantum dots coupled to superconducting electrodes with phases $\Phi_{\mathsf{L}}$ and $\Phi_{\mathsf{R}}$, respectively. The two double quantum dots are coupled to each other by a tunnel barrier controlled by the gate voltage applied to $G$. Red arrows indicate the inhomogenous magnetic field.](figure1.png){width="\columnwidth"} We consider a system of four single-level quantum dots as shown in . The quantum dots form two double dots, $r=\mathsf{L,R}$, each coupled to a grounded superconductor. As we are interested in subgap transport only, we focus on the limit of infinite superconducting gaps. The whole system is subject to a magnetic field that lifts spin degeneracy. In order to have both, direct tunneling between the dots $i=1,2$ within a double dot and a superconducting proximity effect caused by crossed Andreev reflections, we assume the magnetic field acting on the two dots to enclose a finite angle $\alpha_r$. By changing this angle, the relative strength between direct tunneling and the proximity effect can be tuned [@leijnse_parity_2012]. Finally, the two double dots are connected to each other via a tunnel coupling between dot $i=2$ in the left and dot $i=1$ in the right double dot. \[ssec:HEff\]Effective Hamiltonian ---------------------------------- The system is characterized by the total Hamiltonian $$\label{eq:H} H=\sum_r H_{r,\mathsf{eff}}+H_\mathsf{tun}+H_\mathsf{int}.$$ The effective dot Hamiltonian that describes the double quantum dot $r$ is given by $$\label{eq:Heff} H_{r,\mathsf{eff}}=\sum_i \varepsilon_{ri} n_{ri}+t_r c_{r1}^\dagger c_{r2}+\Delta_r\rme^{\rmi\Phi_r} c_{r1}^\dagger c_{r2}^\dagger+\mathsf{H.c.},$$ cf. \[app:Heff\] for a detailed derivation. Here, $\varepsilon_{ri}$ denotes the energy of the single level relevant for transport in the two dots. Direct tunneling between the two dots is characterized by $t_r$. The superconducting proximity effect is described by the induced gap $\Delta_r$ which in our system is proportional to the tunnel coupling between the double dot and the superconductor. Finally, $\Phi_r$ denotes the phase of the superconducting order parameter in the electrode coupled to double dot $r$. The coupling between the two double dots is described by the tunneling Hamiltonian $$H_\mathsf{tun}=t c_{\mathsf{L}2}^\dagger c_{\mathsf{R}1}+\mathsf{H.c.},$$ where the tunnel matrix element $t$ can be controlled by a gate voltage. Finally, interactions are captured by the interaction Hamiltonian, $$H_\mathsf{int}=\sum_{rir'i'} U_{rir'i'} n_{ri} n_{r'i'},$$ where the sum runs over all pairs of different quantum dots and double counting is excluded. \[ssec:Majorana\]Majorana representation ---------------------------------------- ![\[fig:Majorana\]Representation of the quadruple dot in terms of Majorana fermions. Each column denots a quantum dot that can be described in terms of two Majorana fermions $\gamma^r_{iA}$ and $\gamma^r_{iB}$. Dashed lines indicate couplings between Majoranas arising from tunneling and proximity effect within a double dot. The dotted lines indicate couplings due to the tunneling between the double dots. In a) the phase difference is $\Delta\Phi=2n\pi$, in b) it is $\Delta\Phi=(2n+1)\pi$ while c) shows a generic situation.](figure2a.pdf "fig:"){width=".45\columnwidth"} ![\[fig:Majorana\]Representation of the quadruple dot in terms of Majorana fermions. Each column denots a quantum dot that can be described in terms of two Majorana fermions $\gamma^r_{iA}$ and $\gamma^r_{iB}$. Dashed lines indicate couplings between Majoranas arising from tunneling and proximity effect within a double dot. The dotted lines indicate couplings due to the tunneling between the double dots. In a) the phase difference is $\Delta\Phi=2n\pi$, in b) it is $\Delta\Phi=(2n+1)\pi$ while c) shows a generic situation.](figure2b.pdf "fig:"){width=".45\columnwidth"} ![\[fig:Majorana\]Representation of the quadruple dot in terms of Majorana fermions. Each column denots a quantum dot that can be described in terms of two Majorana fermions $\gamma^r_{iA}$ and $\gamma^r_{iB}$. Dashed lines indicate couplings between Majoranas arising from tunneling and proximity effect within a double dot. The dotted lines indicate couplings due to the tunneling between the double dots. In a) the phase difference is $\Delta\Phi=2n\pi$, in b) it is $\Delta\Phi=(2n+1)\pi$ while c) shows a generic situation.](figure2c.pdf "fig:"){width=".45\columnwidth"} We now express the Hamiltonian in terms of Majorana fermions. To this end, we introduce the Majorana operators $$\begin{aligned} \label{eq:Majorana} \gamma_{i\mathsf{A}}^r&=\rme^{-\rmi\Phi_r/2}c_{ri}+\rme^{\rmi\Phi_r/2}c_{ri}^\dagger,\\ \gamma_{i\mathsf{B}}^r&=-\rmi\rme^{-\rmi\Phi_r/2}c_{ri}+\rmi\rme^{\rmi\Phi_r/2}c_{ri}^\dagger,\end{aligned}$$ which satify $(\gamma_{i\mathsf{A/B}}^{r})^\dagger=\gamma_{i\mathsf{A/B}}^r$. The inverse transformation reads $$\begin{aligned} c_{ri}&=\frac{1}{2}\rme^{\rmi\Phi_r/2}\left(\gamma^r_{i\mathsf{A}}+\rmi\gamma^r_{i\mathsf{B}}\right),\\ c_{ri}^\dagger&=\frac{1}{2}\rme^{-\rmi\Phi_r/2}\left(\gamma^r_{i\mathsf{A}}-\rmi\gamma^r_{i\mathsf{B}}\right).\end{aligned}$$ In terms of the Majorana operators, the effective dot Hamiltonian becomes $$\label{eq:HMajorana} H_{r,\mathsf{eff}}=\frac{\rmi}{2}\left[-(t_r+\Delta_r)\gamma_{1\mathsf{B}}^r\gamma_{2\mathsf{A}}^r+(t_r-\Delta_r)\gamma_{1\mathsf{A}}^r\gamma_{2\mathsf{B}}^r+\sum_i\varepsilon_{ri}\gamma_{i\mathsf{A}}^r\gamma_{i\mathsf{B}}^r\right],$$ where we dropped an irrelevant constant. From , we directly read off that the double dot hosts a pair of zero-energy excitations at the special point $t_r=\Delta_r$, $\varepsilon_i=0$ described by the Majorana operators $\gamma_{1\mathsf{A}}^r$ and $\gamma_{2\mathsf{B}}^r$. Expressed in terms of Majorana fermions, the tunnel coupling and the Coulomb interaction become $$H_\mathsf{tun}=\frac{\rmi t}{2}\left[\sin\frac{\Delta\Phi}{2}\left(\gamma^\mathsf{L}_{2\mathsf{A}}\gamma^\mathsf{R}_{1\mathsf{A}}+\gamma^\mathsf{L}_{2\mathsf{B}}\gamma^\mathsf{R}_{1\mathsf{B}}\right)+\cos\frac{\Delta\Phi}{2}\left(\gamma^\mathsf{L}_{2\mathsf{A}}\gamma^\mathsf{R}_{1\mathsf{B}}-\gamma^\mathsf{L}_{2\mathsf{B}}\gamma^\mathsf{R}_{1\mathsf{A}}\right)\right],$$ and $$H_\mathsf{int}=\sum_{rir'i'}\frac{U_{rir'i'}}{4}\left[1+\rmi\left(\gamma^r_{i\mathsf{A}}\gamma^r_{i\mathsf{B}}+\gamma^{r'}_{i'\mathsf{A}}\gamma^{r'}_{i'\mathsf{B}}\right)-\gamma^r_{i\mathsf{A}}\gamma^r_{i\mathsf{B}}\gamma^{r'}_{i'\mathsf{A}}\gamma^{r'}_{i'\mathsf{B}}\right],$$ respectively. Here $\Delta\Phi=\Phi_\mathsf{R}-\Phi_\mathsf{L}$ denotes the phase difference between the two superconductors. A schematic representation of the quadruple dot in terms of its Majorana degrees of freedom for different values of the phase difference $\Delta\Phi$ is shown in . \[sec:Josephson\]Josephson current ================================== The Josephson current through the quantum dots is given by the derivative of the free energy with respect to the phase difference between the superconductors [@beenakker_three_1992], $$\label{eq:JJos1} J_\mathsf{jos}=\frac{2e}{\hbar}\frac{\partial F}{\partial \Delta\Phi}.$$ As the free energy is given by $F=-{k_\mathsf{B}T}\log \sum_i \rme^{-E_i/{k_\mathsf{B}T}}$ where $E_i$ denote the many-body eigenenergies of the system, at low temperatures, ${k_\mathsf{B}T}\ll E_i-E_j$, the Josephson current is essentially determined by the ground state energy $E_0$, $$J_\mathsf{jos}=\frac{2e}{\hbar}\frac{\partial E_0}{\partial \Delta\Phi}.$$ We remark that $E_0$ is a many-body ground state energy, i.e., it is the lowest eigenenergy of the system. This is in contrast to a description in the Bogoliubov-de Gennes picture where the excitation energies close to zero energy determine the Josephson current. Since the Josephson current is completely determined by the spectrum of the quantum dot, in the following, we will discuss the properties of the eigenenergies rather than the Josephson current itself. \[sec:energies\]Eigenenergies ============================= We are now going to discuss the eigenenergies of the quantum dot. We focus on the situation where $t_r=\Delta_r$. As soon as this condition is relaxed, the system will no longer exhibit a fractional Josephson effect. This is in contrast to a nanowire setup where a fractional Josephson effect can occur even when $t_r\neq\Delta_r$ since the induced splitting of the zero-energy states is exponentially small in the length of the wire. First of all, we discuss the “sweet spot” in parameter space where in addition to $t_r=\Delta_r$ we have $\varepsilon_{ri}=0$ and $U_{rir'i'}=0$. We then turn to the effect of detuned levels and analyse the influence of Coulomb interactions. \[ssec:sweet\]Sweet spot ------------------------ ![\[fig:energies\]Eigenenergies as a function of the phase difference $\Delta\Phi$. Parameters are $\Delta_\mathsf{R}=0.7\Delta_\mathsf{L}$, $\varepsilon_{ri}=0$, $t_r=\Delta_r$, $t=0.5\Delta_\mathsf{L}$, $U_{ri,r'i'}=0$. For each eigenenergy, there are two degenerate states with an even and odd number of electrons on the quantum dots, respectively. Note that we discuss the many-body eigenenergies, i.e., the ground state properties are determined by the state with lowest energy.](figure3.pdf){width=".6\columnwidth"} At the sweet spot, $t_r=\Delta_r$, $\varepsilon_{ri}=0$ and $U_{ri,r'i'}=0$, analytical expressions for the eigenenergies can be obtained. However, as they are lengthy and do not provide much insight it is more convenient to discuss the eigenenergies obtained via degenerate perturbation theory in the coupling $t$ between the double dots. Up to second order in $t$, we find $$\begin{aligned} \label{eq:E0} E_{0\pm}=&-\|\Delta_{\mathsf{L}}+\Delta_{\mathsf{R}}\| \pm\frac{t}{2}\cos\frac{\Delta\Phi}{2}\\ &-t^2\frac{(\Delta_{\mathsf{L}}+\Delta_{\mathsf{R}})^2(1-\cos\Delta\Phi)+\Delta_{\mathsf{L}}\Delta_{\mathsf{R}}(1+\cos\Delta\Phi)}{16\Delta_{\mathsf{L}}\Delta_{\mathsf{R}}(\Delta_{\mathsf{L}}+\Delta_{\mathsf{R}})},\nonumber\\ E_{1\pm}=&-\|\Delta_{\mathsf{L}}-\Delta_{\mathsf{R}}\| \pm\frac{t}{2}\cos\frac{\Delta\Phi}{2}\\ &-t^2\frac{(\Delta_{\mathsf{L}}-\Delta_{\mathsf{R}})^2(1-\cos\Delta\Phi)-\Delta_{\mathsf{L}}\Delta_{\mathsf{R}}(1+\cos\Delta\Phi)}{16\Delta_{\mathsf{L}}\Delta_{\mathsf{R}}(\Delta_{\mathsf{L}}+\Delta_{\mathsf{R}})},\nonumber\\ E_{2\pm}=&+\|\Delta_{\mathsf{L}}-\Delta_{\mathsf{R}}\| \pm\frac{t}{2}\cos\frac{\Delta\Phi}{2}\\ &-t^2\frac{(\Delta_{\mathsf{L}}-\Delta_{\mathsf{R}})^2(1-\cos\Delta\Phi)-\Delta_{\mathsf{L}}\Delta_{\mathsf{R}}(1+\cos\Delta\Phi)}{16\Delta_{\mathsf{L}}\Delta_{\mathsf{R}}(\Delta_{\mathsf{L}}+\Delta_{\mathsf{R}})},\nonumber\\ E_{3\pm}=&+\|\Delta_{\mathsf{L}}+\Delta_{\mathsf{R}}\| \pm\frac{t}{2}\cos\frac{\Delta\Phi}{2}\\ &-t^2\frac{(\Delta_{\mathsf{L}}+\Delta_{\mathsf{R}})^2(1-\cos\Delta\Phi)+\Delta_{\mathsf{L}}\Delta_{\mathsf{R}}(1+\cos\Delta\Phi)}{16\Delta_{\mathsf{L}}\Delta_{\mathsf{R}}(\Delta_{\mathsf{L}}+\Delta_{\mathsf{R}})}.\nonumber$$ For each of these eigenenergies, there are two degenerate eigenstates with an even and odd number of electrons on the quantum dots, respectively. Equivalently, for these states the nonlocal fermion $f=(\gamma^{\mathsf{L}}_{1A}+i\gamma^{\mathsf{R}}_{2B})/2$ is occupied or empty. We remark that the above expressions are only valid for $\Delta_{\mathsf{L}}\neq\Delta_{\mathsf{R}}$. For $\Delta_{\mathsf{L}}=\Delta_{\mathsf{R}}$, we instead find $E_{1\pm}=0$ and $E_{2\pm}=\pm t\cos(\Delta\Phi/2)$, respectively. Interestingly, the above analytic expressions are in good agreement with the numerically obtained eigenvalues shown in even for large tunnel couplings $t\sim\Delta$. In order to see whether a fractional Josephson effect can occur in the system, we have to analyze the phase dependence of the eigenenergies $E_{0\pm}$. As shown in , the eigenenergies cross at $\Delta\Phi=(2n+1)\pi$, $n\in\mathbb{Z}$. Due to these crossings, the system stays, e.g., in one of the two states with energy $E_{0+}$ when the phase difference is adiabatically increased and passes through $\Delta\Phi=(2n+1)\pi$. Hence, increasing the phase difference by $2\pi$ exchanges the ground state and the first excited state, thus leading to a $4\pi$ periodic Josephson current. We remark that the observation of the $4\pi$ periodicity requires a conservation of fermion parity over a time scale longer than the time needed to increase the phase difference by $4\pi$. To understand the origin of these crossings, we take a look at the Majorana representation of the quantum dots, cf. . For any phase difference, there are two uncoupled Majorana modes at the end of the quantum dot chain. They are associated with the degeneracy of the even and odd parity states. In addition, for $\Delta\Phi=(2n+1)\pi$, the system hosts two chains consisting of three Majorana fermions. As these chains are described by an antisymmetric $3\times3$ matrix, they must have a zero-energy mode associated with the level crossing. The zero-energy modes are given by $$\begin{aligned} \Gamma_1=\frac{1}{\sqrt{\Delta^2+t^2/4}}\left(\pm\frac{t}{2}\gamma^{\mathsf{L}}_{1B}-\Delta\gamma^{\mathsf{R}}_{1A}\right),\\ \Gamma_2=\frac{1}{\sqrt{\Delta^2+t^2/4}}\left(\pm\frac{t}{2}\gamma^{\mathsf{R}}_{2A}-\Delta\gamma^{\mathsf{L}}_{2B}\right),\end{aligned}$$ where the upper (lower) sign applies at $\Delta\Phi=(4n\pm1)\pi$. Hence, the zero-energy states are localized at the ends of the three-Majorana chains. The nonlocal fermion $f=(\Gamma_1+i\Gamma_2)/2$ associated with these two modes is occupied (empty) for the states with energy $E_{0\pm}$. \[ssec:perturbation\]Influence of perturbations ----------------------------------------------- ![\[fig:perturbation\]Influence of level detunings (a) and b)) and Coulomb interactions (c) and d)) on the eigenenergies of the quadruple quantum dot. In a) we have $\varepsilon_{{\mathsf{L}}1}=\Delta_{\mathsf{L}}/2$, in b) $\varepsilon_{{\mathsf{L}}2}=\Delta_{\mathsf{L}}/2$, in c) $U_{{\mathsf{L}}1{\mathsf{L}}2}=\Delta_{\mathsf{L}}/2$ and in d) $U_{{\mathsf{L}}2{\mathsf{R}}1}=\Delta_{\mathsf{L}}/2$. Other parameters are as in .](figure4a_4d.pdf){width="\columnwidth"} So far, we considered an ideal system tuned to its sweet spot. We now analyze the influence of perturbations away from this point. We discuss both the effect of level detunings as well as of Coulomb interactions. ### Level detunings In the following, we analyse the influence of level detunings on the occurrence of the fractional Josephson effect. We first consider the detuning of the outer dot level ${\mathsf{L}}1$ and ${\mathsf{R}}2$. As can be seen in a), a detuning of one of the outer dots turns the crossings between $E_{0\pm}$ into anticrossings and, hence, destroys the fractional Josephson effect. The reason for this becomes immediately clear when looking at the Majorana representation of the quantum dots. A finite energy $\varepsilon_{{\mathsf{L}}1}$ induces a coupling between the zero-energy Majorana modes $\gamma^{\mathsf{L}}_{1A}$ and $\Gamma_1$. This coupling moves both states to finite energy and, thereby, destroys the level crossing at $\Delta\Phi=(2n+1)\pi$. We next turn to a detuning of the inner dot levels $\varepsilon_{{\mathsf{L}}2}$ and $\varepsilon_{{\mathsf{R}}1}$. In contrast to a detuning of the outer dot levels, we now find that the crossings are preserved and the fractional Josephson effect persists, cf. b). This can again be understood from the Majorana representation of the quantum dots. While a finite $\varepsilon_{{\mathsf{L}}2}$ induces a coupling between $\gamma^{\mathsf{L}}_{2A}$ and $\gamma^{\mathsf{L}}_{2B}$, it does not affect $\Gamma_1$ which therefore stays a zero-energy excitation. More generally, for a detuning of both inner dots, we find the new zero-energy Majorana modes $$\begin{aligned} \Gamma_1=\frac{2}{\sqrt{4\Delta^2+t^2+\varepsilon_{{\mathsf{R}}1}^2}}\left(\pm\frac{t}{2}\gamma^{\mathsf{L}}_{1B}-\Delta\gamma^{\mathsf{R}}_{1A}+\frac{\varepsilon_{{\mathsf{R}}1}}{2}\gamma^{\mathsf{R}}_{2A}\right),\\ \Gamma_2=\frac{2}{\sqrt{4\Delta^2+t^2+\varepsilon_{{\mathsf{L}}2}^2}}\left(\pm\frac{t}{2}\gamma^{\mathsf{R}}_{2A}-\Delta\gamma^{\mathsf{L}}_{2B}+\frac{\varepsilon_{{\mathsf{L}}2}}{2}\gamma^{\mathsf{L}}_{1B}\right).\end{aligned}$$ ### Coulomb interactions We next turn to the influence of Coulomb interactions on the level crossings. As shown in c), Coulomb interactions within a double quantum dot destroy the level crossings at $\Delta\Phi=(2n+1)\pi$. This is due to the fact that the Coulomb interaction induces couplings between the zero-energy modes of the unperturbed system in analogy to the detuning of the outer dot levels discussed above. In contrast, Coulomb interactions $U_{{\mathsf{L}}2{\mathsf{R}}1}$ between the double dots do not affect the occurrence of the fractional Josephson effect, see d). This can be most easily understood by analyzing the system in the many-body picture. The eigenstates of the two uncoupled double dots are given by the product states ${\|\alpha_\pm\rangle}_{\mathsf{L}}\otimes {\|\alpha_\pm\rangle}_{\mathsf{R}}$, ${\|\alpha_\pm\rangle}_{\mathsf{L}}\otimes {\|\beta_\pm\rangle}_{\mathsf{R}}$, ${\|\beta_\pm\rangle}_{\mathsf{L}}\otimes {\|\alpha_\pm\rangle}_{\mathsf{R}}$, ${\|\beta_\pm\rangle}_{\mathsf{L}}\otimes {\|\beta_\pm\rangle}_{\mathsf{R}}$ where the eigenstates of the double dots are [@leijnse_parity_2012] $$\begin{aligned} {\|\alpha_\pm\rangle}_r=\frac{1}{\sqrt{2}}\left(e^{-i\Phi_r/2}{\|00\rangle}_r\pm e^{i\Phi_r/2}{\|11\rangle}_r\right),\quad E_{\alpha_\pm}=\pm \Delta_r,\\ {\|\beta_\pm\rangle}_r=\frac{1}{\sqrt{2}}\left({\|01\rangle}_r\pm{\|10\rangle}_r\right),\quad E_{\beta_\pm}=\pm t_r,\end{aligned}$$ with ${\|11\rangle}\equiv c_{r1}^\dagger c_{r2}^\dagger {\|00\rangle}$. The fourfold degenerate ground states at the sweet spot are given by ${\|\phi_0\rangle}={\|\alpha_-\rangle}_{\mathsf{L}}\otimes{\|\alpha_-\rangle}_{\mathsf{R}}$, ${\|\phi_1\rangle}={\|\alpha_-\rangle}_{\mathsf{L}}\otimes{\|\beta_-\rangle}_{\mathsf{R}}$, ${\|\phi_2\rangle}={\|\beta_-\rangle}_{\mathsf{L}}\otimes{\|\alpha_-\rangle}_{\mathsf{R}}$ and ${\|\phi_3\rangle}={\|\beta_-\rangle}_{\mathsf{L}}\otimes{\|\beta_-\rangle}_{\mathsf{R}}$. To first order in perturbation theory, the Coulomb interaction between the double dots does not mix these states. In addition, it yields the same energy shift for all four states. Hence, the level crossing at $\Delta\Phi=(2n+1)\pi$ is conserved. As can be seen from the numerical analysis in d), the argument remains valid even for large Coulomb interactions where lowest-order perturbation theory is no longer reliable. \[sec:strong\]Strong-coupling limit =================================== ![\[fig:Especial1\]Left: Eigenenergies as a function of phase difference for a symmetric setup $\Delta_{\mathsf{L}}=\Delta_{\mathsf{R}}\equiv\Delta$ with $\varepsilon_{ri}=U_{rir'i'}=0$ and $t=\sqrt{2}\Delta$. Right: Probability $P$ to find the system in the state with energy $E=-2\sqrt{2}\Delta\cos\Delta(\Phi/6)$ for a time-dependent phase difference $\Delta\Phi(\tau)=\omega\tau$.](figure5a.pdf "fig:"){width=".5\columnwidth"} ![\[fig:Especial1\]Left: Eigenenergies as a function of phase difference for a symmetric setup $\Delta_{\mathsf{L}}=\Delta_{\mathsf{R}}\equiv\Delta$ with $\varepsilon_{ri}=U_{rir'i'}=0$ and $t=\sqrt{2}\Delta$. Right: Probability $P$ to find the system in the state with energy $E=-2\sqrt{2}\Delta\cos\Delta(\Phi/6)$ for a time-dependent phase difference $\Delta\Phi(\tau)=\omega\tau$.](figure5b.pdf "fig:"){width=".5\columnwidth"} ![\[fig:Especial2\]Eigenenergies as a function of phase difference. Left: Symmetric system with Coulomb interaction between the double dots $U_{{\mathsf{L}}2{\mathsf{R}}1}=\Delta$ and $t=1.24697\Delta$. Other parameters as in . Right: Asymmetric system $\Delta_{\mathsf{R}}=0.7\Delta_{\mathsf{L}}$ with $t=1.147\Delta_{\mathsf{L}}$. Other parameters as in .](figure6a.pdf "fig:"){width=".5\columnwidth"} ![\[fig:Especial2\]Eigenenergies as a function of phase difference. Left: Symmetric system with Coulomb interaction between the double dots $U_{{\mathsf{L}}2{\mathsf{R}}1}=\Delta$ and $t=1.24697\Delta$. Other parameters as in . Right: Asymmetric system $\Delta_{\mathsf{R}}=0.7\Delta_{\mathsf{L}}$ with $t=1.147\Delta_{\mathsf{L}}$. Other parameters as in .](figure6b.pdf "fig:"){width=".5\columnwidth"} As the tunnel coupling between the double dots is increased, the splitting between the energies $E_{i\pm}$ at $\Delta\Phi=2n\pi$ increases as well. As was first pointed out in Ref. [@nogueira_strong-coupling_2012], for a symmetric system $\Delta_{\mathsf{L}}=\Delta_{\mathsf{R}}\equiv\Delta$ tuned to the sweet spot, this increased splitting leads to the occurrence of new level crossings in the spectrum at $t=\sqrt{2}\Delta$. While this strong-coupling regime might be hard to realize in a nanowire setup, it should be easily realizable in a quantum-dot system. In the following, we demonstrate that these additional crossings give rise to a $12\pi$-periodic Josephson effect. We then discuss the robustness of this effect with respect to asymmetries in the systems and perturbations away from the sweet spot. The spectrum of a symmetric system $\Delta_{\mathsf{L}}=\Delta_{\mathsf{R}}$ with $t=\sqrt{2}\Delta$ is shown in . In addition to the level crossings at $\Delta\Phi=(2n+1)\pi$, new crossings at $\Delta\Phi=2n\pi$ arise, leading to eigenenergies that are $12\pi$ periodic in the phase difference. We emphasize that these additional crossings are of accidential nature and not associated with Majorana degrees of freedom. In order to demonstrate that the system can exhibit a $12\pi$ Josephson effect in the strong-coupling limit, we consider the following situation. Let us assume that at time $\tau=0$ the system has a phase difference $\Delta\Phi=0$ and is prepared in its ground state. We then increase the phase difference with time according to $\Delta\Phi(\tau)=\omega\tau$. By numerically solving the time-dependent Schrödinger equation, we calculate the probability $P$ to find the system in the state with energy $E=-2\sqrt{2}\Delta\cos(\Delta\Phi/6)$. The results shown in clearly indicate that for slow driving, the system indeed stays on one energy branch and, thus, exhibits a $12\pi$-periodic Josephson effect. We finally address the robustness of the additional crossings. As they are of accidential nature, any perturbation will destroy them. This prevents the occurrence of the $12\pi$ Josephson effect and leads to the $4\pi$ effect discussed above. Importantly, there is however the possibility to get at least some of the additional crossings in the presence of perturbations. First of all, additional crossings arise in a symmetric system with Coulomb interaction $U_{{\mathsf{L}}2{\mathsf{R}}1}$ between the two double dots. These give rise to $8\pi$-periodic eigenenergies, cf. . Furthermore, additional crossings can occur for an asymmetric system $\Delta_{\mathsf{L}}\neq\Delta_{\mathsf{R}}$ tuned to the sweet spot. As shown in , here again, the additional crossings lead to the occurrence of an $8\pi$ Josephson effect. \[sec:conclusions\]Conclusions ============================== We investigated the Josephson current through a system of two tunnel-coupled double quantum dots subject to an inhomogenous magnetic field. Similarly to a Majorana nanowire, the system exhibits a crossing of energy levels that gives rise to a $4\pi$ Josephson effect. We find this effect to be robust with respect to a detuning of the inner dot levels as well as to Coulomb interactions between the double dots. In contrast, detunings of the outer dot levels and Coulomb interactions within a double dot convert the crossings into anticrossing and lead to a conventional $2\pi$-periodic effect. In addition, we investigated the regime of strongly couplgy double dots. We found that in this situation additional accidential level crossings can occur. For a symmetric setup, they lead to a $12\pi$-periodic Josephson effect whereas for an asymmetric system an $8\pi$-periodic effect arises. Both effects require a fine-tuning of parameters and are vulnerable to perturbations. We thank Martin Leijnse for reading the manuscript and useful feedback. We acknowledge financial support from the Swiss NSF and NCCR QSIT. \[app:Heff\]Derivation of the effective dot Hamiltonian ======================================================= In the following, we derive the effective dot Hamiltonian . For simplicity, we suppress the index $r=\mathsf{L,R}$ characterizing the two double dots. In the limit of an infinite superconducting gap, the superconductor’s degree of freedom can be integrated out exactly to yield an effective dot Hamiltonian [@eldridge_superconducting_2010] $$H=\sum_i H_i+H_{\mathsf{int}}+H_{\mathsf{tun}}+H_{\mathsf{prox}}.$$ Here, $$\label{eq:Hi} H_i=\sum_{i\sigma} \varepsilon_{i} n_{i\sigma}+\sum_i{\mathbf{B}}_i\cdot{\mathbf{S}}_i$$ describes the relevant single level with energy $\varepsilon_i$ in each quantum dot as well as the influence of an externally applied magnetic field ${\mathbf{B}}_i$ acting on the dot spin ${\mathbf{S}}_i=\sum_{\sigma\sigma'}\frac{1}{2}c_{i\sigma}^\dagger \boldsymbol \sigma_{\sigma\sigma'} c_{i\sigma'}$. The effects of interdot Coulomb interactions $U_i$ as well as of intradot Coulomb interactions $U$ are described by $$\label{eq:Hint} H_{\mathsf{int}}=\sum_i U_in_{i\up}n_{i\down}+\sum_{\sigma\sigma'} U n_{1\sigma}n_{2\sigma'}.$$ Tunneling between the two dots is characterized by the tunneling Hamiltonian $$\label{eq:Htun} H_{\mathsf{tun}}=\sum_\sigma \tilde t c_{1\sigma}^\dagger c_{2\sigma}+{\mathsf{H.c.}}$$ Here, we assumed that tunneling conserves the electron spin, i.e., there is no spin-orbit interaction in the quantum dots. Finally, the superconducting proximity effect on the double dot is captured by $$\label{eq:Hprox} H_{\mathsf{prox}}=-\sum_i \frac{\Gamma_{Si}}{2}\left(c_{i\up}^\dagger c_{i\down}^\dagger+{\mathsf{H.c.}}\right)+\frac{\Gamma_{\mathsf{S}}}{2}\left(c_{2\up}^\dagger c_{1\down}^\dagger-c_{2\down}^\dagger c_{1\up}^\dagger+{\mathsf{H.c.}}\right),$$ where $\Gamma_{\mathsf{S}}=\sqrt{\Gamma_{\mathsf{S1}}\Gamma_{\mathsf{S2}}}$ is the geometric average of the tunnel couplings between the superconductor and dot 1 and 2, respectively. The first term describes a local proximity effect on a single dot whereas the second term captures the nonlocal proximity effect that involves both dots. In the following, we focus on the situation where a large magnetic field is applied to the double dot such that for each dot only a single spin state is relevant for transport. This directly implies that double occupancy of each dot is forbidden and a local proximity effect cannot occur. In the case of a homogenous magnetic field, we conclude from that no proximity effect is possible at all since it requires the pairing of electrons with antiparallel spin. If the magnetic field is pointing in opposite directions in the two dots, we similarly conclude that no tunneling between the two double dots is possible as it conserves spin. For an arbitrary angle $\phi$ enclosed between ${\mathbf{B}}_1$ and ${\mathbf{B}}_2$, the effective tunnel coupling is given by $t=\tilde t\cos\phi/2$ while the effective strength of the proximity effect is characterized by $\Delta=\frac{\Gamma_\mathsf{S}}{2}\sin\phi/2$. As pointed out in [@leijnse_parity_2012], this tunability of the ratio between $t$ and $\Delta$ is crucial to achieve the special point $t=\Delta$ required for the occurrence of Majorana fermions on the double dot. 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